**5.2. Screening of Very Intense Magnetic Fields by chiral
symmetry breaking**

Now we discuss another interesting phenomenon if very strong fields could be created.

In very intense magnetic fields, *B* > 1.5 ×
10^{18} G, the
breaking of the strong interaction *SU*(2) × *SU*(2)
symmetry arranges itself so that instead of the neutral
field acquiring
a vacuum expectation value it is the charged
field that does and
the magnetic field is screened.

In the previous section we discussed that fields with complicated
interactions of non-electromagnetic origin can induce various
instabilities in the presence of very intense magnetic fields. By very
intense we mean 10^{18} G to 10^{24} G. Fields with
anomalous magnetic moments
[223]
or fields coupled by transition
moments [216]
may induce vacuum instabilities. The usual breaking of the strong
interactions, chiral symmetry
(*SM*), is incompatible
with very intense magnetic fields.
Using the standard *SU*(2) × *SU*(2) chiral
model we show
that magnetic fields *B*
*B*_{c} with *B*_{c} = sqrt[2]
*m*_{}
*f*_{} are
screened; *f*_{}
= 132 MeV is the pion decay constant and
*m*_{} is
the mass of the charged pions. This result is opposite to what
occurs in a superconductor; in that case it is weak fields that are
screened and large ones penetrate and destroy the superconducting state.

As the magnetic fields are going to be screened we must be very careful in
how we specify an external field. One way would be to give
*f*_{} a
spatial dependence and take it to vanish outside some large region of
space. In the region that
*f*_{} vanishes
we could specify the external
field and see how it behaves in that part of space where chiral symmetry
is broken. This is the procedure used in studying the behavior of fields
inside superconductors. In the present situation we find this division
artificial and, instead of specifying the magnetic fields, we shall
specify the external currents. Specifically we will look, at first, at the
electromagnetic field coupled to the charged part of the
model
and to the current *I* in a long straight wire. From this result it
will be easy to deduce the behavior in other current configurations. We will
discuss a solenoidal current configuration towards the end of this work.

The Hamiltonian density for this problem is

(5.39) |

**j** is the external current.
We have used cylindrical coordinates with
the two dimensional vector normal to the *z* direction. We will study
this problem in the limit of very large *g*, where the radial degree
of freedom of the chiral field is frozen out and we may write

(5.40) |

In terms of these variables the Hamiltonian density becomes

(5.41) |

The angular field can be eliminated by a gauge transformation. For a current along a long wire we have

(5.42) |

The vector potential will point along the *z* direction, **A** =
*A***z** and the
fields will depend on the radial coordinate only. The equations of
motion become

(5.43) |

In the absence of the chiral field the last of the Eqs. (5.43) gives the classical vector potential due to a long wire

(5.44) |

with *a* an ultraviolet cutoff. The energy per unit length in the
*z* direction associated with this configuration is

(5.45) |

where *R* is the transverse extent of space (an infrared cutoff).

Before discussing the solutions of (5.43) it is instructive to
look at the case where there is no explicit chiral symmetry breaking,
*m*_{} = 0. The
solution that eliminates the infrared divergence in
Eq. (5.45) is =
=
/ 2 and *A* satisfying

(5.46) |

For any current the field *A* is damped for distances
> 1 /
*ef*_{} and there
is no infrared divergence in the energy. (Aside
from the fact that chiral symmetry is broken explicitly, the reason
the above discussion is only of pedagogical value is that the
coupling of the pions to the quantized electromagnetic field does break
the *SU*(2) × *SU*(2) symmetry into *SU*(2) ×
*U*(1) and the charged pions get a light mass,
*m*_{} ~ 35 MeV
[193], even in the
otherwise chiral symmetry limit.)

The term in Eq. (5.41) responsible for the pion mass prevents
us from setting =
/ 2 everywhere; the energy
density would behave
as
*f*^{2}_{}
*m*^{2}_{}
*R*^{2}, an infrared divergence worse than that
due to the wire with no chiral field present. We expect that
will vary from / 2 to 0 as
increases and that
asymptotically we will recover classical electrodynamics. Although we
cannot obtain a closed solution to Eqs. (5.43), if the
transition between =
/ 2 and
= 0 occurs at large
,
we can find an approximate solution. The approximation consists of
neglecting the (
)^{2} term in
Eq. (5.41); we shall return to this shortly. The solution of
these approximate equations of motion is

(5.47) |

_{0} is
a parameter to be
determined by minimizing the energy density of Eq. (5.41).
Note that for >
_{0}
the vector potential as well as the
field return to values these would have in the absence of any chiral
fields and that for
<
_{0}
the magnetic field decreases
exponentially as *B* ~
exp(-*ef*_{}
). The physical
picture is that, as in a superconductor, near
= 0 a
cylindrical current sheet
is set up that opposes the current in the wire and there is a return
current near =
_{0};
Ampère's law insures that the field at
large distances is as discussed above.
The energy density for the above configuration, neglecting
the spatial variation of , is

(5.48) |

where the dots represent infrared and ultraviolet regulated terms
which are, however, independent of
_{0}.
For _{0}
> 1 / *ef*_{}
the term involving the Bessel functions may be neglected and
minimizing the rest with respect to
_{0} yields

(5.49) |

This is the main result of this work.

We still have to discuss the validity of the two approximations we
have made. The neglect of the Bessel functions in Eq. (5.48) is valid for
*ef*_{}
_{0}
> 1 which in
turn provides a condition on the current *I*, *eI* /
*m*_{} > 2
sqrt[2] or more generally

(5.50) |

The same condition permits us to neglect the spatial variation of
around
=
_{0}. Let
vary from
/ 2 to 0 in the region
- *d*/2 to
+ *d*/2,
with 1 / *d* of the order of
*f*_{} or
*m*_{}. The
contribution of the variation of
to the energy density is
*H* =
^{3}
*f*^{2}_{}
_{0}
*d*. Eq. (5.50) insures that
*H* is smaller
than the other terms in Eq. (5.48).

Eq.(5.49) has a very straightforward explanation. It results
from a competition of the magnetic energy density 1 / 2 *B*^{2}
and the energy density of the pion mass term
*m*^{2}_{}
*f*^{2}_{}(1 - cos).
The magnetic field due to the current *I* is
*B* = *I* / 2
and the
transition occurs at *B* = *B*_{c}, with
*B*_{c} =
[sqrt2]*m*_{}
*f*_{}. The
reader may worry that the magnetic
fields very close to such thin wires are so large as to invalidate
completely the use of the chiral model as a low energy effective
QCD theory. In order to avoid this problem we may consider the
field due to a solenoid of radius *R*. The field is zero outside
the solenoid, **B** =
*B*()
**z** inside with *B*(*R*)=*B*_{0}. At
no point does the field become unboundedly large. Using the same
approximations as previously we obtain the following solutions of
the equations of motion (for *B*_{0}
*B*_{c})

(5.51) |

Continuity of the vector potential determines the coefficients
*a*_{1} and *a*_{2},

(5.52) |

*D*(*R*,
_{0})
= *K*_{1}(*e*
*f*_{}
*R*) *I*_{1}(*e*
*f*_{}
_{0})
- *K*_{1}(*e*
*f*_{}
_{0})
*I*_{1}(*e*
*f*_{} *R*)
and _{0}
is determined, once more, by minimizing the energy. For (*R*,
_{0})
> 1/*e* *f*_{}

(5.53) |

For *B*_{0}
*B*_{c},
= 0 everywhere and
*B*()
= *B*_{0} in the interior
of the solenoid. Thus, for any current configuration, the chiral fields
will adjust themselves to screen out fields larger than
*B*_{c}. Topological
excitations may occur in the form of magnetic vortices; the angular field
of Eq. (5.40) will wind
around a quantized flux tube of
radius 1 / *e*
*f*_{}
[171].