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\head{Step-and-Integrate Interferometry\newline
in the Mid-Infrared with\newline
Photothermal Beam Deflection}
\author{R.A. Palmer@1, M.J. Smith@1, C.J. Manning@1, J.L. Chao@2,
A.C. Boccara@3, D.~Fournier@3}
\address{@1Department of Chemistry, Duke University, Durham, NC\ts
27706, USA @2IBM, Research Triangle Park, NC\ts 27713, USA
@3Laboratoire d'Optique Physique, ESPCI, 10, rue Vauquelin,\newline
F-75231 Paris, Cedex 05, France}
%
\headrunning{Step-and-Integrate Interferometry in the Mid-Infrared}%
\authorrunning{R.A. Palmer et al.}%
\titlea{1.}{Introduction}
In this series of three lectures I shall try to present some of the
concepts from elementary particle physics that play a r\^ole in astrophysics.
These lectures shall, together with those given by G.~B\"orner who will
present concepts from astrophysics and cosmology, give an introduction
to this school.
\titlea{2.}{The Standard Model of Electro-Weak\newline
 and Strong Interaction}
In the standard model the electro-weak and strong
interaction of elementary particles is described by an
$SU(3)\times SU(2)\times U(1)$ gauge theory.
In addition there is, of course, the gravitational interaction
described (at least classically) by {\it Einstein's General Relativity}.
The gauge group $SU(2)\times U(1)$ of the electromagnetic and weak
interaction is spontaneously broken; only a $U(1)$ subgroup
(the electromagnetic $U(1)$) remains unbroken.
 
\titlea{3.}{Spontaneous Symmetry Breaking}
We have already encountered one of the many topics in elementary particle
physics where spontaneous symmetry breaking plays a r\^ole,
the standard model.
 
\titleb{3.1}{The $\vec{Z_2}$ Goldstone Model}
The Goldstone model, in its simplest form, describes a self interacting
real scalar field $\varphi(x)$ with the Lagrangean
(I am using space-time coordinates $x^\mu$ ($\mu=0,\ldots,3$) and
a Minkowski metric with signature $(+$,$-$,$-$,$-)$)
$${\cal L}={\textstyle{1\over2}}(\partial_\mu
\varphi\partial^\mu\varphi-\mu^2\varphi^2-{\textstyle{1\over
\scriptstyle 2}}h\varphi^4)={\textstyle{1\over\scriptstyle 2}}
(\partial_\mu\varphi\partial^\mu\varphi-V(\varphi))\eqno(1)$$
and the field equation
$$\partial^\mu\partial_\mu\varphi+\mu^2\varphi+h\varphi^3=0\quad.\eqno(2)$$
The Lagrangean and the field equations are invariant under the symmetry
group $G=Z_2$ generated by $\varphi(x)\to-\varphi(x)$ and under the usual
space-time symmetries (Poincar\'e group, parity $P$ and time reversal
$T$;
see Fig.\ts 1).
The ground state is determined by a classical solution that minimizes
the energy
$$E=\int{\cal H}\;,\qquad
{\cal H}={\textstyle{1\over\scriptstyle 2}}[(\partial_0\varphi)^2
+(\nabla\varphi)^2+V(\varphi)]\quad.\eqno(3)$$
 
\begfig 4.2 cm
\figure{1} {Shape of the potential $V(\varphi)$ for $\mu^2>0$ (symmetric
phase) and for $\mu^2<0$ (spontaneously broken phase)}
\endfig
 
For $\mu^2\ge0$ the only such solution, $\varphi=0$, is the $Z_2$
invariant absolute minimum of the energy.
For $\mu^2<0$ there are, however, the three solutions $\varphi=0$
and $\varphi=\pm f$ with $f=\sqrt{-\mu^2/h}$.
 
\titleb{3.2}{The $\vec{U(1)}$ Goldstone Model}
Let us now consider a self-interacting complex scalar field $\varphi(x)$
(or equivalently two real scalar fields
$\varphi_1=(\varphi+\bar\varphi)/\sqrt{2}$
and $\varphi_2=(\varphi-\bar\varphi)/\sqrt{2}i$)
with the Lagrangean
$${\cal L}=\partial_\mu\bar\varphi\partial^\mu\varphi
    -\mu^2\bar\varphi\varphi-{\textstyle{1\over\scriptstyle 2}}
h(\bar\varphi\varphi)^2=
  \partial_\mu\bar\varphi\partial^\mu\varphi-V(\varphi)$$
and the field equation
$$\partial^\mu\partial_\mu\varphi+\mu^2\varphi
      +h\varphi(\bar\varphi\varphi)=0\;.\eqno(4)$$
 
\titlea{4.}{Topologically Stable Configurations (Defects)}
Here I want to discuss certain field configurations with finite energy
that are stable owing to their topological structure.
Typical such structures are solitons, vortices and monopoles in one, two
and three space dimensions respectively and in four space dimensions
(i.e. in Euclidean space-time) there are instantons.
Since we live in a world with three space dimensions, solitons and vortices
are to be understood as solutions which are (at least approximately)
independent of two or one space coordinate, i.e. as domain walls or
membranes with finite energy per unit surface or as vortex lines or strings
with finite energy per unit length (see Fig.\ts 4).
 
\titleb{4.1}{Solitons in the $\vec{Z_2}$ Goldstone Model}
In the $Z_2$ Goldstone model the potential $V(\varphi)$ has two minima
at $\varphi=\pm f$, and a field configuration with finite energy must
be close to one of these two values almost everywhere.
In particular $\varphi$ must go to one of these two values in the
asymptotic regions $|\vec x|\to\infty$ (Fig.\ts 5).
\begdoublefig 4.7 cm
{\figure{4} {Example of a Higgs potential $V(\varphi)$ with two minima
of different depth. The absolute minimum at $\varphi=-f_1$ corresponds
to a stable ground state (true vacuum); the local minimum at
$\varphi=+f_2$ corresponds to a meta-stable ``ground state'' (false
vacuum)}}
{\figure{5} {Typical form of $\varphi(r)$ for a bubble of true vacuum
inside a background of false vacuum}} \enddoublefig
\bigskip\vdots
\titlea{7.}{Grand Unified Theories}
We have seen that the standard model describes the electro-weak interaction
with a spontaneously broken $SU(2)\times U(1)$ gauge theory
(Glashow Salam Weinberg theory), whereas the strong interaction is described
by an unbroken $SU(3)$ gauge theory.
\begfig 4.6 cm %\FIG\FIGdecay{}
\figure{7} {Possible diagrams for the proton decay reactions
$p\to e^+\pi^0$ and $p\to\bar\nu_e\pi^+$ with the exchange of a heavy
GUT vector boson}
\endfig
 
\titlea{8.}{Photon Splitting in a Strong Magnetic Field}
Next I want to discuss an effect from rather old-fashioned QED that might
have astrophysical implications.
Consider the propagation of photons in a strong external magnetic field.
{\it Adler} $[2]$ has computed the one-loop contributions to this process
exactly, but in order to get estimates it suffices to compute the
lowest-order contribution from box diagrams (Delbr\"uck scattering).
Moreover, for slowly varying external fields and low photon frequencies
one can use the effective Lagrangean derived more than half a
century ago $[3]$
$${\cal L}_{\rm eff}=-{1\over4} F\cdot F
 +{\alpha\over 192 B_{\rm cr}^2}[(F\cdot
F)^2+(F\cdot{}^*\!F)^2]\eqno(9)$$
with $F\cdot F=F_{\mu\nu}F^{\mu\nu}=2(\vec B\cdot\vec B-\vec E\cdot\vec E)$
and $F\cdot{}^*\!F=1/2\varepsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}
=4 \vec E\cdot\vec B$.
The quartic term is due to the creation and annihilation of virtual
electron positron pairs and is negligible for magnetic fields much smaller
than the critical field $B_{\rm cr}=m^2/e=4.41\times10^{13}\,{\rm G}$.
 
According to conventional electrodynamics one computes the response
to the variation of the electric and magnetic field
$$\vec D={\partial{\cal L}\over \vec E}=\vec E+\ldots\;,\quad
  \vec H=-{\partial{\cal L}\over \vec E}=\vec B+\ldots\eqno(10{\rm a})$$
and the tensors of dielectric and magnetic susceptibility
as well as a tensor $\varrho_{ij}$ that can give rise to optical activity
$$\eqalignno{\varepsilon_{ij}&={\partial D_i\over\partial
  E_j}=\delta_{ij}+\ldots,\
  \mu^{-1}_{ij}={\partial H_i\over\partial B_j}=\delta_{ij}+\ldots,\
  \varrho_{ij}\cr
&={\partial D_i\over\partial B_j}
=-{\partial H_j\over\partial E_i}=0+\ldots\;.&{\rm(10b)}\cr}$$
From these tensors one can compute the propagation of photons, and finds
that a strong magnetic field leads to double refraction.
Photons with polarization parallel and perpendicular to the external
magnetic field have slightly different indices of refraction.
 
From the same effective Lagrangean one can compute transition amplitudes
for the splitting of one photon into two photons.
The opening angle for such processes tends to zero for small external fields
but the phase space remains finite provided the splitting process is
kinematically allowed.
Adler has found that there is only one process allowed by this kinematic
constraint
as well as by the CP selection rule: the splitting of one photon with
polarization parallel to the magnetic field into two photons both with
perpendicular polarization.
Thus a very strong magnetic field which is homogeneous over a sufficiently
large region will convert unpolarized photons into polarized ones.
It has been speculated that such magnetic fields might be present in the
vicinity of neutron stars and that they might be responsible for the
observed polarization of photons from pulsars.
 
\titlea{9.}{Candidates for Dark Matter from Particle Physics}
Elementary particle physics theory has a long tradition of predicting
new particles on purely theoretical grounds (mostly on the consistency or
simplicity of the theoretical description).
Some of these postulated new particles have subsequently been confirmed
by experimental observation (e.g. the antiparticles $e^+$ and $\bar p$,
the neutral particles $n$ and $\nu$ and, more recently, the
massive vector mesons $Z^0$ and $W^\pm$)
or are theoretically well established (e.g. quarks and gluons).
Others (e.g. $t$-quarks, gravitinos, axions, \dots) have not (yet?)
been observed.
 
Likewise there is a long tradition and a lot of imagination has been
invested to explain why this or that particle has not yet been observed.
 
Here I want to discuss some of the particles proposed by various theories
that have not yet been observed in experiments.
Some of these somewhat unconventional particles could be the
constituents of dark matter.
 
\titleb{9.1}{Neutrinos with a Non-vanishing Mass}
Let us start rather conventionally with the possibility that
neutrinos could have a non-vanishing rest mass.
The present experimental limits from particle physics experiments
$$
  m_{\nu_e}    \la{\rm 10-15\;eV}\;,\qquad
  m_{\nu_\mu}  \la{\rm 150\;keV}\;,\qquad
  m_{\nu_\tau} \la{\rm a\ few\ keV}\eqno(11)
 $$
leave room for plenty of speculations and will be discussed extensively
in other lectures of this school.
 
\titleb{9.2}{New Particles from Supersymmetry or Supergravity}
Let me start with a short answer to the questions:
What is super supersymmetry (SUSY) and what is supergravity (SUGRA)?
SUSY is a (global) symmetry that puts bosons and fermions (i.e. particles
with different spin) into the same multiplet.
SUGRA is gauged SUSY, i.e. SUSY with space-time-dependent parameters.
Equivalently one might say that SUGRA is SUSY in the presence of gravitons
or SUSY on a curved space-time manifold because each one of these three
definitions implies the other ones.
 
\titlec{9.2.1}{Supersymmetry.}
In the past there have been many attempts to find symmetries which unify
particles with different spins in the sense that they occur in the same
multiplet.
Although this can be achieved in a non-relativistic theory all such attempts
turned out to be incompatible with relativistic QFT.
Later on {\it O'Raifeartaigh} et al. $[4]$ proved ``no-go'' theorems,
which state that the only possible generators of symmetries in a relativistic
local QFT are the generators of the Poincar\'e group plus scalar charges,
and thus all particles in a multiplet must have the same mass {\it  and spin}.
These theorems always (implicitly) assume that the
generators (charges) of the symmetry are bosonic and
therefore transform bosons into bosons and fermions into
fermions.
They do not apply to symmetries such as SUSY with fermionic
generators transforming bosons into fermions and vice versa.
Nevertheless, these theorems impose restrictions on the structure of SUSY
algebras because they do apply to the bosonic part of such algebras $[5]$.
 
\def\Q#1{Q_{#1}}   \def\bQ#1{\bar Q_{\dot #1}}
\def\Qa{\Q\alpha}  \def\bQa{\bQ\alpha}
 
The simplest ($N=1$) SUSY algebra is generated by a
fermionic two-component spinor charge $\Qa$ and its
adjoint $\bQa$ satisfying the (anti)
commutation relations
$$\displaylines{
  \{\Qa,\bQa\}=\sigma_{\alpha\dot\alpha}^\mu P_\mu\;,\quad
  \{Q,Q\}=\{\bar Q,\bar q\}=[Q,P]=[\bar
Q,P]=0\;,\hfil\llap{\rm(12a)}\hfilneg\cr
  [J_{\mu\nu},\Qa]={i\over2}(\sigma_{\mu\nu})_\alpha{}^\beta\Q\beta\;,\quad
  [J_{\mu\nu},\bQa]={i\over2}(\bar\sigma_{\mu\nu})_{\dot\alpha}{}^{\dot\beta}
\bQ\beta\quad.\hfil\llap{\rm(12b)}\hfilneg\cr
}$$
Because of these commutation relations each SUSY multiplet must contain the
same number of bosonic and fermionic states.
The detailed structure is, however, different for $m\ne0$ and for $m=0$.
 
For non-vanishing rest mass ($m\ne0$) a multiplet can be characterized by
an integer or half-integer spin $j=0,1/2,1,\ldots$ and contains
four states with spins $j-1/2$, $j$, $j$ and $j+1/2$, where the two
states with spin $j$ have opposite parity.
As usual, the state with spin $j-1/2$ is absent for $j=0$.
The simplest such multiplet with $j=0$ contains three states with spin
parity $0^+$, $0^-$ and $1/2$ corresponding to the fields $A(x)$, $B(x)$
and $\psi(x)$ of the Wess-Zumino model.
Here, $A$ is a real scalar, $B$ a real pseudo-scalar and $\psi$ a
Majorana spinor, i.e. $\psi$ satisfies a reality condition and
describes particles that are their own anti-particles.
The next multiplet with $j=1/2$ is a massive vector multiplet with states
$0^+$, $1/2$, $1/2$ and $1^-$.
 
For vanishing rest mass ($m=0$) the multiplets are characterized by an
integer or half-integer helicity $\lambda=\ldots,-1,-1/2,0,1/2,1,\ldots$
and contain two states with helicities $\lambda$ and $\lambda+1/2$.
Invariance under P or C requires that the multiplet with helicities
$-\lambda-1/2$ and $-\lambda$ exists as well.
Consider, e.g., the super Yang-Mills (SYM) multiplet, which contains,
for each generator of the gauge group, states with helicities $\pm1/2$ and
$\pm1$ described by a Majorana spinor field $\vec\chi(x)$ and the
gauge field $\vec A_\mu(x)$.
 
Extended SUSY algebras are generated by $N>1$ different super charges
$\Qa^I$ ($I=1,\ldots,N$) satisfying
$$\{\Qa^I,\bQa^J\}=\delta^{IJ}\sigma_{\alpha\dot\alpha}^\mu
P_\mu\quad,\eqno(13)$$
and give rise to larger multiplets.
For $m=0$ a multiplet contains states with helicities in the range
$\lambda_{\rm min}\le\lambda\le\lambda_{\rm max}$, where
$\lambda_{\rm max}-\lambda_{\rm min}=N/2$, and where the state with
helicity $\lambda$ has multiplicity $N\choose 2(\lambda-\lambda_{\rm min})$.
Again we may have to add the P (or C) conjugate multiplet unless the multiplet
is self-conjugate ($\lambda_{\rm max}=N/4$).
If we require $|\lambda|\le1/2$ (absence of gauge fields) $N\le2$ must be
satisfied.
From $|\lambda|\le1$ (gauge fields without gravity) follows $N\le4$
and $|\lambda|\le2$ (theories with gravity) implies $N\le8$.
For $N>8$ (in $d=4$ space-time dimensions), i.e. with $|\lambda|>2$,
no consistent interacting theory has been constructed and there are
indications that no such theory can exist.
 
Unlike for other symmetries (e.g. $SU(2)$, $SU(3)$, \dots),
the trivial representation of SUSY is of no interest in particle physics.
If a (boson or fermion) field $\varphi(x)$ transforms trivially
under SUSY
$$\displaylines{[Q,\varphi(x)]_\pm=[\bar
Q,\varphi(x)]_\pm=0\quad\hbox{, then}\hfil\llap{(14)}\hfilneg\cr
[\{Q,\bar Q\},\varphi(x)]=[\sigma\cdot P,\varphi(x)]
=-i\sigma^\mu\partial_\mu\varphi(x)=0\quad,\hfil\llap{(15)}\hfilneg\cr}$$
i.e. $\varphi(x)$ is constant and describes no particles.
If nature is supersymmetric then each particle must have its SUSY partner.
The SUSY partners of bosons (e.g. the photino and gluino for the photon
and gluon) are fermions, the partners of fermions (e.g. the squark and
selectron for the quark and electron) are bosons; all the internal quantum
numbers are the same for both partners (at least in the case of $N=1$ SUSY).
 
Nature is certainly not supersymmetric. One might assume that SUSY is
explicitely broken, an assumption with very little predictive power.
If SUSY is spontaneously broken there must be a massless Goldstone
fermion with helicity $\lambda=\pm1/2$ (goldstino).
It was once speculated that this could be the neutrino,
but it turned out that such an assignment leads to phenomenologically
unacceptable predictions.
The simplest mechanisms for spontaneous SUSY breaking lead to
predictions for the mass splitting within SUSY multiplets.
 
So far, none of the SUSY partners of known particles has been found.
It might be that SUSY breaking occurs at a very high mass scale
and thus the mass splittings are large. In this situation very
little, if anything, of SUSY would be left at low energies.
 
There are models for, e.g., supersymmetric QED, which are, however,
(in my opinion) not terribly convincing.
 
\titlec{9.2.2}{Supergravity.}
If we consider SUSY with space-time-dependent parameters (gauged SUSY)
we are led to SUGRA.
Gravity becomes involved because space-time-dependent SUSY transformations
imply, via the SUSY algebra, space-time-dependent translations, i.e.
infinitesimal diffeomorphisms.
 
This argument can also be reversed: the combination of SUSY and gravity
yields SUGRA.
First the graviton must have a SUSY partner with helicity
$\lambda=\pm3/2$ (the other possibility $\lambda=\pm5/2$ does not lead to
a consistent theory), the gravitino.
This gravitino is described by a Rarita-Schwinger field $\psi_\mu$,
which is the gauge potential for SUSY.
Moreover, the concept of a constant spinor is meaningless in a generic curved
space-time background and thus the parameters of SUSY transformations
necessarily must be space-time dependent in the presence of gravity.
 
The minimal $N=1$ SUGRA theory describes a graviton and a gravitino.
Various matter multiplets can be coupled to these,
essentially by substituting super covariant derivatives for ordinary ones.
 
The maximal $N=8$ SUGRA theory in $d=4$ space-time dimensions can conveniently
be derived via dimensional reduction from a much simpler $N=2$, $D=10$
or $N=1$, $D=11$ theory.
The $N=8$ multiplet consists of 128~bosonic and 128~fermionic states;
in four dimensions these are
1~graviton with $\lambda=\pm2$,
8~gravitinos with $\lambda=\pm3/2$,
28~vector (gauge) fields with $\lambda=\pm1$,
56~Majorana spinors with $\lambda=\pm1/2$,
35~real scalars and 35~real pseudoscalars with $\lambda^P=0^\pm$.
The 56~vector fields might be the gauge fields of (a subgroup of) $SO(8)$.
Note, however, that the gauge group $SU(3)\times SU(2)\times U(1)$ of the
standard model is not contained in $SO(8)$.
 
If SUGRA is spontaneously broken a supersymmetric version of the
Higgs-Kibble mechanism leads to a massive gravitino instead of
a massless goldstino.
 
Originally it was hoped that SUGRA would solve the problem that gravity
(considered as QFT) is non-renormalizable.
SUGRA, indeed, yields finite one- and two-loop scattering amplitudes
but three-loop divergences cannot be excluded.
By power counting SUGRA is as non-renormalizable as gravity.
 
\titleb{9.3}{New Particles from String Theories}
The fundamental objects of local QFT are point particles.
In string theory they are replaced by elementary (not cosmic) strings
which are extended in one space dimension.
The motion of such a string in time yields a two-dimensional world
sheet instead of the familiar world line.
In order to describe this world sheet embedded into space-time
one uses coordinates $\xi^\alpha$ and a metric $\gamma_{\alpha\beta}(\xi)$
for the two-dimensional world sheet, and coordinates $X^\mu$
and a metric $g_{\mu\nu}(X)$ for $D$-dimensional space-time
(possibly curved).
The dynamics of a bosonic string is described by an action equal to
the area of its world sheet:
$$ S=\int d^2\xi \sqrt{\gamma(\xi)} \gamma^{\alpha\beta}(\xi)
   {\partial X^\mu(\xi)\over\partial\xi^\alpha}
   {\partial X^\nu(\xi)\over\partial\xi^\beta}
g_{\mu\nu}(X(\xi))\quad.\eqno(16)$$
This action has the same form as for a two-dimensional
non-linear $\sigma$-model.
 
If we add SUSY, either in the two-dimensional $\xi$-space or
equivalently in the $D$-dimensional $X$-space, we obtain superstrings.
 
The classical (super) string action is invariant under reparametrizations
of the world sheet and under conformal transformations
$\gamma_{\alpha\beta}(\xi)\to \rho^2(\xi)\gamma_{\alpha\beta}(\xi)$.
The quantized theory has, in general, a conformal anomaly (trace anomaly).
The presence of such an anomaly leads to inconsistencies in the physical
interpretation, e.g. ghost states.
The conformal anomaly is, however, absent if we choose (in the case
of flat $X$-space) $D=26$ for the bosonic string or $D=10$ for
the superstring.
One usually assumes that the 10-dimensional $X$-space of super string
theories can be reduced to a 6-dimensional internal space and a 4-dimensional
space-time through a compactification of the Ka{\l}uza-Klein type.
 
The string action describes both the motion of the center of mass of
a string and string excitations (internal motion).
It is the sum over all these excitations which might lead to a consistent
quantum theory in the presence of gravity.
In the ``zero slope'' approximation the string excitations are extremely heavy.
The remaining ``massless'' excitations are those of a (super) Yang-Mills theory
for open (super) strings and those of (super) gravity for closed
(super) strings.
 
There are, in fact, several versions of superstring theory.
\medskip\item{1.} Type I superstrings have $N=1$ SUSY in $D=10$,
corresponding to $N=4$ in $d=4$.
\item{2.} Type IIa and type IIb superstrings have two different versions
of $N=2$ SUSY in $D=10$, both corresponding to $N=8$ in $d=4$.
\item{3.} Finally heterotic strings have different assignments for left and
right moving excitations.
\medskip
For reasons of consistency the heterotic strings require a gauge group
$E_8\times E_8$.
One $E_8$ may be useful as grand unification group with a breaking scheme
$E_8\supset E_7\supset E_6\supset SO(10)\supset SU(5)$
but the other $E_8$ is phenomenologically completely superfluous.
All known particles or fields transform under representations
${\it any}\times1$ of $E_8\times E_8$.
Why is this so? What is the r\^ole of
the other fields transforming with representations
$1\times{\it any}$ or ${\it any}\times{\it any}$ of $E_8\times E_8$?
One possibility would be that these excitations do exist, but do not
interact with the known particles, except via gravitation.
Does such a ``shadow world'' with ``shadow states'' or ``shadow
particles''
exist? And if so, could they be candidates for dark matter?
 
\titleb{9.4}{Axions}
Let me conclude this discussion of possible dark-matter candidates with
the axion, a light pseudo-scalar particle postulated on purely theoretical
grounds.
 
First consider the ground state (vacuum) in a theory with non-abelian
gauge fields.
It turns out that there are, in fact, infinitely many ``$\theta$-vacua''
that differ by the value of a parameter $\theta$ that is not present
in the classical Lagrangean $[6]$.
If we describe the ground state in terms of a path integral over Euclidean
field configurations this parameter is due to the existence of
(anti) soliton configurations: we can attribute a phase factor
${\rm e}^{{\rm i}n\theta}$ to all configurations with soliton or winding
number $n$.
 
Next consider in addition scalars, pseudo-scalars and spinors coupled
such that the classical action is invariant under a chiral $U(1)$ group.
In the quantized theory there is a chiral anomaly proportional to
$\vec F\cdot{}^*\!\vec F$, i.e. the integrated chiral anomaly is
proportional to the winding number.
A chiral rotation is therefore equivalent to a shift in $\theta$.
Thus if all fermions are massless then all $\theta$-vacua are physically
equivalent.
If the fermions are, however, massive such a chiral rotation
transforms a scalar mass term ($\sim\bf1$) into a pseudo-scalar one
($\sim\gamma_5$).
 
It is experimentally obvious that we live in a world where P and CP are
good symmetries at the level of strong interactions.
This means that in our world we have simultaneously scalar mass terms
and $\theta=0$.
In a general theory this can only be understood as the result of some
fine tuning, an idea that theorists find very unattractive.
One can, however, obtain the same result in a ``natural''
if an additional pseudo scalar particle, the axion, exists $[7]$.
This axion should have a mass in the range 100\ts KeV\ts--\ts1\ts MeV
or could even be nearly massless;
it could be a candidate for dark matter and would have other experimentally
observable consequences $[8]$.
Such an axion has, however, not (yet?) been observed experimentally.
 
\begref
\ref{1.} S. Coleman, F. de Luccia: Phys. Rev.~D {\bf 21}
3305 (1980)
\ref{2.} S.L. Adler, J.N. Bahcall, C.G. Callan,
M.E. Rosenbluth: Phys. Rev. Lett. {\bf 25} 1061 (1970);\newline
S.L. Adler: Ann. Phys. (N.Y.) {\bf 67} 599 (1971)
\ref{3.} H. Euler, B. Kockel:
Naturwissenschaften {\bf 23} 246 (1935);\newline
W. Heisenberg, H. Euler: Z. Phys. {\bf 98} 714 (1936)\newline
V. Wei\ss kopf: Kgl. Danske Vidensk. Selsk.
Mat.-Fys. Medd. {\bf 14} No. 6 (1936).
\ref{4.} S. Coleman, J. Mandula: Phys. Rev. {\bf 159} 1251 (1967)
\ref{5.} R. Haag, J.T. {\L}opuszanski, M. Sohnius:
Nucl. Phys. B {\bf 88} 257 (1975)
\ref{6.} G. 't Hooft: Phys. Rev. Lett. {\bf 37} 8 (1976).
Phys. Rev. D {\bf 14} 3432 (1976)
\ref{7.} R.D. Peccei, H.R. Quinn: Phys. Rev.
Lett. {\bf 38} 1440 (1977)
\ref{8.} S. Weinberg: Phys. Rev. Lett. {\bf 40} 223 (1978);
F. Wilczek: Phys. Rev. Lett. {\bf 40} 279 (1978);\newline
J.E. Kim: Phys. Rev. Lett. {\bf 43} 103 (1979)
\endref
 
\bye
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