8 Beyond the Standard Model
493
proportional only to the logarithem of the cut off and is proportional to the initial
mass of the fermion. The shift vanishes when the initial fermion mass vanishes. The
reason behind the vanishing of the mass of the fermion is that for a massless fermion
the system obtains a new symmetry, chiral symmetry. This symmetry should be
restored in the zero mass limit. As long as that symmetry is not broken the mass
remains zero. If the symmetry is dynamically broken the mass shift can be very
small. In the case of supersymmetrical systems it is the fermionic nature which
prevails and thus the bosonic mass shifts are small, in a supersymmetric theory
one may imagine that the hierarchy problem is solved. (The softer divergences
of the supersymmetric systems are of similar origin as those responsible for the
no renormalization theorems described before) The problem ab initio is a problem
which involves in some manner theoretical taste and in any case it seems from the
LEP data that if one insists on a hierarchy problem it is already present. SUSY has
not come to the rescue in time and if it is a symmetry of nature, solving the hierarchy
problem may well not be its main purpose.
An earlier attempt to solve the hierarchy problem involved the introduction of
a new gauge symmetry similar in many ways to color called technicolor. This is a
specific realization of the idea that the Higgs particle in not a fundamental particle.
Indeed both in describing aspects of superconductivity and superfluidity the Higgs
scalar is only an effective degree of freedom. As of 2007 these line of ideas were
not consistent with some of the experimental data. An often quoted problem is that
such an interaction induces flavor changing neutral interactions at a too high rate.
8.2.9 Effective Theories
One may wonder why the Lagrangians describing the basic interactions are
expressed in terms of a finite number of terms rather then by an infinite one. As
long as the laws of physics allow the decoupling of far away scales this can be
explained. The theory is written down in full generality in the presence of a short
distance/high energy cutoff. The cutoff may reflect the scale below which one
has no knowledge on the interactions. The terms are constrained only by possible
symmetries, their number is a priori infinite. The physics beneath any lower energy
scale,
is obtained by integrating out all the degrees of freedom which are heavier
than
. A new set of terms replaces the original set, it contains generically less
terms. This process can be repeated till a theory approaches a critical surface. The
theory on the critical surface is scale invariant. The resulting theory near the critical
surface is well described by operators whose scaling dimensions, in d space-time
dimensions, can be determined near the surface and happen to be smaller and at
most not much larger than d. In many cases there is only a finite number of such
operators, i.e., only a finite amount of terms in the Lagrangian describe the physics
near the critical surface. This not only explains the concise form of the Lagrangian
but provides one with a systematic method to classify the allowed terms to appear,
the power of the method is further enhanced when symmetries are present, as those
493
proportional only to the logarithem of the cut off and is proportional to the initial
mass of the fermion. The shift vanishes when the initial fermion mass vanishes. The
reason behind the vanishing of the mass of the fermion is that for a massless fermion
the system obtains a new symmetry, chiral symmetry. This symmetry should be
restored in the zero mass limit. As long as that symmetry is not broken the mass
remains zero. If the symmetry is dynamically broken the mass shift can be very
small. In the case of supersymmetrical systems it is the fermionic nature which
prevails and thus the bosonic mass shifts are small, in a supersymmetric theory
one may imagine that the hierarchy problem is solved. (The softer divergences
of the supersymmetric systems are of similar origin as those responsible for the
no renormalization theorems described before) The problem ab initio is a problem
which involves in some manner theoretical taste and in any case it seems from the
LEP data that if one insists on a hierarchy problem it is already present. SUSY has
not come to the rescue in time and if it is a symmetry of nature, solving the hierarchy
problem may well not be its main purpose.
An earlier attempt to solve the hierarchy problem involved the introduction of
a new gauge symmetry similar in many ways to color called technicolor. This is a
specific realization of the idea that the Higgs particle in not a fundamental particle.
Indeed both in describing aspects of superconductivity and superfluidity the Higgs
scalar is only an effective degree of freedom. As of 2007 these line of ideas were
not consistent with some of the experimental data. An often quoted problem is that
such an interaction induces flavor changing neutral interactions at a too high rate.
8.2.9 Effective Theories
One may wonder why the Lagrangians describing the basic interactions are
expressed in terms of a finite number of terms rather then by an infinite one. As
long as the laws of physics allow the decoupling of far away scales this can be
explained. The theory is written down in full generality in the presence of a short
distance/high energy cutoff. The cutoff may reflect the scale below which one
has no knowledge on the interactions. The terms are constrained only by possible
symmetries, their number is a priori infinite. The physics beneath any lower energy
scale,
is obtained by integrating out all the degrees of freedom which are heavier
than
. A new set of terms replaces the original set, it contains generically less
terms. This process can be repeated till a theory approaches a critical surface. The
theory on the critical surface is scale invariant. The resulting theory near the critical
surface is well described by operators whose scaling dimensions, in d space-time
dimensions, can be determined near the surface and happen to be smaller and at
most not much larger than d. In many cases there is only a finite number of such
operators, i.e., only a finite amount of terms in the Lagrangian describe the physics
near the critical surface. This not only explains the concise form of the Lagrangian
but provides one with a systematic method to classify the allowed terms to appear,
the power of the method is further enhanced when symmetries are present, as those