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E. Rabinovici
constrain further the allowed terms. This can also sometimes be turned around. The
collection of marginal and relevant operators may exhibit a symmetry of its own.
This offers a proof that the symmetry should be present at low energies but may well
disappear at higher energies if the irrelevant operators do not respect it. The resulting
Lagrangian is called the effective low energy Lagrangian. It should contain all the
light particles of the system and all the symmetries, may they be realized linearly or
nonlinearly, (when the symmetry at question is spontaneously broken). Integrating
out heavy particles results in a local Lagrangian consisting of a finite number of
terms. Integrating out light degrees of freedom is likely to lead to non-local effects.
The terms whose quantum scaling dimensions are smaller than d are called relevant
terms. They become very large as the system is probed at lower energies and
disappear at high energies, their number is usually finite and in many cases small.
Examples of such terms are mass terms for both bosons and fermions in general and
the non-Abelian Maxwell term for asymptotically free systems such as QCD. The
terms whose scaling dimensions are exactly d are called marginal operators, they
include the full Lagrangian for exactly scale invariant systems. Their number is also
generically finite. They are equally important at all scales. Terms whose dimension
is larger than d are called irrelevant operators. There is an infinite number of them.
Each term on its own becomes insignificant in the low energy region and renders the
theory non-renormalizable for high energies. (One can imagine examples where an
infinite number of such terms collaborate to become relevant but that would most
likely mean that the expansion around the critical surface should be modified).
Those terms which have scaling dimensions not very much larger than d (such
as five and six in four dimensions), can be useful hints for the scale at which the
physics needs to be modified. For example the original four Fermi interactions are
dimension six operators in four space-time dimensions. Their coefficient, the so
called Fermi coupling, has dimension (−2) and hints at the nearly Tev scale of the
weak interactions. Indeed at higher energies this irrelevant term is replaced by the
classically marginal gauge interactions of the standard model. The replacement of
an irrelevant operator by a relevant (or marginal) one at high energy leads to a well
defined theory and it called UV completion. The UV completion is not unique, but
its simplest version seems to do the job in the weak interactions case. This is clearly
to be done when, like for the weak interactions in the 1950s, the leading term in the
effective Lagrangian is irrelevant. When the irrelevant term appears in addition to
marginal and relevant ones it may or may not indicate new physics.
8.2.10 MSSM Lagrangian
All this said one can write down the simplest low energy theory that by definition
contains only marginal and relevant operators. For the case of systems with broken
SUSY which still do not have a hierarchy problem one may allow in addition
only those of the above operators which retain the at most logarithmic divergence
structure of the theory. This requires that the classical marginal operators are all
E. Rabinovici
constrain further the allowed terms. This can also sometimes be turned around. The
collection of marginal and relevant operators may exhibit a symmetry of its own.
This offers a proof that the symmetry should be present at low energies but may well
disappear at higher energies if the irrelevant operators do not respect it. The resulting
Lagrangian is called the effective low energy Lagrangian. It should contain all the
light particles of the system and all the symmetries, may they be realized linearly or
nonlinearly, (when the symmetry at question is spontaneously broken). Integrating
out heavy particles results in a local Lagrangian consisting of a finite number of
terms. Integrating out light degrees of freedom is likely to lead to non-local effects.
The terms whose quantum scaling dimensions are smaller than d are called relevant
terms. They become very large as the system is probed at lower energies and
disappear at high energies, their number is usually finite and in many cases small.
Examples of such terms are mass terms for both bosons and fermions in general and
the non-Abelian Maxwell term for asymptotically free systems such as QCD. The
terms whose scaling dimensions are exactly d are called marginal operators, they
include the full Lagrangian for exactly scale invariant systems. Their number is also
generically finite. They are equally important at all scales. Terms whose dimension
is larger than d are called irrelevant operators. There is an infinite number of them.
Each term on its own becomes insignificant in the low energy region and renders the
theory non-renormalizable for high energies. (One can imagine examples where an
infinite number of such terms collaborate to become relevant but that would most
likely mean that the expansion around the critical surface should be modified).
Those terms which have scaling dimensions not very much larger than d (such
as five and six in four dimensions), can be useful hints for the scale at which the
physics needs to be modified. For example the original four Fermi interactions are
dimension six operators in four space-time dimensions. Their coefficient, the so
called Fermi coupling, has dimension (−2) and hints at the nearly Tev scale of the
weak interactions. Indeed at higher energies this irrelevant term is replaced by the
classically marginal gauge interactions of the standard model. The replacement of
an irrelevant operator by a relevant (or marginal) one at high energy leads to a well
defined theory and it called UV completion. The UV completion is not unique, but
its simplest version seems to do the job in the weak interactions case. This is clearly
to be done when, like for the weak interactions in the 1950s, the leading term in the
effective Lagrangian is irrelevant. When the irrelevant term appears in addition to
marginal and relevant ones it may or may not indicate new physics.
8.2.10 MSSM Lagrangian
All this said one can write down the simplest low energy theory that by definition
contains only marginal and relevant operators. For the case of systems with broken
SUSY which still do not have a hierarchy problem one may allow in addition
only those of the above operators which retain the at most logarithmic divergence
structure of the theory. This requires that the classical marginal operators are all