Renormalization and Lorentz violation

1
Renormalization and Lorentz violation

• Damiano Anselmi
• based on the papers arxiv0707.2480 hep-th
  (PRD), with M. Halat,
• arxiv0801.1216 hep-th (JHEP), arXiv0808.3470
  hep-th,
• arXiv0808.3474 hep-th and arXiv0808.3475
  hep-ph

2
Lorentz symmetry is a basic ingredient of the
Standard Model of particles physics.
3
Lorentz symmetry is a basic ingredient of the
Standard Model of particles physics.
However, several authors have argued that at high
energies Lorentz symmetry and possibly CPT could
be broken.
4
Lorentz symmetry is a basic ingredient of the
Standard Model of particles physics.
However, several authors have argued that at high
energies Lorentz symmetry and possibly CPT could
be broken.
The Lorentz violating parameters of the Standard
Model (Colladay-Kostelecky) extended in the
power-counting renormalizable sector have been
measured with great precision.
5
Lorentz symmetry is a basic ingredient of the
Standard Model of particles physics.
However, several authors have argued that at high
energies Lorentz symmetry and possibly CPT could
be broken.
The Lorentz violating parameters of the Standard
Model (Colladay-Kostelecky) extended in the
power-counting renormalizable sector have been
measured with great precision.
It turns out that Lorentz symmetry is a very
precise symmetry of Nature, at least in
low-energy domain.
6
Lorentz symmetry is a basic ingredient of the
Standard Model of particles physics.
However, several authors have argued that at high
energies Lorentz symmetry and possibly CPT could
be broken.
The Lorentz violating parameters of the Standard
Model (Colladay-Kostelecky) extended in the
power-counting renormalizable sector have been
measured with great precision.
It turns out that Lorentz symmetry is a very
precise symmetry of Nature, at least in
low-energy domain.
Several parameters have bounds
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The set of power-counting renormalizable theories
is considerably small
10
The set of power-counting renormalizable theories
is considerably small
Relaxing some assumptions can enlarge it, but
often it enlarges it too much
11
The set of power-counting renormalizable theories
is considerably small
Relaxing some assumptions can enlarge it, but
often it enlarges it too much
Without locality in principle every theory can be
made finite
12
The set of power-counting renormalizable theories
is considerably small
Relaxing some assumptions can enlarge it, but
often it enlarges it too much
Without locality in principle every theory can be
made finite
Without unitarity even gravity can be renormalized
13
The set of power-counting renormalizable theories
is considerably small
Relaxing some assumptions can enlarge it, but
often it enlarges it too much
Without locality in principle every theory can be
made finite
Without unitarity even gravity can be renormalized
Relaxing Lorentz invariance appears to be
interesting in its own right
14
The set of power-counting renormalizable theories
is considerably small
Relaxing some assumptions can enlarge it, but
often it enlarges it too much
Without locality in principle every theory can be
made finite
Without unitarity even gravity can be renormalized
Relaxing Lorentz invariance appears to be
interesting in its own right
It could be useful to define the ultraviolet
limit of quantum gravity, to study extensions of
the Standard Model, effective field theories,
nuclear physics, and the theory of critical
phenomena
15
The set of power-counting renormalizable theories
is considerably small
Relaxing some assumptions can enlarge it, but
often it enlarges it too much
Without locality in principle every theory can be
made finite
Without unitarity even gravity can be renormalized
Relaxing Lorentz invariance appears to be
interesting in its own right
It could be useful to define the ultraviolet
limit of quantum gravity, to study extensions of
the Standard Model, effective field theories,
nuclear physics, and the theory of critical
phenomena
Here we are interested in the renormalization of
Lorentz violating theories obtained improving
the behavior of propagators with the help of
higher
space derivatives and study under which
conditions no
higher time derivatives are turned on
16
The set of power-counting renormalizable theories
is considerably small
Relaxing some assumptions can enlarge it, but
often it enlarges it too much
Without locality in principle every theory can be
made finite
Without unitarity even gravity can be renormalized
Relaxing Lorentz invariance appears to be
interesting in its own right
It could be useful to define the ultraviolet
limit of quantum gravity, to study extensions of
the Standard Model, effective field theories,
nuclear physics, and the theory of critical
phenomena
Here we are interested in the renormalization of
Lorentz violating theories obtained improving
the behavior of propagators with the help of
higher
space derivatives and study under which
conditions no
higher time derivatives are turned on
Some models are already in use in the theory of
critical phenomena to describe the critical
behavior at Lifshitz points, with a variety of
applications to real physical systems
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Scalar fields
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Scalar fields
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Scalar fields
Start from the free theory
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Scalar fields
Start from the free theory
This free theory is invariant under the
weighted scale transformation
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Scalar fields
Start from the free theory
This free theory is invariant under the
weighted scale transformation
Add vertices constructed
with , and .
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Scalar fields
Start from the free theory
This free theory is invariant under the
weighted scale transformation
Add vertices constructed
with , and . Call
their degrees under
N number of legs, extra label
23
Scalar fields
Start from the free theory
This free theory is invariant under the
weighted scale transformation
Add vertices constructed
with , and . Call
their degrees under
N number of legs, extra label
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Other quadratic terms can be treated as
vertices for the purposes of renormalization
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Other quadratic terms can be treated as
vertices for the purposes of renormalization
Consider a diagram G with L loops, I internal
legs, E external legs and vertices of type (N ,
)
26
Other quadratic terms can be treated as
vertices for the purposes of renormalization
Consider a diagram G with L loops, I internal
legs, E external legs and vertices of type (N ,
)
27
Other quadratic terms can be treated as
vertices for the purposes of renormalization
Consider a diagram G with L loops, I internal
legs, E external legs and vertices of type (N ,
)
is a weighted measure of degree
28
Other quadratic terms can be treated as
vertices for the purposes of renormalization
Consider a diagram G with L loops, I internal
legs, E external legs and vertices of type (N ,
)
is a weighted measure of degree
is a homogeneous weighted function of degree
29
Other quadratic terms can be treated as
vertices for the purposes of renormalization
Consider a diagram G with L loops, I internal
legs, E external legs and vertices of type (N ,
)
is a weighted measure of degree
is a homogeneous weighted function of degree
Its overall divergent part is a homogeneous
weighted polynomial of degree
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Using the standard relations
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Using the standard relations
we get
Where
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Using the standard relations
we get
Where
Renormalizable theories have
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Using the standard relations
we get
Where
Renormalizable theories have
Indeed
implies
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Homogeneous (i.e. strictly renormalizable)
theories have
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Homogeneous (i.e. strictly renormalizable)
theories have
Writing
we see that polynomiality demands
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Homogeneous (i.e. strictly renormalizable)
theories have
Writing
we see that polynomiality demands
and the maximal number of legs is
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Homogeneous (i.e. strictly renormalizable)
theories have
Writing
we see that polynomiality demands
and the maximal number of legs is
E 2 implies 2
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Homogeneous (i.e. strictly renormalizable)
theories have
Writing
we see that polynomiality demands
and the maximal number of legs is
E 2 implies 2
E gt 2 implies lt 2
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Homogeneous (i.e. strictly renormalizable)
theories have
Writing
we see that polynomiality demands
and the maximal number of legs is
Conclusion renormalization does not turn
on higher time derivatives
E 2 implies 2
E gt 2 implies lt 2
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Homogeneous models
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Homogeneous models
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Homogeneous models
They are classically weighted scale invariant,
namely invariant under
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Homogeneous models
They are classically weighted scale invariant,
namely invariant under
The weighted scale invariance is anomalous at the
quantum level
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Homogeneous models
They are classically weighted scale invariant,
namely invariant under
The weighted scale invariance is anomalous at the
quantum level
Case Nmax 4
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Homogeneous models
They are classically weighted scale invariant,
namely invariant under
The weighted scale invariance is anomalous at the
quantum level
Case Nmax 4
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Case d 4
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Case d 4
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Case d 4
Unique solution n 2 Nmax 10
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Case d 4
Unique solution n 2 Nmax 10
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Case d 4
Unique solution n 2 Nmax 10
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Case d 4
Unique solution n 2 Nmax 10
n arbitrary For n 2 we have
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Case d 4
Unique solution n 2 Nmax 10
n arbitrary For n 2 we have
n arbitrary The simplest non-trivial model
has n 3
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Källen-Lehman representation and unitarity
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Källen-Lehman representation and unitarity
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Källen-Lehman representation and unitarity
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Källen-Lehman representation and unitarity
Cutting rules
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Källen-Lehman representation and unitarity
Cutting rules
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Källen-Lehman representation and unitarity
Cutting rules
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Causality
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Causality
Our theories satisfy Bogoliubov's definition of
causality
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Causality
Our theories satisfy Bogoliubov's definition of
causality
which is a simple consequence of the largest time
equation and the cutting rules
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Causality
Our theories satisfy Bogoliubov's definition of
causality
which is a simple consequence of the largest time
equation and the cutting rules
For the two-point function this is just the
statement
if gt0
immediate consequence of
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Fermions
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Fermions
The extension to fermions is straightforward. The
free lagrangian is
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Fermions
The extension to fermions is straightforward. The
free lagrangian is
An example is the four fermion theory with
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Fermions
The extension to fermions is straightforward. The
free lagrangian is
An example is the four fermion theory with
An example of four dimensional scalar-fermion
theory is
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Gauge fields
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Gauge fields
Gauge fields are more tricky. Decompose the gauge
field as and assign weights
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Gauge fields
Gauge fields are more tricky. Decompose the gauge
field as and assign weights
so that the covariant derivative is decomposed
consistently,
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Gauge fields
Gauge fields are more tricky. Decompose the gauge
field as and assign weights
so that the covariant derivative is decomposed
consistently,
The field strength is decomposed in three sets of
components,
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Gauge fields
Gauge fields are more tricky. Decompose the gauge
field as and assign weights
so that the covariant derivative is decomposed
consistently,
The field strength is decomposed in three sets of
components,
The quadratic lagrangian reads
72
Gauge fields
Gauge fields are more tricky. Decompose the gauge
field as and assign weights
so that the covariant derivative is decomposed
consistently,
The field strength is decomposed in three sets of
components,
The quadratic lagrangian reads
where and
are polynomials
of
degrees
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Propagator
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Propagator
The BRST symmetry is unmodified
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Propagator
The BRST symmetry is unmodified
We choose the gauge-fixing
where is a
polynomial of degree n-1
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Propagator
The BRST symmetry is unmodified
We choose the gauge-fixing
where is a
polynomial of degree n-1
The ghost propagator is
where
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On the other hand, the gauge-field propagator is
far more involved
_____
_____
where
_______
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The physical degrees of freedom can be read in
the Coulomb gauge-fixing
which can be reached taking a suitable limit on
the previous one.
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The physical degrees of freedom can be read in
the Coulomb gauge-fixing
which can be reached taking a suitable limit on
the previous one. We get non-propagating ghosts
and
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The physical degrees of freedom can be read in
the Coulomb gauge-fixing
which can be reached taking a suitable limit on
the previous one. We get non-propagating ghosts
and
Writing we find
degrees of freedom with energies
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The physical degrees of freedom can be read in
the Coulomb gauge-fixing
which can be reached taking a suitable limit on
the previous one. We get non-propagating ghosts
and
Writing we find
degrees of freedom with energies
and degrees of freedom with energies

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Regularity of the propagator
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Regularity of the propagator
84
Regularity of the propagator
This component of the propagator is not regular,
in the sense that for large it does not fall off
with the maximal velocity, since depends
only on As a consequence, the -subintegrals
may diverge and there is no way to subtract
such sub-divergences
85
Regularity of the propagator
This component of the propagator is not regular,
in the sense that for large it does not fall off
with the maximal velocity, since depends
only on As a consequence, the -subintegrals
may diverge and there is no way to subtract
such sub-divergences
However, there is one case where such
subdivergences are absent. That is the case
86
Regularity of the propagator
This component of the propagator is not regular,
in the sense that for large it does not fall off
with the maximal velocity, since depends
only on As a consequence, the -subintegrals
may diverge and there is no way to subtract
such sub-divergences
However, there is one case where such
subdivergences are absent. That is the case
Indeed, Feynman integrals in one dimension never
have logarithmic divergences.
87
Thus, to avoid such problems we must assume
that spacetime is
broken into space and time
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Thus, to avoid such problems we must assume
that spacetime is
broken into space and time
A second property of gauge theories is that in
four dimensions gauge interactions
are always super-renormalizable
from the weighted power-counting viewpoint.
89
Thus, to avoid such problems we must assume
that spacetime is
broken into space and time
A second property of gauge theories is that in
four dimensions gauge interactions
are always super-renormalizable
from the weighted power-counting viewpoint.
Indeed, the weight of the gauge coupling is

90
Thus, to avoid such problems we must assume
that spacetime is
broken into space and time
A second property of gauge theories is that in
four dimensions gauge interactions
are always super-renormalizable
from the weighted power-counting viewpoint.
Indeed, the weight of the gauge coupling is

Attaching weightful coupling constants to scalar
fields we can also work in

91
Thus, to avoid such problems we must assume
that spacetime is
broken into space and time
A second property of gauge theories is that in
four dimensions gauge interactions
are always super-renormalizable
from the weighted power-counting viewpoint.
Indeed, the weight of the gauge coupling is

Attaching weightful coupling constants to scalar
fields we can also work in

92
The case is suitable to formulate an extended
Standard Model that contains both the
dimension-5 vertex

93
The case is suitable to formulate an extended
Standard Model that contains both the
dimension-5 vertex

that gives Majorana masses to left-handed
neutrinos after symmetry breaking,
94
The case is suitable to formulate an extended
Standard Model that contains both the
dimension-5 vertex

that gives Majorana masses to left-handed
neutrinos after symmetry breaking,
and four fermion interactions
that can describe proton decay. Such vertices
are renormalizable by weighted power counting
95
The case is suitable to formulate an extended
Standard Model that contains both the
dimension-5 vertex

that gives Majorana masses to left-handed
neutrinos after symmetry breaking,
and four fermion interactions
that can describe proton decay. Such vertices
are renormalizable by weighted power counting
Then the scale of Lorentz violation has the value
96
The (simplified) model
97
The (simplified) model
At low energies we have the Colladay-Kostelecky
Standard-Model Extension
98
The (simplified) model
At low energies we have the Colladay-Kostelecky
Standard-Model Extension
It can be shown that the gauge anomalies vanish,
since they coincide with those of the Standard
Model
99
Conclusions
A weighted power-counting criterion can be used
to renormalize Lorentz violating theories that
contain higher space derivatives, but no higher
time derivatives
100
Conclusions
A weighted power-counting criterion can be used
to renormalize Lorentz violating theories that
contain higher space derivatives, but no higher
time derivatives
The construction of scalar and fermion theories
is relatively simple, but Lorentz violating gauge
theories are more tricky.
101
Conclusions
A weighted power-counting criterion can be used
to renormalize Lorentz violating theories that
contain higher space derivatives, but no higher
time derivatives
The construction of scalar and fermion theories
is relatively simple, but Lorentz violating gauge
theories are more tricky.
The regularity of the gauge-field propagator
demands that spacetime be broken into space and
time.
102
Conclusions
A weighted power-counting criterion can be used
to renormalize Lorentz violating theories that
contain higher space derivatives, but no higher
time derivatives
The construction of scalar and fermion theories
is relatively simple, but Lorentz violating gauge
theories are more tricky.
The regularity of the gauge-field propagator
demands that spacetime be broken into space and
time.
It is possible to renormalize otherwise
non-renormalizable interactions, such as two
scalar-two fermion interactions and four fermion
interactions.
103
Conclusions
A weighted power-counting criterion can be used
to renormalize Lorentz violating theories that
contain higher space derivatives, but no higher
time derivatives
The construction of scalar and fermion theories
is relatively simple, but Lorentz violating gauge
theories are more tricky.
The regularity of the gauge-field propagator
demands that spacetime be broken into space and
time.
It is possible to renormalize otherwise
non-renormalizable interactions, such as two
scalar-two fermion interactions and four fermion
interactions.
We can construct an extended Standard Model that
gives masses to left neutrinos without
introducing right neutrinos or other extra
fields. We can also describe proton decay.
104
Conclusions
A weighted power-counting criterion can be used
to renormalize Lorentz violating theories that
contain higher space derivatives, but no higher
time derivatives
The construction of scalar and fermion theories
is relatively simple, but Lorentz violating gauge
theories are more tricky.
The regularity of the gauge-field propagator
demands that spacetime be broken into space and
time.
It is possible to renormalize otherwise
non-renormalizable interactions, such as two
scalar-two fermion interactions and four fermion
interactions.
We can construct an extended Standard Model that
gives masses to left neutrinos without
introducing right neutrinos or other extra
fields. We can also describe proton decay.
High-energy Lorentz violations could allow us to
define the ultraviolet limit of quantum gravity.
Suitable mechanisms could make the violations
undetectable even in principle.
105
Conclusions
A weighted power-counting criterion can be used
to renormalize Lorentz violating theories that
contain higher space derivatives, but no higher
time derivatives
The construction of scalar and fermion theories
is relatively simple, but Lorentz violating gauge
theories are more tricky.
The regularity of the gauge-field propagator
demands that spacetime be broken into space and
time.
It is possible to renormalize otherwise
non-renormalizable interactions, such as two
scalar-two fermion interactions and four fermion
interactions.
We can construct an extended Standard Model that
gives masses to left neutrinos without
introducing right neutrinos or other extra
fields. We can also describe proton decay.
High-energy Lorentz violations could allow us to
define the ultraviolet limit of quantum gravity.
Suitable mechanisms could make the violations
undetectable even in principle.
Observe that is just right for
gravity!