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Standard Model And High-Energy Lorentz Violation,. Damiano Anselmi 中国科学院理论物理研究所 北京 2010年4月6日. based on the hep-ph papers [0a] D.A., Standard Model Without Elementary Scalars And High Energy Lorentz Violation, Eur. Phys. J. C 65 (2010) 523 and arXiv:0904.1849 [ hep-ph ] - PowerPoint PPT Presentation
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Standard Model And
High-Energy Lorentz Violation,
Damiano Anselmi
中国科学院理论物理研究所 北京 2010年 4月 6日
based on the hep-ph papers
[0a] D.A., Standard Model Without Elementary Scalars And High Energy Lorentz Violation, Eur. Phys. J. C 65 (2010) 523 and arXiv:0904.1849 [hep-ph]
[0b] D.A., Weighted power counting, neutrino masses and Lorentz violating extensions of the Standard Model, Phys. Rev. D 79 (2009) 025017 and arXiv:0808.3475 [hep-ph]
[0c] D.A. and M. Taiuti, Renormalization of high-energy Lorentz violating QED, Phys. Rev. D in press, arxiv:0912.0113 [hep-ph]
[0d] D.A. and E. Ciuffoli, Renormalization of high-energy Lorentz violating four fermion models, Phys. Rev. D in press, arXiv:1002.2704 [hep-ph]
based on the hep-ph papers
[0a] D.A., Standard Model Without Elementary Scalars And High Energy Lorentz Violation, Eur. Phys. J. C 65 (2010) 523 and arXiv:0904.1849 [hep-ph]
[0b] D.A., Weighted power counting, neutrino masses and Lorentz violating extensions of the Standard Model, Phys. Rev. D 79 (2009) 025017 and arXiv:0808.3475 [hep-ph]
[0c] D.A. and M. Taiuti, Renormalization of high-energy Lorentz violating QED, Phys. Rev. D in press, arxiv:0912.0113 [hep-ph]
[0d] D.A. and E. Ciuffoli, Renormalization of high-energy Lorentz violating four fermion models, Phys. Rev. D in press, arXiv:1002.2704 [hep-ph]
and previous hep-th papers
[1] D.A. and M. Halat, Renormalization of Lorentz violating theories, Phys. Rev. D 76 (2007) 125011 and arxiv:0707.2480 [hep-th]
[2] D.A., Weighted scale invariant quantum field theories, JHEP 02 (2008) 05 and arxiv:0801.1216 [hep-th]
[3] D.A., Weighted power counting and Lorentz violating gauge theories. I: General properties, Ann. Phys. 324 (2009) 874 and arXiv:0808.3470 [hep-th]
[4] D.A., Weighted power counting and Lorentz violating gauge theories. II: Classification, Ann. Phys. 324 (2009) 1058 and arXiv:0808.3474 [hep-th]
Lorentz symmetry is a basic ingredient of the Standard Model of particles physics.
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
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 withgreat precision.
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 withgreat precision.
It turns out that Lorentz symmetry is a very precise symmetry of Nature, at least in low-energy domain.
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 withgreat precision.
It turns out that Lorentz symmetry is a very precise symmetry of Nature, at least in low-energy domain.
Several (dimensionless) parameters have bounds
403015 10,10,10
Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken?
The set of power-counting renormalizable theories is considerably “small”
Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken?
The set of power-counting renormalizable theories is considerably “small”
Relaxing some assumptions can enlarge it, but often it enlarges it too much
Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken?
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
Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken?
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
Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken?
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
Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken?
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 and study extensions of the Standard Model
Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken?
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 and study extensions of the Standard Model
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 onto be consistent with unitarity
Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken?
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 and study extensions of the Standard Model
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 onto be consistent with unitarity
The approach that I formulate is based of a modified criterion of power counting,
dubbed weighted power counting
Why is it interesting to consider quantum field theories where Lorentz symmetry is explicitly broken?
We may assume that there exists an energy range
that is well described by a Lorentz violating, but CPT invariant quantum field theory.
and the mentioned range spans at least 4-5 orders of magnitude.
If the neutrino mass has the Lorentz violating origin we propose, then
Scalar fields
Consider the free theory
This free theory is invariant under the “weighted” scale transformation
Break spacetime in two pieces:
Break coordinates and momenta correspondingly:
is the “weighted dimension”
The propagator
behaves better than usual in the barred directions
Adding “weighted relevant’’ terms we get a free theory
that flows to the previous one in the UV and to the Lorentz invariant free theory in the infrared
(actually, the IR Lorentz recovery is much more subtle, see below)
Add vertices constructed with , and .
Call their degrees under
N = number of legs, = extra label
Other quadratic terms canbe treated as “vertices” for thepurposes of renormalization
Consider a diagram G with L loops, I internal legs, E external legs andvertices of type (N , )
is a weighted measure of degree
we see that is a homogeneous weighted function of degree
Its overall divergent part is a homogeneous weighted polynomial of degree
Performing a “weighted rescaling” of external momenta,
together with a change of variables
Using the standard relations
we get
Renormalizable theories have
Where
Indeed implies
Writing
we see that polynomiality demands
and the maximal number of legs is
E = 2 implies 2
E > 2 implies < 2
Conclusion:
renormalization does not turn onhigher time derivatives
Examples
Strictly-renormalizable models are classically weighted scale invariant, namely invariant under
The weighted scale invariance is anomalous at the quantum level
n=2: six-dimensional -theory
Four dimensional examples
n = 2
n = 2
n = 3
Källen-Lehman representation and unitarity
Cutting rules
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 >0
immediate consequence of
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
High-energy Lorentz violating QED
High-energy Lorentz violating QED
Gauge symmetry is unmodified
High-energy Lorentz violating QED
Gauge symmetry is unmodified
A convenient gauge-fixing lagrangian is
Integrating the auxiliary field B away we find
Integrating the auxiliary field B away we find
Propagators
Integrating the auxiliary field B away we find
Propagators
This gauge exhibits the renormalizability of the theory, but not its unitarity
Coulomb gauge
Coulomb gauge
Coulomb gauge
Two degrees of freedom with dispersion relation
Coulomb gauge
Two degrees of freedom with dispersion relation
The Coulomb gauge exhibits the unitarity of the theory,
but not its renormalizability
Coulomb gauge
Two degrees of freedom with dispersion relation
The Coulomb gauge exhibits the unitarity of the theory,
but not its renormalizability
Correlation functions of gauge invariant objects
are both unitary and renormalizable
Weighted power counting
Weighted power counting
The theory is super-renormalizable
Counterterms are just one- and two-loops
Weighted power counting
The theory is super-renormalizable
Counterterms are just one- and two-loops
High-energy one-loop renormalization
High-energy one-loop renormalization
High-energy one-loop renormalization
Low-energy renormalization
Low-energy renormalization
Low-energy renormalization
Two cut-offs, with the identification
Low-energy renormalization
Two cut-offs, with the identification
Logarithmic divergences give
More generally, the theory can have 2n spatial derivatives, at most, which can
improve the UV behavior even more.
More generally, the theory can have 2n spatial derivatives, at most, which can
improve the UV behavior even more.
The number n is crucial for the weighted power counting, together with the
weighted dimension
More generally, the theory can have 2n spatial derivatives, at most, which can
improve the UV behavior even more.
The number n is crucial for the weighted power counting, together with the
weighted dimension
Our previous case had n=3
n=odd is necessary to describe chiral fermions
A general property of gauge theories is that in four dimensions gauge interactions
are always super-renormalizable
from the weighted power-counting viewpoint
More generally, the theory can have 2n spatial derivatives, at most, which can
improve the UV behavior even more.
The number n is crucial for the weighted power counting, together with the
weighted dimension
Our previous case had n=3
n=odd is necessary to describe chiral fermions
A general 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
More generally, the theory can have 2n spatial derivatives, at most, which can
improve the UV behavior even more.
The number n is crucial for the weighted power counting, together with the
weighted dimension
Our previous case had n=3
n=odd is necessary to describe chiral fermions
The case
allows us to formulate a consistent Lorentz violating extended Standard Model
that contains both the dimension-5 vertex
that gives Majorana masses to left-handed neutrinos after symmetry breaking,
The case
allows us to formulate a consistent Lorentz violating extended Standard Model
that contains both the dimension-5 vertex
that gives Majorana masses to left-handed neutrinos after symmetry breaking,
that can describe proton decay.
Such vertices are renormalizable by weighted power counting
and the four fermion interactions
The case
allows us to formulate a consistent Lorentz violating extended Standard Model
that contains both the dimension-5 vertex
that gives Majorana masses to left-handed neutrinos after symmetry breaking,
that can describe proton decay.
Such vertices are renormalizable by weighted power counting
Matching the vertex with estimates of the electron neutrino Majorana
mass the scale of Lorentz violation has roughly the value
and the four fermion interactions
2)(LH
The (simplified) model reads
The (simplified) model reads
where
The (simplified) model reads
where
The (simplified) model reads
where
The (simplified) model reads
where
The (simplified) model reads
At low energies we have the Colladay-Kostelecky Standard-Model Extension
where
The (simplified) model reads
It can be shown that the gauge anomalies vanish, since they coincide with those of
the Standard Model
At low energies we have the Colladay-Kostelecky Standard-Model Extension
where
The (simplified) model reads
The model simplifies enormously at high energies.
Indeed, both gauge and Higgs interactions are super-renormalizable, so
asymptotically free.
The model simplifies enormously at high energies.
Indeed, both gauge and Higgs interactions are super-renormalizable, so
asymptotically free.
At very high energies gauge fields and Higgs boson decouple and the theory
becomes a four fermion model in two weighted dimensions.
Since four fermion vertices can trigger a Nambu—Jona-Lasinio mechanism,
It is natural to enquire if we can do without the elementary Higgs field
Indeed we can and the model then simplifies enormously
The model simplifies enormously at high energies.
Indeed, both gauge and Higgs interactions are super-renormalizable, so
asymptotically free.
At very high energies gauge fields and Higgs boson decouple and the theory
becomes a four fermion model in two weighted dimensions.
The scalarless model reads
The scalarless model reads
where the kinetic terms are
To illustrate the Nambu—Jona-Lasinio mechanism in our case,
consider the t-b model
In the large Nc limit, where
To illustrate the Nambu—Jona-Lasinio mechanism in our case,
consider the t-b model
We can prove that
In the large Nc limit, where
is a minimum of the effective potential and gives masses to the fermions
The Lorentz violating fermionic mass terms are not turned on, so the Lorentz
violation remains highly suppressed.
The Lorentz violating fermionic mass terms are not turned on, so the Lorentz
violation remains highly suppressed.
We can read the bound states from the effective potential. We find
where
The Lorentz violating fermionic mass terms are not turned on, so the Lorentz
violation remains highly suppressed.
We can read the bound states from the effective potential. We find
where
Studying the poles we find
where
When gauge interactions are turned on, the Goldstone bosons are “eaten” by
W’s and Z, as usual. Precisely, the effective potential becomes
where
When gauge interactions are turned on, the Goldstone bosons are “eaten” by
W’s and Z, as usual. Precisely, the effective potential becomes
Choosing the unitary gauge fixing
we find the gauge-boson mass terms. The masses are
In particular, such relations give
In particular, such relations give
Using our estimated value
and the measured value of the Fermi constant, we find the top mass
PDG gives 171.2 2.1GeV
In particular, such relations give
Using our estimated value
and the measured value of the Fermi constant, we find the top mass
Moreover,
for
PDG gives 171.2 2.1GeV
ConclusionsA weighted power-counting criterion can be used to renormalize Lorentz violating theories that contain higher space derivatives, but no higher time derivatives
ConclusionsA weighted power-counting criterion can be used to renormalize Lorentz violating theories that contain higher space derivatives, but no higher time derivativesRenormalizable theories can contain two scalar-two fermion vertices and four fermion vertices
ConclusionsA weighted power-counting criterion can be used to renormalize Lorentz violating theories that contain higher space derivatives, but no higher time derivativesRenormalizable theories can contain two scalar-two fermion vertices and four fermion vertices
We can construct an extended Standard Model that gives masses to left neutrinoswithout introducing right neutrinos or other extra fields. We can also describe proton decay
ConclusionsA weighted power-counting criterion can be used to renormalize Lorentz violating theories that contain higher space derivatives, but no higher time derivativesRenormalizable theories can contain two scalar-two fermion vertices and four fermion vertices
We can construct an extended Standard Model that gives masses to left neutrinoswithout introducing right neutrinos or other extra fields. We can also describe proton decay
At very high energies gauge and Higgs interactions become negligible, since theyare super-renormalizable, so the model becomes a four fermion model in twoweighted dimensions.
ConclusionsA weighted power-counting criterion can be used to renormalize Lorentz violating theories that contain higher space derivatives, but no higher time derivativesRenormalizable theories can contain two scalar-two fermion vertices and four fermion vertices
We can construct an extended Standard Model that gives masses to left neutrinoswithout introducing right neutrinos or other extra fields. We can also describe proton decay
It is thus natural to ask ourselves if the scalarless variant of the model is able to reproduce all known low-energy physics without the ambiguities of usual Nambu—Jona-Lasinio non-renormalizable approaches.
At very high energies gauge and Higgs interactions become negligible, since theyare super-renormalizable, so the model becomes a four fermion model in twoweighted dimensions.
ConclusionsA weighted power-counting criterion can be used to renormalize Lorentz violating theories that contain higher space derivatives, but no higher time derivativesRenormalizable theories can contain two scalar-two fermion vertices and four fermion vertices
We can construct an extended Standard Model that gives masses to left neutrinoswithout introducing right neutrinos or other extra fields. We can also describe proton decay
It is thus natural to ask ourselves if the scalarless variant of the model is able to reproduce all known low-energy physics without the ambiguities of usual Nambu—Jona-Lasinio non-renormalizable approaches.
We have shown, in the leading order of the large Nc limit and with gauge interactions switched off, that the effective potential admits a Lorentz invariant (local) minimum, that gives masses to fermion and gauge bosons, and produces composite Higgs bosons.
At very high energies gauge and Higgs interactions become negligible, since theyare super-renormalizable, so the model becomes a four fermion model in twoweighted dimensions.
We have shown that the scalarless model predicts relations among the parameters of the Standard Model. Our approximation has a good 50% oferror, but one prediction, the relation between the top mass and the Fermi constantturns out to be in even too good agreement with experiment.
We have shown that the scalarless model predicts relations among the parameters of the Standard Model. Our approximation has a good 50% oferror, but one prediction, the relation between the top mass and the Fermi constantturns out to be in even too good agreement with experiment.
So far, we learn that Lorentz symmetry in flat space is not necessaryfor the consistency of quantum field theory. This leads us to question is the Universe exactly Lorentz invariant? If yes, WHY?
Maybe only quantum gravity can resolve this issue
High-energy Lorentz violations could allow us to define the ultraviolet limit of quantum gravity. Hopefully suitable mechanisms could make the violations undetectable even in principle
We have shown that the scalarless model predicts relations among the parameters of the Standard Model. Our approximation has a good 50% oferror, but one prediction, the relation between the top mass and the Fermi constantturns out to be in even too good agreement with experiment.
So far, we learn that Lorentz symmetry in flat space is not necessaryfor the consistency of quantum field theory. This leads us to question is the Universe exactly Lorentz invariant? If yes, WHY?
Maybe only quantum gravity can resolve this issue
High-energy Lorentz violations could allow us to define the ultraviolet limit of quantum gravity. Hopefully suitable mechanisms could make the violations undetectable even in principle
Observe that is the weighted dimension where Lorentz violating gravity becomes strictly renormalizable. The real challenge is to break the local Lorentz symmetry without destroying unitarity.
We have shown that the scalarless model predicts relations among the parameters of the Standard Model. Our approximation has a good 50% oferror, but one prediction, the relation between the top mass and the Fermi constantturns out to be in even too good agreement with experiment.
So far, we learn that Lorentz symmetry in flat space is not necessaryfor the consistency of quantum field theory. This leads us to question is the Universe exactly Lorentz invariant? If yes, WHY?
Maybe only quantum gravity can resolve this issue
High-energy Lorentz violations could allow us to define the ultraviolet limit of quantum gravity. Hopefully suitable mechanisms could make the violations undetectable even in principle
Observe that is the weighted dimension where Lorentz violating gravity becomes strictly renormalizable. The real challenge is to break the local Lorentz symmetry without destroying unitarity.
We have shown that the scalarless model predicts relations among the parameters of the Standard Model. Our approximation has a good 50% oferror, but one prediction, the relation between the top mass and the Fermi constantturns out to be in even too good agreement with experiment.
If this were shown to be impossible, we would have a good reason to think thatLorentz invariance (and therefore CPT) must be exact at arbitrarily high energies.
So far, we learn that Lorentz symmetry in flat space is not necessaryfor the consistency of quantum field theory. This leads us to question is the Universe exactly Lorentz invariant? If yes, WHY?
Maybe only quantum gravity can resolve this issue