Standard Model And
High-Energy Lorentz Violation
DAMIANO ANSELMI
CORTONA 27/05/2010
The papers about the Lorentz violating Standard Model are
[1] 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]
[2] 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]
The papers about the Lorentz violating Standard Model are
[1] 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]
[2] 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]
Then QED: [3] D.A. and M. Taiuti, Renormalization of high-energy Lorentz violating QED, Phys. Rev.
D 81 (2010) 085042 arxiv:0912.0113 [hep-ph]
The papers about the Lorentz violating Standard Model are
[1] 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]
[2] 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]
Then QED: [3] D.A. and M. Taiuti, Renormalization of high-energy Lorentz violating QED, Phys. Rev.
D 81 (2010) 085042 arxiv:0912.0113 [hep-ph]
Four fermion models:[4] D.A. and E. Ciuffoli, Renormalization of high-energy Lorentz violating four fermion
models, Phys. Rev. D 81, (2010) 085043 and arXiv:1002.2704 [hep-ph].
The papers about the Lorentz violating Standard Model are
[1] 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]
[2] 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]
Then QED: [3] D.A. and M. Taiuti, Renormalization of high-energy Lorentz violating QED, Phys. Rev.
D 81 (2010) 085042 arxiv:0912.0113 [hep-ph]
Four fermion models:[4] D.A. and E. Ciuffoli, Renormalization of high-energy Lorentz violating four fermion
models, Phys. Rev. D 81, (2010) 085043 and arXiv:1002.2704 [hep-ph].
and previous, more technical papers
[5] D.A. and M. Halat, Renormalization of Lorentz violating theories, Phys. Rev. D 76 (2007) 125011 and arxiv:0707.2480 [hep-th]
[6] D.A., Weighted scale invariant quantum field theories, JHEP 02 (2008) 05 and arxiv:0801.1216 [hep-th][7] D.A., Weighted power counting and Lorentz violating gauge theories. I: General
properties, Ann. Phys. 324 (2009) 874 and arXiv:0808.3470 [hep-th][8] D.A., Weighted power counting and Lorentz violating gauge theories. II: Classification, Ann. Phys. 324 (2009) 1058 and arXiv:0808.3474 [hep-th]
Outlook of the talk
Outlook of the talk
If we assume that Lorentz symmetry is violated at high energies we can renormalize otherwise non-renormalizable interactions, such as two-scalar-two-fermion vertices and four-fermion vertices.
Outlook of the talk
If we assume that Lorentz symmetry is violated at high energies we can renormalize otherwise non-renormalizable interactions, such as two-scalar-two-fermion vertices and four-fermion vertices.
It is possible to give Majorana masses to left-handed neutrinos without adding extra fields.
Indeed, the interaction is renormalizable as a fundamental vertex. A match with the
expected Majorana masses gives a scale of Lorentz violation (with preserved CPT symmetry) of about 10^14 GeV.
L
LH
2)(
L
Outlook of the talk
If we assume that Lorentz symmetry is violated at high energies we can renormalize otherwise non-renormalizable interactions, such as two-scalar-two-fermion vertices and four-fermion vertices.
It is possible to give Majorana masses to left-handed neutrinos without adding extra fields.
Indeed, the interaction is renormalizable as a fundamental vertex. A match with the
expected Majorana masses gives a scale of Lorentz violation (with preserved CPT symmetry) of about 10^14 GeV.
It is possible to explain proton decay.
Lorentz violating quantum field theory in flat space is perfectly consistent. All basic properties, such as causality, unitarity, stability, the Kallen-Lehman representation, etc., as well as renormalizability, locality and polynomiality, therefore predictivity, can be proved without assuming Lorentz symmetry.
L
LH
2)(
L
Outlook of the talk
If we assume that Lorentz symmetry is violated at high energies we can renormalize otherwise non-renormalizable interactions, such as two-scalar-two-fermion vertices and four-fermion vertices.
It is possible to give Majorana masses to left-handed neutrinos without adding extra fields.
Indeed, the interaction is renormalizable as a fundamental vertex. A match with the
expected Majorana masses gives a scale of Lorentz violation (with preserved CPT symmetry) of about 10^14 GeV.
It is possible to explain proton decay.
Lorentz violating quantum field theory in flat space is perfectly consistent. All basic properties, such as causality, unitarity, stability, the Kallen-Lehman representation, etc., as well as renormalizability, locality and polynomiality, therefore predictivity, can be proved without assuming Lorentz symmetry.
There exists a Lorentz violating Standard Model extension with these features. The inclusion of four-fermion vertices allows us to give masses to fermions and gauge-bosons by means of a Nambu-Jona-Lasinio mechanism, even if scalars, such as the Higgs field, are absent at the elementary level. Higgs composite fields arise as low-energy effects. The scalarless model is quite simple and predictive.
L
LH
2)(
L
Several new phenomena appear when Lorentz symmetry is violated, such as Cherenkov radiation in vacuum, pair production from a single photon, and so on [Coleman-Glashow]. Such processes allow us to make comparisons with experimental tests and put bounds on the Lorentz violation.
Several new phenomena appear when Lorentz symmetry is violated, such as Cherenkov radiation in vacuum, pair production from a single photon, and so on [Coleman-Glashow]. Such processes allow us to make comparisons with experimental tests and put bounds on the Lorentz violation.
The predictions of my models can be used to make comparisons with presently known experimental data and astrophysical observations, or propose new experiments and observations, to test the models, search for signals of the symmetry violation or extend the present bounds on the validity of the symmetry.
Several new phenomena appear when Lorentz symmetry is violated, such as Cherenkov radiation in vacuum, pair production from a single photon, and so on [Coleman-Glashow]. Such processes allow us to make comparisons with experimental tests and put bounds on the Lorentz violation.
The predictions of my models can be used to make comparisons with presently known experimental data and astrophysical observations, or propose new experiments and observations, to test the models, search for signals of the symmetry violation or extend the present bounds on the validity of the symmetry.
We may assume that there exists an energy range
that is well described by a Lorentz violating, but CPT invariant quantum field theory.
Several new phenomena appear when Lorentz symmetry is violated, such as Cherenkov radiation in vacuum, pair production from a single photon, and so on [Coleman-Glashow]. Such processes allow us to make comparisons with experimental tests and put bounds on the Lorentz violation.
The predictions of my models can be used to make comparisons with presently known experimental data and astrophysical observations, or propose new experiments and observations, to test the models, search for signals of the symmetry violation or extend the present bounds on the validity of the symmetry.
We may assume that there exists an energy range
that is well described by a Lorentz violating, but CPT invariant quantum field theory.
Glast: GeVM 1810
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
Conclusions
Is our Universe exactly Lorentz invariant at arbitrarily high energies?
Conclusions
Is our Universe exactly Lorentz invariant at arbitrarily high energies? If yes,
why?
Conclusions
Is our Universe exactly Lorentz invariant at arbitrarily high energies? If yes,
why?If this were not true, our perspective about high-energy physics would change completely, in particular physics around the Planck scale.
Conclusions
Is our Universe exactly Lorentz invariant at arbitrarily high energies? If yes,
why?If this were not true, our perspective about high-energy physics would change completely, in particular physics around the Planck scale.
We face two possibilities:
1) It is possible to construct a unitary, renormalizable theory of quantum gravity that takes advantage of the violation of Lorentz symmetry at high energies,
Conclusions
Is our Universe exactly Lorentz invariant at arbitrarily high energies? If yes,
why?If this were not true, our perspective about high-energy physics would change completely, in particular physics around the Planck scale.
We face two possibilities:
1) It is possible to construct a unitary, renormalizable theory of quantum gravity that takes advantage of the violation of Lorentz symmetry at high energies,
2) quantum gravity is not compatible with the Lorentz-symmetry violation.
Conclusions
Is our Universe exactly Lorentz invariant at arbitrarily high energies? If yes,
why?If this were not true, our perspective about high-energy physics would change completely, in particular physics around the Planck scale.
We face two possibilities:
1) It is possible to construct a unitary, renormalizable theory of quantum gravity that takes advantage of the violation of Lorentz symmetry at high energies,
2) quantum gravity is not compatible with the Lorentz-symmetry violation.
If 2) can be proved, this would give us a convincing argument to believe that Lorentz symmetry must beexact at arbitrarily high energies.
Conclusions
Is our Universe exactly Lorentz invariant at arbitrarily high energies? If yes,
why?If this were not true, our perspective about high-energy physics would change completely, in particular physics around the Planck scale.
We face two possibilities:
1) It is possible to construct a unitary, renormalizable theory of quantum gravity that takes advantage of the violation of Lorentz symmetry at high energies,
2) quantum gravity is not compatible with the Lorentz-symmetry violation.
If 2) can be proved, this would give us a convincing argument to believe that Lorentz symmetry must beexact at arbitrarily high energies.
The existence of an absolute reference frame raises questions at the fundamental level that can help us understand aspects about the nature of space and time.