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.