Standard Model And High-Energy Lorentz Violation,

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