Particle–Antiparticle Asymmetry in Relativistic Deformed Kinematics

Round 1

Reviewer 1 Report

The authors discuss how quantum gravity effect encoded in a modified relativistic kinematics can provide a framework to explain matter antimatter asymmetry. The particular form of relativistic kinematics discussed features non-commutativity that results in violation of CPT symmetry. From this argument is natural to argue that the modifications discussed in the paper have an effect on the generation of matter-anti matter asymmetry. The authors also study the effect of their framework on muon decay.

The paper is clearly written and the results well presented. I am in favor of its publication in Symmetry.

Author Response

We thank the referee for his/her positive comments of our work.

Reviewer 2 Report

Dear Authors

The paper considers Relativistic deformed kinematics in attempt to explain the observed baryon asymmetry in the early Universe.
The main idea is to use a non-commutative addition law for the momenta which appears, for example, for kappa-Poincare algebra and doubly special relativity theories. In such theories the total momentum of two particles
depends on the ordering in which the momenta are composed. Authors show that a non-commutative addition law may be used to generate an asymmetry between particles and antiparticles which results in CPT violation. Authors made calculations for muon decay and have  obtained a difference in the lifetimes of the particle and the antiparticle which depends on the energy cutoff scale and therefore maybe put in the agreement with measurements.
 
The physical systems in which such a symmetry is broken are not something very unusual, e.g., in solids electrons and vacations possess different masses and are not symmetric. The basic advantage of the present approach is that it actually doesn't violate the underlying Lorentz symmetry. 
However if the mathematically all constructions and results are clear, in order to expand the audience I would recommend to present also a simple/idealized physical model in which such a non-commutative addition law works.

my only comment is as follows.

If it is possible, please  present a simple physical/mathematical  model which may produce eqs. (2), (3) for the deformed composition.

This will essentially enlarge the audience of readers. The problem is that references cited, e.g., [50] which use the same equations do not contain such model as well.

Author Response

We thank the referee for his/her comments about our work, and try to answer here to the single point he/she raised, that is:

"If it is possible, please present a simple physical/mathematical  model which may produce eqs. (2), (3) for the deformed composition. This will essentially enlarge the audience of readers. The problem is that references cited, e.g., [50] which use the same equations do not contain such model as well."

A modified composition law of energy and momentum is a general ingredient of any deformation of SR compatible with the relativistic invariance. The modified composition law that we have used in our work is just one example where there is no modification of the energy-momentum relation and then the strong constrains on the energy scale of the deformation due to observable consequences of a modification of the energy-momentum relation do not apply. 

It is a conjecture that the low energy limit of a quantum theory of gravity will include a deformation of SR but there is at present no derivation of this result in four dimensional spacetime. This is the reason why we can not provide a phyisical model with the modified composition of energy and momentum used in the present work. 

However, there is a mathematical model which produces such deformed compositions, which is Hopf algebras. In this case, it is the mathematical structure known as "coproduct" which is associated to this deformed composition.

We have tried to clarify this in the new version of the manuscript, with a new paragraph before the end of Section 2. It reads:

"In the language of Hopf algebras, a deformed composition law such as Eq. (2) corresponds to a non-trivial form of the mathematical structure know as coproduct [51]. Although it is a conjecture that the low-energy limit of a quantum theory of gravity will include such mathematical construction, a modified composition law of energy and momentum is a necessary ingredient in any attempt to deform special relativity in a way compatible with relativistic invariance. The modified composition law that we are using in the present work is just one example where there is no modification of the energy-momentum relation and then the strong constrains on the energy scale of the deformation due to observable consequences of a modification of the energy-momentum relation do not apply." 

Reviewer 3 Report

The article is of great interest because it provides an interesting approach to solving an important problem of cosmology regarding baryon asymmetry, but there are several clarifying questions.

The following sentence (which refers to the equations (6)) should be clarified. "Written in this way, one sees that only in the second alternative one has an expression for the total energy-momentum of the final state with a definite ordering of the four-momenta of the particles and antiparticles."

In the "5. Search for observable effects" why it is possible to neglect the electron mass?

To improve readability, it is necessary to reformulate a number of sentences and divide them into two or more. For example: (1) "This is a simple enough process not to involve irrelevant complications and on the other hand is an
interesting process from a phenomenological perspective to look for possible observable effects of a deformed relativistic kinematics." (2) "But one should take into account that all the bounds on the scale of departures from the SR kinematics which are of the order of the Planck scale are obtained assuming a modification of the dispersion relation, and then they do not apply in the proposal of relativistic deformed kinematics considered in this work." etc.

After working on these shortcomings, the article can be accepted for publication.

Author Response

We thank the referee for his/her comments, which we answer here:

1) The following sentence (which refers to the equations (6)) should be clarified. "Written in this way, one sees that only in the second alternative one has an expression for the total energy-momentum of the final state with a definite ordering of the four-momenta of the particles and antiparticles."

We have tried to clarify it by adding a more detailed explanation before equation (6):

"In this way, the total momentum of a particle-antiparticle system is defined as the composition of the momentum of the particle (which appears on the left of the composition law) with the momentum of the antiparticle (which appears to the right)."

and also after equation (7):

"The first choice has on the right hand side the composition of the momentum of an antiparticle (the momentum $q'$ of the electron antineutrino) with the momentum of a particle (the momentum $p'$ of the muon neutrino). This goes against the rule selected for the total momentum as composing the momenta of particles with the momenta of antiparticles and not the other way round. The second alternative, however, leads to an expression for the total energy-momentum of the final state with a definite ordering of the momenta, with the momenta of particles appearing to the left of the momenta of antiparticles."

2) In the "5. Search for observable effects" why it is possible to neglect the electron mass?

The dependence on the electron mass on the zero-th term (the standard physics part) is the same for the particle and antiparticle and then it cancels out in the quotient of lifetimes; we are interested in the "new physics" terms, which will involve m_e/Lambda and m_\mu/Lambda, but since m_e << m_\nu, the dominant contribution in the new physics is the one proportional to m_\mu/Lambda. We have clarified the text with the sentence at the beginning of section 5:

"For the purpose of this work, which is the determination of the term proportional to $(1/\Lambda)$ in the ratio of lifetimes of particles and antiparticles, one can neglect the dependence on the electron mass, since $m_e \ll m_\mu$. "

3) To improve readability, it is necessary to reformulate a number of sentences and divide them into two or more. For example: (1) "This is a simple enough process not to involve irrelevant complications and on the other hand is an
interesting process from a phenomenological perspective to look for possible observable effects of a deformed relativistic kinematics." (2) "But one should take into account that all the bounds on the scale of departures from the SR kinematics which are of the order of the Planck scale are obtained assuming a modification of the dispersion relation, and then they do not apply in the proposal of relativistic deformed kinematics considered in this work." etc.

We have followed the advice by the referee and have reformulated these two sentence, dividing them into shorter sentences, and we have done the same with other particular long sentences throughout the text. In particular, the changes are in the sentences:

• Page 1: "Indeed, there must be a difference between two charge-conjugated processes, that is, those related by changing all particles into antiparticles (and viceversa). If this were not the case, the net baryonic number generated in one of the processes would be compensated by the generation of an equal amount of net antibaryonic number in the second one."
• Page 3: "As we will see, this non-commutativity of the composition law will provide us with a natural way of introducing an asymmetry between particles and antiparticles. The purpose of the present paper is just to illustrate this possibility in a particular example, leaving the potential consequences for the problem of baryogenesis for further work."
• Page 3: "In the following, we will consider a deformed relativistic kinematics such that the dispersion relation, and then the relation between the energy and the momentum of each particle, does not depend on the new scale $\Lambda$. This choice will allow us to illustrate in the simplest way the idea that a relativistic deformation of the kinematics provides a new way to introduce a particle-antiparticle asymmetry."
• Page 4: "Compatibility with the invariance under boosts, however, requires that they act nonlinearly on at least one of the momenta (that can be taken, for example, as the one which is on the right in the composition law). The momentum of the other can be taken to transform linearly, as well as the total momentum"
• Page 4: "This is a simple enough process not to involve irrelevant complications. Moreover, it is an interesting process from a phenomenological perspective to look for possible observable effects of a deformed relativistic kinematics." 
• Page 5: "The deformation of the composition of four-momenta can be associated with the locality of the interaction responsible for the decay. Hence, it is natural to express the energy-momentum conservation relation in terms of the composition of the momenta $(q, q')$ and in terms of..."
• Page 7: "In fact (see [43]), the  nonlinear implementation of the Lorentz transformations on the momenta is fixed by the condition that both the composition law between momenta and the dispersion relation of the momenta which are being composed (which, in this case, is simply their square) have to be invariant (relativity principle). This means that the square of the composition of momenta $(p'\oplus q)$ in the $\mu^-$ decay, and $(q\oplus p')$ in the $\mu^+$ case, are invariant under Lorentz transformations. Therefore, the deformed matrix elements defined in this way are the simplest choice of a deformation of the SR matrix element compatible with the deformed relativistic invariance."
• Page 9: "But one should take into account that all the bounds on the scale of departures from the SR kinematics which are of the order of the Planck scale are obtained assuming a modification of the dispersion relation. Therefore, they do not apply in the proposal of relativistic deformed kinematics considered in this work, which does not modify the dispersion relation."

Reviewer 4 Report

 The paper "Particle-antiparticle asymmetry in a relativistic
deformed kinematics" is very interesting and very well written and can be published as it is.

Author Response

We thank the referee for his/her positive comments of our work.