Past and Current Attempts to Explain Baryogenesis
Why Neutron-Antineutron Oscillations?
Advanced Quantum Mechanics II
Taught by Dr. George Siopsis
Written by Joshua Barrow
Abstract
Our current understanding of the universe is shaped from a quite impressive fundamental fact:
it exists. Theoretically, there can be no more surprising statement, as its ramifications hint upon
the need for some new way of understanding the most fundamental interactions in nature.
Within the Standard Model (SM), the quantities of baryon number 𝑩 and lepton number 𝑳 are
considered to be conserved. However, as detailed in the 1960’s by Sakharov in his three
conditions governing the development of the early universe, we know that violation of baryon
number is a requirement in order to explain any observed matter-antimatter asymmetry.
This research was pursued in earnest by figures like t’Hooft, Rubakov, Shaposhnikov, and
Kuzmin, where efforts were made to use the sphaleron mechanism and nonpertubative
conservation of 𝑩 − 𝑳 within the SM to generate realistic baryon abundance ratios for the
universe. However, these attempts either failed entirely, or required an inordinate amount of
fine tuning to produce even slightly plausible quantities of matter over antimatter.
Due to this, the observation of neutrino oscillations, along with failure to observe possible
beyond SM (BSM) processes such as proton decay, theorists have recently proposed extensions
to the SM with new heavy scalar fields which decay into quarks or antiquarks only, violating
𝑩 − 𝑳 with 𝚫𝑩 = 𝟐 operators (usually of dimension nine), while suppressing most all possible
proton decay modes. It has been shown that these operator, with their corresponding tree and
one-loop level diagrams, are able to be constrained by current experimental data enough to
make plausible, indeed, precise predictions of the baryon abundance. With appropriate
crossing symmetries, these diagrams lead to neutron-antineutron oscillations, where normal
quarks are spontaneously transformed into their antiquark cousins by a six-quark operator.
Interestingly, some models are developed and restrictive enough that observation of the
neutrino mass hierarchy could allow or disallow the existence of neutron-antineutron
oscillation, which itself can be predicted to have an absolute transition-time maximum given a
range of important parameters constrained by current experimental data. The particular works
reviewed in this paper, which properly reproduce the matter-antimatter asymmetry, put the
observability of such oscillations just within reach of the next generation of experiments.
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Introduction and Generalities
Matter-Antimatter Symmetry Broken
Given what we know of the big bang and the plausible mechanisms by which it arose, we
believe that the universe today is flat (zero net energy density), expanding, and made
principally of matter. Most believe that the universe began from a charge (C) symmetric state,
wherein all masses of particles and antiparticles were balanced, the decay widths of those
particles identical, and the electric charges of those particles opposite. However, while such a
symmetric state would require equivalent number densities of particles and their antiparticles,
we instead see (albeit in our local neighborhood) that the universe if predominately made of
matter rather than antimatter.
Here, electrons, protons, and neutrons dominate the density of their respective antiparticles
cousins. This fact is solidified by the observation of little to no flux of energetic 𝛾-radiation due
to annihilation events (for example, in a 𝑝 − 𝑝̅ collision, production of 𝜋 0 ’s can occur, which can
subsequently decay into 2𝛾). This observation, along with data showing the near perfect
isotropy of the cosmic microwave background (CMB) radiation allows us to calculate a
convenient, dimensionless number characterizing the magnitude of baryon asymmetry in the
universe:
𝛽=
𝑛𝐵 − 𝑛𝐵̅
≈ 10−10
𝑛𝛾
where 𝛽 is the baryon abundance, 𝑛𝐵 the number density of baryonic charge, 𝑛𝐵̅ the number
density of antibaryonic charge, and 𝑛𝛾 the number density of photons. Here, 𝑛𝛾 depends
implicitly on the radiation temperature of the CMB photons and a determined constant.
Accurate estimates of 𝛽, which is (assumed to be) constant in time, are confirmed by the
astronomical observation of light elements (nuclei) throughout the universe, such as 4He, 3He,
7Li, and especially 2H, all thought to be made during the first few minutes of the big bang. In
models of such nucleosynthesis at the beginning of the universe, these observations are key
and sensitive inputs, and offer a good understanding of the baryonic number density during this
epoch.
It should be noted that in the case of a pure, symmetric state of the universe, one would expect
for there to be effectively no baryons (or antibaryons) at all due to mutual annihilation
occurring down to a temperature of roughly 1 𝐺𝑒𝑉 (where they would “freeze” out), giving rise
to a surviving baryon abundance of
𝑛𝐵 = 𝑛𝐵̅ ≈
𝑛𝛾
≈ 10−19 𝑛𝛾
𝜎𝑎𝑛𝑛 𝑚𝐵 𝑚𝑝𝑙
Here, 𝜎𝑎𝑛𝑛 is the cross-section of nucleon annihilation, 𝑚𝐵 is the baryon mass, and 𝑚𝑃𝑙 is the
Planck mass. This quantity, of course, is far too small, and compared to the actual value, shows
an effectively 100% asymmetric state when considering the universe today [8]. Understanding
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this asymmetry could hold important clues to physics beyond the standard model of particle
physics [3].
The Sakharov (Boundary) Conditions and CPT
Let the universe be created from a C-symmetric vacuum state with total baryon and lepton
numbers of zero. Consider an arbitrarily heavy particle that existed close to the beginnings of
our universe, 𝑋, and its antiparticle, 𝑋̅. Recall by consequence of the charge-parity-time
inversion (CPT) theorem that all decay rates of particle and antiparticles are identical. If 𝑋 were
to decay with a branching ratio 𝑓 into a state with baryon number 𝐵1, and into another possible
state 𝐵2 with branching (1 − 𝑓), then we would have that
̅ 1 + [(1 − 𝑓) − (1 − 𝑓)]𝐵
̅ 2 = (𝑓 − 𝑓)(𝐵
̅ 1 − 𝐵2 ) ≠ 0
0 = ∆𝐵 = (𝑓 − 𝑓 )𝐵
where 𝑓 ̅ and (1 − 𝑓)̅ pertain to the same processes for the antiparticle. From this, we must
conclude that 1) 𝑓 ≠ 𝑓 ̅ means we have CP non-conservation, and 2) 𝐵1 ≠ 𝐵2 implies baryon
number (𝐵) non-conservation. This argument permits one of the three suggested hypotheses of
Sakharov in 1967, pertaining to an explanation of the baryon asymmetry of the universe (and
thus its mere existence):
1. CP non-conservation (C non-conservation; different interactions of particles and
antiparticles)
2. B charge non-conservation
3. Departure from thermal equilibrium (provided by the expansion of the universe)
All three of these arguments are central to our understanding of the evolution of the universe,
and are key boundary conditions any permissible model must be consistent with [8,13,14,15].
The Bell-Jackiw-Adler Anomaly
We are ready now to consider the intricacies of fermionic fields of quarks and leptons that, in
the SM, interact via vector bosons. We will find that the renormalizability of the model requires
the introduction of vector boson fields through local gauge symmetry by the addition of
counter terms in the Lagrangian density. It is this addition that allows for the renormalization of
the SM, but has the effect of allowing simultaneous baryon and lepton number violation in nonperturbative ways. This fact lead many to investigate such a mechanism for generating baryon
asymmetry in the universe.
Bell and Jackiw, and separately Adler, were the first to find an “anomaly” in a field theory when
considering the axial vector current.
The Axial Current
Consider the Lagrangian density
1
ℒ = 𝜓̅[𝛾 𝜇 (𝑖𝜕𝜇 − 𝑞𝐴𝜇 ) − 𝑚]𝜓 − 𝐹𝜇𝜈 𝐹𝜇𝜈
4
4
which we see has the local gauge symmetry needed for electromagnetism. Here, the
transformations are
𝜓(𝑥) → 𝜓 ′ (𝑥) = 𝑒 𝑖𝑞𝜒(𝑥) 𝜓(𝑥)
𝐴𝜇 (𝑥) → 𝐴𝜇′ (𝑥) = 𝐴𝜇 (𝑥) + 𝜕𝜇 𝜒(𝑥)
Now, if 𝑚 = 0, then we say that the Lagrangian has a global chiral symmetry, meaning that if
we choose a transformation such as
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𝜓(𝑥) → 𝜓 ′ (𝑥) = 𝑒 𝑖𝛼(𝑥)𝛾 𝜓(𝑥)
then ℒ remains invariant (due to properties of the gamma matrices). If this is treated as an
infinitesimal transformation (so that 𝛼 is small) and apply this to ℒ, we find that
𝜇
ℒ → ℒ + 𝛿ℒ = ℒ + 𝛼(𝑥)[𝜕𝜇 𝑗𝐴 − 2𝑖𝑚𝜓̅𝛾 5 𝜓]
𝜇
where 𝑗𝐴 = 𝜓̅𝛾 𝜇 𝛾 5 𝜓 is called the axial current.
The Axial Anomaly
However, we find that (as was shown in class) this current is not, in fact, generally conserved;
𝜇
this is the case because the four-divergence leads to 𝜕𝜇 𝑗𝐴 = 2𝑖𝑚𝜓̅𝛾 5 𝜓, implying it is conserved
if and only if 𝑚 = 0 (and so becomes divergenceless).
Secondarily, from results of currents being calculated via perturbation theory and requiring that
electric charge be conserved, along with local gauge symmetry, we know that this divergence in
general becomes
𝜇
𝜕𝜇 𝑗𝐴 = 2𝑖𝑚𝜓̅𝛾 5 𝜓 −
𝑒 2 𝜇𝜈𝜆𝜌
𝜀
𝜕𝜇 𝐴𝜈 𝜕𝜆 𝐴𝜌
2𝜋 2
Thus, even if 𝑚 = 0, we maintain the existence of the second term: the Adler-Bell-Jackiw axial
anomaly. Interestingly, this hairy second term has an elegant, explicitly gauge-invariant form if
we choose to write the derivatives of the electromagnetic potentials in terms of the electric and
magnetic fields (if 𝑚 = 0):
𝜇
𝜕𝜇 𝑗𝐴
𝑒2
=− 2𝐄⋅𝐁
𝜋
Note that, from this perturbative formulation, we can construct a total current which is
conserved:
𝜇
𝑗𝑡𝑜𝑡𝑎𝑙 = −
𝑒 2 𝜇𝜈𝜆𝜌
𝒆𝟐 𝝁𝝂𝝀𝝆
𝜇
𝜀
𝜕
𝐴
𝜕
𝐴
+
𝜺
𝑨𝝂 𝑭𝝀𝝆 ⟹ 𝜕𝜇 𝑗𝑡𝑜𝑡𝑎𝑙 = 0
𝜇 𝜈 𝜆 𝜌
2𝜋 2
𝟒𝝅𝟐
Of course, such a total current can be used to define a time-independent charge in the usual
manner. Given the 𝒔𝒆𝒄𝒐𝒏𝒅 𝒕𝒆𝒓𝒎 in such a current, this formulation is obviously gauge
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dependent due to its proportionality to the field 𝐴𝜈 ; because to this, we cannot draw specific
physical significance from it [6].
t’Hooft’s Renormalization
It is clear that in the preceding models of fermions (and indeed in the entirety of the SM), that,
when coupled with gauge fields, anomalies arise such that certain current conservations laws
are violated. From perturbation theory (Feynman diagrams), it is clear that the total charges
and corresponding currents are conserved, so this is an entirely non-perturbative effect. It is the
mix of vector and axial vector currents that give rise to these anomalies, threatening the
renormalizability of the SM, especially in the electroweak sector.
Indeed, if a theory were to have no quarks and just leptons (also shown in class), anomalies
spoil any conservation laws relating currents to the necessary bosons; similarly, if one has
quarks, but no leptons, anomalies again crop up. However, if we include both in our theory, the
anomalies cancel exactly in detailed calculations, if and only if the number of lepton families
equals the number of quark families. This leads to strict conservation of electroweak gauge
currents, and thus the renormalizability of the SM, even though individual anomalies exist for
baryons and leptons [6,12].
Lepton and Baryon Number Conservation and the BAU
We will now endeavor to fill in some specifics relating to t’Hoof’s renormalization, requiring that
the number of lepton families be equal to the number of quark families. From this, it was thought
that a viable mechanism to generate the BAU could be that, in non-pertubative regimes, baryon
non-conservation exists even within the confines of the standard model; but, as mentioned in
the abstract of this paper, this is most likely not the case.
Lepton and Baryon Currents
Currents of course arise from symmetries of the Lagrangian; we seek to find such a current that
is able to conserve the number of leptons and the number of quarks. If we consider the case of
neutrinos being Dirac fermions, it is possible to write down a classically conserved leptonic
current which upon quantization leads to the divergence
𝜇
𝜕𝜇 𝐽𝑙𝑒𝑝𝑡𝑜𝑛 =
3
1
𝜖 𝜇𝜈𝜆𝜌 ( 𝑔22 Tr[𝐖𝜇𝜈 𝐖𝜆𝜌 ] − 𝑔12 𝐁𝜇𝜈 𝐁𝜆𝜌 )
2
64𝜋
2
which should bear a striking resemblance to the previous anomaly discussed, where
𝜇
𝜕𝜇 𝑗𝐴 = 2𝑖𝑚𝜓̅𝛾 5 𝜓 −
𝑒 2 𝜇𝜈𝜆𝜌
𝜀
𝜕𝜇 𝐴𝜈 𝜕𝜆 𝐴𝜌
2𝜋 2
However, here we instead have 𝐖𝜇𝜈 and 𝐁𝜇𝜈 , which are the 𝑆𝑈(2) × 𝑈𝑌 (1) fields (as discussed
in class), and 𝑔1,2 are the usual coupling constants. While the total quark number is also classically
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conserved, the same anomaly arises for its fields when each color is considered. If we in turn sum
over all three colors, we find the remarkable statement that is
𝜇
𝜇
𝜕𝜇 𝐽𝑞𝑢𝑎𝑟𝑘 = 3𝜕𝜇 𝐽𝑙𝑒𝑝𝑡𝑜𝑛
or, if the leptonic current also sums over all flavors,
𝜇
𝜇
𝜕𝜇 𝐽𝑏𝑎𝑟𝑦𝑜𝑛 = 𝜕𝜇 𝐽𝑙𝑒𝑝𝑡𝑜𝑛
Recall that this only holds if neutrinos are indeed Dirac, for if they are Majorana, we find instead
that with all of these anomalies, there can be no global conservation laws.
The preceding divergence equation for baryon and lepton currents effectively shows that the
anomalies reduce these two classically conserved currents of the SM to a single conserved total
current
𝜇
𝜇
𝜕𝜇 (𝐽𝑏𝑎𝑟𝑦𝑜𝑛 − 𝐽𝑙𝑒𝑝𝑡𝑜𝑛 ) = 0
𝜇
𝜇
However, it should be noted that the current 𝐽𝑏𝑎𝑟𝑦𝑜𝑛 + 𝐽𝑙𝑒𝑝𝑡𝑜𝑛 is not conserved. From this, we see
that the total changes in Δ𝐵 = Δ𝐿 for non-perturbative processes within the SM; this is precisely
the “good” quantum number of the nonpertubative SM, 𝐵 − 𝐿 [6].
Failures
Matter Asymmetry Generated by an Anomaly?
It can be shown from such baryonic and fermionic currents that a respective topological number
exists which effectively separates different possible vacuum state configurations for leptons and
baryons. It was thought that tunneling through an energy “barrier” separating these vacuum
states via instantons could, with simultaneous baryon and lepton number violation, account for
the matter-antimatter asymmetry. If one believes that the barrier separating the various vacuum
states is able to disappear at high temperatures, one could conclude that baryon number
violation is no longer suppressed. At energies above the electroweak phase transition in the early
universe, it has been suggested that the rates of processes with Δ𝐵 ≠ 0 are faster than the
expansion rate of the universe, meaning that any asymmetry between baryons and antibaryons
would be removed. To be clear, it could be the case that in electroweak interactions at high
temperature, one may conserve 𝐵 − 𝐿, but instead 𝐵 + 𝐿 is erased. However, such tunneling is
incredibly inefficient, and even classical motion over the energy barrier can be shown to be
ineffective.
If this is the case then, the anomalies of the SM, especially in the electroweak sector, while
capable of rendering the model renormalizable, are ineffective at generating any asymmetry
(they would, in fact, be a terminator). Precisely, if the electroweak phase transition is of second
order and thermal equilibrium is not disturbed, then the asymmetry is not generated. This turns
out to depend critically on the mass of the Higgs boson, where for a high mass (≿ 100 𝐺𝑒𝑉) the
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transition will be of second order. However, for a low mass (≾ 50 𝐺𝑒𝑉), it will be of first order,
and regions of asymmetry could be generated. However, considering the Higgs is heavy, we know
now the transition must be second order, resulting in effectively no asymmetry whatsoever
[6,8,9,11,12,13].
The Sphaleron
The effect of the electroweak phase transition and its ability to generate the BAU is recognized
to be far, far too weak of an effect to act as an adequate explanation of the universe we see
today. In order to mitigate these facts, the sphaleron mechanism was proposed. Speaking
roughly, it is known that processes with a non-zero change in baryonic charge are, at high
temperatures, accompanied by changes in the structure of the Higgs field. First, let us consider
the classical field configurations (present during the transition from one vacua to another)
𝐴𝑘𝑠𝑝ℎ𝑎𝑙𝑒𝑟𝑜𝑛 =
𝑖𝜖𝑘𝑙𝑚 𝑥 𝑙 𝜏 𝑚 𝑓𝐴 (𝑔𝜂𝑟)
𝑟2
&
𝜙 𝑠𝑝ℎ𝑎𝑙𝑒𝑟𝑜𝑛 =
𝑖𝜂 𝜏 𝑖 𝑥𝑖
(
) 𝑓𝜙 (𝑔𝜂𝑟)
√2 𝑟
where 𝜂 is the vacuum expectation value of the Higgs field, and 𝑓 varies from 1 to 0 on ℝ+ . From
these, it is possible to deduce that the sphalerons are objects that, if assumed to be in thermal
equilibrium with one another (and so described by a Boltzmann exponent dependent upon the
Gibbs free energy) at 𝑇𝑒𝑉 scales, then baryon number violating process again may not be
suppressed at high temperature.
However, the rate of production of these classical field states is not known and one cannot say if
they would even be in thermal equilibrium or not. From these configurations, it is thought that
one would need to create quite a special coherent field, which is quite improbable considering
the chaos of the early universe; if such is the case, we again end up with a situation where the
process does not produce an asymmetry in a significant enough amount. This would imply that
the quantum number of 𝐵 − 𝐿 is itself not a good quantum number, and must too be violated at
some point to generate the BAU. This points to the plausible need of a model which goes beyond
the standard model of particle physics.
It should be noted that these conclusions are not from analytical solutions to this problem, but
instead stem from numerical lattice calculations. Considering such effects are non-perturbative
and multi-particle, this leads to many different results from different groups [7,8,9,12,14].
Going Beyond the Standard Model of Particle Physics
Rumors of Revolution
It cannot be stressed enough the incredible descriptive power the SM holds. Using nearly only
the language of mathematical symmetries (indeed, requiring them where none were previously
believed to exist), new particle after new particle have together formed a zoo of predictions
backed up by mountains of experimental evidence. However, we know such predictions have
more room to grow (especially in consideration of the preceding discussion).
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One of the most well-known yet little understood phenomena that effectively confirms the
hypothesis that the SM is incomplete is the presence of neutrino oscillations. Within the SM,
such weakly interacting particles should remain massless, in blatant violation of such oscillation
data. This has led to the discovery of the two possible neutrino mass hierarchies (normal and
inverted), whereby the ordering of the neutrino masses can be determined based upon certain
oscillation parameters gathered from multiple sources, such as the CMB and ground-based
neutrino detectors of short and far baseline lengths. The most popular mechanism to explain
neutrino masses is the “see-saw”, which are a class of models which can be added to the SM
Lagrangian in such a way as to produce light neutrino masses for all three flavors. For instance,
in a type-1 see-saw mechanism, this usually implies that there exists a corresponding heavy
neutrino for each flavor, possibly with masses near the grand unification scale (and so most
likely impossible to observe). This see-saw phenomena can be illustrated by the calculation of
the eigenvalues of a (mass) matrix like
0
(
𝑀
𝑀
)
𝐵
where we assume that 𝐵 ≫ 𝑀; calculating eigenvalues yields
𝐵 ± √𝐵 2 + 4𝑀2
𝑀2
𝜆± =
≈ {𝐵, − }
2
𝐵
which implies that as one eigenvalue increases, the other decreases (or see-saws), and viceversa, in turn allowing for the observed low neutrino masses.
Other interesting forays into grand unification theories (GUTs) at high energy scales include
extensions to the SM (usually, but not always, in the form of supersymmetric theories) in low
dimension operators which can generate proton decay. Such an operator is usually of
dimension six, and could take the form of
𝒪𝑝 ~𝜆𝑝
𝑞𝑞𝑞𝑙
2
𝑀𝐺𝑈𝑇
where 𝑞’s represent quarks, and 𝑙 a lepton; thus, all proton decays violate baryon number by
one unit, while technically lepton number can be violated by any odd unit (if these operators
take different forms) [1]. A possible diagram of such a process can be seen here:
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However, this process, such as the reaction 𝑝 → 𝑒 + 𝜋 0 can be severely constrained given very
strong experimental evidence: it has historically never been observed, no matter how big the
detector volume has become (sources of background make this difficult, but even despite
~1030 protons available for decay within the fiducial volume at Super-Kamiokande, no high
confidence level observation has yet been performed).
Moving On?
Lack of experimental evidence for proton decay has been a setback of sorts for the particle
physics community; this, coupled with lack of any viable candidates for supersymmetry at the
LHC, have prompted some physicists to abandon the theoretical aspects of proton decay
entirely. For our purposes focusing on the discussion of the baryon abundance, it should be
further noted that the “golden channel” of observation of the proton decay (thought to be the
most easily distinguishable from background candidates in large mass detectors), where 𝑝 →
𝐾 + 𝜈̅ , does not itself violate 𝐵 − 𝐿, and so cannot help account for the baryon abundance. This
process, if it exists, may indeed be interesting in its own right, but such a discovery would have
little impact on our more fundamental (and arguably important) questions.
In light of this, models of baryon number violation have been formulated wherein the proton
decay plays little to no role (and similarly for a bound neutron). Importantly, models such as
these utilize decays of new, high mass, colored, Higgs-like, (diquark) scalar fields which couple
to quark (and possibly lepton) bilinear terms.This new field can lead to baryon number violation
through trilinear or quartic scalar interactions such as 𝑋𝑌𝑍, 𝑋𝑌𝑌, 𝑋𝑍𝑍, et cetera [1,2,3]. It
should be noted that, while these models are not explicitly supersymmetric (though they can
be), the cubic scalar interaction is similar to renormalizable terms in the superpotential that
gives rise to 𝐵 − 𝐿 violation in supersymmetric extensions to the SM [Wise]. This decaying
scalar couples to the SM via a high-dimensional operator 𝒪, which can take the form of
𝒪~𝜆
𝑞𝑞𝑞𝑞𝑞𝑞
5
𝑀~𝐸𝑊𝑃𝑇
which we see to be of dimension nine (𝑑 = 9); unlike the immense mass scale seen in the
proton decay operator, this one is far more manageable, and potentially visible, given it would
be closer to that of the electroweak phase transition temperature. We will soon come to the
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proposed diagrams for these decays, whose crossing symmetry we will see to be an ideal
candidate for neutron-antineutron oscillation.
Post-Sphaleron Baryogenesis
To account for the baryon asymmetry of the universe and lack of symmetry breaking ability of
the sphaleron mechanism given conservation of 𝐵 − 𝐿, these new extensions to the SM must
make use of post-sphaleron baryogenesis (PSB), where the dynamics occur at or below the TeV
scale, critically following the epoch of the electroweak phase transition, and when the
sphaleron’s have gone out of thermal equilibrium.
One popular PSB model makes use of higher-order gauge groups which utilize spontaneous
symmetry breaking at various scales down to the current SM group. The chief candidate for
such a theory is due to Mohapatra et al., where PSB is realized through an upper limit
symmetry group of
𝑆𝑈(2)𝐿 × 𝑆𝑈(2)𝑅 × 𝑆𝑈(4)𝑐 ,
utilizing a quark-lepton unified field theory generalization of the seesaw mechanism at the TeV
scale; from here, this representation has its symmetries broken by a Higgs field—which splits
the 𝑆𝑈(4)𝑐 mass scale from the remaining ones above ≳ 1000 𝑇𝑒𝑉 (satisfying constraints from
kaon decay)—down to the gauge group
𝑆𝑈(2)𝐿 × 𝑆𝑈(2)𝑅 × 𝑈(1)𝐵−𝐿 × 𝑆𝑈(3)𝑐 .
This surviving group is then broken down again by another Higgs field to
𝑆𝑈(2)𝐿 × 𝑈(1)𝐼3𝑅 × 𝑈(1)𝐵−𝐿 ,
which in turn eventually breaks to the group of the SM. The 𝑑 = 9 operator 𝒪 from this model
couples to a TeV-scale scalar field 𝑆 from the exchange of color-sextet fields; 𝒪 also leads to
baryon violation through neutron-antineutron oscillation.
The additional interaction Lagrangian for this model takes the form of
ℒ𝐼 =
𝑓𝑖𝑗
ℎ𝑖𝑗
𝑔𝑖𝑗
𝜆
Δ𝑑𝑑 𝑑𝑖 𝑑𝑗 +
Δ𝑢𝑢 𝑢𝑖 𝑢𝑗 +
Δ𝑢𝑑 (𝑢𝑖 𝑑𝑗 + 𝑢𝑗 𝑑𝑖 ) + Δ𝜈𝜈 Δ𝑑𝑑 Δ2𝑢𝑑 + 𝜆′ Δ𝜈𝜈 Δ𝑢𝑢 Δ2𝑑𝑑 + 𝐻𝐶
2
2
2
2√2
Here, Δ𝑞𝑞 are colored, scalar, diquark fields, while the Yukawa couplings obey boundary
conditions of 𝑓𝑖𝑗 = ℎ𝑖𝑗 = 𝑔𝑖𝑗 in the higher gauge group symmetry limit; note that all fermions
are right handed at this stage (before symmetry breaking), and the chiral projection operators
have been neglected for simplicity. The field Δ𝜈𝜈 is a neutral complex field whose real part
acquires vacuum expectation value (VEV) 𝑣𝐵𝐿 in the ground state such that the field can be
written as
Δ𝜈𝜈 = 𝑣𝐵𝐿 +
1
√2
(𝑆 + 𝑖𝜒)
11
where the field 𝜒 is eventually absorbed by the 𝐵 − 𝐿 gauge boson of the appropriate ~𝑈(1)
symmetry subgroup (above), while the real scalar 𝑆 remains as a physical Higgs; the decay of 𝑆
is what leads to the baryon abundance [3].
A Plausible Baryon Abundance?
The 𝑆 scalar field is contained within Δ𝜈𝜈 as a real, physical Higgs particle, and therefore can
decay into six quarks or antiquarks (or, with appropriate crossing symmetries, a mixture of
both), and therefore violate baryon number by two units with no associated change in the
lepton number of the universe. These same interactions, upon insertion of VEV for 𝑆, leads to
neutron-antineutron oscillations.
Note that if the diquarks Δ𝑞𝑞 have TeV range masses, then they will lead to large rates for
baryon violating processes, allowing neutron-antineutron oscillation to remain in equilibrium
until the TeV scale (near the electroweak phase transition). Thus, as previously discussed, this
would only continually erase any preexisting matter-antimatter asymmetry through the
sphaleron process; therefore, a new process (the PSB) must be commissioned, which can
simultaneously satisfy all of Sakharov’s conditions. If we assume that 𝑆 is lighter than the
diquark fields, then it will go out of equilibrium and decay after the electroweak phase
transition, producing six quarks (antiquarks), and so asymmetrically generating the baryon
asymmetry.
Let 𝑇 be the average temperature of the universe. Considering that the 𝑆 particle decay rate
decreases as ~𝑇 13, compared to the universe’s expansion rate going down as 𝑇 2 , then the two
will meet at some temperature 𝑇𝑑 ; up until this point, we will say that the particles of the 𝑆
field will simply drift about the expanding volume of the universe. When the Hubble rate
approaches the temperature crossing point 𝐻(𝑇𝑑 ), which in turn becomes of the same order as
12
the decay width of the 𝑆 particles, then the particle decays will begin; if 𝑇𝑑 ≤ 100 𝐺𝑒𝑉 (below
which the sphalerons have definitely gone out of thermal equilibrium), and yet 𝑇𝑑 > 200 𝑀𝑒𝑉
(otherwise, nucleosynthesis would be spoiled), which we treat as the effective range for the
mass of the 𝑆 boson, we will generate an appropriate baryon abundance. This properly arises
from the complicated interplay of the Hubble rate
1
1.66𝑔∗2 𝑇𝑑
𝐻(𝑇𝑑 ) =
,
𝑀𝑃𝑙𝑎𝑛𝑐𝑘
where 𝑔∗ is the number of (relativistic) degrees of freedom at temperature 𝑇𝑑 , and the decay
width of the 𝑆 particle (with appropriate phase-space factor for the six-quark final state)
𝑃
12 2
𝑀𝑆13
†
†
2
|𝜆| Tr[𝑓 𝑓] ⋅ (Tr[𝑔̂ 𝑔̂]) 8
Γ𝑆 = Γ(𝑆 → 6𝑞) + Γ(𝑆 → 6𝑞̅ ) = 9 25
.
𝜋 ⋅ 2 ⋅ 45 4
𝑀Δ𝑢𝑑 ⋅ 𝑀Δ4𝑑𝑑
Here, 12 comes from the number of final states with different combinations of 𝑆𝑈(3)𝑐 color,
and 𝑃 is the numerical value of the phase space integral. The model is also able to calculate,
using a unitary gauge, a primordial CP asymmetry (not shown) arising from loop diagrams such
as
The authors contend that the limits on all necessary couplings and masses (when examined in
MonteCarlo) will produce both proper CP asymmetry of the order of ~10−8 , which in turn can
be further tuned (with 𝑀𝑆 < 10 𝑇𝑒𝑉, and so not extremely limiting in the ranges of all
variables) such that a calculation of the baryon abundance yields values close to those of
𝑛𝑏 − 𝑛𝑏̅
= (6.04 ± 0.08) × 10−10 ,
𝑛𝛾
the astronomically observed value [2,3].
A Toy Model
A good way to understand this particular phenomenon is to use a toy model to calculate and
recognize the hurdles that must be jumped in order to fit the true extension into the SM. To
understand this, we must look back on the history of particle physics, where in the 1960’s
13
Nanopoulos and Weinberg (NW) discussed one-loop contributions to the baryon asymmetry in
a theorem. This theorem held that if 𝛼𝐵 is the baryon number violating coupling, then nonzero
baryon asymmetry could arise from one-loop contributions, with the asymmetry being
proportional to 𝛼𝐵3 (two powers coming from tree level, one from the loop); however, with the
current theory considered here, it is shown that at an order of 𝛼𝐵2 , the asymmetry can begin to
arise. This is because the assumptions that were used in proving the NW theorem are not
applicable to this work, and so this PSB model is one of a new class that could explain
baryogenesis.
Consider a toy model with an interaction Lagrangian of real scalar field 𝑋 and complex scalar
field 𝑌, each with baryon number zero. Let 𝑌 mimic the effects of the W-bosons in loops
pertinent to the toy model. Let 𝑋 decay into baryons and or antibaryons with couplings 𝑓1∗ 𝑓2
and 𝑓1 𝑓2∗ . The interaction Lagrangian takes the form
ℒ1 = 𝑔1 𝑋𝑓2† 𝑓1 + 𝑔2 𝑋𝑓4† 𝑓3 + 𝑔3 𝑌𝑓3† 𝑓1 + 𝑔4 𝑌𝑓4† 𝑓2 + 𝐻𝐶
where the couplings 𝑔𝑖 have dimensions of mass, not all of which are real, as one phase can be
permitted between them, thus violating CP symmetry explicitly.
Some diagrams pertinent to the discussion of baryon number violation within this toy theory
are at tree and one-loop level (triangle), and consist of:
If we assume that the mass of 𝑌 is much more than the mass of 𝑋, then, in the early universe,
when the 𝑋 particles decay, we know that the 𝑌’s have also decayed away. Two baryon
violating final states can occur for the 𝑋 boson: 𝑋 → 𝑓1∗ + 𝑓2 or 𝑋 → 𝑓3∗ + 𝑓4 , each of which
have baryon number 𝐵 = −1. Thus, the conjugate of these decays have baryon number 𝐵 =
+1. Taking these all together, we can construct the net baryon asymmetry due to 𝑋 decays as:
𝜖𝐵 =
Γ(𝑋 → 𝑓1 + 𝑓2∗ ) + Γ(𝑋 → 𝑓3 + 𝑓4∗ ) − Γ(𝑋 → 𝑓1∗ + 𝑓2 ) − Γ(𝑋 → 𝑓3∗ + 𝑓4 )
Γ(𝑋 → 𝑓1 + 𝑓2∗ ) + Γ(𝑋 → 𝑓3 + 𝑓4∗ ) + Γ(𝑋 → 𝑓1∗ + 𝑓2 ) + Γ(𝑋 → 𝑓3∗ + 𝑓4 )
Thus, the interference between tree and one-loop decays lead to a net baryon asymmetry of
the universe within this toy model. This asymmetry is nonvanishing, even with the heavy 𝑌
boson being utilized (and with a baryon conserving vertex, no less), at only at 𝛼𝐵2 order, due to
the differences in masses of the flavor states between these particles. It should be noted that
this toy model is not a gauge model, and so there are no issues of gauge invariance to worry
about [3].
14
Restrictions and Further Explanation of the Model
Going back to our consideration of the true SM extension for PSB, such a model must,
importantly, make use of experimental inputs for all necessary variables in order to properly vet
its viability, as well as to attest to the prospect that its new phenomena could be observed in
next or next-to-next generation experiments.
One of the most important things to understand is that such a model has not been entirely
ruled out yet simply due to the presumably light 𝑆 mass; this is because of the fact that 𝑆 does
not couple to quarks or electrons directly, but through virtual diquark intermediaries. Its
primary coupling is instead to right handed neutrinos, which, thanks to the type-II seesaw
mechanism, have masses in the TeV range.
The specific scheme considered in the 2013 work by Mohapatra et al. considered the type-II
seesaw, wherein they assumed that the mass hierarchy of observed neutrinos would be the
inverted type; while there are rumblings in the neutrino community that the true hierarchy may
indeed be normal type, this is not necessarily a death-spell for the model. In general, there are
always two contributions that enter the model, and, if cancellation is assumed, it is not
impossible to fit normal mass hierarchy neutrinos to it; however, such work is forthcoming.
To be sure, all current limits on the understanding of oscillation phenomena (kaons and the
like) are considered as important inputs, where it happens that the diquark fields generate new
contributions with box diagrams. Thus, the model is able to reproduce all of the experimentally
known phenomena within the appropriate parameters with constraints listed in Table I and II,
above, from Mohapatra et al’s 2013 work; what we will see is that what really matters for
neutron-antineutron oscillations, specifically, is the mass scales of the new scalar fields [3].
Neutron-Antineutron Oscillations
The general diagram that yields neutron-antineutron (𝑛 − 𝑛̅) oscillations can be shown as:
15
Here, three quarks turn into (through some unknown, shaded reaction region) three
antiquarks; particularly, we have the transition 𝑢𝑑𝑑 → 𝑢̅𝑑̅ 𝑑̅ for our purposes. As previously
discussed, the Mohapatra model uses the exchange of high mass, color sextet diquark fields to
generate this Δ𝐵 = 2 oscillation. It is found in the authors’ 2013 work that there are two
contributions to the amplitude for this process at tree level
𝑡𝑟𝑒𝑒
𝒜𝑛−𝑛
̅ =
2
2
𝑓11 𝑔11
𝜆𝑣𝐵𝐿 𝑓11 ℎ11
𝜆′𝑣𝐵𝐿
+
,
2
4
4
𝑀Δ𝑑𝑑 𝑀Δ𝑢𝑑
𝑀Δ𝑑𝑑 𝑀Δ2𝑢𝑑
where 𝑓11 is a vanishingly small coupling (satisfying flavor changing neutral current constraints),
and the extra mass factor in the denominator of each term is due to the dimensionfull VEV for
the neutral complex field Δ𝜈𝜈 which gives rise to the process at higher energy scales. A limit on
the strength of 𝑓11 can be found by studying the one-loop contribution to the amplitude, whose
diagram can be seen as:
The contribution of this one-loop diagram to the overall 𝑛 − 𝑛̅ amplitude can be shown to be
proportional to the transition amplitude ~𝜆𝑣𝐵𝐿 ⟨𝑛̅|𝒪2 |𝑛⟩, where various masses, constants, and
coupling are ignored for simplicity [3]. The operator 𝒪 (called 𝒪𝑅𝐿𝑅 to denote chiral properties
of the quarks in this work) is the operator which governs 𝑛 − 𝑛̅, as mentioned previously;
precisely, this operator takes the form of
𝑠
T
T
2
T
𝒪𝑅𝐿𝑅
= (𝑢𝑖𝑅
𝐶𝑑𝑗𝑅 )(𝑢𝑘𝐿
𝐶𝑑𝑙𝐿 )(𝑑𝑚𝑅
𝐶𝑑𝑛𝑅 )Γ𝑖𝑗𝑘𝑙𝑚𝑛
16
𝑠
which we see to be 𝑑 = 0, and where Γ𝑖𝑗𝑘𝑙𝑚𝑛
is similar to a Gell-Mann matrix, containing color
information in the form of linear combinations of Levi-Civita symbols. From work in the 1980’s
2 |𝑛⟩
using the MIT bag model, this work is able to use the amplitude ⟨𝑛̅|𝒪𝑅𝐿𝑅
= −0.314 ×
−5
6
10 𝐺𝑒𝑉 to predict an upper limit on the mean oscillation time within this model when
properly weighting parameters within permissible ranges with MonteCarlo. To ascertain the
oscillation time, we consider
1−𝑙𝑜𝑜𝑝
−1
𝜏𝑛−𝑛
|
̅ = 𝐶(𝛼𝑠 , 𝜇Δ , 1𝐺𝑒𝑉) ⋅ |𝒜𝑛−𝑛̅
where 𝐶 is the renormalization group running factor bringing the scale down from the original
Δ scale to that of the mass of the neutron (𝜇Δ is the geometric mean of the two diquark fields,
and is on the order of a TeV), giving it an approximate value for 𝐶~0.18.
Varying the parameters available to PSB yield the following distributions in terms of the
pertinent mass scales:
These, in turn, can give an absolute upper limit to the free oscillation time of 𝜏𝑛−𝑛̅ ≤ 4.7 ×
1010 𝑠 for the most relevant parameters (such as the diquark field masses); note the
experimental lower limit has a value of ~108 𝑠. This same model is also able to generate a
likelihood probability for the oscillation time given these parameters, where there are implicit
assumptions requiring that the VEV 𝑣𝐵𝐿 ≰ 200 𝑇𝑒𝑉, or otherwise there are no allowed points
17
within the model, as below such a scale, the decay rate of the 𝑆 particle would no longer be
dominant, thus not contributing to the baryon abundance in the proper way (even while
satisfying all other constraints) [3].
Lattice Quantum Chromodynamics Studies
We have and will chiefly consider only one model in this article for conciseness; however, many
other types can be considered. Thus, in order to compare many of the various models with
differing rates between hadronic states (which can include non-perturbative QCD effects), it is
important to consider model-independent calculations of operator matrix elements with
studies in lattice quantum chromodynamics (LQCD). Truly, experimental tests are necessary to
constrain the parameter space of all possible BSM theories to determine which could describe
our material universe. Luckily, in many attractive BSM theories with suppression of high energy
Δ𝐵 = 1 processes, such as proton decay, we can instead search for low energy Δ𝐵 = 2
processes, such as 𝑛 − 𝑛̅, utilizing a six-quark operator. The characteristically low mass scale
permits the possibility of next or next-to-next generation experimental detection.
Baryon number violating processes can be treated in a model independent way by thinking of
the SM as an effective field theory (EFT) which includes parametrizations of baryon violating
local operators using quarks and other SM particles [4]. The Hamiltonian governing 𝑛 − 𝑛̅
oscillations can be thought of as a linear combination of all permitted six-quark, dimension nine
operators built from the SM. The only way to make reliable predictions for experimental testing
is by using LQCD, whose controlled uncertainties (which get smaller the higher in order one
computes) allow easy comparison with data, unlike the uncontrolled uncertainties present in
older MIT bag model calculations [5].
The free 𝑛 − 𝑛̅ oscillation time can be predicted given the matrix elements of the effective
Hamiltonian density
𝑛−𝑛̅
ℋ𝑒𝑓𝑓
= ∑ 𝐶𝐼 (𝜇)𝑄𝐼 (𝜇) ∝
𝐼
1
1
𝑛−𝑛̅
⇒
= ⟨𝑛̅|ℋ𝑒𝑓𝑓
|𝑛⟩
5
𝑀
𝜏𝑛−𝑛̅
18
where 𝑄𝐼 forms a complete basis of dimension nine operators with nonvanishing transition
2
matrix elements between 𝑛 and 𝑛̅ states (with roughly identical structure to 𝒪𝑅𝐿𝑅
above), 𝐶𝐼
are Wilson coefficients (which are scale and renormalization scheme dependent), and 𝜇 is a
renormalization scale. Thus, this formula, and thus the oscillation time, will differ from theory
to theory. How to regularize and renormalize these operators is a complicated business, and
not the subject of this article, but calculations of the oscillation time at up to the two-loop level
continue today using suitably generalized four-quark operator renormalization techniques with
projectors which simplify two-loop tensor decompositions [4,5].
19
In all, up to two-loop level, over 320 diagrams must be considered and calculated, then
properly renormalized. Preliminary results simulate 𝑛 − 𝑛̅ on anisotropic Wilson lattices (which
lead to spacing artifacts, breaking chiral symmetry in a non-physical way) with 390 MeV pions
(far heavier than a truly realistic calculation would necessitate), and preliminary results suggest
these and others lead to significant sources of systemic error, which are not currently
quantified explicitly; in principle, all of these would be reduced with increased computational
time.
Currently, given the Super-Kamiokande lower limit for the oscillation time of
1
𝑂 16
𝜏𝑛−𝑛
̅
< 2 × 10−33 𝐺𝑒𝑉
we expect the LQCD value to be even smaller; reliable predictions for this can be made by
perturbatively matching BSM calculations to EFT SM operators. Current work, assuming a 𝑀 =
500 𝑇𝑒𝑉 mass for the pertinent scalar bosons yields
1
𝐿𝑄𝐶𝐷
𝜏𝑛−𝑛
̅
= ⋯ = (3.68 ± 0.16) × 10−34 𝐺𝑒𝑉
thanks to LO, NLO, and NNLO (matching and running) contributions, and taking into account
statistical lattice errors. What is quite interesting is that this LQCD method can actually derive a
limit on the mass 𝑀 of 𝑀 > 357 ± 3 𝑇𝑒𝑉, in keeping with previous limits of ≥ 200 𝑇𝑒𝑉 set by
Mohapatra et al [5].
In all, this burgeoning technique will provide the best comparisons with future experimental
values, and, if no oscillation is observed, implement the most heavy-handed constraints on
models of post-sphaleron baryogenesis, possibly excluding it entirely.
Conclusions
The baryon asymmetry of the universe is one of the most fundamental problems plaguing
physics today. Current standard model theoretical results imply that fine-tuning of the initial
conditions of the universe is a requirement to explain this asymmetry, where otherwise the
sphaleron process erases any observable asymmetry. Thus, it is thought that the baryon
abundance should have been produced in a dynamic way; any mechanism describing such a
dynamic process is inherently beyond the standard model, and must of course satisfy all of the
Sakharov conditions. One such mechanism utilizes high mass, Higgs-like scalar fields which
decay into virtual diquark fields, which can produce neutron-antineutron oscillations around
the TeV scale. Such a model can appropriately predict the baryon abundance of the universe,
and current work is continuing on loop-level calculations and computations of pertinent
dimension nine operator matrix elements which contribute to the oscillation amplitude. Such
predictions are important, along with future experimental findings, to distinguish between
various extensions to the standard model, supersymmetric and not.
20
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