Topological Leptogenesis

The gist: Topological leptogenesis proposes a new mechanism to explain lepton asymmetry by introducing a gapped topological order sector that can cancel the baryon minus lepton (B − L) mixed gauge-gravitational anomaly of the Standard Model and mediate decaying processes from highly entangled excitations

Introduction and Summary

Leptogenesis is a hypothetical physical process that produces lepton and antilepton number asymmetry in the very early universe Baryogenesis is yet another hypothetical process during the early universe to produce baryon and antibaryon asymmetry. The two syntheses result in the present-day dominance of leptons over antileptons and baryon over antibaryons in the observed universe. Sakharov proposed three necessary conditions to produce an imbalance of baryons and antibaryons: (1) Baryon number B violation – namely the continuous U(1)B breaking (2) Discrete charge conjugation C symmetry and CP (or time-reversal T) symmetry violation. (3) These symmetry violations happen when the universe is out of thermal equilibrium. The standard Standard Model does not contribute enough to the condition (2) and (3), thus the leptogenesis requires some beyond the Standard Model (BSM) effect.

Major Scenarios for Leptogenesis

One of the most popular scenarios is Majorana leptogenesis [2], which introduces hypothetical heavy sterile right-handed neutrino νR that are sterile to the Standard Model (SM) gauge force, while νR is paired with itself with Majorana mass term in the lagrangian form MνR TνR + h.c. that breaks the fermion number conservation U(1)F down to the fermion parity ZF2 subgroup. The decay of heavy νR can convert to Higgs boson and other experimentally observed less-heavy leptons — including U(1EM charged leptons such as electron e −, muon µ − and tauon τ −, and U(1EM neutral left-handed neutrinos νL,e, νL,µ and νL,τ. Some of these leptons can further decay to quarks through the electroweak sphaleron [4, 5] via the Adler-Bell-Jackiw (ABJ) anomaly [6, 7] under the Yang-Mills field instanton [8, 9] of the weak gauge force, which is a popular scenario of baryogenesis. Another scenario without necessarily introducing any sterile right-handed neutrino νR is gravitational leptogenesis [10, 11]. The curved spacetime ripple locally may give rise to a source of gravitational instanton globally. So the mixed gravitational anomaly [12–14] between the lepton number U(1)L and the gravitational source from the triangle Feynman diagram U(1)L-gravity can generate an unbalanced new lepton number out of the curved spacetime ripple.

Topological Leptogenesis Mechanism

The purpose of this work is to propose a new leptogenesis mechanism — topological leptogenesis — by introducing a hidden BSM topological sector that couples to the SM via topological gauge interaction [15–18]. Precisely, the BSM topological sector is a Topological Quantum Matter sector whose low energy has a Topological Quantum Field Theory (TQFT, in the sense of Schwarz-type Chern-Simons-Witten-like theory [19–21]) and whose zero temperature phase (in terms of quantum many-body condensed matter sense) is topological order [22] (here sometimes abbreviated as TO). We propose that the fractionalized anyon like excitations [23, 24] with an energy E above the topological order gap ∆TO (so E ≥ ∆TO) can decay into the SM’s “elementary” particles via topological discrete gauge interaction (later in eq. (17)) or via the mixed baryon minus lepton (B−L) gauge-gravitational anomaly process (later in eq. (16)).

Anomalies and Symmetry Structure

Regardless of which types of leptogenesis out of the three scenarios that we decide to study, all three can be motivated by the same common theme on the quantum anomaly structure of the SM. The continuous baryon B minus lepton L number symmetry, U(1)B−L is preserved within the SM. To have a properly quantized charge, it is better to scale U(1)B−L by a factor of color number Nc as the quark Q number minus Nc lepton L number symmetry, U(1)Q−NcL, where the color number is Nc = 3 in the SM. The U(1)Q−NcL symmetry has a ’t Hooft anomaly in 4d spacetime captured by two triangle Feynman diagrams. This 4d ’t Hooft anomaly is also captured by a 5d invertible topological field theory (iTFT) in one extra dimension with the following invertible complex U(1) valued functional via the anomaly inflow. The reason we prefer to focus on U(1)Q−NcL instead of U(1)L symmetry is because only the former has mixed anomaly-free with the SM gauge group but the latter is anomalous with the SM gauge group.

Comparison of Scenarios

Table II summarizes, compares, and contrasts the three scenarios. Majorana leptogenesis [2] requires νR, while Gravitational leptogenesis [10] does not require νR. Topological leptogenesis [17] requires no νR. The comparison shows that for Majorana leptogenesis the UV symmetry is (B − L) and U(1)X alike, and Z4,X explicitly broken by νR’s Majorana mass term. For Gravitational leptogenesis, (B − L), U(1)X, and Z4,X are preserved. For Topological leptogenesis, a A X-symmetric gapped topological order with a low-energy TQFT is present.

Conclusion

The three scenarios studied here are general and universal in the sense of anomaly cancelation consideration given the systematic classification of anomalies in Refs. [35, 38, 44]. Regardless of which scenario nature may choose or not, The three scenarios will need to welcome new experimental signatures for verification.

Dark Matter and Baryogenesis

For Topological leptogenesis as a dark matter candidate, the characteristic properties of topological order are that the energy spectrum above the TQFT ground state is gapped, with an energy gap size denoted ∆TO. The energetic excitations with energy E above the gap (E ≥ ∆TO) contain fractionalized excitations with anyon statistics [24] (that can be neither boson nor fermion statistics). Baryogenesis in all scenarios can use the sphaleron from the electroweak su(2)w × u(1)Y sector to generate baryon asymmetry.

Comments

Pros for Majorana leptogenesis include a Seesaw mechanism with tiny νL mass, and then the sphaleron generates the baryon asymmetry. Cons are that both continuous and discrete X (U(1)X and Z4,X) are explicitly broken at deep UV, but re-emerge at IR then to be SSB again by Higgs vev ⟨ϕH⟩. Pros for Gravitational leptogenesis include the U(1) of (B − L) is not exact nor dynamically gauged. No additional new gauge U(1) of (B − L), thus no new photon is needed. Cons are that the model itself is gravitational anomalous. Pros for Topological leptogenesis include a consistent discrete Z4,X gauge symmetry preserving from deep UV to IR, until below ΛEW scale, the Z4,X SSB by ⟨ϕH⟩ condensate. Cons are that topological quantum matter as dark matter and its topological discrete gauge interactions with the SM are difficult to be detected by the conventional experiments.

Dark Matter Candidate Details

The Dark Matter candidate for Topological leptogenesis includes fractionalized 0d particle (1d worldline) and anyon 1d loop (2d worldsheet) gapped excitations above ∆TO in the 4d spacetime. It also includes Gapped SPTs with low-energy iTFT in the 5d spacetime and Gapped topological order with low-energy noninvertible TQFT in the 5d spacetime when Z4,X becomes dynamically gauged at UV high-energy.

Baryogenesis Details

Baryogenesis can occur through the sphaleron doing the conversion from the anomalous lepton current via L-su(2)2 w anomaly to the anomalous baryon current via B-su(2)2 w anomaly in the flat spacetime. It is possible to have the conversion from the anomalous lepton current via L-u(1)2 Y anomaly to the anomalous baryon current via B-u(1)2 Y anomaly in the nontrivial spacetime topology. The (B − L) current is conserved and non-anomalous, because (B − L)-su(2)2 w and (B − L)-u(1)2 Y are anomaly-free, at this energy scale range.

This work proposes topological leptogenesis as an alternative scenario to resolve the challenges of other mechanisms by introducing a new gapped topological order sector.

1. Bold Dark Matter Candidate Identification: The improved system can identify dark matter candidates as heavy νR with Majorana mass (from Table II) or Fractionalized 0d particle (1d worldline) and anyon 1d loop (2d worldsheet) gapped excitations above ∆TO in the 4d spacetime.

2. Bold Baryogenesis Mechanism Selection: The system can distinguish between three distinct baryogenesis paths: Majorana leptogenesis, Gravitational leptogenesis, and Topological leptogenesis, based on whether the scenario requires or necessitates a sterile right-handed neutrino (νR).

3. Bold Anomaly Cancellation Verification: The system can verify anomaly cancellation by checking specific conditions, such as how adding νR affects anomalies: Adding νR cancels the continuous (B − L)-grav2 and (B − L)3 anomalies, and the discrete Z4,X-gravity anomaly.

4. Bold Symmetry Pattern Mapping: The system can map the symmetry behavior across energy scales for each scenario: Majorana leptogenesis [2] Gravitational leptogenesis [10] Topological leptogenesis [17] by comparing whether symmetries like (B − L) and U(1)X alike, and Z4,X explicitly broken by νR’s Majorana mass term or if they are preserved (no Majorana mass term).

5. Bold Dark Matter Signature Prediction: The system can predict experimental signatures for dark matter based on the scenario: Gapped SPTs with low-energy iTFT in the 5d spacetime for topological leptogenesis, or Heavy νR with Majorana mass additional axion φ like in φTr[R ∧ R] [10] for gravitational leptogenesis.

6. Bold IR Symmetry Preservation Check: The system can assess the stability of symmetries at low energies: The continuous and discrete X (U(1)X and Z4,X) are spontaneously broken down to ZF2 due to Higgs condensate ⟨ϕH⟩ carries X and electroweak hyper Y charges for Majorana leptogenesis.

In the standard lore, the baryon asymmetry of the present universe is attributed to the leptogenesis from the sterile right-handed neutrino with heavy Majorana mass decaying into the Standard Model's leptons at the very early universe -- called the Majorana fermion's leptogenesis; while the electroweak sphaleron causes baryogenesis at a later time. In this work, we propose a new mechanism, named topological leptogenesis, to explain the lepton asymmetry. Topological leptogenesis replaces some of the sterile neutrinos by introducing a new gapped topological order sector (whose low-energy exhibits topological quantum field theory with long-range entanglement) that can cancel the baryon minus lepton (B - L) mixed gauge-gravitational anomaly of the Standard Model. Then the Beyond-the-Standard-Model dark matter consists of topological quantum matter, such that the gapped non-particle excitations of extended line and surface defect with fractionalization and anyon charges can decay into the Standard Model particles. In addition, gravitational leptogenesis can be regarded as an intermediate step (between Majorana particle leptogenesis and topological non-particle leptogenesis) to mediate such decaying processes from the highly entangled gapped topological order excitations.