Real versus Complex Fermion Bare Mass and Multi-Wall String Defect Annihilation

The gist: The authors propose a novel mechanism for triggering the annihilation of cosmological domain walls in theories with global U(1) symmetries by introducing a tiny bare fermion mass term, which generates a temperature-dependent bias through radiative corrections.

Formation of Domain Walls in a global U(1) framework

Domain walls can arise when spontaneous symmetry breaking occurs in the early Universe, leading to topological defects such as domain walls (DWs) [2, 3]. These DWs are cosmologically problematic because their slow scaling behavior can eventually dominate the energy density of the Universe, giving rise to the well known cosmological DW problem [11]. Viable particle physics models must provide mechanisms that ensure their annihilation before the temperature drops below TBBN ∼ O(1 MeV) [10].

The Mechanism for Annihilation

The proposed mechanism involves a continuous global U(1) symmetry being explicitly broken to a discrete subgroup by gravitational effects, leading to string-wall networks [2]. If the U(1) theory contains fermions coupled to the symmetry-breaking scalar and also admits a small bare mass term, loop corrections can radiatively generate a temperature-dependent bias capable of triggering DW annihilation [17–21]. This works irrespective of the number of DWs attached to a string [2].

Detailed Dynamics in Majoron Framework

In the context of the global U(1)L majoron framework, gravitational effects induce Planck-suppressed operators that explicitly break the continuous symmetry down to a discrete subgroup, leading to a string-wall network [3]. The resulting low-energy effective potential for the majoron field acquires a form resembling a standard Sine-Gordon potential [7]. As the temperature decreases, the discrete vacuum structure becomes cosmologically relevant and domain walls form between each adjacent domains, with each string becoming attached to n domain walls [8].

The Role of Bare Fermion Mass

The annihilation dynamics are driven by the one-loop Coleman-Weinberg (CW) effective potential generated by a bare Right-Handed Neutrino (RHN) mass term [41, 42]. The leading k-dependent contribution to the CW bias is approximated as proportional to cos(2πk/n) [13]. This structure is similar to the Planck-suppressed bias discussed in Ref. [36, 43] [15]. The total bias between two vacua labeled by k1 and k2 can be expressed as the difference between the total potentials, where the first term is U(1) invariant and cancels out [16].

Annihilation Outcome

The analysis shows that for any integer k, implying that Vmajoron = constant for all vacua [5], the CW and thermal corrections generate nonzero contributions to ∆V due to the nontrivial structure of the potentials [13] and [14]. The condition k1 + k2 = n causes the CW contribution in Eq. (13) to vanish, and similarly, both CW and thermal contributions become identical, yielding ∆V = 0 for adjacent vacua satisfying this condition [5]. This implies that at most a single wall configuration survives, which is topologically unstable and self-annihilates [2].

Conclusion

The authors conclude that the formalism automatically generates the required bias for arbitrary values of n without requiring explicit ad hoc bias terms in the Lagrangian [44]. Furthermore, they find that even with friction effects from RHNs, the dominant source of vacuum bias arises from the CW term [15]. The study resolves the DW problem by showing that no stable DW network survives [2].

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HRI-RECAPP-2026-06

A New Route to the Annihilation of Multi-Wall String Topological Configurations

Utsav Atta,1, 2, ∗ Tathagata Ghosh,

1, 2, † and Sudip Manna 1, 2, ‡

1Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Prayagraj 211019, India

2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India

Particle physics models beyond the Standard Model often contain global symmetries to address various unanswered questions. However, a common criticism of theories based on global symmetries is that such symmetries are generally expected to be explicitly violated by gravitational effects at the Planck scale. In the case of a global U(1) symmetry, this explicit breaking can reduce the symmetry to a discrete subgroup of U(1), leading to the formation of cosmic strings attached to multiple domain walls (DWs). These DWs are usually cosmologically problematic, since their slow scaling behavior can eventually dominate the energy density of the Universe, giving rise to the wellknown cosmological DW problem, which is strongly constrained by Big Bang Nucleosynthesis. In this letter, we propose a new annihilation mechanism of such DWs in theories with the simplest continuous global symmetry, U(1), in the presence of gravitational effects. The mechanism is as follows: if a fermion coupled to the symmetry-breaking scalar possesses a small bare mass term, radiative corrections can generate a temperature-dependent bias for triggering DW annihilation. As a representative example, we study a majoron framework containing right-handed neutrinos with small bare mass terms, in which a wall-string network can arise once gravitational effects are taken into account. Within this setup, we show that the small bare masses of the right-handed neutrinos provide the origin of the bias responsible for triggering the annihilation of the DW network.

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explored more recently in the case of Z2 DWs generated by a real scalar field, where a small bare fermion mass term is introduced in the Lagrangian, which can radiatively generate the required bias [17–21]

1. Here, in this Letter, we apply this recently proposed idea of introducing a bare fermion mass term in Z2 DW scenarios to a more general and nontrivial setting of wallstring networks, which arise from the explicit breaking of a global U(1) symmetry to a discrete subgroup by gravitational effects and constitute the primary focus of our study. We show that if the U(1) theory contains fermion(s) coupled to the symmetry-breaking scalar and also admits a small bare mass term, loop corrections can radiatively generate a temperature-dependent bias capable of triggering DW annihilation. Importantly, we find that this mechanism works irrespective of the number of DWs attached to a string. Furthermore, owing to the nontrivial vacuum structure induced by gravitational effects in the global U(1) theory, we observe that the annihilation dynamics are considerably more intricate than in the simple Z2 case and therefore give rise to a significantly richer pattern of DW annihilation and the subsequent evolution of the vacua toward the true vacuum. A schematic illustration of this DW annihilation mechanism in a string-wall system with multiple attached DWs is shown in Fig. 1.

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Formation of Domain Walls in a global U(1) framework.– As a representative example, we consider the well known global U(1)L majoron framework [23], in which the SM is extended by three singlet RHNs and a complex scalar Φ charged under the global U(1)L symmetry. When this symmetry is spontaneously broken, the RHNs acquire Majorana masses, while the light neutrino masses can be generated via the type-I seesaw mechanism [24–30]. Here, L denotes the lepton number. Although we illustrate our discussion within this framework, our results apply more generally to all theories in which a global U(1) symmetry is spontaneously broken by a complex scalar field that has Yukawa interactions with fermions. As mentioned earlier, an important aspect of such scenarios is that global symmetries are generally expected to be violated by quantum gravitational effects near the Planck scale [12, 31]. For instance, such explicit breaking may arise due to the nucleation of wormholes. Following the wormhole solution presented in Ref. [12], for a field Φ carrying a nonzero charge under the global symmetry, integrating out wormhole configurations induces a low1 Ref. [22] proposed an interesting alternative mechanism in which effective operators induce linear couplings between the PecceiQuinn field and Standard Model particles, which ultimately generate a thermal potential that dynamically drives the axion field toward a universal value at high temperatures, thereby preventing DW formation in postinflationary axion models with discrete symmetry. energy effective term in the Lagrangian of the form LQG = X∞ n=1 Yngn Φ n + h.c., (1) where Yn are complex parameters associated with the choice of vacuum state of the model and n denotes the mass dimension of the operator. While, the effective couplings gn take the approximate form gn ≈ λ(n−4)/3n(n−16)/3M(n−4)Pl!e−λ1/3n4/3≡g n M(n−4)Pl. (2) Since the effective couplings ˜gn are exponentially small due to the combined gravitational and scalar Euclidean action in the wormhole background, they are expected to be extremely small. For operator dimensions in the range n = 5-20 and for λ = 1, one typically finds ˜gn≃ 10−(n+2), where λ being the self coupling of the scalar Φ [12]. In principle, as shown in Eq. (1), one can introduce operators starting from n = 1 up to infinity. However, allowing n ≤ 4 operators with O(1) values of ˜gn leads to hard breaking of the global U(1) operator at MP l itself and therefore such scenarios are not particularly interesting from a phenomenological perspective. For this reason, in our analysis we retain only operators with n > 4.

1. Dynamical DW Annihilation Mechanism The improved AI system can dynamically trigger DW annihilation by utilizing a tiny bare RHN mass term in the Lagrangian to generate a temperature-dependent bias, as described in Section 3.2 of the paper. This moves beyond introducing an ad hoc bias term in the scalar potential by leveraging loop corrections to generate the required bias.

2. Vacuum Structure Mapping The system can map the discrete vacuum structure by calculating the k-dependent effective total bias potential, defined as Vbias, total(k, T) = Vo(Φ) Φ=⟨Φ⟩k + Vmajoron(χ) χ=2πkv/n + VCW (k) bias + VJF MR,k2 T2 bias. This allows for precise prediction of which vacua survive annihilation.

3. Annihilation Epoch Prediction The system can determine the annihilation temperature Tann by solving the condition ∆V (Tann) ≃ ρw(Tann), providing a quantitative prediction for when DWs will annihilate, such as finding Tann = 1.78 × 10 11 GeV for BP1 in the n=5 case.

4. Scaling Behavior Analysis The AI can analyze the scaling behavior of wall energy density, noting that "numerical simulations suggest that the wall energy density scales as ρw ∼ a −r with 1 < r < 2," allowing it to predict how long-lived walls will behave relative to matter and radiation.

5. Topological Stability Assessment The system can assess topological stability by determining if the condition k1 + k2 = n is satisfied, noting that no stable DW network survives because this condition leads to a configuration where the relevant homotopy group is trivial.

Particle physics models beyond the Standard Model often contain global symmetries to address various unanswered questions. However, a common criticism of theories based on global symmetries is that such symmetries are generally expected to be explicitly violated by gravitational effects at the Planck scale. In the case of a global U(1) symmetry, this explicit breaking can reduce the symmetry to a discrete subgroup of U(1), leading to the formation of cosmic strings attached to multiple domain walls (DWs). These DWs are usually cosmologically problematic, since their slow scaling behavior can eventually dominate the energy density of the Universe, giving rise to the well-known cosmological DW problem, which is strongly constrained by Big Bang Nucleosynthesis. In this work, we propose a mechanism for the annihilation of these DWs. If a fermion coupled to the symmetry-breaking scalar has a small bare mass, radiative corrections generate a temperature-dependent vacuum bias that drives DW collapse. As an example, we consider a majoron model with right-handed neutrinos, where gravity-induced operators produce a string-wall network. We show that a real bare Majorana mass is sufficient to induce complete DW annihilation for even-dimensional gravity-induced operators, whereas odd-dimensional operators require a complex bare Majorana mass whose phase lifts the residual vacuum degeneracy, enabling complete annihilation.