Nonlocality enters through the covariant Dirac operator
The paper builds a nonlocal spin-1/2 field theory in curved spacetime by replacing the local kinetic term with i Psi-bar /nabla f(/nabla) Psi, where /nabla is the covariant Dirac operator and f is a nonpolynomial entire function. This differs from earlier fermionic nonlocal constructions that used form factors depending on the d'Alembertian, because the first-order, matrix-valued Dirac operator carries its own analytic structure.
Squaring the resulting nonlocal Dirac equation gives [K f^2(/nabla) + m^2] Psi = 0 with K = box - R/4, so the spinor components do not generally satisfy independent scalar nonlocal equations. The paper also imposes a Hermiticity condition on the kinetic operator that was not considered in the earlier Minkowski-space construction, and it analyzes coupling to gravity and to a U(1) gauge field, where nonlocal corrections are organized through the Zassenhaus expansion.
Each branch maps onto a local Dirac equation with effective mass
Because the form factor is an analytic function of the Dirac operator, it acts diagonally on a local Dirac mode satisfying i /nabla Psi_M = M Psi_M, giving f(/nabla) Psi_M = f(-iM) Psi_M. The nonlocal equation is then satisfied provided M f(-iM) = m, so nonlocal Bunch-Davies modes follow from local solutions by the replacement m -> M_n, where M_n solves that condition.
For the exponential form factors considered here, the effective masses are expressed in terms of branches of the Lambert-W function. This branch structure is the central technical object of the paper: it determines which solutions can be treated as physical modes and which are formal analytic continuations.
Only the principal branch of Model II carries physical modes
The paper studies two exponential form factors, Model I and Model II, and classifies their propagator poles and residues. Model I is non-Hermitian and has generically complex poles at finite Lambda. Model II satisfies the Hermiticity condition and admits a real, positive-residue principal branch n = 0 for 0 < m/Lambda < 1/e; its second real branch n = -1 has negative residue and is ghostlike, and the remaining branches are complex.
The physical analysis is therefore restricted to the principal branch of Model II, where M_0 = -Lambda W_0(-m/Lambda). No independent creation and annihilation operators are assigned to the negative-residue or complex branches, and no tensor-product Hilbert space over the different Lambert-W branches is assumed. The physical nonlocal Bunch-Davies vacuum is the branchwise vacuum evaluated at M_n = M_0.
Euclidean analyticity of the Bunch-Davies state survives nonlocality
The local construction selects Bunch-Davies modes through Euclidean analyticity of the spinor modes and the Bogoliubov transformation between the Euclidean and R/L open chart bases. In the nonlocal theory this structure is preserved: positive-frequency modes are still selected by analyticity on the lower hemisphere of the Euclidean continuation, and the mass-dependent quantities entering the construction are modified by m -> M_n.
The paper states that the Euclidean analyticity argument carries over directly to real branches, whereas complex branches are understood as formal analytic continuations. The radial functions, normalization factor, and analytic-continuation coefficients A_n and B_n are obtained from the local expressions with the replacement m -> M_n.
No physical fermionic supercurvature modes appear
The local theory was known to have no physical fermionic supercurvature modes, and the paper shows this feature is preserved. Setting p = i p-tilde with p-tilde > 0, the analytic-continuation coefficient on the principal branch becomes B_0(i p-tilde) proportional to Gamma(1/2 + p-tilde)^2 / Gamma(1/2 + p-tilde + i mu_0)^2, which is nonzero.
Because that coefficient does not vanish, the divergent contribution at the opposite boundary cannot be eliminated and the corresponding mode is not normalizable. The absence of physical fermionic supercurvature modes therefore survives on the principal branch of Model II.
Occupation number and entanglement entropy shift with the mass
On the physical principal branch, the mass shift in the large-Lambda regime is delta M = M_0 - m = m^2/Lambda + 3 m^3/(2 Lambda^2) + O(Lambda^-3). The analytic-continuation coefficients expand as A_0 = A(m) + delta M A'(m) + O(delta M^2) and B_0 = B(m) + delta M B'(m) + O(delta M^2), with derivatives taken at fixed p and H.
Defining q = |gamma_{p,0}|^2 and q_p = |gamma_p(m)|^2, the paper finds q = q_p + delta M dq_p/dm + O(delta M^2). The nonlocal correction to the mean occupation number is delta N_p = (m^2/Lambda)(1 + q_p)^2 dq_p/dm + O(Lambda^-2), and the correction to the entanglement entropy is delta S_p = -(2 ln q_p)/(1 + q_p)^2 (m^2/Lambda) dq_p/dm + O(Lambda^-2). Since delta M > 0 in the large-Lambda regime, both leading corrections are controlled by the mass dependence of the local squeezing parameter and originate from the same nonlocal mass shift.
For the parameters considered, nonlocality suppresses R-L entanglement
The paper defines delta S_p = S_p^NL - S_p^local, so delta S_p > 0 means enhanced entanglement between the R and L open charts and delta S_p < 0 means suppression. Figure 1 plots the leading nonlocal correction delta S_p as a function of m/H for p = 0.1 and Lambda/H = 5, 10, and 20 on the physical principal branch of Model II.
For those parameters, delta S_p < 0, indicating suppression of the R-L entanglement. The suppression increases as Lambda/H decreases and vanishes in the local limit Lambda/H -> infinity. The paper also notes that the local Dirac theory is smoothly recovered in the limit Lambda -> infinity.
The physical branch, not the form factor, sets the priority
The paper's practical priority is the branch selection: among the Lambert-W branches generated by the exponential form factors, only the principal branch n = 0 of Model II is a real, positive-residue physical sector for 0 < m/Lambda < 1/e. The ghostlike second real branch and the complex branches are not assigned independent creation and annihilation operators, so their contributions should not be read as additional physical particles or as extra entanglement channels.
Within that physical sector, the decision-relevant comparison is between the local result and the nonlocal correction at fixed p and H: the same mass shift m -> M_0 drives both the occupation-number correction and the entanglement-entropy correction, and the sign of the entanglement change follows from the sign of dq_p/dm. The paper's own continuation plans point to near-horizon black-hole geometry and braneworld scenarios, which are stated as future work rather than as results of this study.
Sources used in this answer
Nonlocal Dirac theory in curved spacetime and the Bunch–Davies vacuum
The paper formulates a nonlocal spin-1/2 theory in curved spacetime with an entire-function form factor of the covariant Dirac operator, maps the nonlocal dynamics branch by branch onto the local Dirac equation with a Lambert-W effective mass M_n, and shows that on the physical positive-residue principal branch of Model II the Bunch-Davies construction keeps its Euclidean analyticity while the Bogoliubov coefficients, squeezing parameter, occupation number, and entanglement entropy acquire nonlocal corrections through m -> M_0, with R-L entanglement suppressed for the parameters considered.