arxiv: 2604.18030 · v1 · submitted 2026-04-20 · ✦ hep-th · gr-qc

Recognition: unknown

Yukawa scalar self energy at two loop and langle φ² rangle in the inflationary de Sitter spacetime

Sourav Bhattacharya , Moutushi Dutta Choudhury

Authors on Pith no claims yet

Pith reviewed 2026-05-10 04:43 UTC · model grok-4.3

classification ✦ hep-th gr-qc

keywords Yukawa theoryde Sitter spacetimetwo-loop self-energysecular logarithmsscalar varianceinflationary fluctuationsBunch-Davies vacuum

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The pith

Two-loop Yukawa corrections in de Sitter space generate ln^4 a secular growth in scalar variance, which resums to a bounded value that decreases with coupling strength.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the two-loop scalar self-energy in Yukawa theory for a massless minimally coupled scalar and massless fermion in inflationary de Sitter spacetime. Internal scalar lines in the two diagrams break the conformal invariance present at one loop and produce infrared secular logarithms. After renormalization, the authors evaluate the loop-corrected coincident correlator in the Bunch-Davies vacuum and argue that local ultraviolet contributions dominate late-time behavior because they contain higher powers of logarithms of the scale factor. This leads to leading secular terms of ln^3 a at one loop and ln^4 a at two loops. A non-perturbative resummation then yields a bounded expression for that falls monotonically as the Yukawa coupling increases, implying a dynamically generated scalar mass that grows with the coupling.

Core claim

In Yukawa theory with massless minimally coupled scalar and massless fermion in inflationary de Sitter spacetime, the two-loop self-energy diagrams each contain one internal scalar line and thereby generate infrared secular logarithms. The renormalized late-time leading secular contribution to <phi^2> is ln^4 a, a hybrid of ultraviolet and infrared logarithms. The resummed <phi^2> remains bounded and decreases monotonically with increasing Yukawa coupling, so the dynamically generated scalar mass increases with the coupling.

What carries the argument

The local ultraviolet self-energy contributions that dominate the non-local infrared ones at late times purely through higher powers of logarithms of the scale factor, enabling resummation of the secular series in <phi^2>.

If this is right

The coincident two-point function <phi^2> grows as ln^3 a at one loop and ln^4 a at two loops.
Resummation keeps <phi^2> finite at late times instead of letting it diverge.
The variance <phi^2> decreases as the Yukawa coupling increases.
The effective mass dynamically generated for the scalar rises with the Yukawa coupling.
Fermion lines remain subdominant because they are conformally invariant and produce no secular infrared logs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Stronger Yukawa coupling could prevent scalar fluctuations from growing large enough to back-react on the inflationary background.
The same log-power dominance argument may apply to other interactions involving non-conformally invariant fields in de Sitter space.
Extending the calculation to three loops would test whether the pattern of increasing log powers continues or eventually saturates.

Load-bearing premise

Local ultraviolet self-energy terms dominate non-local infrared terms at late times solely because they contain higher powers of logarithms of the scale factor.

What would settle it

An explicit two-loop or higher computation of the non-local infrared pieces of the self-energy that produces equal or higher powers of ln a than the local ultraviolet pieces would falsify the dominance assumption and the resulting resummed expression.

Figures

Figures reproduced from arXiv: 2604.18030 by Moutushi Dutta Choudhury, Sourav Bhattacharya.

**Figure 1.** Figure 1: One loop scalar and fermion self energy diagrams for the Yukawa interaction. The solid and broken lines respectively represent the scalar and the fermions. −iΣ(x, x′ ) 1−loop ss′ = λss′g 2 (aa′ ) dTriSss′ (x, x′ )iSss′ (x ′ , x), (no sum on s, s′ ) (14) where s, s′ = ±, and λ++ = λ−− = 1 and λ+− = λ−+ = −1. Using Eq. (58), the ++-self energy reads −iΣ(x, x′ ) 1−loop ++ = − g 2dΓ 2 (d/2)(aa′ ) 2 2π d 1 ∆ 6−… view at source ↗

**Figure 2.** Figure 2: Two loop scalar self energy diagrams for the Yukawa interaction. where the subscripts 1, 2, 3, 4 respectively stand for the intervals x − x ′′ , x ′′ − x ′′′ , x ′′′ − x ′ and x ′ − x. For the purpose of doing renormalisation and finding local part of the self energy, the relevant part of the scalar propagator would be A(x, x′ ), Eq. (8). This gives −iΣf (x, x′ ) 2−loop,1 ++,loc = g 4 (aa′ )Γ5 (1 − ϵ/2)(1 … view at source ↗

**Figure 3.** Figure 3: One loop vertex diagram for the Yukawa interaction. Let us now come to the second diagram, −iΣf (x, x′ ) 2−loop,2 ++ = g 4 (aa′ ) dTr Z (a ′′a ′′′) d d dx ′′d dx ′′′iS++(x, x′′)iS++(x ′′, x′ )iS++(x ′ , x′′′)iS++(x ′′′, x)i∆++(x ′′, x′′′) = g 4dΓ 4 (2 − ϵ/2)(aa′ ) 2 4π 8−2ϵ Z a ′′a ′′′d dx ′′d dx ′′′i∆++(x ′′, x′′′) " −i∆x µ 1 ∆x 4−ϵ 1++ −i∆x4µ ∆x 4−ϵ 4++ −i∆x ν 2 ∆x 4−ϵ 2++ −i∆x3ν ∆x 4−ϵ 3++ + (1, 2 and 3… view at source ↗

**Figure 4.** Figure 4: The tadpoles at any order of the perturbation theory vanish for the Yukawa theory carrying a massless fermion, for it always involves traces of odd number of γ-matrices. The primed vertices appearing above are integrated. 4 Computation of ⟨ϕ 2 ⟩, leading secular behaviour and resummation In this section we wish to compute the coincident correlation function, ⟨ϕ 2 (x)⟩, owing to the loop effects due to the … view at source ↗

**Figure 5.** Figure 5: Series summation for the propagator due to infinite number of self energy loop insertions. For our present purpose, the two endpoints will be identified. The shaded circle represent the one plus two loop self energies containing leading secular terms (i.e. local in this case). We may assign a non-perturbative value to ⟨ϕ¯2 ⟩resum, by assigning the same to N as follows. We compute d⟨ϕ¯2 ⟩resum dN = 1 − g 2N… view at source ↗

**Figure 6.** Figure 6: The plot of the late time non-perturbative ⟨ϕ¯2 ⟩resum with respect to the Yukawa coupling. The same divergences for g = 0, as is indicated by Eq. (53) or Eq. (54). See main text for detail. generated mass of the scalar field through the non-perturbative value of ⟨ϕ¯2 ⟩resum. It is well known that for a light scalar field, we have ⟨ϕ 2 ⟩ = 3H2 8πm2 Substituting for the resummed value of ⟨ϕ¯2 ⟩resum, and tr… view at source ↗

**Figure 7.** Figure 7: Schematic diagram for the effective potential for the massless Yukawa theory in the absence of any scalar self interaction. For field vales away from zero, the potential becomes unbounded from below. See main text for discussion. increasing value of the coupling parameter. Finally, we note that if one computes the effective potential for the Yukawa theory, the same becomes unbounded from below if there is … view at source ↗

read the original abstract

We have considered the Yukawa theory at two loop in the inflationary de Sitter spacetime, for a massless minimally coupled scalar and a massless fermion. The one loop computation for the same has been investigated in detail in the earlier literatures. The chief motivation behind this study is the fact that at one loop, the scalar self energy contains only fermions, which are conformally invariant. At two loop, there are two diagrams, each containing one internal scalar line, thereby breaking the conformal invariance. This should result in the appearance of infrared secular logarithms in the scalar self energy. The renormalisation of this two loop self energy has been performed. We next compute the loop corrected coincident two point correlation function, $\langle \phi^2\rangle$, due to the self energies. The expectation value has been taken with respect to the initial Bunch-Davies vacuum. We argue that the late time secular contribution from the local or UV self energy must dominate the non-local or IR ones in the present case, from the point of view of the powers of these large logarithms of the scale factor. This corresponds to the fact that fermion lines do not show any IR secular effect. The leading behaviour of $\langle \phi^2\rangle$ at one and two loop are respectively found to be $\ln^3 a$ and $\ln^4 a$. These are hybrids of UV and IR logarithms, where the latter originate from the massless and minimal two external scalar lines. A resummed expression for $\langle \phi^2\rangle$ has also been computed. The same is found to be bounded and decreasing monotonically with the increasing magnitude of the Yukawa coupling. Accordingly, the dynamically generated scalar mass increases with the increasing coupling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Two-loop Yukawa self-energy in de Sitter yields ln^4 a secular growth in <φ²> with a bounded resummation, but the UV-over-IR dominance rests on log-power counting without explicit leading coefficients.

read the letter

Hi, the main thing to know about arXiv:2604.18030 is that the authors compute the two-loop scalar self-energy in a massless Yukawa theory on de Sitter, renormalize it, and extract a leading ln^4 a secular term in the late-time coincident correlator <φ²> that they then resum to a bounded expression decreasing with the Yukawa coupling y. This extends the one-loop literature where only fermion loops appear and conformal invariance keeps IR logs absent. At two loops the diagrams with internal scalar lines break that invariance, and the paper shows how the resulting hybrid UV-IR logs produce the higher power. The resummation treats the local part as dominant and solves the integral equation for the correlator, yielding a finite <φ²> whose effective mass grows with y. The diagram evaluation and renormalization follow standard perturbative methods in curved space and look reproducible for readers already working in this area. The log-power argument is laid out clearly: fermion lines suppress IR secular effects, so only the scalar-internal contributions reach the highest powers. That logic is reasonable on counting grounds and gives a concrete next step beyond the cited one-loop results. The softer part is the dominance claim itself. After renormalization the finite non-local pieces could in principle carry coefficients large enough to compete at O(ln^4 a) or shift the monotonicity of the resummed result, yet the paper does not display the explicit leading coefficients from both local and non-local contributions to confirm the separation. If those coefficients work out as assumed, the boundedness holds; without that check the conclusion is plausible but not fully verified. This is targeted at the small group doing loop calculations in inflationary backgrounds who already know the one-loop secular effects. It is worth sending to referees because the calculation is direct, the result is specific, and the gap on coefficients is fixable with existing methods rather than a load-bearing flaw.

Referee Report

2 major / 1 minor

Summary. The paper computes the two-loop scalar self-energy in Yukawa theory (massless minimally coupled scalar + massless fermion) in de Sitter spacetime. After renormalization it evaluates the coincident correlator ⟨ϕ²⟩ in the Bunch-Davies vacuum, argues that local/UV self-energy pieces dominate non-local/IR pieces at late times purely by virtue of higher powers of ln a, identifies the leading secular growth as ln⁴a (a hybrid of UV and IR logarithms), and presents a resummed expression for ⟨ϕ²⟩ that remains bounded and decreases monotonically with the Yukawa coupling y.

Significance. If the dominance argument is verified, the result supplies a concrete example of how Yukawa interactions generate a dynamical scalar mass in inflation, yielding a stabilized, coupling-dependent variance. The direct diagrammatic evaluation, absence of free parameters, and explicit resummation constitute methodological strengths. The findings bear on back-reaction and effective potentials in inflationary cosmology.

major comments (2)

[Abstract and late-time secular contribution discussion] The central claim that local (UV) self-energy contributions dominate non-local (IR) ones at late times, thereby producing a leading ln⁴a secular term in ⟨ϕ²⟩ while preserving monotonicity of the resummed result, rests on log-power counting alone. After renormalization the finite parts of local and non-local diagrams are not automatically guaranteed to maintain strict power dominance; explicit coefficient verification for the O(ln⁴a) term is required to confirm that non-local pieces remain subdominant and do not alter the sign or monotonicity of the resummed ⟨ϕ²⟩.
[Resummation of ⟨ϕ²⟩] The resummation procedure that converts the secular series into a bounded, monotonically decreasing function of y assumes that the UV-local terms control the integral equation for the coincident correlator. The manuscript should supply the explicit integral equation and demonstrate that the neglected non-local kernels do not compete at the same logarithmic order.

minor comments (1)

The abstract states that the one-loop result is ln³a and the two-loop result is ln⁴a; a brief table or equation summarizing the coefficient of each power would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We address each major comment point by point below, providing clarifications and committing to revisions where appropriate to strengthen the presentation.

read point-by-point responses

Referee: [Abstract and late-time secular contribution discussion] The central claim that local (UV) self-energy contributions dominate non-local (IR) ones at late times, thereby producing a leading ln⁴a secular term in ⟨ϕ²⟩ while preserving monotonicity of the resummed result, rests on log-power counting alone. After renormalization the finite parts of local and non-local diagrams are not automatically guaranteed to maintain strict power dominance; explicit coefficient verification for the O(ln⁴a) term is required to confirm that non-local pieces remain subdominant and do not alter the sign or monotonicity of the resummed ⟨ϕ²⟩.

Authors: We appreciate the referee's emphasis on rigorous verification beyond power counting. Our argument relies on the fact that fermion propagators remain conformally invariant with no IR secular logarithms, so non-local diagrams involving internal fermion lines cannot generate ln⁴a terms at the same order as the local scalar self-energy insertions. Nevertheless, to address the concern directly, we will add an explicit extraction of the leading coefficients for the O(ln⁴a) contributions from both local and non-local diagrams in the revised manuscript. This will confirm the dominance, the positive sign of the leading term, and the preservation of monotonicity in the resummed ⟨ϕ²⟩. revision: yes
Referee: [Resummation of ⟨ϕ²⟩] The resummation procedure that converts the secular series into a bounded, monotonically decreasing function of y assumes that the UV-local terms control the integral equation for the coincident correlator. The manuscript should supply the explicit integral equation and demonstrate that the neglected non-local kernels do not compete at the same logarithmic order.

Authors: We agree that including the explicit integral equation will improve clarity. The resummation follows from the Dyson-Schwinger equation for the coincident two-point function, where the self-energy is inserted iteratively. We will present this integral equation explicitly in the revised version and demonstrate that non-local kernels, arising from diagrams with internal fermion lines, contribute only subleading logarithms (at most ln³a) due to conformal invariance of the fermions. Consequently, they do not compete with the leading UV-local terms at late times and do not affect the bounded, monotonically decreasing behavior of the resummed ⟨ϕ²⟩ with respect to the Yukawa coupling y. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit diagram evaluation and log-power argument yield independent secular terms

full rationale

The paper computes the two-loop Yukawa self-energy diagrams directly, performs renormalization, and extracts secular logarithms from the internal scalar lines that break conformal invariance. The leading late-time terms ln³a (one-loop) and ln⁴a (two-loop) for ⟨ϕ²⟩ follow from the structure of the coincident correlator integral equation with the computed self-energy insertions. The claim that local/UV pieces dominate non-local/IR pieces rests on explicit power counting of large logarithms of the scale factor (higher powers from scalar propagators versus fermion conformal invariance), not on any fitted parameter, self-definition, or load-bearing self-citation. The resummation that produces a bounded ⟨ϕ²⟩ decreasing with Yukawa coupling is performed on these independently derived logarithmic coefficients. No step reduces the central result to an input by construction; the derivation remains self-contained against the explicit perturbative calculation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculation rests on standard background assumptions of QFT in curved spacetime without introducing new free parameters or postulated entities.

axioms (2)

domain assumption The spacetime is de Sitter and the state is the Bunch-Davies vacuum.
Defines the propagators and the expectation value used throughout the computation.
standard math Perturbative renormalization techniques developed for curved spacetime apply to the two-loop diagrams.
Underpins the removal of divergences and extraction of the finite secular logarithms.

pith-pipeline@v0.9.0 · 5627 in / 1379 out tokens · 50439 ms · 2026-05-10T04:43:29.102236+00:00 · methodology

discussion (0)

Reference graph

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