Dynamical generation of fermion mass in a scalar-fermion theory with ${\lambda}{\phi}^4$ interaction

Krishnendu Mukherjee; Somnath Majumder

arxiv: 2602.16425 · v2 · submitted 2026-02-18 · ✦ hep-th · hep-ph

Dynamical generation of fermion mass in a scalar-fermion theory with {λ}{φ}⁴ interaction

Somnath Majumder , Krishnendu Mukherjee This is my paper

Pith reviewed 2026-05-15 21:28 UTC · model grok-4.3

classification ✦ hep-th hep-ph

keywords dynamical fermion masseffective potentialCJT formalism2PI diagramsYukawa couplingphi^4 interactionsymmetry breaking

0 comments

The pith

In a scalar-fermion theory with quartic interaction the fermion acquires dynamical mass when the coupling constant lies outside an open interval of positive values.

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

The authors compute the effective potential in a theory of a scalar field with λφ⁴ interaction coupled to a massless fermion by Yukawa coupling. They sum an infinite series of two-particle-irreducible diagrams using the Cornwall-Jackiw-Tomboulis method. For coupling values outside an open set of positive numbers, the potential shows maxima and minima away from zero, so the vacuum can choose a positive non-zero value and break the φ to -φ symmetry. The fermion then gains mass from the Yukawa term in this broken phase. Inside the interval the only minimum is at zero and the fermion stays massless.

Core claim

The effective potential obtained by summing infinite 2PI diagrams of two different types and one of a third type via the CJT formalism possesses an inversion symmetry φ → -φ. When the coupling constant lies outside an open interval of positive reals, this potential develops maxima above and minima below the zero line on both sides of the φ=0 minimum. The system can settle at a positive non-zero φ, breaking the symmetry, at which point the originally massless fermion acquires a mass through the Yukawa interaction.

What carries the argument

The Cornwall-Jackiw-Tomboulis effective potential constructed from the sum of all two-particle-irreducible diagrams in the theory.

If this is right

The vacuum expectation value of the scalar field becomes non-zero for couplings beyond the critical range.
The fermion mass is generated dynamically without any explicit mass term.
The inversion symmetry of the potential is spontaneously broken in the stable vacuum.
The theory exhibits a phase transition-like behavior at the boundaries of the open interval of couplings.

Where Pith is reading between the lines

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

This approach could be applied to similar models in quantum field theory to study dynamical symmetry breaking.
Comparison with perturbative calculations might reveal the importance of the non-perturbative contributions.
Lattice field theory simulations could test whether the predicted minima actually occur.

Load-bearing premise

The infinite sum of 2PI diagrams in the CJT formalism accurately determines the shape and location of the minima in the effective potential.

What would settle it

A calculation using a different non-perturbative method such as lattice regularization that shows no non-zero minima for couplings outside the specified interval would falsify the result.

Figures

Figures reproduced from arXiv: 2602.16425 by Krishnendu Mukherjee, Somnath Majumder.

**Figure 1.** Figure 1: (a) Lobe (b) Trilinear line kind is shown in FIG.2 which involves two lobes each containing one trilinear line. The contribution reads as [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: 2PI diagram with two lobes each containing one trilinear line [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: 2PI diagram with three lobes each containing one trilinear line [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: 2PI diagram with n lobes each containing one trilinear line [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: 2PI diagram with n lobes γ (n,0) b = −iλ 4! n Z d 4p (2π) 4 d 4 q (2π) 4 d 4k1 (2π) 4 · · · d 4kn−1 (2π) 4 4G(p)3G(q)4G(k1)3G(p + q − k1)· · · 4G(kn−2) ×3G(p + q − kn−2)2G(kn−1)G(p + q − kn−1) = − iλ 12 Z d 4p (2π) 4 d 4 q (2π) 4 G(p)G(q) −iλ 2 I(p + q) n−1 , (19) where I(p) = Z d 4k (2π) 4 G(k)G(p − k). (20) We now place r trilinear lines in n − 1 lobes (1 ≤ r ≤ n − 1) of FIG.5 such that each lobe c… view at source ↗

**Figure 6.** Figure 6: 2PI diagram with n lobes and r trilinear lines γ (n,r) b = −iλ 4! n −iλφ 3! 2r Z d 4p (2π) 4 d 4 q (2π) 4 4G(p)3G(q)(3.3.2.4.3 J(p + q))r (4.3 I(p + q))n−2−r 2I(p + q) = − iλ 12 Z d 4p (2π) 4 d 4 q (2π) 4 G(p)G(q) iλ3φ 2 4 J(p + q) r −iλ 2 I(p + q) n−1−r (21) 4 [PITH_FULL_IMAGE:figures/full_fig_p004_6.png] view at source ↗

**Figure 7.** Figure 7: 2PI diagram of type c. V (c) 2 (φ, G) = λ 8 Z d 4p (2π) 2 G(p) 2 . (24) Therefore, the contributions of 2PI diagrams of type-a, b and c to the effective potential are given by V2(φ, G) = V (a) 2 (φ, G) + V (b) 2 (φ, G) + V (c) 2 (φ, G) (25) 4 Solutions of ∂Vef f ∂S(k) = 0 and ∂Vef f ∂G(k) = 0 We use the stationary condition Eq.(12) under the variation of S(k), we obtain the equation as iS−1 (φ, k) = iS −… view at source ↗

**Figure 8.** Figure 8: Plots of Mˆ2 1 (φˆ) versus φˆ for different values of λˆ. Here, λˆ = λ/16π 2 and φˆ = φ/p 4πµ2. increases rapidly with φ for λˆ λˆ ≤ 3.0. When λ >ˆ 3.0, M2 1 attains negative value for φˆ ≥ 0 (shown in the inset plot). The indications are of two-fold: On the one hand it suggests that the result of first iteration may be invalid for λ >ˆ 3.0. On the other hand it may be assumed that apart from those of a, b… view at source ↗

**Figure 9.** Figure 9: Plots of Vˆ ef f versus φˆ for ˆm = 15.0 and for different values of λˆ. -10 -5 0 5 10 -3.0x109 -2.5x109 -2.0x109 -1.5x109 -1.0x109 -5.0x108 0.0 5.0x108 1.0x109 1.5x109 2.0x109 2.5x109 3.0x109 -0.10 -0.05 0.00 0.05 0.10 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Veff 49.2 5.2 51.2 52.2 76 sgm 10,109.2,0.15 52.2 51.2 5.2 49.2 Veff [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗

**Figure 10.** Figure 10: Plots of Vˆ ef f versus φˆ for ˆm = 15.0 and for different values of λˆ. In [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗

read the original abstract

The effective potential for a scalar theory with $\lambda\phi^4$ interaction, coupled to a massless fermion through Yukawa interaction is calculated by summing over infinite number of two particle irreducible (2PI) diagrams of two different types and a 2PI diagram of a third type using Cornwall, Jackiw and Tomboulis (CJT) method. There is an inversion symmetry present in the effective potential under $\phi\rightarrow -\phi$. When the value of coupling constant falls beyond an open set of positive real numbers, the effective potential exhibits both maxima and minima above and below the zero potential line respectively on either side of its minimum at $\phi=0$. The fermion acquires a mass in this region of coupling constant when the system settles into the minimum at positive, non-zero $\phi$ breaking the inversion symmetry of the vacuum. However, the effective potential exhibits a minimum only at $\phi=0$ and also the fermion remains massless when the coupling constant assumes any value from this open set.

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

The paper uses selective 2PI resummation in the CJT potential to claim a specific open interval of λ where dynamical fermion mass appears, but the truncation lacks any control or validation.

read the letter

The main takeaway is that the authors resum two infinite classes of 2PI diagrams plus one extra diagram in the CJT effective potential for a λφ⁴ scalar coupled to a massless fermion via Yukawa. They report that outside an open set of positive λ values the potential develops nonzero minima that break the φ to -φ symmetry, giving the fermion a mass, while inside that interval the minimum stays at zero and the fermion stays massless. This pins down a concrete range for the transition in this model. The calculation is new in that narrow sense and shows the potential shape that follows from their diagram choice. The work is straightforward in laying out which diagrams enter the sum. The real limitation is the truncation. Nothing in the paper shows that the omitted 2PI diagrams are small in the relevant regime or that their inclusion would not shift the location or depth of the reported minima. The balance that creates the nonzero minima depends directly on the resummed pieces versus the tree-level quartic, so uncontrolled omissions move the critical λ values. No convergence tests, perturbative limits, or external benchmarks are supplied to check the result. This leaves the claimed open set of couplings on shaky ground. The paper is useful mainly to readers already comfortable with CJT methods who want to see the technique applied to a simple mass-generation example. Anyone looking for a reliable new statement about dynamical symmetry breaking will hit the same truncation issue. I would send it to peer review. Referees who know the 2PI formalism can judge whether the diagram selection can be justified or whether the central claim needs more support before it is taken seriously.

Referee Report

2 major / 1 minor

Summary. The manuscript computes the effective potential in a scalar-fermion theory with λφ⁴ and Yukawa interactions by resumming infinite 2PI diagrams of two types plus one additional 2PI diagram within the CJT formalism. It reports that for λ outside an open interval of positive values, the effective potential develops minima at nonzero φ, breaking the φ → -φ inversion symmetry and generating a dynamical mass for the fermion, while inside the interval the minimum remains at φ=0 with massless fermions.

Significance. If the resummation accurately captures the non-perturbative dynamics, the result would demonstrate a mechanism for dynamical fermion mass generation via vacuum symmetry breaking in a renormalizable model. This could have implications for understanding mass generation in beyond-Standard-Model scenarios or condensed-matter analogs. However, the absence of convergence checks or comparisons to known limits limits the immediate impact.

major comments (2)

The central claim that minima at φ ≠ 0 appear only for λ outside a specific open set rests on the explicit resummation of two infinite classes of 2PI diagrams and one additional diagram. No argument is provided that the omitted 2PI diagrams are parametrically suppressed, nor is a convergence proof or numerical error estimate supplied for the truncation. This directly affects the location of the reported critical values of λ and the depth of the minima.
The abstract states the outcome of the diagram resummation but supplies no explicit equations, truncation scheme, or numerical checks. It is therefore impossible to verify whether the claimed minima and mass generation follow rigorously from the stated method.

minor comments (1)

Clarify the precise definition of the 'open set of positive real numbers' for λ (e.g., via an explicit inequality or numerical bounds) in the results section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the changes we will make in the revised version.

read point-by-point responses

Referee: The central claim that minima at φ ≠ 0 appear only for λ outside a specific open set rests on the explicit resummation of two infinite classes of 2PI diagrams and one additional diagram. No argument is provided that the omitted 2PI diagrams are parametrically suppressed, nor is a convergence proof or numerical error estimate supplied for the truncation. This directly affects the location of the reported critical values of λ and the depth of the minima.

Authors: We acknowledge that the manuscript does not supply a formal argument for the parametric suppression of the omitted 2PI diagrams or a convergence proof for the truncation. The resummation was chosen to include the infinite classes of diagrams that can be summed in closed form within the CJT formalism while retaining the essential non-perturbative structure responsible for the symmetry breaking. In the revision we will add a dedicated subsection that (i) explicitly lists the diagram classes retained and omitted, (ii) provides a qualitative rationale for the truncation based on the structure of the 2PI effective action, and (iii) includes numerical checks that compare the location and depth of the minima obtained with the present truncation against partial sums that incorporate a finite number of additional diagrams for representative values of λ. revision: yes
Referee: The abstract states the outcome of the diagram resummation but supplies no explicit equations, truncation scheme, or numerical checks. It is therefore impossible to verify whether the claimed minima and mass generation follow rigorously from the stated method.

Authors: We agree that the abstract, while concise, does not indicate the methodological framework. We will revise the abstract to include a brief statement that the effective potential is obtained via the CJT formalism by resumming two infinite classes of 2PI diagrams together with one additional diagram. The explicit equations, truncation details, and numerical results remain in the body of the paper; we will also add a short summary paragraph at the end of the introduction that points the reader to the relevant sections. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation is a direct resummation calculation

full rationale

The paper computes the effective potential explicitly via the CJT formalism by resumming specified infinite classes of 2PI diagrams starting from the given scalar-fermion Lagrangian with λφ⁴ and Yukawa terms. The reported minima, symmetry breaking, and dynamical fermion mass follow as consequences of analyzing this computed V_eff(φ), without any parameter fitting to external data, self-referential definitions, or load-bearing self-citations that reduce the result to its inputs. The approach is a standard (if truncated) non-perturbative technique whose validity is an assumption about convergence rather than a definitional tautology. No step equates output to input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the assumption that the CJT 2PI resummation provides a reliable non-perturbative effective potential for this theory; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The Cornwall-Jackiw-Tomboulis formalism with infinite 2PI diagram summation correctly determines the effective potential and its minima in this scalar-fermion theory.
The entire analysis is built on applying the CJT method to sum the diagrams.

pith-pipeline@v0.9.0 · 5479 in / 1388 out tokens · 27753 ms · 2026-05-15T21:28:21.529008+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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A. M. Jaﬀe, Divergence of perturbation theory for bosons, Co mm. Math. Phys. 1 (1965) 127

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R. F. Dashen, B. Hasslacher and A. Neveu, Nonperturbative me thods and extended-hadron models in ﬁeld theory. II. Two-dimensional models and extended hadrons, Phys. Rev. D 1 0 (1974) 4114

work page 1974

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L. N. Lipatov, Divergence of the perturbation-theory series a nd the quasi-classical theory, Sov. Phys. JETP 45 (1977) 216

work page 1977

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Coleman and E

S. Coleman and E. Weinberg, Radiative Corrections as the Origin of Spontaneous Symmetry Breaking, Phys. Rev. D 7 (1973) 1888. 16

work page 1973

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Brezin, J

E. Brezin, J. C. Le Guillou and J. Zinn-Justin, Perturbation theor y at large order. I. The φ 2N interaction, Phys. Rev. D 15 (1977) 1544

work page 1977

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J. M. Cornwall and R. Jackiw and E. Tomboulis, Eﬀective action for composite operators, Phys. Rev. D 10 (1974) 2428

work page 1974

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Amelino-Camelia and S-Young Pi, Self-consistent improvement o f the ﬁnite-temperature eﬀective potential, Phys

G. Amelino-Camelia and S-Young Pi, Self-consistent improvement o f the ﬁnite-temperature eﬀective potential, Phys. Rev. D 47 (1993) 2356

work page 1993

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Amelino-Camelia, On the CJT formalism in multi-ﬁeld theories, Nucl

G. Amelino-Camelia, On the CJT formalism in multi-ﬁeld theories, Nucl. Phys. B 476 (1996) 255

work page 1996

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M. G. Alford, M. Braby, and A. Schmitt, Critical temperature fo r kaon condensation in color-ﬂavor locked quark matter, J. Phys. G 35 (2008) 025002

work page 2008

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J. O. Andersen and L. E. Leganger, Kaon condensation in the c olor-ﬂavor-locked phase of quark matter, the Goldstone theorem, and the 2PI Hartree approximation, Nucl. Phy s. A 828 (2009) 360

work page 2009

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H. P. Tran, V. Nguyen, T. A. Nguyen and V. H. Le, Kaon conden sation in the linear sigma model at ﬁnite density and temperature, Phys. Rev. D 78 (2008) 105016

work page 2008

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Suppl.) 63A-C (1998) 637-639

A.Agodi, G.Andronico, P.Cea, M.Consoli and L.Cosmai, The φ 4 theory on the lattice: eﬀective potential and triviality, Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 637-639

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J.M.Chung and B.K.Chung, Three-loop renormalization of the eﬀec tive potential, Phys. Rev. D 56 (1997) 6509

work page 1997

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G. N. J. A˜ na˜ nos, Scalar ﬁeld theory at ﬁnite temperature in D = 2 + 1, J. Math. Phys. 47 (2006) 012301

work page 2006

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Collins, Renormalization, Cambridge Monographs on Math ematical Physics Cambridge University Press (1984)

John C. Collins, Renormalization, Cambridge Monographs on Math ematical Physics Cambridge University Press (1984)

work page 1984

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Ramond, Field Theory: A Modern Primer, Addison-Wesley Publis hing Company, USA (1990)

P. Ramond, Field Theory: A Modern Primer, Addison-Wesley Publis hing Company, USA (1990). 17

work page 1990