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Allowed and excluded coefficients in one gapped scalar channel

Physical scope

A finite algebraic comparison of two coefficient values for one hypothetical gapped identical-real-scalar channel under the same exact amplitude and matching assumptions. Passing the necessary inequality is not evidence that a full amplitude or physical QFT exists.

Example

Fix $m > 0$ , $Λ = 4 m$ , and one common matching scale $0 < μ_{*} < 4 m$ in the fixed local representative of this page's channel definition. Impose all its amplitude hypotheses and the exact matching identity $c_{1} = 4 C_{8} (μ_{*}) / Λ^{4} + Δ_{m a t c h}$ with $Δ_{m a t c h} = 0$ at this scale. Compare the candidate values $C_{8} (μ_{*}) = + 1 / 2$ and $C_{8} (μ_{*}) = - 1 / 2$ under these same conditions. The first coefficient passes the necessary forward-positivity inequality; the second is excluded by it. Passing the inequality is not sufficient for a full amplitude.

Example references

Gapped scalar EFT forward channel and matched coefficient data

Forward positivity for a matched scalar EFT coefficient

de Rham et al., Positivity Bounds for Scalar Theories, §§ 2.1–2.2

Basile et al., Lectures in Quantum Gravity, § 3.3.1

Facts & Assumptions

Given: A gapped scalar channel with $m > 0$ , $Λ = 4 m$ , one fixed common matching scale $0 < μ_{*} < 4 m$ , exact $Δ_{m a t c h} = 0$ at that scale, and the two candidate coefficients $C_{8} (μ_{*}) = + 1 / 2$ and $C_{8} (μ_{*}) = - 1 / 2$ .

[A1]

Gapped scalar EFT forward channel and matched coefficient data: $c_{1} = B^{''} (0) / 2!$ and the declared match is $c_{1} = 4 C_{8} (μ_{*}) / Λ^{4} + Δ_{m a t c h}$ for the fixed operator representative at the common scale $μ_{*}$ .

[F1]

Forward positivity for a matched scalar EFT coefficient: under the same mass-gap, analytic, crossing, pole-subtraction, spectral, and matching assumptions with $Δ_{m a t c h} = 0$ , the necessary bound is $C_{8} (μ_{*}) \geq 0$ .

Verification

1.1givenA1algebra

For $C_{8} (μ_{*}) = + 1 / 2$ and $Λ = 4 m$ , [A1] gives $c_{1} = 4 (1 / 2) / (4 m)^{4} = 1 / (128 m^{4}) > 0$ . This value passes the necessary forward-positivity test.

1.2givenA1F1algebra

For $C_{8} (μ_{*}) = - 1 / 2$ under the same $m$ , $Λ$ , $μ_{*}$ , and $Δ_{m a t c h}$ , [A1] gives $c_{1} = 4 (- 1 / 2) / (4 m)^{4} = - 1 / (128 m^{4}) < 0$ . This violates [F1], so the value is excluded under the stated channel assumptions.

2.1F1step 1.1step 1.2algebra∎

Since [F1] permits equality, $C_{8} (μ_{*}) = 0$ lies on the boundary of the region that passes this necessary inequality. The positive point and boundary pass do not establish that an amplitude realizing them exists.

Remarks

At the fixed common matching scale $μ_{*}$ , the horizontal coordinate is the dimensionless coefficient $C_{8} (μ_{*})$ in the fixed representative. Red shading denotes values excluded by this one necessary inequality; blue shading denotes values that pass it, including the zero boundary. The diagram encodes only $C_{8} (μ_{*}) \geq 0$ when $Δ_{m a t c h} = 0$ and does not assert amplitude existence.

Allowed and excluded coefficients in one gapped scalar channel

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources