arxiv: 2605.04010 · v1 · submitted 2026-05-05 · ✦ hep-ph

Recognition: unknown

Real and Complex Singlet-Scalar Benchmarks with a Vector-Like Down Quark for Bto X_sγ and B_s-bar B_s Mixing

Qazi Maaz Us Salam

Authors on Pith no claims yet

Pith reviewed 2026-05-07 15:09 UTC · model grok-4.3

classification ✦ hep-ph

keywords vector-like down quarksinglet scalarB to Xs gammaBs mixingWilson coefficientsnew physicsradiative decay

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The pith

New physics from a vector-like down quark and singlet scalar shifts the B to Xs gamma coefficient by only 0.4 percent of the SM value.

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

The paper examines a minimal extension of the Standard Model that adds a vector-like down-type quark D together with a neutral singlet scalar, taken either as a real field or as a complex field. The scalar couples to D and the right-handed down quarks, generating one-loop diagrams for the radiative b to s gamma transition. Because the scalar is electrically neutral and colorless, the emitted photon originates from the internal D line, which fixes the ratio of the resulting magnetic and chromomagnetic dipole Wilson coefficients to minus one-third. For benchmark masses of 1 TeV and order-one couplings the new-physics piece in the low-scale effective coefficient reaches only 0.4 percent of the Standard Model value. The same diagrams contribute to Bs mixing, where the real-scalar case cancels exactly between direct and crossed boxes while the complex-scalar case leaves a non-vanishing box that bounds the product of flavor couplings.

Core claim

In these benchmarks the new-physics contribution to the dipole operator for b to s gamma is |C7 gamma NP,eff| approximately 1.1 times 10 to the minus 3, or 0.4 percent of the SM value, at 1 TeV masses with absolute value of lambda s star lambda b equal to 1; B s mixing is unaffected in the real scalar case due to exact cancellation of direct and crossed box diagrams but provides the dominant constraint in the complex scalar case, limiting the flavor coupling product to a few tenths.

What carries the argument

The one-loop diagram in which the neutral singlet scalar couples the vector-like quark D to a right-handed down quark, with the gauge boson radiated from the D line, which fixes the dipole operator ratio C7 gamma NP over C8G NP equal to minus one-third at the matching scale.

Load-bearing premise

The scalar carries neither electric charge nor color so the emitted gauge boson must come from the internal D line, together with the exact minimal limit in which direct and crossed box diagrams cancel for the real scalar.

What would settle it

A precision measurement of the B to Xs gamma branching ratio that deviates by more than one percent from the Standard Model prediction, or the absence of any constraint on the coupling product from Bs mixing data when the complex scalar mass is near 1 TeV.

Figures

Figures reproduced from arXiv: 2605.04010 by Qazi Maaz Us Salam.

**Figure 1.** Figure 1: Feynman rule vertices for the benchmark model. In panels (a) and (b), view at source ↗

**Figure 2.** Figure 2: Feynman one-loop diagrams for the electromagnetic and chromomagnetic dipole operators view at source ↗

**Figure 3.** Figure 3: One-loop box diagrams generating the ∆B = 2 transition b b → s s in the real and complex singlet-scalar with vector-like-quark benchmark. The coupling product λ ∗ sλb also enters ∆B = 2 box diagrams with two internal D and two internal S, where S = SR, Φ as shown in view at source ↗

**Figure 4.** Figure 4: Loop function F(x) entering the matching contributions to C NP 7γ and C NP 8G . The smooth decrease with x = m2 S /M2 D shows the decoupling of a heavier singlet scalar S relative to the vector-like down quark D. Using Eqs. (4.2) and (4.3), the magnitudes at the matching scale are then given by view at source ↗

**Figure 5.** Figure 5: Magnitude of the low scale NP contribution view at source ↗

**Figure 6.** Figure 6: Approximate percentage shift in the branching-ratio ratio, view at source ↗

**Figure 7.** Figure 7: Illustrative radiative-only upper limit on the coupling product view at source ↗

**Figure 8.** Figure 8: 2D scan of the relative NP contribution to SM, view at source ↗

**Figure 9.** Figure 9: shows that the box loop function Gbox(x) entering the ∆B = 2 WCs for Bs − B¯ s mixing. The function is positive and decreases as x increases, showing that a heavier scalar suppresses the box contribution. For the equal-mass benchmark mS = MD, one has x = 1 and Gbox(1) = 1/3. 0 1 2 3 4 5 x = m2 S /M2 D 0.0 0.2 0.4 0.6 0.8 1.0 G b o x(x) Gbox(0) = 1 Gbox(1) = 1/3 m S = M D Gbox(x) view at source ↗

**Figure 10.** Figure 10: Relative Bs − B¯ s mixing contribution hs = |C bs,NP RR /Cbs,SM LL | as a function of MD for mS = MD and |λ ∗ sλb| = 1. Using Eqs. (5.3), (5.7), and (6.18), then the condition from Eq. (6.19) giving |λ ∗ sλb| Bs max = " δBs |C SM LL (µb)| 128π 2M2 D U∆B=2 Gbox(x) #1/2 . (6.21) 16 view at source ↗

**Figure 11.** Figure 11: Upper limit on |λ ∗ sλb| from Bs − B¯ s mixing using the conservative criterion hs < 0.20, shown for representative mass ratios mS/MD = 0.5, 1, 2. The Comparison of the B → Xsγ bound, the Bs − B¯ s mixing bound, and the combined upper limit on |λ ∗ sλb| along the equal-mass line mS = MD is given in view at source ↗

**Figure 12.** Figure 12: Comparison of the radiative-only B → Xsγ bound, the Bs − B¯ s mixing bound, and the combined upper limit on |λ ∗ sλb| view at source ↗

**Figure 13.** Figure 13: 2D scan of the Bs − B¯ s mixing ratio hs = |C bs,NP RR /Cbs,SM LL | in the (MD, mS) plane for |λ ∗ sλb| = 1. 19 view at source ↗

**Figure 14.** Figure 14: Profiled 95% upper limit on Λ = |λ ∗ sλb| in the (MD, mS) plane from the simplified flavor likelihood based on B(B¯ → Xsγ), ∆Ms, and ϕs. The phase φλ is profiled over, and the conservative perturbativity condition Λ ≤ 1 is imposed. /2 0 /2 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 = | s b| 0.17 MD = mS = 1 TeV 1 2 3 Best fit 0.00 1.51 3.02 4.53 6.04 7.55 9.06 10.57 12.08 13.59 2 flav or view at source ↗

**Figure 15.** Figure 15: Complex-scalar benchmark likelihood contours in the view at source ↗

read the original abstract

We study a simple extension of the Standard Model with a vector-like down-type quark $D$ and a neutral singlet scalar ${\cal S}=S_R,\Phi$. The scalar is considered in two forms, a real field $S_R=S_R^\dagger$ and a complex field $\Phi\neq\Phi^\dagger$. The interaction $-\lambda_i{\cal S}\bar D_L d_{Ri}+{\rm h.c.}$ generates the radiative transitions $b\to s\gamma$ and $b\to sg$ at one loop. Since ${\cal S}$ has no electric charge or color, the gauge boson is emitted from the internal $D$ line, giving $C_{7\gamma}^{\rm NP}/C_{8G}^{\rm NP}=Q_D=-1/3$ at the matching scale for $\Delta B=1$ dipole transitions. For $M_D=m_{\cal S}=1~{\rm TeV}$ and $|\lambda_s^\ast\lambda_b|=1$, the low scale contribution is $|C_{7\gamma}^{\rm NP,eff}(\mu_b)|\simeq 1.1\times10^{-3}$, This is about $0.4\%$ of the Standard Model value $|C_{7\gamma}^{\rm SM,eff}(\mu_b)|\simeq 0.30$. We also discuss $B_s-\bar B_s$ mixing. In the real-scalar case, the direct and crossed box diagrams cancel in the exact minimal limit. In the complex-scalar case, the direct box contribution remains and gives a bound on the flavor $|\lambda_s^\ast\lambda_b|$ at the level of a few tenths for TeV-scale masses. Thus, in these minimal benchmarks, $B\to X_s\gamma$ is radiatively safe, while $B_s-\bar B_s$ mixing gives the stronger constraint in the complex-scalar $\Phi$ benchmark.

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

This paper sets up two minimal benchmarks with a vector-like down quark and neutral singlet scalar, showing the b to s gamma dipole gets only a 0.4% shift while Bs mixing supplies the stronger bound for the complex case.

read the letter

The core result is straightforward: with MD = mS = 1 TeV and order-one couplings, the new physics piece of C7 at the b scale is about 1.1 times 10 to the minus 3, or 0.4 percent of the SM value, because the neutral colorless scalar forces the photon to come off the internal D line and fixes the C7 over C8 ratio to minus one third. For the real scalar the direct and crossed boxes cancel exactly in the minimal limit, while the complex scalar keeps a direct box that bounds the coupling at a few tenths. These are standard one-loop Passarino-Veltman results with the assumptions written out clearly up front. The work is useful because it gives concrete numbers rather than leaving the reader to redo the integrals for this specific setup. The calculations line up with the usual loop techniques in the literature and avoid any post-hoc fitting to the observables themselves. The soft spots are limited. Everything is quoted at fixed benchmark points, so rescaling with mass or coupling is left to the reader, and the exact cancellation for the real scalar is fragile once small extra terms appear. No error bars or two-loop estimates are mentioned in the abstract, which keeps the support for the 0.4 percent figure at the moderate level. The paper does not claim broader generality beyond these choices. This is for flavor model builders who need quick reference points when adding vector-like quarks and singlets. A reader already working on Bs mixing constraints will get the most direct value. I would send it to peer review because the assumptions are explicit, the method is standard, and the numerical claims are internally consistent even if the overall scope stays narrow.

Referee Report

0 major / 3 minor

Summary. The manuscript studies a minimal SM extension with a vector-like down-type quark D and a neutral singlet scalar (real S_R or complex Φ). The Yukawa interaction generates one-loop contributions to b→sγ and b→sg, with the gauge boson emitted from the internal D line due to the scalar's neutrality, fixing C_{7γ}^{NP}/C_{8G}^{NP} = Q_D = -1/3 at the matching scale. For the benchmark M_D = m_S = 1 TeV and |λ_s^* λ_b| = 1, the low-scale |C_{7γ}^{NP,eff}(μ_b)| ≃ 1.1×10^{-3} (∼0.4% of SM), so B→X_sγ is radiatively safe. For B_s−B¯s mixing, direct and crossed boxes cancel exactly in the real-scalar minimal limit, while the complex-Φ case yields a non-vanishing contribution that constrains |λ_s^* λ_b| at the level of a few tenths.

Significance. If the one-loop results hold, the paper supplies well-defined, reproducible benchmark points that cleanly separate the dipole and mixing constraints in this class of models. The explicit charge-assignment argument for the dipole ratio and the cancellation mechanism for real scalars are model-defining strengths that make the small NP effect in B→X_sγ and the stronger mixing bound in the complex case falsifiable and useful for future phenomenology.

minor comments (3)

[Abstract] Abstract and the numerical claim |C_{7γ}^{NP,eff}(μ_b)| ≃ 1.1×10^{-3}: the result is presented without an accompanying equation, table of Passarino-Veltman integrals, or error estimate, even though the benchmark parameters are fully specified. Adding the explicit loop expression or a short numerical table would allow independent verification.
[B_s mixing discussion] The phrase 'exact minimal limit' in which direct and crossed boxes cancel for the real scalar is stated but not defined by an equation; a brief condition on the mass or coupling relations that enforce exact cancellation would remove ambiguity.
[One-loop matching section] The manuscript refers to 'standard one-loop Passarino-Veltman reduction' without citing the specific reduction formulas or software used; a short appendix or reference to the employed loop library would improve reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript. The description correctly captures the central results: the fixed ratio C_{7γ}^{NP}/C_{8G}^{NP} = -1/3 arising from the neutral scalar, the small ~0.4% NP contribution to B→X_sγ at TeV scales, the exact cancellation of box diagrams in the real-scalar minimal limit, and the non-vanishing direct-box contribution that yields stronger constraints in the complex-Φ case. We are pleased that the benchmark points are regarded as useful and falsifiable for future phenomenology.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines an SM extension via the Lagrangian term −λi S D̄L dRi + h.c. with neutral color-singlet scalar, then computes one-loop dipole operators and box diagrams using standard Passarino-Veltman reduction for chosen benchmark values MD = mS = 1 TeV and |λs∗λb| = 1. The ratio C7NP/C8NP = QD = −1/3 follows directly from the scalar being electrically neutral (photon emitted only from the D line). The quoted 0.4 % correction to |C7γeff(μb)| is the explicit numerical output of that calculation, not a fit to B→Xsγ data. Cancellation of direct and crossed boxes for the real scalar is stated as a property of the minimal limit (only the given interaction present), and the complex-scalar case supplies an independent bound on |λs∗λb|. No parameters are adjusted to the observables being constrained, no load-bearing self-citations appear, and the derivation remains self-contained against the model Lagrangian and standard loop integrals.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 2 invented entities

The central claim rests on the introduction of two new particles and a specific interaction term whose loop effects are evaluated at chosen benchmark scales; no independent evidence for the new particles is provided.

free parameters (2)

M_D = m_S = 1 TeV
Benchmark mass scale selected for numerical evaluation of the Wilson coefficients
|lambda_s^* lambda_b| = 1
Assumed coupling product used to obtain the quoted 0.4 percent contribution

axioms (2)

domain assumption The scalar has no electric charge or color
Invoked to fix the ratio C7gamma^NP / C8G^NP = Q_D = -1/3 at the matching scale
standard math One-loop matching for Delta B=1 dipole operators
Standard effective-field-theory procedure used to obtain the low-scale effective coefficient

invented entities (2)

Vector-like down-type quark D no independent evidence
purpose: Mediates the one-loop radiative b to s transitions
New postulated fermion whose charge fixes the dipole ratio
Neutral singlet scalar S_R or Phi no independent evidence
purpose: Provides the flavor-violating interaction with D and down quarks
New scalar field introduced to generate the desired loop diagrams

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