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Testing Minimal $U(2)^3$ Right-Handed Flavor Breaking in a Singlet-Scalar Vector-Like Quark Portal

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Qazi Maaz Us Salam

High Energy Physics - Phenomenology · Submitted 11 Oct 2026 · 92 references

Pith's verdict: A solid, conservative flavor analysis of a right-handed portal model; main conclusion stands but the uncorrelated-observable assumption deserves a robustness check.

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arXiv ID2610.15911 (version 1)
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CitationQazi Maaz Us Salam. "Testing Minimal $U(2)^3$ Right-Handed Flavor Breaking in a Singlet-Scalar Vector-Like Quark Portal." arXiv:2610.15911 (2026). https://pith.science/paper/EMZRH6IB
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The paper

The paper claims

Current data cannot distinguish flavor structures in a right-handed portal

The reading

This paper tests three flavor structures—generic spurion, minimal $U(2)^3$, and CKM alignment—in a portal where a $Z_2$-odd singlet scalar and a vector-like down quark couple to right-handed down quarks. Using a profile likelihood of seven observables ($B_s$ and $B_d$ mixing, $\epsilon_K$, and radiative decays) with CKM, lattice, and short-distance inputs as nuisance parameters, the author finds that the Standard Model is compatible with the data ($p=0.20$). No direction of the light-quark couplings can be excluded at 95% CL. For mediator masses near 1 TeV, the relative coupling to light quarks is bounded by $0.227$ (generic) or $0.211$ (minimal $U(2)^3$). The key conclusion is that the present data cannot yet test the flavor structure; more precise measurements are needed.

What carries the argument

The model is defined by a singlet scalar $S$ and a vector-like down quark $D$, both $Z_2$-odd, coupling to right-handed down quarks through a single coupling vector $(\lambda_d, \lambda_s, \lambda_b)$. This generates one-loop primed dipole operators and, for a complex scalar, $\Delta F = 2$ box amplitudes proportional to $(\lambda_i^* \lambda_j)^2$. For a real scalar the direct and crossed boxes cancel exactly. The flavor-changing products obey the identity $\Lambda_{ds} = \Lambda_{db} \Lambda_{sb}^* / |\lambda_b|^2$, forcing the kaon, $B_d$, and $B_s$ systems to probe one common structure. The analysis uses a profile likelihood that treats CKM parameters, lattice inputs, and short-distance factors as fifteen nuisance parameters and compares nested hypotheses through the ratio $h_d/h_s$ of the new-physics contributions to $B_d$ and $B_s$ mixing.

Extended reading notes

Core claim

The central claim is that the singlet-scalar vector-like quark portal with a single coupling vector cannot be discriminated in flavor: a generic right-handed spurion, a minimal $U(2)^3$ hypothesis that suppresses the $b\to d$ coupling by $m_d/m_s$, and a CKM-aligned ansatz are all compatible with the current $B$-mixing, kaon, and radiative data. The best-fit complex scalar at the 1 TeV benchmark gives $\Delta\chi^2 = 5.66$ relative to the SM, but a pseudo-experiment yields $p = 0.20$, so this is not evidence for new physics. For a real scalar the $\Delta F = 2$ boxes cancel, leaving no meaningful constraint. The paper demonstrates that the rank-one relation $\Lambda_{ds} = \Lambda_{db} \Lambda_{sb}^* / |\lambda_b|^2$ links all three down-type transitions, yet the observational power is insufficient to test whether the right-handed spurion follows the minimal $U(2)^3$ pattern or a different one.

Load-bearing premise

The seven observables are treated as uncorrelated because a full experimental covariance matrix is not publicly available; if significant correlations exist, the profile likelihood results and confidence intervals could change.

Editorial extensions

If this is right

  • If the portal exists, all three flavor hypotheses (generic, minimal $U(2)^3$, CKM-aligned) remain viable with current data; the 95% CL upper limits on $\epsilon_R$ range from $0.112$ to $0.227$ depending on the hypothesis.
  • Minimal $U(2)^3$ predicts $h_d/h_s \approx (m_d/m_s)^2 \approx 2.5\times10^{-3}$, so new physics would appear almost exclusively in $B_s$ mixing, whereas CKM alignment predicts $h_d/h_s = 1$. A future measurement of a deviation in $B_s$ mixing without a corresponding deviation in $B_d$ mixing would favor the minimal structure.
  • For a real scalar, the $\Delta F = 2$ boxes cancel, leaving only radiative constraints that are too weak to produce an upper limit below $\epsilon_R = 1$. Thus the real scalar can escape all current flavor bounds.
  • Improved lattice QCD inputs for $f_B\sqrt{\hat{B}_B}$ and $\xi$, together with more precise tree-level CKM parameters, will sharpen the upper limit on $\epsilon_R$ by reducing nuisance uncertainties in the profile likelihood.
  • The pseudo-experiment calibration at the SM boundary shows that the asymptotic $\chi^2$ approximation fails when the portal phase is not identified; the critical value at $\epsilon_R = 0$ is $9.00$ instead of $3.841$, meaning the usual Wilks theorem does not apply in the null.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's explicit calibration of the profile likelihood null distribution serves as a cautionary example for model-specific fits with parameters that become unidentifiable at the SM point. Similar issues may arise in other searches for weakly-coupled new physics.
  • The cancellation of $\Delta F = 2$ boxes for a real scalar implies that if the dark matter candidate in this model is a real scalar, the strongest flavor bounds come from radiative decays, which are presently weak. This opens the possibility of a flavored dark matter scenario with suppressed meson-mixing signals.
  • Comparing the three flavor hypotheses with the same number of free parameters (minimal $U(2)^3$ and CKM alignment) shows that the preference between them depends on the CKM input scheme (e.g., tree-level Wolfenstein parameters vs. $|V_{cb}|, |V_{ub}|, \gamma$). This highlights the need for a global analysis that consistently treats all CKM constraints before drawing conclusions about the flavor st
  • The rank-one relation $\Lambda_{ds} = \Lambda_{db} \Lambda_{sb}^*/|\lambda_b|^2$ is a generic prediction of any single-mediator portal with only three couplings; it is not a consequence of flavor symmetry. This feature could be tested independently if a deviation is seen in both $b\to s$ and $b\to d$ transitions.
Original abstract

We study a portal of a $Z_2$-odd singlet scalar and a vector like down quark, in which a right handed flavor breaking spurion extends the portal coupling from $b_R$ to $d_R$ and $s_R$. This single coupling vector links the $b\to s$, $b\to d$, and $s\to d$ transitions. We compute the one loop dipole and $\Delta F=2$ Wilson coefficients, and show that a real scalar generates no $\Delta F=2$ amplitude at one loop. Using a seven observable profile likelihood of $B$ meson mixing, $\epsilon_K$, and radiative decays, with the CKM, lattice, and short distance inputs treated as nuisance parameters, we compare a generic spurion, minimal $U(2)^3$ with a free size of the light quark couplings, and a CKM-aligned ansatz. We have found that the present data cannot yet test the flavor structure: the Standard Model is compatible with the data, with a pseudo experiment $p$-value of $0.20$, and no direction of the light quark couplings can be excluded at 95% CL. For $M_D\simeq m_{\mathcal{S}}=1$ TeV and $|\lambda_b|=1$, the relative light quark coupling $\epsilon_R$ is bounded at 95% CL by $0.227$ for a generic spurion and by $0.211$ for minimal $U(2)^3$. More precise measurements of the $B_s$ mixing phase at the LHC and of $\sin2\beta$ at LHCb and Belle-II, together with improved lattice and CKM inputs, can make the minimality of the flavor structure testable if a deviation from the Standard Model is observed.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-14: 1 request, 6,128 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.95 · human participants 0.01 · controlled comparison 0.51 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.82 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We study a portal of a $Z_2$-odd singlet scalar and a vector like down quark, in which a right handed flavor breaking spurion extends the portal coupling from $b_R$ to $d_R$ and $s_R$. This single coupling vector links the $b\to s$, $b\to d$, and $s\to d$ transitions. We compute the one loop dipole and $\Delta F=2$ Wilson coefficients, and show that a real scalar generates no $\Delta F=2$ amplitude at one loop. Using a seven observable profile likelihood of $B$ meson mixing, $\epsilon_K$, and radiative decays, with the CKM, lattice, and short distance inputs treated as nuisance parameters, we compare a generic spurion, minimal $U(2)^3$ with a free size of the light quark couplings, and a CKM-aligned ansatz. We have found that the present data cannot yet test the flavor structure: the Standard Model is compatible with the data, with a pseudo experiment $p$-value of $0.20$, and no direction of the light quark couplings can be excluded at 95% CL. For $M_D\simeq m_{\mathcal{S}}=1$ TeV and $|\lambda_b|=1$, the relative light quark coupling $\epsilon_R$ is bounded at 95% CL by $0.227$ for a generic spurion and by $0.211$ for minimal $U(2)^3$. More precise measurements of the $B_s$ mixing phase at the LHC and of $\sin2\beta$ at LHCb and Belle-II, together with improved lattice and CKM inputs, can make the minimality of the flavor structure testable if a deviation from the Standard Model is observed.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study a portal of a $Z_2$-odd singlet scalar and a vector like down quark, in which a right handed flavor breaking spurion extends the portal coupling from $b_R$ to $d_R$ and $s_R$. contribution 0.93 0.48 0.02 asserted 0.96
s1 This single coupling vector links the $b\to s$, $b\to d$, and $s\to d$ transitions. contribution 0.57 0.27 0.03 asserted 1.00
s2 We compute the one loop dipole and $\Delta F=2$ Wilson coefficients, and show that a real scalar generates no $\Delta F=2$ amplitude at one loop. result 0.79 0.94 0.05 asserted 1.00
s3 Using a seven observable profile likelihood of $B$ meson mixing, $\epsilon_K$, and radiative decays, with the CKM, lattice, and short distance inputs treated as nuisance parameters, we compare a generic spurion, minimal $U(2)^3$ with a free size of the light quark couplings, and a CKM-aligned ansatz. method 0.98 0.92 0.04 asserted 0.94
s4 We have found that the present data cannot yet test the flavor structure: the Standard Model is compatible with the data, with a pseudo experiment $p$-value of $0.20$, and no direction of the light quark couplings can be excluded at 95% CL. result 0.91 0.96 0.75 asserted 0.96
s5 For $M_D\simeq m_{\mathcal{S}}=1$ TeV and $|\lambda_b|=1$, the relative light quark coupling $\epsilon_R$ is bounded at 95% CL by $0.227$ for a generic spurion and by $0.211$ for minimal $U(2)^3$. result 1.00 0.92 0.12 asserted 1.00
s6 More precise measurements of the $B_s$ mixing phase at the LHC and of $\sin2\beta$ at LHCb and Belle-II, together with improved lattice and CKM inputs, can make the minimality of the flavor structure testable if a deviation from the Standard Model is observed. future work 0.99 0.30 0.12 hedged possibility 0.98

What kind of paper the classifier reports this is: theoretical result 0.92 · evidence from analysis of existing data 0.82.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. One-loop Feynman diagram for b → qγ and b → qg in the portal. The dashed line is the scalar S, the double line is the vector-like quark D, and the photon or gluon is emitted from D. and PL,R = (1 ∓ γ5)/2. The unprimed operators O7γ and O8G, with PR in place of PL, are the ones generated in the SM. The portal generates only the primed operators, because the interaction in eq. (2.7) contains a single chirality. The incoming b quark enters through DP¯ Rb and is therefore right-handed. The outgoing quark leaves through qP¯ LD, and since qP¯ L = PRq, it is also right-handed. The…
Figure 2. Direct and crossed box diagrams for ∆F = 2 transitions. A complex scalar allows only the direct diagram, while a real scalar allows both. The labels give the momenta of the two D lines. that is, the electromagnetic WC equals the chromomagnetic one multiplied by the electric charge of D. This ratio is changed by the QCD evolution to low scales, discussed in section 3.3. 3.2 Neutral-meson mixing We now turn to the ∆F = 2 transitions. For i ̸= j, the portal generates only the right-handed operator O ij RR = ( ¯d α i γµPRd α j )( ¯d β i γ µPRd β j ), (3.7) with H∆F =2 eff
Figure 3. Profiled ∆χ 2 in the complex hd/hs plane at MD ≃ mS = 1 TeV; the solid line is the 68% contour. The star is the CKM-aligned ansatz, the diamond is minimal U(2)3 , and the open circle is the generic best fit. and profile ϵR, the common phase, and all fifteen nuisances at each point. Both signs of eqs. (2.15) and (2.17) map to the same point, since hd/hs depends on w 2 . The aligned point lies inside the 68% region at ∆χ 2 = 1.51, and the minimal U(2)3 point lies just outside it at ∆χ 2 = 2.65. These values agree with eq. (6.6); the small difference for the aligned point arises be
Figure 4. Profile in ϵR for the complex and real scalar; the dotted line is the asymptotic 95% threshold. 2βeff constrain it. The two restricted endpoints use the asymptotic threshold of eq. (5.8); only the generic endpoint is coverage-calibrated, see section 6.4. For minimal U(2)3 , the best fit ϵR = 0.135 and the upper limit ϵ 95,minimal R = 0.211 are larger than the natural size in eq. (2.16) by factors of about 180 and 280, respectively. A portal effect of the fitted size with the minimal direction therefore requires the product of the coefficient in eq. (2.12) and |λV |/|λb| to be of…
Figure 5. Asymptotic 95% upper endpoint on ϵR for the complex scalar with |λb| = 1. The shaded region mS ≥ MD has a stable colored relic; crosses mark points with no endpoint below ϵR = 1. isotopes, which cover masses from 0.1 to 10 TeV [112]. This region is shaded in figure 5, and the one-TeV benchmark is understood with mS slightly below MD, so that D → q S is allowed. For mS < MD, the scalar is stable. For a real scalar, the annihilation into light-quark pairs through t-channel D exchange vanishes in both the s and p waves, and a thermal relic abundance requires either coannihilation with…
Figure 6. Loop functions F(x), Gbox(x), and G9(x). The dots mark the values at x = 1, and the dotted lines the limits x → 0. In particular, F(1) = 1/24 and F(0) = 1/12. For the box function, Gbox(x) = 2 Z 1 0 dz z(1 − z) z + x(1 − z) , (B.3) and Gbox(1 + u) = X n≥0 2(−1)nu n (n + 2)(n + 3) = 1 3 − u 6 + u 2 10 − u 3 15 + u 4 21 − · · · , (B.4) so that Gbox(1) = 1/3 and Gbox(0) = 1. The photon-penguin function of eq. (4.14), with the same argument x = m2 S /M2 D, is G9(x) = −16x 3 + 45x 2 − 36x + 7 + 6(2x − 3)x 2 ln x 36(x − 1)4 , (B.5a) G9(1 + u) = X n≥0 (
Figure 7. Profiled theory-nuisance pulls at the best complex-scalar point for MD = mS = 1 TeV and |λb| = 1. The dashed lines mark ±1 standard deviation. – 36 –
Figure 8. Empirical cumulative distributions of the profile-likelihood statistic q(ϵR) at the three calibration points. The dotted curve is the χ 2 1 distribution; the vertical dashed line marks 3.841 and the horizontal dashed line marks 95%. References [1] S.L. Glashow, J. Iliopoulos and L. Maiani, Weak Interactions with Lepton-Hadron Symmetry, Phys. Rev. D 2 (1970) 1285. [2] G. Isidori, Y. Nir and G. Perez, Flavor Physics Constraints for Physics Beyond the Standard Model, Ann. Rev. Nucl. Part. Sci. 60 (2010) 355 [1002.0900]. [3] A.J. Buras and J. Girrbach, Towards the Identification of New

Figures reproduced from arXiv:2610.15911 by the authors.

References

Discussion

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