Gluino Decays in Split Supersymmetry

This is the paper you want on your desk if you care about the long-lived gluino in Split Supersymmetry. Gambino, Giudice and Slavich do the full effective-theory job: match the G-odd dimension-six operators at the squark scale, compute the one-loop anomalous dimensions in α_s and α_t, run down to the gluino mass, rotate to chargino/neutralino mass eigenstates, and give explicit decay widths in the appendix. The new content is real — the anomalous-dimension matrices, the RG-evolved Wilson coefficients, and the resulting widths and branching ratios are not in Toharia–Wells (tree-level three-body, non-resummed one-loop two-body) or in the Arvanitaki et al. appendix, whose O(α_s) anomalous dimensions they explicitly dispute. The cross-check in the appendix, recovering the tree-level results of ref. [7] when resummation is switched off, is the right sanity test and it passes. The goldstino section is a sensible extension with a clean signature: for √F ≈ m̃ the radiative goldstino channel dominates and can cut the lifetime by a large factor, and it falls below 1% once √F/m̃ ~ 10. That is falsifiable and useful for direct-mediation model building. The scaling law in eq. (63) is a nice packaging of the result. Where I would push back is the framing of the headline number. Fig. 4 quotes Γ/Γ0 ≈ 1.5–2 for the three-body channels at large m̃, but the 'without resummation' baseline for three-body is literally tree level (sect. 4), so that ratio contains the entire leading O(α_s) correction, not just the exponentiated logs. The paper is honest about this in remark (ii) — it neglects the finite O(αⁿ⁺¹ lnⁿ) terms — but it never shows a fixed-order NLO comparison, so a reader cannot separate 'resummation buys this' from 'the first correction is this big.' At m̃ = 10⁹ GeV, α_s ln(m̃/m_g) ≈ 0.7, so a good chunk of the effect is the leading log itself. This is a quantification gap, not an error in the RG derivation, and it does not sink the qualitative conclusion that resummation matters at large m̃. The degenerate-squark choice (remark v) zeroes C_B7 and makes the two-body baseline loop-induced; it is disclosed and mainly shapes the two-body curve. The μ-from-WMAP input via ref. [2] is a modeling choice, not a circularity. Bottom line: a solid, well-executed precision calculation that will be the standard reference for gluino decays in Split Supersymmetry. Send it to referees; ask them to check the disputed anomalous dimensions against ref. [5] and to press on the NLO baseline question.

The central claim [abstract:s2] is that resummation has a sizeable numerical impact. Fig. 4 quantifies this as Γ/Γ0 ≈ 1.5–2 for the 3-body channels at large m~. But per sect. 4, the 'without resummation' baseline replaces the four-fermion Wilson coefficients with their tree-level values (eqs. 9–13) and the magnetic coefficient with its one-loop value. For the 3-body channels this baseline is literally tree level, so Γ/Γ0 contains the entire leading O(α_s) correction, not just the summation of logs. The paper's own accuracy statement (sect. 2, remark ii) says it neglects O(α^{n+1} ln^n) with n≥0, i.e., the finite NLO terms. Thus the quoted 50–100% is 'resummed vs tree,' and a reader cannot tell how much survives against a proper fixed-order NLO calculation, which would already capture the α_s ln(m~/m_g) piece. For m~ = 10^9 GeV, α_s ln(m~/m_g) ≈ 0.7, so a large fraction of the effect is the first correction rather than the exponentiation. The paper is transparent that fixed-order would miss higher logs (sect. 1), but it never shows the NLO comparison, so the headline number overstates what resummation per se buys. This is a quantification/framing gap, not an error in the RG derivation. Separately, the degenerate-squark choice (remark v) zeroes C_B7 (eq. 10) and makes the 2-body baseline loop-induced; this is acknowledged and mainly affects the 2-body curve, not the 3-body claim.

The paper is a careful, standard EFT calculation with explicit cross-checks (reproducing ref. [7]'s tree-level results from eqs. 88–89), disclosed approximations, and a falsifiable gravitino signature. The RG derivation and the anomalous-dimension matrices are given explicitly and are checkable; nothing here looks internally inconsistent. The reader's weakest assumption (degenerate squarks, remark v) is real but acknowledged, and it mainly reshapes the 2-body baseline rather than the 3-body resummation claim. The more load-bearing unacknowledged gap is the choice of baseline for the headline 50–100% number: fig. 4 compares resummed vs tree-level for 3-body decays, so the quoted effect conflates the leading radiative correction with the summation of logs. This does not undermine the calculation's correctness, but it qualifies the abstract's framing of what resummation buys. Because the concern is about quantification and presentation rather than a flaw in the result, I would not move the verdict; the paper's central claim survives, and the recommended check is a clarifying addition rather than a correction.

The calculation is a timely and useful effective-theory treatment of a signature relevant to long-lived-gluino searches at the LHC and to gluino cosmology in Split Supersymmetry. The analytic appendix and the compact scaling law of eq. (63) are practical deliverables, and the explicit cross-check that the non-resummed limit reproduces the tree-level results of ref. [7] is a genuine strength. The paper also provides the full one-loop anomalous-dimension matrices, which is the reusable technical core of the result.

In Split Supersymmetry the squarks are pushed to a very high mass scale $m$, while the gluino stays near the TeV scale. Because the gluino can then only decay through operators suppressed by $1/m^2$, it lives long enough to travel macroscopic distances, and the logarithm $m/m_g$ becomes large enough that a fixed-order calculation is unreliable. This paper computes the anomalous dimensions of the relevant dimension-six operators and resums the leading logarithms controlled by the strong coupling and the top Yukawa coupling. The authors find that resummation changes the partial widths by 50 to 100 percent for large $m$, enhancing the three-body decays and suppressing the radiative one. They also treat decays into gravitinos, which can dominate entirely if supersymmetry breaking is mediated directly at the squark scale.

What carries the argument. The carrying device is an effective Lagrangian of G-odd dimension-six operators suppressed by $1/m^2$, matched to the underlying supersymmetric theory at the squark scale and then evolved with renormalization-group equations. The operator mixing is the substantive part: the anomalous-dimension matrices in eqs. (29)-(33) and (73)-(74) couple four-fermion operators to each other and, through penguin-type diagrams, to the magnetic gluon operator. For most coefficients the running integrates analytically into powers of $m/m_g$; for the higgsino pair $C_{H2}$ and $C_{H5}$ it is done numerically because of a mixing term that prevents a closed form. A second ingredient is the matching of the magnetic operator: the coefficient $C_{B7}$ vanishes for degenerate squarks, so the radiative decay is generated instead by the loop-induced $C_{H5}$ and by the matrix element in which the two top quarks in $Q_{H2}$ close into a gluon, eq. (62).

Core claim. The paper's central claim is that the gluino decay width and branching ratios in Split Supersymmetry receive corrections from large logarithms of $m/m_g$ that are big enough to matter numerically, and that these must be summed to all orders rather than truncated. Working in an effective theory below the squark scale, the authors list the G-parity-violating dimension-six operators that let the gluino decay, fix their Wilson coefficients at the matching scale $m$, and derive the one-loop anomalous-dimension matrices for the B-ino, W-ino, and higgsino operator sets. Solving the renormalization-group equations down to $m_g$ and rotating to chargino and neutralino mass eigenstates, they obtain the partial widths in closed form. Comparing with the same calculation without resummation, they report corrections of order 50-100 percent in the range $10^8$ GeV $< m < 10^{13}$ GeV, where neither the two-body nor the three-body channel clearly dominates. The lifetime is packaged as $4$ s $N (m/10^9~{GeV})^4 (1~{TeV}/m_g)^5$, with $N$ of order unity but with a slope that records the size of the corrections.

The premise it rests on. All squark masses are taken degenerate at the matching scale, which sets the tree-level magnetic coefficient to zero and leaves the radiative decay generated only by loop-induced terms; with split squark masses the relative weight of the two-body channel, and hence the quoted size of the resummation effect on branching ratios, would move.

A long-lived gluino at the LHC would have a lifetime that can be converted into a measurement of the supersymmetry-breaking scale $m$, since the scaling law in eq. (63) is explicit in $m$ and $m_g$.

Cosmological bounds on a late-decaying gluino must use the resummed width rather than the tree-level one, because the corrections reach a factor of order two in the region of interest.

The branching ratio of the radiative decay $\tilde g \to \chi^0 g$ falls with increasing $m_g$ and rises with increasing $m$, so the two-body channel is a diagnostic of the hierarchy rather than a fixed prediction.

If supersymmetry breaking is mediated directly at the squark scale, the decay $\tilde g \to \tilde G g$ dominates over all chargino and neutralino channels, suppressing the lifetime by a further large factor.

The authors report disagreement with an earlier computation of the four-fermion anomalous dimensions, which they say omitted penguin mixing and gave different higgsino entries; the resummed result supersedes that treatment.

The reported 50-100 percent corrections are largest precisely in the window $10^8$-$10^{13}$ GeV that the scenario needs, so a discovery of a long-lived gluino would be a case where leading-log resummation is not optional bookkeeping but a leading-order effect.

Because the magnetic coefficient vanishes for degenerate squarks, the predicted size of the two-body channel is a direct probe of squark mass splitting; a future measurement of $\tilde g \to \chi^0 g$ at the percent level would test the degeneracy assumption rather than the resummation itself.

The same operator-mixing machinery applies to any scenario in which a coloured fermion decays through a heavy scalar mediator, so the anomalous-dimension matrices here could be reused for other long-lived-particle cosmologies.

The gravitino branch turns on sharply: the branching ratio falls below one percent once $\sqrt{F}/m$ exceeds about ten, which makes the absence of anomalous heavy isotopes a fairly tight filter on direct-mediation models.

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