Sudakov-Sommerfeld Interplay in Dark Matter

• Sudakov/Sommerfeld interplay is the combined effect of electroweak Sudakov double logarithms and nonperturbative Sommerfeld enhancement that modifies annihilation rates of heavy electroweak multiplets.
• It involves a cancellation of infrared divergences in the dominant s-wave channels while leaving residual velocity-suppressed corrections weighted by channel-specific Sommerfeld factors.
• Numerical analyses for Higgsino-like, wino-like, and quintuplet multiplets show that the interplay induces percent-level shifts in annihilation cross sections, impacting dark matter relic predictions.

Searching arXiv for the specified paper to ground the article in the current record. Sudakov/Sommerfeld interplay denotes the combined effect of electroweak Sudakov double logarithms and nonperturbative Sommerfeld enhancement in annihilation processes of heavy electroweak multiplets. In the treatment of fermionic Minimal Dark Matter multiplets, one-loop weak corrections leave the dominant inclusive $s$-wave annihilation rates infrared finite, while infrared-enhanced corrections survive in velocity-suppressed channels once different isospin channels are weighted by different Sommerfeld factors. This mechanism modifies the subleading annihilation rate at the percent level in phenomenologically motivated cases such as the Higgsino-like doublet, the wino-like triplet, and the stable quintuplet (Buttazzo et al., 16 Jun 2026).

1. Minimal Dark Matter setting and partial-wave organization

The framework considered is a fermionic SU(2)$_L$ multiplet of dimension $n$, mass $m_\chi$, and gauge coupling $\alpha_2=g_2^2/(4\pi)$. In the unbroken SU(2)$_L$ limit, the dominant tree-level $s$-wave annihilations proceed through total isospin channels $I=1,3,5$. Summing over final states and averaging over the $n$ multiplet components gives the tree-level rates

$$\begin{aligned} \sigma v^{WW}_{I=1}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)^2}{24\,m_\chi^2},\ \sigma v^{WW}_{I=5}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-4)(n^2-1)}{12\,m_\chi^2},\ \sigma v^{f\bar f}_{I=3}\Big|_{\rm tree} &= n_L\,\frac{\pi\,\alpha_2^2\,(n^2-1)}{8\,m_\chi^2},\ \sigma v^{HH^\dagger}_{I=3}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)}{16\,m_\chi^2}, \end{aligned}$$

with $_L$0 counting the SM fermion doublets (Buttazzo et al., 16 Jun 2026).

The nonrelativistic expansion in the relative velocity $_L$1 separates the leading $_L$2-wave contribution from the $_L$3 terms. The leading $_L$4-wave pieces are

$_L$5

while the velocity-suppressed corrections to the $_L$6-wave channels are

$_L$7

Accordingly, the tree-level cross section up to $_L$8 may be organized as

$_L$9

This partial-wave decomposition is central because the Sudakov/Sommerfeld interplay does not primarily alter the dominant $n$0 terms; it enters through the velocity-suppressed sector.

2. One-loop weak corrections and cancellation in the dominant $n$1-wave

At one loop, each isospin channel receives virtual and real corrections,

$n$2

For the inclusive $n$3-wave rates, the leading infrared divergences and associated Sudakov double logarithms cancel channel by channel, in accord with the KLN theorem. The resulting correction factors for the dominant $n$4-waves are finite. An example is

$n$5

where $n$6 is the SM SU(2)$n$7 beta-function and $n$8 (Buttazzo et al., 16 Jun 2026).

A key point is that no large Sudakov $n$9 terms survive in these dominant $m_\chi$0-wave $m_\chi$1. In this formulation, the inclusive and isospin-resolved treatment is sufficient to render the leading $m_\chi$2 contribution infrared finite. This distinguishes the present setting from cases in which Sudakov logarithms remain explicit already in the leading annihilation rate.

The significance of this cancellation is conceptual as well as computational. It implies that the phenomenologically dominant piece of the thermal annihilation cross section is not the locus of the Sudakov enhancement. The nontrivial infrared structure is instead displaced to the $m_\chi$3 sector.

3. Sudakov double logarithms in velocity-suppressed channels

The velocity-suppressed contributions do not exhibit a complete KLN cancellation once different isospin channels are weighted by different Sommerfeld factors. The resulting real-plus-virtual analysis yields the master formula

$m_\chi$4

where $m_\chi$5 is the numerical mixing matrix given in eq. (4.10) of the source paper, and

$m_\chi$6

The factor $m_\chi$7 is the Sudakov double-logarithmic enhancement in the velocity-suppressed channels (Buttazzo et al., 16 Jun 2026).

To double-logarithmic accuracy, the same structure may be written in terms of a quadratic Casimir,

$m_\chi$8

This form makes explicit the growth of the effect with the multiplet size through $m_\chi$9. Since $\alpha_2=g_2^2/(4\pi)$0 increases with $\alpha_2=g_2^2/(4\pi)$1, the logarithmically enhanced weak correction becomes progressively more important for higher-dimensional electroweak representations.

This structure clarifies the meaning of “Sudakov” in the interplay: the logarithms are not merely ultraviolet-running artifacts, but infrared-sensitive double logarithms associated with soft and collinear electroweak radiation, surviving in the subleading channels after the leading inclusive cancellation has occurred.

4. Sommerfeld enhancement from long-range electroweak exchange

Sommerfeld enhancement arises because long-range electroweak gauge-boson exchange distorts the two-body wavefunction of the annihilating state. In a channel of total isospin $\alpha_2=g_2^2/(4\pi)$2, the relative-coordinate wavefunction $\alpha_2=g_2^2/(4\pi)$3 obeys

$\alpha_2=g_2^2/(4\pi)$4

with Yukawa-screened potential

$\alpha_2=g_2^2/(4\pi)$5

The Sommerfeld factor is defined by

$\alpha_2=g_2^2/(4\pi)$6

In a single-channel Coulomb-like approximation,

$\alpha_2=g_2^2/(4\pi)$7

(Buttazzo et al., 16 Jun 2026).

For $\alpha_2=g_2^2/(4\pi)$8, one has

$\alpha_2=g_2^2/(4\pi)$9

Thus the Sommerfeld effect enhances low-velocity annihilation by a factor scaling parametrically as $_L$0 in the attractive regime. In the present context this nonperturbative amplification multiplies the partial-wave cross sections channel by channel, so any residual channel dependence in the perturbative correction structure can become physically relevant even when it is attached to a formally subleading $_L$1 term.

A plausible implication is that the interplay is intrinsically tied to the noncommutativity of two operations: taking the low-velocity limit and summing over weakly corrected isospin channels after Sommerfeld weighting. That is the sense in which the source derives a distinct Sudakov/Sommerfeld effect rather than two separable corrections applied independently.

5. Combined corrected cross section and the $_L$2 limit

Including both one-loop weak corrections and Sommerfeld enhancement, the partial-wave cross section in channel $_L$3 is

$_L$4

Equivalently, with $_L$5,

$_L$6

in the notation used in the derivation (Buttazzo et al., 16 Jun 2026).

As $_L$7, the pure $_L$8-wave part proportional to $_L$9 remains infrared finite because $s$0 contains no $s$1 terms, and it dominates the annihilation rate. The velocity-suppressed part scales as

$s$2

Therefore the suppressed contributions vanish in the strict zero-velocity limit.

The interplay nonetheless remains physically nontrivial at intermediate velocities. The source emphasizes that $s$3 while $s$4, so the two factors do not simply decouple. The result is that the Sudakov correction leaves a nonzero imprint on the $s$5 channel at the percent level even though the dominant annihilation rate is infrared finite and even though the entire $s$6 sector vanishes asymptotically as $s$7.

This behavior addresses a common misconception. It is not the case that Sommerfeld enhancement universally promotes all subleading annihilation structures to leading importance at low velocity. In the formulation at hand, the $s$8 Sommerfeld growth is insufficient to overcome the explicit $s$9 suppression, so the corrected subleading contribution still vanishes as $I=1,3,5$0.

6. Numerical pattern for doublet, triplet, and quintuplet multiplets

Using a reference freeze-out velocity $I=1,3,5$1 and $I=1,3,5$2, the source gives explicit estimates for the most studied fermionic multiplets (Buttazzo et al., 16 Jun 2026).

Multiplet	One-loop and Sudakov estimates	Sommerfeld estimate
$I=1,3,5$3 (Higgsino-like doublet)	$I=1,3,5$4-wave one-loop correction: $I=1,3,5$5; Sudakov in $I=1,3,5$6-suppressed: $I=1,3,5$7	$I=1,3,5$8
$I=1,3,5$9 (Wino-like triplet)	$n$0-wave one-loop: $n$1; $n$2	$n$3
$n$4 (stable quintuplet)	$n$5-wave one-loop: $n$6; $n$7	$n$8

These values show two systematic tendencies. First, the overall weak correction to the dominant $n$9-wave contribution is already at the few-percent level. Second, the Sudakov contribution to the velocity-suppressed terms becomes more pronounced as the multiplet size increases. This is consistent with the Casimir scaling in the double-logarithmic approximation.

The same numerical pattern indicates that the interplay is not restricted to extreme or highly resonant Sommerfeld regimes. Even modest Sommerfeld factors such as $$\begin{aligned} \sigma v^{WW}_{I=1}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)^2}{24\,m_\chi^2},\ \sigma v^{WW}_{I=5}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-4)(n^2-1)}{12\,m_\chi^2},\ \sigma v^{f\bar f}_{I=3}\Big|_{\rm tree} &= n_L\,\frac{\pi\,\alpha_2^2\,(n^2-1)}{8\,m_\chi^2},\ \sigma v^{HH^\dagger}_{I=3}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)}{16\,m_\chi^2}, \end{aligned}$$0 for the doublet are sufficient for the corrected subleading structure to be tracked quantitatively. For the triplet and quintuplet, the larger $$\begin{aligned} \sigma v^{WW}_{I=1}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)^2}{24\,m_\chi^2},\ \sigma v^{WW}_{I=5}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-4)(n^2-1)}{12\,m_\chi^2},\ \sigma v^{f\bar f}_{I=3}\Big|_{\rm tree} &= n_L\,\frac{\pi\,\alpha_2^2\,(n^2-1)}{8\,m_\chi^2},\ \sigma v^{HH^\dagger}_{I=3}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)}{16\,m_\chi^2}, \end{aligned}$$1 values make the effect more visible.

Because the relic abundance scales as $$\begin{aligned} \sigma v^{WW}_{I=1}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)^2}{24\,m_\chi^2},\ \sigma v^{WW}_{I=5}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-4)(n^2-1)}{12\,m_\chi^2},\ \sigma v^{f\bar f}_{I=3}\Big|_{\rm tree} &= n_L\,\frac{\pi\,\alpha_2^2\,(n^2-1)}{8\,m_\chi^2},\ \sigma v^{HH^\dagger}_{I=3}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)}{16\,m_\chi^2}, \end{aligned}$$2, a few-percent change in $$\begin{aligned} \sigma v^{WW}_{I=1}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)^2}{24\,m_\chi^2},\ \sigma v^{WW}_{I=5}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-4)(n^2-1)}{12\,m_\chi^2},\ \sigma v^{f\bar f}_{I=3}\Big|_{\rm tree} &= n_L\,\frac{\pi\,\alpha_2^2\,(n^2-1)}{8\,m_\chi^2},\ \sigma v^{HH^\dagger}_{I=3}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)}{16\,m_\chi^2}, \end{aligned}$$3 shifts the preferred dark-matter mass by $$\begin{aligned} \sigma v^{WW}_{I=1}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)^2}{24\,m_\chi^2},\ \sigma v^{WW}_{I=5}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-4)(n^2-1)}{12\,m_\chi^2},\ \sigma v^{f\bar f}_{I=3}\Big|_{\rm tree} &= n_L\,\frac{\pi\,\alpha_2^2\,(n^2-1)}{8\,m_\chi^2},\ \sigma v^{HH^\dagger}_{I=3}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)}{16\,m_\chi^2}, \end{aligned}$$4–$$\begin{aligned} \sigma v^{WW}_{I=1}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)^2}{24\,m_\chi^2},\ \sigma v^{WW}_{I=5}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-4)(n^2-1)}{12\,m_\chi^2},\ \sigma v^{f\bar f}_{I=3}\Big|_{\rm tree} &= n_L\,\frac{\pi\,\alpha_2^2\,(n^2-1)}{8\,m_\chi^2},\ \sigma v^{HH^\dagger}_{I=3}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)}{16\,m_\chi^2}, \end{aligned}$$5. This establishes the practical significance of electroweak one-loop and Sommerfeld effects in precision studies of Minimal Dark Matter (Buttazzo et al., 16 Jun 2026).

7. Interpretation, scope, and relation to infrared structure

Within this formulation, the Sudakov/Sommerfeld interplay is not a statement that the leading annihilation rate contains uncanceled double logarithms. Rather, the dominant inclusive $$\begin{aligned} \sigma v^{WW}_{I=1}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)^2}{24\,m_\chi^2},\ \sigma v^{WW}_{I=5}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-4)(n^2-1)}{12\,m_\chi^2},\ \sigma v^{f\bar f}_{I=3}\Big|_{\rm tree} &= n_L\,\frac{\pi\,\alpha_2^2\,(n^2-1)}{8\,m_\chi^2},\ \sigma v^{HH^\dagger}_{I=3}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)}{16\,m_\chi^2}, \end{aligned}$$6-wave channels are infrared safe after real-virtual cancellation. The distinctive effect appears because the velocity-suppressed terms are reorganized by channel-dependent Sommerfeld factors, preventing a complete cancellation pattern analogous to the one operative in the leading $$\begin{aligned} \sigma v^{WW}_{I=1}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)^2}{24\,m_\chi^2},\ \sigma v^{WW}_{I=5}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-4)(n^2-1)}{12\,m_\chi^2},\ \sigma v^{f\bar f}_{I=3}\Big|_{\rm tree} &= n_L\,\frac{\pi\,\alpha_2^2\,(n^2-1)}{8\,m_\chi^2},\ \sigma v^{HH^\dagger}_{I=3}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)}{16\,m_\chi^2}, \end{aligned}$$7-wave sector.

The topic therefore sits at the interface of two standard nonrelativistic effects in heavy electroweak annihilation. The first is perturbative electroweak Sudakov structure, associated with weak virtual and real emissions and scaling as $$\begin{aligned} \sigma v^{WW}_{I=1}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)^2}{24\,m_\chi^2},\ \sigma v^{WW}_{I=5}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-4)(n^2-1)}{12\,m_\chi^2},\ \sigma v^{f\bar f}_{I=3}\Big|_{\rm tree} &= n_L\,\frac{\pi\,\alpha_2^2\,(n^2-1)}{8\,m_\chi^2},\ \sigma v^{HH^\dagger}_{I=3}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)}{16\,m_\chi^2}, \end{aligned}$$8. The second is nonperturbative threshold dynamics from repeated gauge-boson exchange, encoded in $$\begin{aligned} \sigma v^{WW}_{I=1}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)^2}{24\,m_\chi^2},\ \sigma v^{WW}_{I=5}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-4)(n^2-1)}{12\,m_\chi^2},\ \sigma v^{f\bar f}_{I=3}\Big|_{\rm tree} &= n_L\,\frac{\pi\,\alpha_2^2\,(n^2-1)}{8\,m_\chi^2},\ \sigma v^{HH^\dagger}_{I=3}\Big|_{\rm tree} &= \frac{\pi\,\alpha_2^2\,(n^2-1)}{16\,m_\chi^2}, \end{aligned}$$9. Their combination is especially relevant when annihilation rates are decomposed into isospin channels and partial waves before thermal averaging.

The source’s analysis also delimits the scope of the phenomenon. The dominant relic-density contribution remains the infrared-finite $_L$00-wave annihilation. The interplay modifies the velocity-suppressed sector and is numerically at the $_L$01 level in the most motivated cases: the Higgsino-like doublet, the wino-like triplet, and the stable quintuplet (Buttazzo et al., 16 Jun 2026). This suggests that the effect belongs to precision electroweak dark-matter phenomenology rather than to a qualitative restructuring of the leading annihilation mechanism.