Dijet Angular Distributions in Collider Physics

• Dijet angular distributions are defined using rapidity differences and invariant mass bins to characterize the scattering geometry in collider events.
• They employ normalized observables like χ_dijet to differentiate between forward and central scattering patterns, providing a clear test of QCD and potential BSM signals.
• Studies using these distributions set stringent limits on contact interactions, extra dimensions, and other nonstandard dynamics, enhancing our search for new physics.

Dijet angular distributions are differential characterizations of two-jet final states that encode the scattering geometry of the underlying hard interaction. In hadron-collider practice they are most often formulated for the two highest-$p_T$ jets in bins of dijet invariant mass $M_{jj}$, using rapidity-based observables such as $\chi_{\text{dijet}}=\exp(|y_1-y_2|)$, normalized distributions $$(1/\sigma_{\text{dijet}})\,d\sigma_{\text{dijet}}/d\chi_{\text{dijet}}$$, and related central fractions. They are classic tests of QCD because $t$-channel gluon exchange yields a characteristic forward-peaked partonic pattern that becomes nearly flat in $\chi_{\text{dijet}}$ after the standard fiducial cuts and normalization, while many extensions of the Standard Model enhance central scattering at low $\chi_{\text{dijet}}$ (0906.4819, Collaboration, 2015, Goldouzian et al., 2020).

1. Kinematic definition and standard observables

At leading order, dijet production is identified with $2\to2$ parton scattering,

$ab \to cd,\qquad a,b,c,d\in\{q,\bar q,g\}.$

In the partonic center-of-mass frame, the scattering angle $\theta^*$ is the natural angular variable. For a massless process,

$M_{jj}$0

Experimentally, the directly accessible quantities are the rapidities of the two leading jets,

$M_{jj}$1

from which one constructs the rapidity difference and the boost of the dijet system. Common definitions are

$M_{jj}$2

For massless collinear scattering,

$M_{jj}$3

The standard hadron-collider angular variable is

$M_{jj}$4

which is equivalent, in the massless limit, to

$M_{jj}$5

Hence $M_{jj}$6 corresponds to central scattering, while large $M_{jj}$7 corresponds to forward or backward scattering. Measurements are usually organized in bins of

$M_{jj}$8

the invariant mass of the two leading jets, and reported as normalized shapes,

$M_{jj}$9

Several related observables have also been used. Early ATLAS analyses employed the centrality ratio

$\chi_{\text{dijet}}=\exp(|y_1-y_2|)$0

while later ATLAS studies introduced the central fraction

$\chi_{\text{dijet}}=\exp(|y_1-y_2|)$1

with the central region defined by $\chi_{\text{dijet}}=\exp(|y_1-y_2|)$2, corresponding to $\chi_{\text{dijet}}=\exp(|y_1-y_2|)$3. These ratios compress the angular information into observables that remain highly sensitive to central enhancements from non-QCD dynamics (Goldouzian et al., 2020, Collaboration, 2012, DeViveiros, 2010).

2. QCD structure and the rationale for normalized shapes

At high energies, many QCD subprocesses are dominated by $\chi_{\text{dijet}}=\exp(|y_1-y_2|)$4-channel gluon exchange. The propagator structure generates terms of the form

$\chi_{\text{dijet}}=\exp(|y_1-y_2|)$5

so the partonic distribution is strongly enhanced at small scattering angle. This forward peaking is the underlying reason that dijet angular distributions are sensitive to nonstandard dynamics: contact-like or isotropic contributions populate the central region much more strongly than QCD.

After convolution with PDFs and the standard rapidity cuts, however, the normalized QCD prediction in $\chi_{\text{dijet}}=\exp(|y_1-y_2|)$6 is approximately flat over the measured range. This near-flatness is a central experimental advantage. It makes deviations in shape, especially low-$\chi_{\text{dijet}}=\exp(|y_1-y_2|)$7 excesses, visible against a comparatively simple baseline. CMS and ATLAS analyses at 8 and 13 TeV therefore used NLO QCD predictions from NLOJET++ or nlojet++, combined with modern PDF sets and electroweak corrections, as the standard reference for normalized $\chi_{\text{dijet}}=\exp(|y_1-y_2|)$8 spectra (Goldouzian et al., 2020, Collaboration, 2014, Collaboration, 2017).

The residual theoretical uncertainties are dominated by scale variation rather than PDFs. In the CMS 8 TeV analysis, PDF effects on the normalized $\chi_{\text{dijet}}=\exp(|y_1-y_2|)$9 distributions were at most about $$(1/\sigma_{\text{dijet}})\,d\sigma_{\text{dijet}}/d\chi_{\text{dijet}}$$0 to $$(1/\sigma_{\text{dijet}})\,d\sigma_{\text{dijet}}/d\chi_{\text{dijet}}$$1, while scale variations reached up to about $$(1/\sigma_{\text{dijet}})\,d\sigma_{\text{dijet}}/d\chi_{\text{dijet}}$$2 in the highest mass region. In the CMS 13 TeV analysis with $$(1/\sigma_{\text{dijet}})\,d\sigma_{\text{dijet}}/d\chi_{\text{dijet}}$$3, PDF uncertainties remained below $$(1/\sigma_{\text{dijet}})\,d\sigma_{\text{dijet}}/d\chi_{\text{dijet}}$$4 and scale variations reached up to $$(1/\sigma_{\text{dijet}})\,d\sigma_{\text{dijet}}/d\chi_{\text{dijet}}$$5 in the highest $$(1/\sigma_{\text{dijet}})\,d\sigma_{\text{dijet}}/d\chi_{\text{dijet}}$$6 bin. Non-perturbative corrections were found to be negligible or at the percent level in these normalized shapes (Collaboration, 2014, Collaboration, 2017).

This behavior explains why dijet angular distributions have remained competitive across collider generations. The observable is driven primarily by matrix-element structure, while normalization to unity in each $$(1/\sigma_{\text{dijet}})\,d\sigma_{\text{dijet}}/d\chi_{\text{dijet}}$$7 interval suppresses luminosity dependence and reduces sensitivity to many detector and PDF systematics.

3. Experimental realization at hadron colliders

The experimental implementation has been remarkably stable. ATLAS and CMS reconstruct jets with anti-$$(1/\sigma_{\text{dijet}})\,d\sigma_{\text{dijet}}/d\chi_{\text{dijet}}$$8, using $$(1/\sigma_{\text{dijet}})\,d\sigma_{\text{dijet}}/d\chi_{\text{dijet}}$$9 in early ATLAS studies, $t$0 in 7 and 8 TeV CMS analyses, and $t$1 in later CMS 13 TeV analyses. Fiducial selections typically require the two leading jets to define the dijet system, a restricted boost such as $t$2 or $t$3, and an angular acceptance such as $t$4 or $t$5, which together keep both jets in the central detector region (DeViveiros, 2010, Collaboration, 2011, Collaboration, 2015, Collaboration, 2018).

Early measurements emphasized robust detector-level comparisons and ratio observables. ATLAS, with $t$6 at 7 TeV, measured $t$7 in two $t$8 bins and the centrality ratio $t$9, already exploiting the weak dependence of angular shapes on the absolute jet energy scale. CMS, with $\chi_{\text{dijet}}$0 at 7 TeV, organized normalized $\chi_{\text{dijet}}$1 spectra across nine $\chi_{\text{dijet}}$2 intervals and used a modified frequentist procedure to set compositeness limits. Later analyses moved to multi-bin unfolding in $\chi_{\text{dijet}}$3, explicit NLO QCD baselines, and bin-by-bin or matrix-based propagation of detector response (DeViveiros, 2010, Collaboration, 2011, Collaboration, 2014, Collaboration, 2017).

A parallel methodological development was the use of detector-level fits for BSM interpretation even when unfolded spectra were published for SM comparison. CMS at 13 TeV with $\chi_{\text{dijet}}$4 unfolded the normalized angular spectra to particle level for presentation, but set limits using detector-level distributions obtained by folding the SM and SM+BSM predictions through the same response matrix. This avoided introducing additional correlations from unfolding into the likelihood used for exclusion (Collaboration, 2018).

A recurring feature across experiments is the use of high-$\chi_{\text{dijet}}$5 control of trigger efficiency. Mass bins were defined so that the relevant single-jet or $\chi_{\text{dijet}}$6-based triggers were fully efficient. As a result, the dominant uncertainties in the highest bins became statistical and theoretical rather than trigger-related.

4. Searches for short-distance interactions and broad new-physics signals

Dijet angular distributions have long complemented dijet mass-spectrum searches. They are especially sensitive to broad or nonresonant effects that do not produce a narrow bump in $\chi_{\text{dijet}}$7, but do alter the balance between central and forward scattering. D0 performed the first Tevatron measurement of dijet angular distributions in $\chi_{\text{dijet}}$8 collisions at $\chi_{\text{dijet}}$9 TeV, over dijet masses from $\chi_{\text{dijet}}$0 TeV to above $\chi_{\text{dijet}}$1 TeV, and used the agreement with perturbative QCD to set the most stringent direct limits of that time on quark compositeness and extra-dimensional models. Early LHC analyses by ATLAS and CMS then established the same program at 7 TeV with $\chi_{\text{dijet}}$2 and $\chi_{\text{dijet}}$3, respectively (0906.4819, DeViveiros, 2010, Collaboration, 2011).

With larger datasets, the limits strengthened rapidly. CMS at 7 TeV with $\chi_{\text{dijet}}$4 reported 95% CL lower limits on contact-interaction scales ranging from $\chi_{\text{dijet}}$5 up to $\chi_{\text{dijet}}$6 TeV, and incorporated exact NLO QCD corrections for benchmark CI models. ATLAS at 8 TeV with $\chi_{\text{dijet}}$7 excluded a compositeness scale below $\chi_{\text{dijet}}$8 TeV in the destructive-interference scenario and $\chi_{\text{dijet}}$9 TeV in the constructive-interference scenario. CMS at 13 TeV with $2\to2$0 excluded left-handed contact interactions up to $2\to2$1 or $2\to2$2 TeV, while the $2\to2$3 analysis extended the corresponding limits to $2\to2$4 and $2\to2$5 TeV and, for the first time in this channel, set lower limits between $2\to2$6 and $2\to2$7 TeV on the mass of a dark matter mediator for (axial-)vector mediators with universal quark coupling $2\to2$8 (Collaboration, 2012, Collaboration, 2015, Collaboration, 2017, Collaboration, 2018).

The same observables constrain extra-dimensional and threshold-like scenarios. D0 derived direct limits on large extra dimensions in both GRW and HLZ conventions and on TeV$2\to2$9-scale extra dimensions. CMS at 13 TeV with $ab \to cd,\qquad a,b,c,d\in\{q,\bar q,g\}.$0 obtained lower limits on the ADD cutoff scale in the range $ab \to cd,\qquad a,b,c,d\in\{q,\bar q,g\}.$1–$ab \to cd,\qquad a,b,c,d\in\{q,\bar q,g\}.$2 TeV and excluded quantum black holes below $ab \to cd,\qquad a,b,c,d\in\{q,\bar q,g\}.$3 or $ab \to cd,\qquad a,b,c,d\in\{q,\bar q,g\}.$4 TeV, depending on the model. The later $ab \to cd,\qquad a,b,c,d\in\{q,\bar q,g\}.$5 dataset pushed the GRW cutoff to $ab \to cd,\qquad a,b,c,d\in\{q,\bar q,g\}.$6 TeV and excluded quantum black holes below $ab \to cd,\qquad a,b,c,d\in\{q,\bar q,g\}.$7 and $ab \to cd,\qquad a,b,c,d\in\{q,\bar q,g\}.$8 TeV, depending on the model (0906.4819, Collaboration, 2017, Collaboration, 2018).

These exclusions reflect a common dynamical pattern. Contact interactions, virtual graviton exchange, quantum black holes, and broad dijet mediators all tend to produce more isotropic partonic scattering than QCD. In normalized $ab \to cd,\qquad a,b,c,d\in\{q,\bar q,g\}.$9 spectra, that translates into characteristic distortions concentrated at low $\theta^*$0 and high $\theta^*$1. The continued absence of such distortions is what makes dijet angular distributions one of the most durable search channels for short-distance physics.

5. Effective-field-theory and precision-QCD developments

A notable extension of the classical contact-interaction program is the reinterpretation of dijet angular distributions within the Standard Model Effective Field Theory. The dimension-six triple-gluon operator,

$\theta^*$2

modifies the three-gluon vertex, the four-gluon vertex, and generates effective higher-multiplicity gluon interactions. In inclusive dijet production, the dominant $\theta^*$3 helicity amplitudes involving $\theta^*$4 are orthogonal to the SM QCD helicity amplitudes, so the leading nonzero contribution arises at order $\theta^*$5 rather than $\theta^*$6. Even so, the normalized $\theta^*$7 distribution from the pure $\theta^*$8 term is enhanced at small $\theta^*$9, and the effect grows rapidly with $M_{jj}$00. Reinterpreting a CMS 13 TeV search with $M_{jj}$01, the resulting 95% CL bound was

$M_{jj}$02

observed, with an expected limit of $M_{jj}$03. The same study argued that this bound is sufficiently strong that $M_{jj}$04 can be neglected in global SMEFT analyses focused on top and Higgs sectors (Goldouzian et al., 2020).

On the theory side, dijet angular observables have also been generalized from interjet scattering geometry to jet-internal angular structure. In Soft-Collinear Effective Theory, boost-invariant jet angularities for dijet events at hadron colliders are defined as

$M_{jj}$05

with $M_{jj}$06 corresponding, up to power corrections, to the squared jet mass normalized to $M_{jj}$07. The SCET factorization formula for dijet production with such measurements contains hard, jet, and soft functions together with previously unstudied unmeasured beam functions, which are present for finite rapidity cuts around the beams. At NLL accuracy in regions where non-global logarithms are not large, the factorized cross section can be evolved consistently; the same framework also implements soft-collinear refactorization to resum logarithms of the jet size $M_{jj}$08. For the explicit $M_{jj}$09 example, the refactorized soft function reduces normalization and scale uncertainty and makes the dependence on $M_{jj}$10, on $M_{jj}$11, and on the angularity parameter $M_{jj}$12 explicit (Hornig et al., 2016).

Taken together, these developments show that dijet angular distributions now occupy two complementary roles: they remain precision shape observables for broad new-physics searches, and they have become controlled probes of operator-level and resummed-QCD structure.

6. Extensions beyond inclusive $M_{jj}$13 dijets

The term “dijet angular distributions” has acquired broader meanings outside inclusive $M_{jj}$14 scattering. In $M_{jj}$15Pb collisions at $M_{jj}$16 TeV, CMS studied the azimuthal separation

$M_{jj}$17

and the pseudorapidity of the dijet system,

$M_{jj}$18

for jets reconstructed with anti-$M_{jj}$19, $M_{jj}$20, and cuts $M_{jj}$21 GeV, $M_{jj}$22 GeV, $M_{jj}$23. The $M_{jj}$24 distribution was sharply peaked near $M_{jj}$25, with an inclusive fitted width

$M_{jj}$26

and was essentially insensitive to forward-activity selections. By contrast, $M_{jj}$27 shifted monotonically with increasing forward calorimeter energy, and the inclusive $M_{jj}$28 distribution matched CT10+EPS09 more closely than CT10 alone, indicating sensitivity to nuclear PDFs and to correlations between forward soft activity and the longitudinal motion of the dijet frame (Collaboration, 2014).

A related NLO perturbative-QCD analysis of cold nuclear matter effects in $M_{jj}$29Pb and PbPb collisions formulated the angular observable

$M_{jj}$30

and the normalized nuclear modification factor

$M_{jj}$31

The calculation found $M_{jj}$32 and nearly flat over the full $M_{jj}$33 range for EPS09, EKS98, HKN, and DS nuclear PDFs. This implies that normalized dijet angular distributions are largely insensitive to initial-state cold nuclear matter effects and therefore provide particularly clean probes of final-state hot-medium phenomena such as jet quenching (He et al., 2011).

At much smaller $M_{jj}$34, exclusive diffractive dijet production in electron–proton scattering introduces another angular structure. In the Color Glass Condensate description, the cross section for exclusive dijets at a future Electron Ion Collider exhibits an elliptic modulation in the angle between the average dijet transverse momentum $M_{jj}$35 and the recoil momentum $M_{jj}$36,

$M_{jj}$37

This modulation is attributed to non-trivial angular correlations between transverse coordinate and transverse momentum in the gluon Wigner or Husimi distribution. Small-$M_{jj}$38 evolution reduces the elliptic modulation in the EIC kinematics because of the growth of the proton with decreasing $M_{jj}$39, so the dijet angular distribution becomes a form of gluon phase-space tomography rather than a purely hard-scattering test (Mäntysaari et al., 2019).

Across these settings, the common thread is unchanged: dijet angular observables isolate geometric information that is not redundant with the mass spectrum. In inclusive $M_{jj}$40 collisions they test short-distance QCD and constrain broad new-physics scenarios; in SMEFT they resolve operator-specific deformations of multigluon interactions; in nuclear collisions they separate longitudinal and transverse correlations from medium effects; and in diffractive DIS they encode the angular structure of small-$M_{jj}$41 gluon distributions.