USD-DPS: Ultraviolet-Consistent Double Parton Scattering

• USD-DPS is a formulation that regulates double parton scattering by explicitly subtracting ultraviolet overlap with single parton scattering contributions.
• It employs double parton distributions with explicit transverse dependence and evolves them via homogeneous double-DGLAP equations.
• The framework distinguishes splitting–splitting, splitting–intrinsic, and intrinsic–intrinsic contributions to ensure consistent removal of double counting.

Searching arXiv for USD-DPS and related double parton scattering papers. arxiv_search(query="USD-DPS double parton scattering ultraviolet subtraction", max_results=10, sort_by="relevance") USD-DPS, an Editor’s term for the ultraviolet-consistent formulation of double parton scattering (DPS), denotes a treatment in which the DPS contribution is regulated in the short-distance region and combined with an explicit overlap subtraction and the genuine single parton scattering (SPS) term. In this framework, the central theoretical problem is that parton pairs can be generated perturbatively via $1\to2$ splitting, which creates a double counting with SPS loop corrections unless the overlap is removed consistently (Diehl et al., 2016). The formalism preserves the use of double parton distributions (DPDs) for individual hadrons, while making their ultraviolet behavior explicit. It is best understood against the broader phenomenology of DPS, where factorized “pocket-formula” descriptions have been applied to four-jet production and double-charm production in $k_T$-factorization [(Maciula et al., 2016); (Szczurek et al., 2013)].

1. DPS as a two-hard-scattering mechanism

DPS describes the production of two hard final states in a single hadron–hadron collision through two distinct partonic interactions. In the conventional factorized ansatz, one assumes that the two hard scatterings are uncorrelated, so that the DPS cross section factorizes into a product of two SPS cross sections divided by an effective overlap parameter. For production of final states $A$ and $B$, the naive DPS formula takes the form

$$\sigma_{\rm DPS} = \frac1{C}\, \sum_{i,j,k,l} \int dx_1\,dx_2\,d\bar x_1\,d\bar x_2 \int d^2\bm y\; F^{ij}(x_1,x_2,\bm y;\mu)\, F^{kl}(\bar x_1,\bar x_2,\bm y;\mu)\, \hat\sigma_{ik\to A}\, \hat\sigma_{jl\to B},$$

with $C=2$ if $A=B$, otherwise $C=1$ (Diehl et al., 2016).

In phenomenological applications, a simplified “pocket formula” is often used. In four-jet production, the DPS contribution is written as a product of two independent $2\to2$ scatterings with a combinatorial factor $m=1/2$ for identical scatterings and a phenomenological parameter $k_T$0 (Maciula et al., 2016). In double-charm production, the fully differential cross section for $k_T$1 is likewise written as

$k_T$2

again under the factorized ansatz (Szczurek et al., 2013).

This simplified picture is adequate for many exploratory analyses, but it obscures the ultraviolet structure of DPS. The central issue is that the same final-state topology can arise either from two independent partons in each proton or from perturbative splitting of a single parton into two. The latter mechanism populates the small-transverse-separation region and overlaps with SPS loop topologies, making a naive sum of SPS and DPS inconsistent (Diehl et al., 2016).

2. Double parton distributions and their ultraviolet behavior

The ultraviolet-consistent formulation is built around DPDs. For a hadron of momentum $k_T$3, the unpolarized DPD for finding two partons of flavors $k_T$4 with momentum fractions $k_T$5 and transverse separation $k_T$6 is defined by the matrix element

$k_T$7

where each $k_T$8 is a renormalized twist-two field operator and $k_T$9 is the $A$0 factorization/renormalization scale (Diehl et al., 2016).

The crucial feature is the short-distance limit $A$1. In that limit the operator product develops an ultraviolet divergence of order $A$2. After $A$3 subtraction, the DPDs depend on $A$4 and obey the homogeneous double-DGLAP equation

$A$5

with the usual DGLAP splitting kernels without virtual pieces (Diehl et al., 2016).

This renormalized DPD framework differs conceptually from the “zero-correlation” approximation used in some phenomenological studies, where longitudinal, spin, colour, and flavour correlations are neglected and one approximates

$A$6

(Maciula et al., 2016). The latter is operationally useful, but the DPD formalism makes the transverse-separation dependence and the perturbative small-$A$7 singularity explicit.

3. Regulated DPS, subtraction, and the avoidance of double counting

The defining feature of USD-DPS is the regulated short-distance treatment. To suppress the ultraviolet region, one introduces a smooth cutoff function $A$8 satisfying

$A$9

and defines the regulated DPS contribution as

$B$0

The total cross section is then organized as

$B$1

Here $B$2 is the usual SPS cross section at the desired perturbative order, and $B$3 is obtained from the same regulated DPS integral but with each full DPD replaced by its fixed-order perturbative $B$4 approximation (Diehl et al., 2016).

The subtraction term reproduces the small-$B$5 limit of the DPS contribution and cancels the double counting with SPS diagrams of the same topology, referred to as “1v1” graphs. At small $B$6, one has $B$7, so that $B$8. At large $B$9, one finds $$\sigma_{\rm DPS} = \frac1{C}\, \sum_{i,j,k,l} \int dx_1\,dx_2\,d\bar x_1\,d\bar x_2 \int d^2\bm y\; F^{ij}(x_1,x_2,\bm y;\mu)\, F^{kl}(\bar x_1,\bar x_2,\bm y;\mu)\, \hat\sigma_{ik\to A}\, \hat\sigma_{jl\to B},$$0, so that $$\sigma_{\rm DPS} = \frac1{C}\, \sum_{i,j,k,l} \int dx_1\,dx_2\,d\bar x_1\,d\bar x_2 \int d^2\bm y\; F^{ij}(x_1,x_2,\bm y;\mu)\, F^{kl}(\bar x_1,\bar x_2,\bm y;\mu)\, \hat\sigma_{ik\to A}\, \hat\sigma_{jl\to B},$$1. The unphysical cutoff parameter $$\sigma_{\rm DPS} = \frac1{C}\, \sum_{i,j,k,l} \int dx_1\,dx_2\,d\bar x_1\,d\bar x_2 \int d^2\bm y\; F^{ij}(x_1,x_2,\bm y;\mu)\, F^{kl}(\bar x_1,\bar x_2,\bm y;\mu)\, \hat\sigma_{ik\to A}\, \hat\sigma_{jl\to B},$$2 drops out in the sum (Diehl et al., 2016).

This construction addresses a frequent misconception: the overlap between perturbative splitting in DPS and SPS loop corrections is not a higher-order technicality that can be ignored once $$\sigma_{\rm DPS} = \frac1{C}\, \sum_{i,j,k,l} \int dx_1\,dx_2\,d\bar x_1\,d\bar x_2 \int d^2\bm y\; F^{ij}(x_1,x_2,\bm y;\mu)\, F^{kl}(\bar x_1,\bar x_2,\bm y;\mu)\, \hat\sigma_{ik\to A}\, \hat\sigma_{jl\to B},$$3 is fitted. In the ultraviolet-consistent framework, it is a structural issue of the factorization formula itself.

4. Perturbative splitting, intrinsic two-parton structure, and model DPDs

At small transverse separation, the DPD is dominated by perturbative splitting. To first order in $$\sigma_{\rm DPS} = \frac1{C}\, \sum_{i,j,k,l} \int dx_1\,dx_2\,d\bar x_1\,d\bar x_2 \int d^2\bm y\; F^{ij}(x_1,x_2,\bm y;\mu)\, F^{kl}(\bar x_1,\bar x_2,\bm y;\mu)\, \hat\sigma_{ik\to A}\, \hat\sigma_{jl\to B},$$4,

$$\sigma_{\rm DPS} = \frac1{C}\, \sum_{i,j,k,l} \int dx_1\,dx_2\,d\bar x_1\,d\bar x_2 \int d^2\bm y\; F^{ij}(x_1,x_2,\bm y;\mu)\, F^{kl}(\bar x_1,\bar x_2,\bm y;\mu)\, \hat\sigma_{ik\to A}\, \hat\sigma_{jl\to B},$$5

so the short-distance singularity has a calculable perturbative origin (Diehl et al., 2016).

More generally, at a scale $$\sigma_{\rm DPS} = \frac1{C}\, \sum_{i,j,k,l} \int dx_1\,dx_2\,d\bar x_1\,d\bar x_2 \int d^2\bm y\; F^{ij}(x_1,x_2,\bm y;\mu)\, F^{kl}(\bar x_1,\bar x_2,\bm y;\mu)\, \hat\sigma_{ik\to A}\, \hat\sigma_{jl\to B},$$6 one writes

$$\sigma_{\rm DPS} = \frac1{C}\, \sum_{i,j,k,l} \int dx_1\,dx_2\,d\bar x_1\,d\bar x_2 \int d^2\bm y\; F^{ij}(x_1,x_2,\bm y;\mu)\, F^{kl}(\bar x_1,\bar x_2,\bm y;\mu)\, \hat\sigma_{ik\to A}\, \hat\sigma_{jl\to B},$$7

where $$\sigma_{\rm DPS} = \frac1{C}\, \sum_{i,j,k,l} \int dx_1\,dx_2\,d\bar x_1\,d\bar x_2 \int d^2\bm y\; F^{ij}(x_1,x_2,\bm y;\mu)\, F^{kl}(\bar x_1,\bar x_2,\bm y;\mu)\, \hat\sigma_{ik\to A}\, \hat\sigma_{jl\to B},$$8 parametrizes the nonperturbative intrinsic two-parton content. The sum is then evolved from $$\sigma_{\rm DPS} = \frac1{C}\, \sum_{i,j,k,l} \int dx_1\,dx_2\,d\bar x_1\,d\bar x_2 \int d^2\bm y\; F^{ij}(x_1,x_2,\bm y;\mu)\, F^{kl}(\bar x_1,\bar x_2,\bm y;\mu)\, \hat\sigma_{ik\to A}\, \hat\sigma_{jl\to B},$$9 to the hard scale $C=2$0 with the homogeneous double-DGLAP equation (Diehl et al., 2016).

For numerical studies, Diehl and Gaunt introduce model DPDs at the scale

$C=2$1

A convenient ansatz writes $C=2$2 with Gaussian transverse factors involving

$C=2$3

and defines a DPS “luminosity”

$C=2$4

which enters the regulated DPS master formula (Diehl et al., 2016).

The sample luminosities in the paper separate “1v1” (splitting–splitting), “2v1” (splitting–intrinsic), and “2v2” (intrinsic–intrinsic) contributions. This suggests that $C=2$5, while phenomenologically useful, compresses several physically distinct mechanisms into a single number.

5. Phenomenological realizations in proton–proton collisions

The practical relevance of DPS is illustrated by LHC studies in which the factorized ansatz is implemented in $C=2$6-factorization. In four-jet production, the SPS Born-level inclusive four-jet cross section is computed with transverse-momentum-dependent PDFs $C=2$7, off-shell incoming partons, and gauge-invariant off-shell matrix elements $C=2$8 evaluated numerically in AVHLIB. The scale choice is $C=2$9, real higher-order loop corrections are not included, and for the SPS component a constant $A=B$0-factor with $A=B$1 is applied, whereas no $A=B$2-factor is used for the DPS term (Maciula et al., 2016).

With optimized symmetric cuts,

$A=B$3

the DPS fraction $A=B$4 increases strongly in selected regions of phase space. At $A=B$5 TeV, representative values are $A=B$6 inclusively, $A=B$7 for forward jets with $A=B$8, $A=B$9 for $C=1$0, $C=1$1 for $C=1$2, $C=1$3 for $C=1$4, and $C=1$5 for $C=1$6 (Maciula et al., 2016). The DPS-sensitive observables are the maximum rapidity separation

$C=1$7

the azimuthal angle between the most forward and most backward jets,

$C=1$8

and the three-jet minimal azimuthal sum

$C=1$9

Regions of large rapidity separation and small azimuthal correlation are identified as the “sweet spots” where DPS constitutes $2\to2$0 of the sample at $2\to2$1 TeV (Maciula et al., 2016).

In double-charm production, each step of DPS is calculated in $2\to2$2-factorization using off-shell $2\to2$3 matrix elements and unintegrated gluon distribution functions. The Kimber–Martin–Ryskin prescription is found to give the best description and typically the largest DPS rates, whereas other UGDFs such as KMS and Jung–CCFM tend to reduce the cross section by up to a factor $2\to2$4 (Szczurek et al., 2013). In the LHCb fiducial region,

$2\to2$5

with a veto on opposite-charge pairs, the same-sign $2\to2$6 channel is DPS-dominated: $2\to2$7–$2\to2$8 with $2\to2$9 uncertainties, while the SPS contribution is negligible, $m=1/2$0. These values compare well to the measured $m=1/2$1 (Szczurek et al., 2013). The distinguishing correlations are a broad plateau in $m=1/2$2, a softer $m=1/2$3 spectrum, and an almost flat $m=1/2$4 distribution for DPS, versus small-$m=1/2$5 and back-to-back structure for SPS (Szczurek et al., 2013).

These proton–proton studies are not themselves ultraviolet subtraction formalisms. Rather, they exemplify the kinematic domains in which a refined treatment of DPDs, perturbative splitting, and subtraction is most consequential.

6. Nuclear generalizations and interpretive scope

DPS can also be used to probe impact-parameter dependent nuclear PDFs in proton–nucleus and nucleus–nucleus collisions. A key assumption is that all spatial dependence of the nPDFs enters only through the nuclear thickness function

$m=1/2$6

with $m=1/2$7 (Shao, 2020).

Under the usual factorization ansatz for double parton distributions, and neglecting inter-parton correlations beyond geometry, the DPS cross section in an $m=1/2$8–$m=1/2$9 collision is written in terms of the overlap function

$k_T$00

together with spatial nuclear modification factors $k_T$01 multiplying free-proton PDFs. If one sets $k_T$02, the result reduces to the purely geometric Glauber expression; the additional spatial-nPDF terms become important whenever $k_T$03 is large, as in small-$k_T$04 shadowing or moderate-$k_T$05 anti-shadowing and EMC regions at scales $k_T$06 (Shao, 2020).

The proposed strategy is to use minimum-bias DPS data, together with known single-inclusive nuclear modifications, to fit the profile function $k_T$07 that determines the impact-parameter dependence of the nPDFs. The practical steps are: choose a DPS-dominated final state with suppressed SPS background; measure the DPS yield in $k_T$08 and extract $k_T$09; measure the single-inclusive suppressions $k_T$10 and $k_T$11; measure the total $k_T$12 yield; and fit $k_T$13 using the DPS master formula and the known thickness function $k_T$14 (Shao, 2020). Because everything is integrated over $k_T$15, the method has the stated virtue of independence of Glauber modeling.

A plausible implication is that ultraviolet-consistent DPS and geometry-dependent nuclear DPS address complementary layers of the same problem: the former resolves the small-$k_T$16 overlap with SPS, whereas the latter organizes the transverse structure of the nuclear initial state.

7. Conceptual significance and limitations

The main significance of USD-DPS is formal rather than merely phenomenological. It preserves the DPD language, incorporates perturbative $k_T$17 splitting, and avoids double counting by construction through the combination

$k_T$18

(Diehl et al., 2016). This distinguishes it from approaches in which DPS is represented only through a process-independent $k_T$19.

At the same time, phenomenological studies indicate why simpler approximations remain useful. In four-jet production, the pocket-formula ansatz and $k_T$20-factorization already identify high-purity DPS regions through $k_T$21 and $k_T$22 selections (Maciula et al., 2016). In same-sign charm production, the factorized ansatz reproduces a fiducial cross section compatible with LHCb and explains the near-flat azimuthal correlations (Szczurek et al., 2013). For heavy-ion collisions, minimum-bias DPS observables can be recast as probes of spatially dependent nPDFs without reference to centrality modeling (Shao, 2020).

The limitation is that these phenomenological frameworks often neglect correlations or compress them into a single effective parameter. By contrast, the ultraviolet-consistent formulation shows explicitly that DPS contains splitting–splitting, splitting–intrinsic, and intrinsic–intrinsic components, and that only the full combination with overlap subtraction is free of the SPS/DPS double counting problem (Diehl et al., 2016). In that sense, USD-DPS is less a replacement for DPS phenomenology than the field-theoretic completion required when the ultraviolet region and perturbative splitting are treated with full consistency.