--- title: 'USD-DPS: Ultraviolet-Consistent Double Parton Scattering' url: https://www.emergentmind.com/topics/usd-dps type: topic --- # USD-DPS: Ultraviolet-Consistent Double Parton Scattering 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 [1611.01316]. 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 [1606.09067; 1306.6837]. ## 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 $C=1$ [1611.01316]. 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 $\sigma_{\rm eff}\simeq 15\,\mathrm{mb}$ [1606.09067]. In double-charm production, the fully differential cross section for $pp\to c\bar c\,c\bar c$ is likewise written as \[ d\sigma^{\rm DPS} = \frac{1}{2\,\sigma_{\rm eff}}\; d\sigma^{\rm SPS}(pp\to c\bar c;1)\, d\sigma^{\rm SPS}(pp\to c\bar c;2), \] again under the factorized ansatz [1306.6837]. 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 [1611.01316]. ## 2. Double parton distributions and their ultraviolet behavior The ultraviolet-consistent formulation is built around DPDs. For a hadron of momentum $p$, the unpolarized DPD for finding two partons of flavors $i,j$ with momentum fractions $x_1,x_2$ and transverse separation $\bm y$ is defined by the matrix element \[ F^{ij}(x_1,x_2,\bm y;\mu) = \int\frac{dz_1^-}{2\pi}\,\frac{dz_2^-}{2\pi}\, e^{\,i(x_1 p^+z_1^- + x_2 p^+ z_2^-)} \langle p|\mathcal O_i(z_1^-,\bm y)\,\mathcal O_j(z_2^-,\bm 0)|p\rangle_\mu , \] where each $\mathcal O_i$ is a renormalized twist-two field operator and $\mu$ is the $\overline{\rm MS}$ factorization/renormalization scale [1611.01316]. The crucial feature is the short-distance limit $\bm y\to \bm 0$. In that limit the operator product develops an ultraviolet divergence of order $1/y^2$. After $\overline{\rm MS}$ subtraction, the DPDs depend on $\mu$ and obey the homogeneous double-DGLAP equation \[ \frac{d}{d\ln\mu^2}\,F^{ij}(x_1,x_2,y;\mu) = \sum_k \int_{x_1}^{1-x_2}\frac{du}{u}\, P_{i\to k}\!\left(\frac{x_1}{u}\right)\, F^{kj}(u,x_2,y;\mu) +(i\leftrightarrow j), \] with the usual DGLAP splitting kernels without virtual pieces [1611.01316]. 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 \[ D_{ij}(x_1,x_2)\approx f_i(x_1)\,f_j(x_2) \] [1606.09067]. The latter is operationally useful, but the DPD formalism makes the transverse-separation dependence and the perturbative small-$y$ 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 $\Phi(\nu y)$ satisfying \[ \Phi(u)\to0\quad (u\to0),\qquad \Phi(u)\to1\quad (u\gg1), \] and defines the regulated DPS contribution as \[ \sigma_{\rm DPS}^{\rm reg} = \frac1{C} \sum_{i,j,k,l} \int dx_1\,dx_2\,d\bar x_1\,d\bar x_2 \int d^2\bm y\; [\Phi(\nu y)]^2\, F^{ij}(x_1,x_2,\bm y;\mu)\, F^{kl}(\bar x_1,\bar x_2,\bm y;\mu)\, \hat\sigma_{ik}\,\hat\sigma_{jl}. \] The total cross section is then organized as \[ \boxed{ \sigma_{\rm tot} = \sigma_{\rm DPS}^{\rm reg} - \sigma_{\rm sub} + \sigma_{\rm SPS}. } \] Here $\sigma_{\rm SPS}$ is the usual SPS cross section at the desired perturbative order, and $\sigma_{\rm sub}$ is obtained from the same regulated DPS integral but with each full DPD replaced by its fixed-order perturbative $1\to2$ approximation [1611.01316]. The subtraction term reproduces the small-$y$ limit of the DPS contribution and cancels the double counting with SPS diagrams of the same topology, referred to as “1v1” graphs. At small $y\sim 1/Q$, one has $\sigma_{\rm DPS}^{\rm reg}\simeq \sigma_{\rm sub}$, so that $\sigma_{\rm tot}\simeq \sigma_{\rm SPS}$. At large $y\gg 1/Q$, one finds $\sigma_{\rm SPS}\simeq \sigma_{\rm sub}$, so that $\sigma_{\rm tot}\simeq \sigma_{\rm DPS}^{\rm reg}$. The unphysical cutoff parameter $\nu\sim Q$ drops out in the sum [1611.01316]. 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 $\sigma_{\rm eff}$ 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 $\alpha_s$, \[ F^{ij}(x_1,x_2,\bm y;\mu) \longrightarrow F^{ij}_{\rm spl}(x_1,x_2,y;\mu) = \frac{1}{\pi y^2}\, \frac{\alpha_s(\mu)}{2\pi} \sum_k \frac{f_k(x_1+x_2;\mu)}{x_1+x_2}\, P_{k\to i}\!\left(\frac{x_1}{x_1+x_2}\right) +\mathcal O(\alpha_s^2), \] so the short-distance singularity has a calculable perturbative origin [1611.01316]. More generally, at a scale $\mu_y\sim 1/y$ one writes \[ F^{ij}(x_1,x_2,y;\mu_y) = F^{ij}_{\rm spl}(x_1,x_2,y;\mu_y) + F^{ij}_{\rm int}(x_1,x_2,y;\mu_y), \] where $F_{\rm int}$ parametrizes the nonperturbative intrinsic two-parton content. The sum is then evolved from $\mu_y$ to the hard scale $\mu$ with the homogeneous double-DGLAP equation [1611.01316]. For numerical studies, Diehl and Gaunt introduce model DPDs at the scale \[ \mu_y=\frac{b_0}{y^*},\qquad y^*=\frac{y}{\sqrt{1+(y/y_{\max})^2}},\qquad y_{\max}=0.5\,\mathrm{GeV}^{-1},\qquad b_0=2e^{-\gamma_E}. \] A convenient ansatz writes $F^{ij}=F^{ij}_{\rm spl}+F^{ij}_{\rm int}$ with Gaussian transverse factors involving \[ h_{ij}=h_i+h_j,\qquad h_g=2.33\,\mathrm{GeV}^{-2},\qquad h_q=h_{\bar q}=3.53\,\mathrm{GeV}^{-2}, \] and defines a DPS “luminosity” \[ \mathcal L^{ijkl} = \int d^2\bm y\;[\Phi(\nu y)]^2\, F^{ij}(x_1,x_2,\bm y;\mu)\, F^{kl}(\bar x_1,\bar x_2,\bm y;\mu), \] which enters the regulated DPS master formula [1611.01316]. The sample luminosities in the paper separate “1v1” (splitting–splitting), “2v1” (splitting–intrinsic), and “2v2” (intrinsic–intrinsic) contributions. This suggests that $\sigma_{\rm eff}$, 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 $k_T$-factorization. In four-jet production, the SPS Born-level inclusive four-jet cross section is computed with transverse-momentum-dependent PDFs $\mathcal F_i(x,k_T,\mu_F)$, off-shell incoming partons, and gauge-invariant off-shell matrix elements $\mathcal M(i^*,j^*\to 4\,{\rm part.})$ evaluated numerically in AVHLIB. The scale choice is $\mu_F=\mu_R=\tfrac12\,\hat H_T\equiv \tfrac12\sum_{l=1}^4 k_T^l$, real higher-order loop corrections are not included, and for the SPS component a constant $K$-factor with $K<1$ is applied, whereas no $K$-factor is used for the DPS term [1606.09067]. With optimized symmetric cuts, \[ p_{T,i}>20\,\mathrm{GeV},\qquad |\eta_i|<4.7,\qquad \Delta R_{ij}>0.5, \] the DPS fraction $f_{\rm DPS}=\sigma_{\rm DPS}/(\sigma_{\rm SPS}+\sigma_{\rm DPS})$ increases strongly in selected regions of phase space. At $13$ TeV, representative values are $15\%$ inclusively, $25\%$ for forward jets with $|\eta_{\rm jet}|>3.0$, $50\%$ for $\Delta Y>4$, $75\%$ for $\Delta Y>6$, $65\%$ for $\Delta\phi_{jj}<\pi/2$, and $45\%$ for $\Delta\varphi_{3j}^{\min}<2$ [1606.09067]. The DPS-sensitive observables are the maximum rapidity separation \[ \Delta Y=\max_{i\neq j}|\eta_i-\eta_j|, \] the azimuthal angle between the most forward and most backward jets, \[ \Delta\phi_{jj}=|\phi_i-\phi_j| \quad\text{for}\quad |\eta_i-\eta_j|=\Delta Y, \] and the three-jet minimal azimuthal sum \[ \Delta\varphi_{3j}^{\min} = \min_{i\neq j\neq k} \Bigl(|\phi_i-\phi_j|+|\phi_j-\phi_k|\Bigr). \] Regions of large rapidity separation and small azimuthal correlation are identified as the “sweet spots” where DPS constitutes $50\!-\!80\%$ of the sample at $13$ TeV [1606.09067]. In double-charm production, each step of DPS is calculated in $k_t$-factorization using off-shell $g^*g^*\to c\bar c$ 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 $\sim2$ [1306.6837]. In the LHCb fiducial region, \[ 2