# Proof of sum rules for double parton distributions in QCD

- 98 Downloads

## Abstract

Double hard scattering can play an important role for producing multiparticle final states in hadron-hadron collisions. The associated cross sections depend on double parton distributions, which at present are only weakly constrained by theory or measurements. A set of sum rules for these distributions has been proposed by Gaunt and Stirling some time ago. We give a proof for these sum rules at all orders in perturbation theory, including a detailed analysis of the renormalisation of ultraviolet divergences. As a by-product of our study, we obtain the form of the inhomogeneous evolution equation for double parton distributions at arbitrary perturbative order.

## 1 Introduction

In high-energy hadron hadron collisions, two or more partons in each incoming hadron may simultaneously take part in hard-scattering subprocesses. The importance of such multiparton interactions increases with the collision energy, because the density of partons in a hadron increases as their momentum fraction becomes smaller. Theoretical predictions with highest possible accuracy are required for a wide range of final states and kinematic regions in order to use the full potential of the LHC and of possible future hadron colliders. It is therefore highly desirable to develop a theory and phenomenology of multiparton interactions based on first principles in QCD. The most important contribution typically comes from double parton scattering (DPS), which is the subject of this work. To compute DPS cross sections one needs double parton distributions, which give the probability density for finding two specified partons inside a hadron. Our knowledge of these distributions is still very poor, given their dependence on several variables and the difficulty to separate the DPS contribution to a physical process from the contribution due to single hard scattering. Often one makes the simplest assumption that the two partons are entirely uncorrelated. This can at best be a first approximation. In the region of relatively large momentum fractions many studies in dynamical models find that correlations are actually strong (see [1] for a recent review).

In such a situation, theoretical constraints on double parton distributions (DPDs) are valuable. One type of constraint comes from the mechanism in which the two partons originate from the short-distance splitting of a single parton. This mechanism, which we will call \(1\rightarrow 2\) splitting, dominates DPDs at small distance between the two partons and can be expressed in terms of perturbative splitting functions and the usual single parton distributions (PDFs). The impact of \(1\rightarrow 2\) splitting on DPDs and DPS cross sections has been investigated from several points of view, see e.g. [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. A different constraint is provided by the DPD sum rules proposed by Gaunt and Stirling [14], which express the conservation of quark flavour and of momentum and nicely fit together with the interpretation of DPDs as probability densities. Attempts to construct DPDs satisfying these sum rules were made in [14, 15]. A study of \(W^+ W^+\) and \(W^- W^-\) production with the DPDs proposed in [14] can be found in [16].

DPDs depend on a renormalisation scale, just like ordinary PDFs. This dependence is described by a generalisation of the DGLAP evolution equations. Due to the \(1\rightarrow 2\) splitting mechanism, these equations contain an inhomogeneous term whose form at leading order (LO) has been extensively discussed in the literature [17, 18, 19, 20]. Gaunt and Stirling noted that if the sum rules they postulated are valid at some scale, then their validity is preserved by LO evolution to any other scale [14]. More detailed analyses of how the momentum sum rule remains valid in the presence of parton splitting can be found in [8, 21]. Still lacking is however a full proof that the sum rules hold at some starting scale. A derivation using the light-cone wave function representation is given in appendix C of [22]. This framework formalises the physical picture of the parton model and in particular allows one to keep track of kinematic and combinatorial factors. An explicit analysis of ultraviolet (UV) divergences and of the associated scale dependence is however missing in [22]. The same holds for rapidity divergences that are present in light-cone wave functions (but cancel in the DPDs appearing in the sum rules).

The aim of the present paper is to provide an explicit proof of the DPD sum rules in QCD. After defining DPDs and stating the sum rules in Sect. 2, we show in Sect. 3 how the sum rules arise at the first nontrivial order in a simple perturbative toy model. We recall the essentials of light-cone perturbation theory in Sect. 4 and then use this formalism in Sect. 5 to give an all-order proof of the sum rules for bare (i.e. unrenormalised) DPDs. In Sect. 6 we show how DPDs are renormalised and establish that the sum rules remain valid if UV divergences are subtracted in a suitable scheme. In Sect. 7 we explore the consequences of our analysis on the evolution of DPDs: we obtain the general form of the inhomogeneous term beyond LO (confirming the NLO result given in [20]), we derive sum rules for the associated evolution kernels, and we cross check that the sum rules are preserved by evolution at any order in \(\alpha _s\). Our findings are briefly summarised in Sect. 8.

## 2 Definitions and sum rules

*j*is a fermion, \(n_j = 1\) if it is a gluon, and \(n_j = -1\) if one considers scalar partons. We use light-cone coordinates \(v^\pm = (v^0 \pm v^3) /\sqrt{2}\) for any four-vector \(v^\mu \) and write its transverse part in boldface, \(\varvec{v} = (v^1, v^2)\). The twist-two operators read

*i*in (2) is to be summed over. We only consider colour-singlet DPDs here, so that the colour indices of the quark or gluon fields in (2) are implicit and to be summed over. For scalar partons we have

*B*and have an additional dependence on the renormalisation scale \(\mu \). We postpone a detailed discussion to Sect. 6 but already note here that the integral over \(\varvec{y}\) in (4) diverges at \(\varvec{y} = \varvec{0}\) in \(D=4\) dimensions and hence requires additional renormalisation. This divergence is due to the \(1\rightarrow 2\) splitting mechanism mentioned in the introduction and leads to the inhomogeneous term in the evolution equations for DPDs. It only appears for DPDs depending on the momentum \(\varvec{\Delta }\), but not for their counterparts depending on the distance \(\varvec{y}\), as was already noted in [2].

*j*in a hadron is denoted by \(N_{j_v}\), so that e.g. \(N_{u_v} = \int \mathrm {d}x\, f^{u_v}(x) = 2\) in a proton. Using the convention that \(\overline{\jmath }\) denotes a quark if

*j*is an antiquark, we have \(N_{\overline{u}_v} = 2\) in an antiproton.

## 3 Analysis of low-order graphs and its limitations

In this section, we show for a simple example how the DPD sum rules for bare distributions can be obtained from Feynman graphs in covariant perturbation theory. We consider all graphs at lowest order in \(\alpha _s\). In a second example, we exhibit the limitations of this covariant approach. This leads us to use light-cone perturbation theory in the following sections, where we formulate a proof that is valid at all orders in \(\alpha _s\). It is clear that neither covariant nor light-cone perturbation theory are suitable for actually computing parton distributions, which are non-perturbative objects. We must thus assume that general properties of Green functions – in our case the sum rules – remain valid beyond perturbation theory. This is similar to the spirit of perturbative proofs of factorisation in QCD [25, 26].

We use a toy model with scalar “quarks” of two flavours, *u* and *d*, which we take to be mass degenerate. The coupling between these quarks and the gluons is as required by gauge invariance. We consider a scalar “hadron” that has a pointlike coupling to *u* and \(\bar{d}\) and compute parton distributions at lowest order in perturbation theory.

### 3.1 Sum rules with a gluon PDF

*u*). They lead to identical expressions due to the symmetries of our model, viz. charge conjugation and the identical masses of the two quark flavours. The contributions of the graphs to the gluon PDF are

*g*denotes the strong coupling and

*m*the quark mass. For simplicity we set the coupling between the hadron and the quarks to 1. A factor of two for the diagrams with reversed arrows on the quark lines is included in these expressions. For brevity, we omit the subscript

*B*for bare distributions throughout this section.

*u*quarks, we first combine all contributions from graphs 1.1 and 1.2:

*u*and \(\bar{d}\) in our model, and in the second step we performed a change of variables in \(F^{g \bar{d}}\). The last equality is easily seen from the explicit expressions in (21) and (23). One readily derives the analogue of (27) for the contributions from graphs 2.1 and 2.2 and hence for the sum over all graphs. Since \(N_{u_v} = 1\) in our model, this shows that the number sum rule for

*u*quarks is fulfilled. In the same manner, one can show the number sum rule for \(\bar{d}\) quarks. For the momentum sum rule, we start again with the contributions from graphs 1.1 and 1.2:

We have seen that both sum rules hold individually for each PDF graph and the set of corresponding DPD graphs. This is already suggested by Fig. 2 and will remain true in the all-order proof in Sect. 5. However, a crucial step in the preceding derivation was that for each DPD graph we could perform the integrations over minus momenta in such a way that, after applying the theorem of residues, the momentum of parton 2 to the left and to the right of the final state cut was set on shell. This is not readily possible for other graphs, as we shall now see.

### 3.2 Sum rules with a quark PDF

We consider now the case in which parton 1 in the sum rules is a *u* quark (the expressions for \(\overline{d}\) quarks are identical for symmetry reasons). At order \(\alpha _s\) the number of graphs contributing to each PDF is much higher than in the previous section. There are the real emission graphs in Fig. 4, as well as graphs with a cut quark loop and a vertex or propagator correction to the left or to the right of the cut. For the graph in Fig. 4a and the corresponding graphs for DPDs, one can establish the validity of the sum rules exactly as in the previous section. This situation is however different for the remaining graphs of Fig. 4.

This situation bears some similarity to the proof of cancellation of Glauber gluons in single [25, 26] or double hard scattering [27]. For simple cases, this cancellation can be established in covariant perturbation theory, using the theorem of residues for integrations over minus momenta in a similar way as here, but for more complicated graphs, that method turns out to be cumbersome [27]. It is not clear whether a general proof could be given at all in that framework. In the next sections we will see that light-cone perturbation theory provides a powerful tool for proving the sum rules at all orders in the strong coupling, as it is for establishing Glauber gluon cancellation [25, 26, 27] .

## 4 Light-cone perturbation theory

Light-cone perturbation theory (LCPT, also called light-front perturbation theory) is quite similar to old-fashioned time ordered perturbation theory, with the difference that the vertices of a graph are ordered in “light-cone time” \(x^{+} = (x^{0} + x^{3}) /\sqrt{2}\) instead of ordinary time \(x^{0}\). The rules of LCPT can be derived from regular covariant perturbation theory by performing the integrations over all internal minus momenta, thus setting all internal lines on-shell. This is for instance shown in chapter 7.2.3 in [26], whose normalisation conventions we adopt in the following. Further discussion of LCPT can be found in [28, 29, 30, 31, 32, 33, 34, 35].

- 1.
Starting from a given Feynman graph, one assigns a light-cone time \(x_j^+\) to each vertex and considers all possible orderings of the \(x_j^+\). When drawing LCPT graphs, we follow the convention that \(x^+\) increases from left to right on the l.h.s. of the final state cut, while on the r.h.s. it increases from right to left.

- 2.
Coupling constants and vertex factors are the ones known from covariant perturbation theory. An exception are momentum dependent vertices, which are discussed below.

- 3.
Plus and transverse momentum components, \(k_l^+\) and \(\varvec{k}_l^{}\), of a line

*l*are conserved at the vertices. - 4.
For each propagating line

*l*in a graph, one has a factor \(1 / (2 k_l^+)\) together with a Heaviside step function \(\Theta (k_l^+)\) if the routing of \(k_l\) is from smaller to larger values of \(x^+\). - 5.Loop momenta \(\ell \) are integrated over their plus and transverse components with measure$$\begin{aligned} \int \frac{\mathrm {d}\ell ^+\mathrm {d}^{D-2}\varvec{\ell }}{(2\pi )^{D-1}}. \end{aligned}$$(32)
- 6.For each state
*i*between two vertices at consecutive light-cone times \(x_i^+\) and \(x_{i+1}^+\) there is a factorwhere \(P_i^-\) is the minus component of the sum of all external momenta entering the graph before \(x_i^+\). The sum is over the on-shell values of the minus components$$\begin{aligned} \frac{1}{P_i^--\sum _{l\in i}k_{l, \mathrm {os}}^-+i\epsilon }, \end{aligned}$$(33)of all lines$$\begin{aligned} k_{l, \mathrm {os}}^- = \frac{\varvec{k}_l^2+m_l^2}{2 k_l^+} \end{aligned}$$(34)*l*between \(x_i^+\) and \(x_{i+1}^+\).

*k*and \(k_{\mathop {os}}\) only differ in their minus components, one can rewrite the covariant fermion propagator asThe first term describes the propagation of a fermion and the second term the propagation of an antifermion, both with positive plus momentum as stated in point 4 above. The third term is independent of \(k^-\) and hence instantaneous. In LCPT graphs it gives rise to a vertical fermion line whose ends are associated with the same light-cone time \(x^+\). For the gluon propagator in light-cone gauge, we have

*n*is the light-like vector projecting on plus components, \(k n = k^+\). In analogy with (36), the first term in (37) can be further decomposed into parts with \(\Theta (k^+)\) or \(\Theta (-k^+)\). The last term in (37) is again instantaneous.

The decomposition (35) has to be made for all Feynman rules in covariant perturbation theory that have a dependence on minus momenta in the numerator, in particular for the three-gluon vertex. We will however only be concerned with gluon lines that are either internal to a graph or associated with the twist-two operator (2) for an observed parton. Any gluon vertex in a graph then has all its Lorentz indices contracted with a gluon propagator. The difference \((k - k_{\mathrm {os}})^\mu \propto n^\mu \) between a covariant and an on-shell momentum gives zero when contracted with \(G_g^{\mu \nu }\), so that in the three-gluon vertex we can simply replace *k* with \(k_{\mathrm {os}}\) in the numerator factor.

## 5 All-order proof for bare distributions

In this section we derive the DPD sum rules for bare, i.e. unrenormalised, distributions. We consider graphs in LCPT at arbitrary fixed order in \(\alpha _s\), using perturbation theory in the same spirit as discussed at the beginning of Sect. 3. Having established the sum rules for any fixed order in \(\alpha _s\), one immediately obtains their validity for the sum over all perturbative orders.

### 5.1 Representation of PDFs and DPDs in LCPT

*g*now specifies a cut LCPT graph, which has a definite light-cone time ordering of vertices. We label the independent momenta in a graph by \(k_i\), always starting with \(k_1\) for the observed parton. The total number of independent momenta is \(N(g)-1\), and \(M(g)-1\) of these go across the final state cut. We collectively write \(\{ x \}\) and \(\{ \varvec{k} \}\) for the set of light-cone momentum fraction and transverse momentum arguments of a graph.

*l*on which the operator for parton

*j*can be inserted, with \(\delta _{j, f(l)}\) selecting the parton type and \(\delta (x_l - z)\) the plus momentum fraction. Notice that both the plus momentum and the transverse momentum components of the two observed partons are equal on both sides of the final state cut. The former is always the case for a DPD, whilst the latter holds because for the sum rules we are considering the case \(\varvec{\Delta } = \varvec{0}\).

*j*are associated with the

*same*light-cone time \(x^+\). This is not surprising, because the two operators \(\mathcal {O}_{j_1}\) and \(\mathcal {O}_{j_2}\) in the definition (1) of a DPD are taken at the same light-cone time.

*I*, \(I'\) and

*F*are over intermediate states \(\xi \) either before or after the light-cone time of the operator insertion. Let us now take the sum over all graphs

*g*that differ only by the state \(F_A\) where the cut is made but are otherwise identical. Numbering the states in \(F(F_A)\) from 1 to

*N*, we can write

*H*and \(H'\) on each side of the final state cut.

*g*can be restricted to the time orderings just discussed. We integrated the DPD over its second momentum fraction

*z*with weight \(z^m\), as is required for the sum rules (where \(m=0\) or \(m=1\)).

### 5.2 Equality between PDF and DPD graphs

*g*and all partons

*l*that contribute in (48) and (49). This includes the statement that there is a unique correspondence between the LCPT graphs

*g*that contribute to \(f^{j_1}\) and those that contribute to \(F^{j_1 j}\), as indicated in Fig. 7.

We already showed in the last section that in a PDF graph the vertex *H* of the twist-two operator insertion must be later in light-cone time than the vertex \(V_2\) where the final state line *l* leaves the graph. Let us now show that there can be no instantaneous propagator attached to a twist-two operator insertion. The vertex \(V_1\) where parton 1 leaves the graph must then come before *H* as well, and in a DPD graph both \(V_1\) and \(V_2\) must come before *H*. In both PDF and DPD graphs, \(V_1\) may come before or after \(V_2\). Corresponding statements hold for \(\tilde{V}_1\), \(\tilde{V}_2\) and \(\tilde{H}\) to the right of the cut.

To show this, we identify the vertex rules (in momentum space) associated with the unpolarised twist-two operators (2). For quarks we get a factor \(\gamma ^+ /2\) that connects the Dirac indices of the parton to the left and the right of the cut, and for antiquark we get a factor \(- \gamma ^+ /2\). This gives zero when multiplied with the instantaneous part of the fermion propagator (36) because \((\gamma ^+)^2 = 0\). In light-cone gauge \(A^+ = 0\), the twist-two operator for gluons gives a factor \((k^+)^2\, \delta ^{i i'}\). Here \(k^+\) denotes the gluon plus momentum, which is equal on both sides of the cut. The index *i* (\(i'\)) denotes the gluon polarisation index to the left (right) of the cut and is restricted to be transverse. This gives zero when contracted with the instantaneous part of the gluon propagator (37).

*l*selected by the operator for parton 2 in the DPD and the factor associated with the corresponding final state line in the PDF. In the DPD, the momenta

*k*and \(k'\) carried by

*l*to the left and the right of the cut have the same plus and transverse components, as already noted after (40). Their on-shell values are hence identical as well, i.e.

*k*is equal to the momentum of the corresponding final state line in the PDF. If

*l*is a quark, we getin the DPD graph. Here a factor Open image in new window appears for each propagating line (see (36)), the factor \(\gamma ^+/2\) comes from the twist-two quark operator, and the factor 2 on the l.h.s. is taken from the l.h.s. of (50). On the r.h.s. of (52) we recognise the factor for the final state line

*l*in the PDF graph, which proves (50) for quarks. The same argument is easily repeated for antiquarks. If

*l*is a gluon, we get

### 5.3 Number sum rule

*l*on the l.h.s. gives the number of partons with flavour \(j_2\) crossing the final state cut in the PDF graph, minus the corresponding number of partons with flavour \(\overline{\jmath _2}\). If the observed parton \(j_1\) in the PDF has flavour \(j_2\) (\(\overline{\jmath _2}\)), that number is increased (decreased) by 1. The result is obviously equal to the difference of partons with flavour \(j_2\) and those with flavour \(\overline{\jmath _2}\) in the hadron, which is indeed \(N_{j_{2,v}}\). The number sum rule is thus verified.

### 5.4 Momentum sum rule

## 6 Renormalisation

Up to now, we have essentially shown that the parton model interpretation of the DPD sum rules is reflected in the graphs that represent single or double parton distributions in QCD, where quark number and parton momentum are conserved quantities. However, these graphs have short-distance singularities that must be renormalised. It is known that the literal interpretation of PDFs as probability densities can be invalidated by renormalisation. This is most obvious for the positivity of the distributions, because one has to *subtract* terms that become infinite if the UV regulator is removed. It is hence important to establish whether the sum rules retain their validity after renormalisation in a specified scheme. We will show that this is indeed the case for the \(\overline{\mathrm {MS}}\) scheme. As in the previous section, our arguments are valid at arbitrary order in \(\alpha _s\).

### 6.1 Convolution integrals

*D*be a function of two momentum fractions, while \(A,\,B,\,C\) are functions of one momentum fraction only. We define

*A*, is a PDF we have \(A(x)=0\) if \(x<0\) or \(x>1\), and when a two-variable function is a DPD we have \(D(x_{1},x_{2})=0\) if \(x_{1}<0\) or \(x_{2}<0\) or \(x_{1}+x_{2}>1\). The same properties hold also for the associated evolution kernels and renormalisation factors (which may include delta and plus distributions at the endpoints of their support). The convolution with respect to the second argument of

*D*is introduced in analogy to (57) and denoted by \(\underset{2}{\otimes }\). We then have a combined convolution

### 6.2 Renormalisation of DPDs

*Z*factor for each parton, i.e.

*y*and produces an additional short-distance singularity, which is due to the \(1\rightarrow 2\) splitting mechanism. As discussed in [13, 36], this mechanism dominates the DPD at small

*y*and gives

*y*via powers of \((y \mu )^{2\varepsilon } \alpha _s(\mu )\). The Fourier transform of (69) w.r.t. the transverse distance

*y*has a logarithmic singularity in \(D=4\) dimensions and gives a simple pole \(1/\varepsilon \) for \(D = 4 - 2\varepsilon \). Notice the difference between this and the UV divergences in the twist-two operators, which lead to higher powers of \(1/\varepsilon \) with increasing powers of \(\alpha _s\). The splitting singularity in \(F_{B}(\Delta )\) is renormalised additively with a renormalisation factor \(Z_{i_{1} i_{2}, j}(x_{1},x_{2};\mu , \epsilon )\) depending on two momentum fractions:

**Implementation of the**\(\overline{\mathrm {MS}}\)

**scheme**The derivations in the remainder of this paper will be significantly simplified by using a particular implementation of the \(\overline{\mathrm {MS}}\) renormalisation scheme. We start with the definition of this scheme given in Sect. 3.2.6 of [26], where the bare and the renormalised couplings are related by

*M*(

*n*) of the highest pole depends on the quantity being renormalised. The tree-level value \(Z^{(0)}\) is not important here. The coefficients \(B_{n m}\) and \(Z_{n m}\) are independent of \(\varepsilon \), but

*Z*and thus \(Z^{(0)}\) and \(Z_{n m}\) may depend on additional variables like momentum fractions. The standard choice for the factor \(S_{\varepsilon }\) is \(S_{\varepsilon }= (4\pi e^{-\gamma })^{\varepsilon }\), where \(\gamma \) is the Euler–Mascheroni constant. The alternative \(S_{\varepsilon } = (4\pi )^{\varepsilon } / \Gamma (1-\varepsilon )\) was proposed in [26], and the following arguments are valid in both cases. The counterterms in (77) and (78) contain finite parts that result from multiplying powers of \(1/\epsilon \) with powers of \(S_\epsilon \).

*R*is

For the standard choice \(S_{\varepsilon } = (4\pi e^{-\gamma })^{\varepsilon }\), the relation (79) takes the form of a minimal subtraction scheme with \(\overline{\mu }^2 = \mu ^2 / (S_{\varepsilon })^{1/\varepsilon } = \mu ^2\, e^{\gamma } /(4\pi )\). This way of implementing \(\overline{\mathrm {MS}}\) subtraction is in fact well known in the literature. Our above argument shows that one can also use the implementation of (79) and (80) for different choices of \(S_{\varepsilon }\). In the remainder of this work, we will use this implementation, omitting the primes on \(\alpha _s\), \(B_{n m}\) and \(Z_{n m}\).

### 6.3 Number sum rule

*i*is a gluon we define \(\overline{\imath } = i\), so that the above relation is valid for all parton labels. For the first term in (89) we thus have

### 6.4 Momentum sum rule

*j*, as shown in Sect. 8.6 of [26]. Using this and the momentum sum rule for bare DPDs, we have

*x*and again using (64), we see that \(x Z^{-1}_{i,j}(x)\) is the inverse of \(x Z^{}_{i,j}(x)\) w.r.t. Mellin convolution and matrix multiplication. This allows us to write

## 7 DPD evolution and its consequences

*D*dimensions implies that there are only terms of order \(\epsilon ^{n}\) with \(n \le 0\) on the l.h.s. of (105) and (108).

## 8 Conclusion

The sum rules proposed by Gaunt and Stirling [14] present one of the few general constraints on double parton distributions that are currently known. This has motivated us to give a detailed proof for them in QCD. We saw in Sect. 3 that an analysis of Feynman graphs in covariant perturbation theory yields the sum rules in simple cases but quickly becomes complicated for certain types of graphs, which makes this technique unsuitable for a general proof. Instead, we used light-cone perturbation theory in Sect. 5 to show the validity of the DPD sum rules for bare, i.e. unrenormalised, distributions at any order in the coupling. In Sect. 6 we analysed the renormalisation of DPDs and showed that in the \(\overline{\mathrm {MS}}\) scheme this procedure yields renormalised distributions that again satisfy the sum rules.

As by-products of our analysis, we derived in Sect. 7 an all-order evolution equation for DPDs in momentum space and obtained sum rules for the kernel \(P_{i_1 i_2, j}\) that appears in the inhomogeneous term of that equation. These sum rules can be used to verify explicitly that the DPD sum rules are consistent with evolution at any order in perturbation theory. They will also provide valuable cross-checks for the calculation of \(P_{i_1 i_2, j}\) beyond the known order \(\alpha _s\). Introducing a compact notation and deriving a number of relations for convolution integrals in one or two variables (Sect. 6.1) allowed us to keep the computations for renormalised DPDs reasonably short and transparent.

To construct DPD models that fulfil the sum rules – exactly or approximately – is by far not an easy task [14, 15]. The results of the present work provides an additional motivation for further efforts in this direction. In a forthcoming numerical study [40] we will show how the sum rules can be used to improve existing models for DPDs in position space.

## Notes

### Acknowledgements

We gratefully acknowledge discussions with Jonathan Gaunt.

## References

- 1.T. Kasemets, S. Scopetta, Adv. Ser. Direct. High Energy Phys.
**29**, 49 (2018). arXiv:1712.02884 CrossRefGoogle Scholar - 2.
- 3.M.G. Ryskin, A.M. Snigirev, Phys. Rev. D
**83**, 114047 (2011). arXiv:1103.3495 ADSCrossRefGoogle Scholar - 4.B. Blok, Yu. Dokshitser, L. Frankfurt, M. Strikman, Eur. Phys. J. C
**72**, 1963 (2012). arXiv:1106.5533 ADSCrossRefGoogle Scholar - 5.A.V. Manohar, W.J. Waalewijn, Phys. Lett. B
**713**, 196 (2012). arXiv:1202.5034 ADSCrossRefGoogle Scholar - 6.M.G. Ryskin, A.M. Snigirev, Phys. Rev. D
**86**, 014018 (2012). arXiv:1203.2330 ADSCrossRefGoogle Scholar - 7.
- 8.B. Blok, Yu. Dokshitzer, L. Frankfurt, M. Strikman, Eur. Phys. J. C
**74**, 2926 (2014). arXiv:1306.3763 ADSCrossRefGoogle Scholar - 9.A.M. Snigirev, N.A. Snigireva, G.M. Zinovjev, Phys. Rev. D
**90**, 014015 (2014). arXiv:1403.6947 ADSCrossRefGoogle Scholar - 10.K. Golec-Biernat, E. Lewandowska, Phys. Rev. D
**90**, 094032 (2014). arXiv:1407.4038 ADSCrossRefGoogle Scholar - 11.J.R. Gaunt, R. Maciula, A. Szczurek, Phys. Rev. D
**90**, 054017 (2014). arXiv:1407.5821 ADSCrossRefGoogle Scholar - 12.M. Rinaldi, S. Scopetta, M.C. Traini, V. Vento, JHEP
**10**, 063 (2016). arXiv:1608.02521 ADSCrossRefGoogle Scholar - 13.
- 14.
- 15.K. Golec-Biernat, E. Lewandowska, M. Serino, Z. Snyder, A.M. Stasto, Phys. Lett. B
**750**, 559 (2015). arXiv:1507.08583 ADSCrossRefGoogle Scholar - 16.J.R. Gaunt, C.-H. Kom, A. Kulesza, W.J. Stirling, Eur. Phys. J. C
**69**, 53 (2010). arXiv:1003.3953 ADSCrossRefGoogle Scholar - 17.R. Kirschner, Phys. Lett. B
**84**, 266 (1979)ADSCrossRefGoogle Scholar - 18.V.P. Shelest, A.M. Snigirev, G.M. Zinovev, Phys. Lett. B
**113**, 325 (1982)ADSCrossRefGoogle Scholar - 19.
- 20.
- 21.
- 22.Gaunt, J.R., Ph.D. thesis, University of Cambridge (2012). https://doi.org/10.17863/CAM.16589
- 23.M.G.A. Buffing, M. Diehl, T. Kasemets, JHEP
**01**, 044 (2018). arXiv:1708.03528 ADSCrossRefGoogle Scholar - 24.
- 25.J.C. Collins, D.E. Soper, G.F. Sterman, Nucl. Phys. B
**308**, 833 (1988)ADSCrossRefGoogle Scholar - 26.J. Collins, Foundations of perturbative QCD. Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol.
**32**, 1 (2011)MathSciNetGoogle Scholar - 27.M. Diehl, J.R. Gaunt, D. Ostermeier, P. Plößl, A. Schäfer, JHEP
**01**, 076 (2016). arXiv:1510.08696 ADSCrossRefGoogle Scholar - 28.S.-J. Chang, S.-K. Ma, Phys. Rev.
**180**, 1506 (1969)ADSCrossRefGoogle Scholar - 29.J.B. Kogut, D.E. Soper, Phys. Rev. D
**1**, 2901 (1970)ADSCrossRefGoogle Scholar - 30.T.-M. Yan, Phys. Rev. D
**7**, 1780 (1973)ADSCrossRefGoogle Scholar - 31.S.J. Brodsky, G.P. Lepage, Adv. Ser. Direct. High Energy Phys.
**5**, 93 (1989)ADSCrossRefGoogle Scholar - 32.W.-M. Zhang, A. Harindranath, Phys. Rev. D
**48**, 4881 (1993)ADSCrossRefGoogle Scholar - 33.N.E. Ligterink, B.L.G. Bakker, Phys. Rev. D
**52**, 5954 (1995). arXiv:hep-ph/9412315 ADSCrossRefGoogle Scholar - 34.Y.V. Kovchegov, E. Levin, Quantum Chromodynamics at High Energy, vol. 33 (Cambridge University Press, Cambridge, 2012). https://doi.org/10.1017/CBO9781139022187
- 35.J. Collins. arxiv:1801.03960
- 36.
- 37.J. Kalinowski, K. Konishi, T.R. Taylor, Nucl. Phys. B
**181**, 221 (1981)ADSCrossRefGoogle Scholar - 38.J. Kalinowski, K. Konishi, P.N. Scharbach, T.R. Taylor, Nucl. Phys. B
**181**, 253 (1981)ADSCrossRefGoogle Scholar - 39.M. Diehl, J. R. Gaunt, P. Plößl, A. Schäfer. arxiv:1902.08019
- 40.M. Diehl, J.R. Gaunt, D. Lang, P. Plößl, A. Schäfer
**(in preparation)**Google Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}