Transverse momentum in double parton scattering: factorisation, evolution and matching

We give a description of double parton scattering with measured transverse momenta in the final state, extending the formalism for factorisation and resummation developed by Collins, Soper and Sterman for the production of colourless particles. After a detailed analysis of their colour structure, we derive and solve evolution equations in rapidity and renormalisation scale for the relevant soft factors and double parton distributions. We show how in the perturbative regime, transverse momentum dependent double parton distributions can be expressed in terms of simpler nonperturbative quantities and compute several of the corresponding perturbative kernels at one-loop accuracy. We then show how the coherent sum of single and double parton scattering can be simplified for perturbatively large transverse momenta, and we discuss to which order resummation can be performed with presently available results. As an auxiliary result, we derive a simple form for the square root factor in the Collins construction of transverse momentum dependent parton distributions.


Colour structure of DPDs
The analysis of the colour structure of two-gluon DPDs is incomplete in the original manuscript and needs to be extended. Consider a quantity with four adjoint indices that transforms like an overall colour singlet. One can first couple the indices pairwise to one of the combinations R = 1, A, S, 10, 10, 27. An overall singlet is then obtained if the colour representations two index pairs are in conjugate colour representations. This includes the cases where one pair is in the antisymmetric octet and the other in the symmetric octet. These cases were omitted in [1], see equations (6.a), (6.b) and (7) in that work. This mistake was repeated in equation (2.121) of [2].
In appendix A of [3] it was shown that the two-gluon distributions corresponding to R = 10 and 10 in the previous construction are equal. The proof given there makes use of the colour decompositions given in [1,2] and breaks down once the missing combinations with one symmetric and one antisymmetric octet are included. We hence must extend the colour decompositions in the present work by adding the mixed octet combinations and restoring separate quantities for R = 10 and 10.

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In the list of projection operators P R in (4.3), we hence replace P D = P 10 + P 10 with the separate projectors for the decuplet and antidecuplet. These projectors agree with P 10 aa bb and P 10 aa bb in equation (1.19) of [4]. They also agree with the corresponding projectors in equation (12) of [1], provided that we identify a b |P R |ab = P ab a b R in analogy to equation (2) of that paper. Note that the form of these projectors is correct for SU(N ) and not restricted to N = 3 [4]. The symmetry relation (4.7) is not valid for decuplet projectors and must be corrected to read P r s R = P s r R , (1.2) where R is the conjugate representation of R. It is understood that R = 10 for R = 10 and that R = R for the singlet, all octets, and for R = 27. The projector relation (4.8) remains valid. For the mixed gluon octet channels, we introduce the tensors which satisfy the relations They are not projectors, since they do not satisfy (4.8).
We also note that (4.8) is incorrect for the octet tensors with mixed fundamental and adjoint indices: the contraction P aa ii A P i i bb S is not zero but equal to P aa bb AS . The octet tensors in (4.4) are therefore not projectors either. Fortunately, the incorrect instances of (4.8) were not used in our calculations.
To have a uniform notation for calculations, we write the projectors as where the double indices r and s are both in the fundamental or both in the adjoint representation. We furthermore write P i a for the tensors in (4.4), with R 1 = 1, 8 for the fundamental indices and R 2 = 1, A, S for the adjoint ones. We then have the simple rules The corrected form of (4.9) hence reads where the definition (4.10) of the multiplicity m(R) remains valid as it stands. 1

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The tensor P R 1 R 2 couples two double-index objects in the representations R 1 and R 2 to an overall singlet. Hence, the two representations must have the same multiplicity. Formally one can impose this by defining P RR = 0 for m(R) = m(R ). One should then also define P 1010 = P 10 10 = 0.
An object with four colour indices that transforms as an overall colour singlet can now be represented in terms of the tensors P RR . The corrected form of the decomposition (4.11) reads (1.8) A simple example illustrating the necessity to include RR = AS in the sum over representation pairs is M aa bb = f aa c d bb c . The relation (4.12) correctly reads (1.9) Note that, for each contracted double index, the corresponding representations in M 1 and M 2 are conjugate to each other as a consequence of (1.7). Various derivations and results in the paper must be adjusted to reflect the change from one to two representation labels and the conjugation of representations (which is trivial for all representations apart from the decuplet and antidecuplet). We present the necessary changes in the order in which they appear in the manuscript.
The corrected form for the colour decomposition (4.13) of a DPD is where instead of ε a (R) from (4.15) we now use (1.12) With our new notation, the definition (4.18) of colour projected twist-two operators reads and (4.19) must be modified to (1.14) (1.15)

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The same holds for the colour space matrices S −1 , s, s −1 , and K. Together with (1.9), this leads to a modification of the multiplication rule for matrices in representation space, which we define in general by where the projections of M 1 and M 2 on representations are defined as in (1.15). The correct form of (4.21) then reads and corresponding changes are to be made in (4.22) and (4.23). Again as a consequence of (1.9), the expression (4.24) for the cross section is modified to We now discuss the necessary adjustments to the symmetry properties derived in section 4.3. Using that P aa bb one finds that (4.27) and (4.28) must be corrected to (1.20) An analogous correction is to be made for (4.29). The result that the soft matrix in representation space is real valued remains true, so that the complex conjugation in (1.20) and (1.21) can be omitted. Doing this in the first line of (1.21), one finds that 10 10,10 10 S gg = 10 10,10 10 S gg , 10 10,10 10 S gg = 10 10,10 10 S gg , which is expected because charge conjugation interchanges the representations 10 and 10. Furthermore, one finds that R 1 R 2 ,R 1 R 2 S gg = 0 if there is an odd number of representations A and no representation is equal to 10 or 10. This reflects the fact that the antisymmetric and the symmetric gluon octets transform under charge conjugation with a relative minus sign. Corresponding results hold for the Collins-Soper Kernel K gg . Using that ε(R 1 ) ε(R 2 )P aa bb is real valued in all colour channels except for the decuplets, for which one has 10 10 F gg (x i , y) = 10 10 F gg (x i , y) * .

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The soft factor for collinear DPDs. The relations (4.39) and (4.40) become so that (4.41) generalises to Note that these relations do not involve conjugation of representation labels. As a corollary, one obtains and its analogue for the second parton. Using these results and the rules in (1.6), one can generalise equations (4.41) to (4.48) to all valid combinations of representations. We find that for the collinear soft factor the first two representations must be conjugate to the second two, and that for singlet and octet channels, the soft factors are equal for the quark and gluon representations. We thus get unity in the singlet sector, 1 S = 11,11 S qq = 11,11 S qg = 11,11 S gq = 11,11 S gg = 1 , where for the decuplet sector we used (1.22). Analogous results hold for the Collins-Soper Kernel J for collinear DPDs. On the left-hand side of (1.28), we have labelled the soft factors by one representative of the different labels on the right-hand sides. Taking a different representative, one has 8 S = A S = S S and 10 S = 10 S. The extended soft factor can be expressed in terms of the one for colour singlet production, and in generalisation of (4.43), we have where the combination of tensors on the left and Kronecker deltas on the right implies that all representations must have the same multiplicity. This can be used to correct the JHEP07(2021)046 argument in section 4.5 for the production of final states with net colour. In the definition (4.51), the order of colour indices in H ab must be interchanged (see figure 3), so that Correcting (4.52) to include all relevant colour channels, we obtain as a replacement of (4.53) the final expression with hard-scattering factors Evolution of DPDs. The colour structure of DPDF and DTMD evolution equations and their solutions must be adjusted for the number of representation labels and the difference between a representation and its conjugate. The Collins-Soper kernel for DTMDs must have four representation labels, as does the soft factor from which it is derived. The Collins-Soper equation (5.11) then becomes with a kernel satisfying and a corresponding equation for µ 2 . The matrix exponential in (5.14) is now defined as the exponential series with the matrix multiplication in (1.16). Using the analogue of (1.21) for K a 1 a 2 , the relation (5.18) becomes where we have abbreviated the logarithm by L. The matrix M a 1 a 2 is defined by the updated version of (5.16),

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where 11 K a is the Collins-Soper kernel for a single-parton TMD. The corrected form of (5.19) will be given below.
The kernel for the rapidity evolution of DPDFs derives from the collinear soft factor and hence is labelled by a single representation. We thus have as update of (5.22), where instead of R 1 J we could also write R 2 J. By contrast, the DGLAP kernels for DPDFs carry two colour indices, and the evolution equation for the first parton scale has the structure (1.38) The Collins-Soper equation (5.25) for the evolution kernel should be modified to and corresponding changes are to be made in (5.26). As a consequence of charge conjugation invariance, one has 10 10 P gg = 10 10 P gg , instead of (6.4). The matching of the twist-two operators at small z reads

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one obtains the corrected form of (6.6) as for the colour projected operators defined in (1.13). The possibility of quark-gluon mixing requires two colour labels R and R for C us,ab , in contrast to C S,a . Adjusting the following steps in the manuscript and defining , (1.46) one finds that the correct colour structure for the matching of a DTMD onto a DPDF is where for brevity, we only give the transverse position arguments. The full version is given in (2.11) below. Due to charge conjugation invariance, one has 10 10 C gg = 10 10 C gg , and analogous relations for C us,gg . The definitions of the kernel R K a for the rapidity dependence of R C S,a and of the associated anomalous dimension R γ a remain as given in (6.14) and (6.18). Equation (6.15) hence becomes whereas the limiting expression (6.16) for small z i should be replaced with Charge conjugation invariance implies 10 C S,g = 10 C S,g and hence 10 K g = 10 K g . The representation labels in (6.20) and (6.21) should be adapted as in (1.50) and (1.49), respectively. The corrected forms of (6.19), (6.22) and (6.24) will be given below. The functions g F and g K introduced in section 6.2.3 must have separate colour labels for the two partons, corresponding to the functions on the r.h.s. of their definitions in (6.31). The result (6.36) for the expansion of the soft factor at small z 1 , z 2 and y correctly reads a 1 a 2 (z i , y; µ i , Y ) . (1.51) The kernel T a 0 →a 1 a 2 for parton-level splitting in (6.38) and (6.42) should have separate labels R 1 , R 2 for a 1 , a 2 , as do the DPDs. The same holds for the twist-four function G a 1 a 2 in (6.39) and (6.42).

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In section 7.3.1, one should replace (7.22) with , y, z) , y, z) whilst the second equation in (7.23) remains valid as it stands. The adjustment of the remaining equations in that section is straightforward.
One-loop kernels. The kernel matrix (7.12) for Collins-Soper evolution of two-gluon DTMDs should be extended to include all colour channels. We refrain from giving the corresponding expressions here. In the limit |z 1 |, |z 2 | |y|, the elements of this matrix can be expressed in terms of the kernels R K g and R J g according to (1.50). We find that 10 K g = 0 and 10 J = D J with D J given in (7.17).
The colour factors c ab in (7.87) and c a 0 →a 1 a 2 in (7.97) now depend on two representation labels (R 1 R 2 ) referring to the parton pairs (ab) or (a 1 a 2 ). In addition to the results given in the manuscript, we find that both factors are zero for R 1 R 2 = 10 10 and 10 10. This holds for general number of colours N . Both factors are also zero for R 1 R 2 = AS and SA due to charge conjugation invariance.

Definition and rescaling of the rapidity parameter ζ
The definition (5.8) of the rapidity parameter in DPDs refers to the plus-momenta x 1 p + and x 2 p + of the two extracted partons, in generalisation of the definition (3.50) for singleparton TMDs. This leads to a problem in convolutions of DPDFs, which was overlooked and led to a number of mistakes in the original manuscript. The problem concerns the evolution equation (5.24) of DPDFs and the matching equation (6.11) of DTMDs onto DPDFs. The DPDFs on the right-hand-sides of these equations are to be taken at a fixed central rapidity Y C . This is explicit in the steps leading to (6.11), and a corresponding derivation can be given for (5.24). This implies that in integrals over a momentum fraction of a DPD, the variable ζ does not remain constant but must change such that Y C remains constant. Such a rescaling in all convolution integrals is possible, but we find it rather awkward. To avoid it, we express the rapidity dependence of DPDFs by the parameter which refers to the proton momentum. We use this parameter also for DTMDs, given their close connection with DPDFs. The handling of the rapidity dependence in DTMDs is then JHEP07(2021)046 different from the one for single-parton TMDs in the modern literature, notably in [5]. We note, however, that (2.1) corresponds to the definition of the ζ parameter in the original work of Collins and Soper [6]. The parameter ζ p is then to be used instead of ζ from (5.8) as argument for DTMDs and DPDFs throughout the paper. In the following, we point out equations that change in a non-trivial way.
Defining the analogue of (2.1) for the left-moving proton as the product of rapidity parameters is fixed to where s = 2p +p− is the squared c.m. energy of the proton-proton collision (neglecting proton mass corrections). In terms of the invariant masses produced by the two hard scatters, one has s = Q 2 1 (x 1x1 ) = Q 2 2 (x 2x2 ). The definitions (2.1) and (2.2) refer to the Collins regulator for rapidity divergences. If one works in a different scheme, such as the δ regulator discussed in appendix B, the replacement of ζ by ζ p = ζ/(x 1 x 2 ) as argument of DPDs must be implemented accordingly.

DTMD evolution.
Equations that involve DTMDs but no DPDFs are correct in the original manuscript and just need to be rewritten to reflect the change of argument in the distributions. In terms of the new rapidity parameter, the renormalisation group equation (5.7) for DTMDs reads and in analogy for the derivative w.r.t. log µ 2 . The rescaled rapidity parameters x 1 ζ/x 2 and x 2 ζ/x 1 in (5.13) should hence be replaced with x 2 1 ζ p and x 2 2 ζ p , respectively. Some attention is needed when choosing initial conditions for Collins-Soper evolution. In the matching relations for small y discussed in section 6.3, the rapidity parameter dependence of the DTMD arises from a short-distance matching kernel, which does not know about the proton momentum and hence can depend on x 1 x 2 ζ p but not on ζ p . Taking that kernel at a fixed scale ζ 0 so as to minimise corrections from higher orders thus gives the DTMD at ζ p = ζ 0 /(x 1 x 2 ). Assuming that a corresponding initial condition is also useful at large y, we therefore rewrite (5.17) in the form

DPDF evolution. The corrected DGLAP equation (5.24) for DPDFs is
where the rapidity argument of the evolution kernel P is rescaled in the same way as the anomalous dimensions in (2.4), thus referring to the plus-momentum of the parton associated with the renormalization scale µ 1 . An equation analogous to (2.7) holds of course for the derivative w.r.t. log µ 2 .
With the corrected DGLAP equations for DPDFs, equations (5.27) to (5.29) must be modified as well. A correct way to separate the ζ dependence is where F obeys the evolution equation

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and its analogue for µ 2 , with the initial condition (2.10) In (2.8) it is essential that ζ 0 does not depend on x 1 or x 2 , so that the initial condition (2.10) is taken at constant ζ p = ζ 0 . In the second line of that equation, we have indicated that this can of course be related to an initial condition at ζ p = ζ 0 /(x 1 x 2 ) by another step of Collins-Soper evolution. Reflecting these changes, the last sentence in section 7.3.5 should be modified as follows: "One can thus adapt numerical code for the one-loop evolution of colour singlet DPDs by rescaling the evolution kernels and adding the term with R 1 γ J from (2.9)." Matching DTMDs to DPDFs. The corrected master formula (6.11) for the matching of a DTMD at large y reads (2.11) Corresponding substitutions for the rescaled rapidity parameters must be made in (6.21) and (G.1). Moreover, equation ( which involves a special rescaling of the rapidity parameter in the evolution kernel P . The corrected form of (6.22) is This is quite similar to (2.5), but now the initial condition for the rapidity parameter depends on the momentum fractions x 1 and x 2 of the DPD under the convolution integrals JHEP07(2021)046 with the matching kernels C. In (6.23) one should replace ζ with ζ p and ζ 0 with ζ 0 /(x 1 x 2 ) in the function arguments and in the explicit logarithm. In the cross-section level result (6.24), one should then replace the last line by where as in (2.13) the constant ζ 0 is rescaled by the momentum fractions of the DPDs under the convolution integrals with the matching kernels. Given the above modifications regarding F a 1 a 2 , µ 0 ,ζ 0 , we do not see a corrected form of (6.27) that would be particularly useful, so this equation and its discussion should be discarded.