Combining TMD factorization and collinear factorization

This talk was based on Collins et al. (2016), and provides a summary of the results there. More detailed references to earlier literature can be found in that paper.

The issue addressed is the matching of transverse-momentum-dependent (TMD) and collinear factorization for processes like the Drell-Yan process, with both a hard scale $Q$ and a separate measured transverse momentum $q_{\text{T}}$. TMD factorization applies when $q_{\text{T}}\ll Q$, and its accuracy degrades as $q_{\text{T}}$ increases towards $Q$. It involves TMD parton densities (pdfs) $f_{\text{parton}/\text{hadron}}(x,k_{\text{T}})$, and in more general process also TMD fragmentation functions.

In contrast, collinear factorization applies at large $q_{\text{T}}$ (i.e., of order $Q$), and it also applies to the cross section integrated over all $q_{\text{T}}$ (and hence for the integral over $q_{\text{T}}$ up to a maximum of order $Q$). Collinear factorization involves “collinear pdfs” $f_{\text{parton}/\text{hadron}}(x)$. Its accuracy degrades as $q_{\text{T}}$ decreases, and collinear factorization by itself provides unphysically singular cross sections as $q_{\text{T}}\to 0$. But the nature of the degradation is constrained by the fact the collinear factorization also applies to the cross section integrated over $q_{\text{T}}$.

To get full accuracy over all $q_{\text{T}}$ one may combine both methods suitably. Collins, Soper, and Sterman (CSS) Collins and Soper (1982); Collins et al. (1985) implemented this as a kind of simple matched asymptotic expansion. But it has become increasingly clear—e.g., Boglione et al. (2015a, b)—that improved matching methods are needed to get adequate performance in practice, especially at the relatively low $Q$ used in many experiments on semi-inclusive deep-inelastic scattering (SIDIS). These are of particular relevance to this conference, since these experiments measure important transverse spin asymmetries analyzed with the aid of TMD factorization, as can be seen from other contributions at the conference.

We will summarize the issues and then our proposed improvements in the matching methods.

The essential measure of the quality of the applicability of a matching method is an evaluation of its accuracy. This is determined by the accuracy of the approximations used in deriving factorization from the exact cross section. The approximations can be understood from an examination of the derivations, as in Collins (2011). A simple example is given by the Feynman-graphical structure of the basic parton-model form, Fig. 1, where the Drell-Yan pair is created by quark-antiquark annihilation, with the quark and antiquark arising from structures that are collinearly moving with respect to the incoming hadrons.

We use light-front coordinates, where the parton momenta are

The incoming hadrons $p_{A}$ and $p_{B}$ have large 3-momenta in the $+z$ and $-z$ directions, with no transverse momentum.

CSS implemented the combination of TMD and collinear factorization by

where $W$ is the TMD factorized form

and $Y$ is a collinear correction term

In $W$, the convolution over the two TMD pdfs is rewritten as a Fourier transform over a transverse position variable $\boldsymbol{b}_{\text{T}}$.

The errors in $W$ and $Y$ caused by the approximations in the derivations can be estimated as:

for $W$ when $q_{\text{T}}\lesssim Q$,

for $Y$ when $\Lambda\lesssim q_{\text{T}}\lesssim Q$. Here $a$ is some fixed positive number determined by QCD and the power-law errors in the approximations used in deriving factorization. To these errors are to be added truncation errors of perturbative calculations. Hence the error in Eq. (2) is estimated to be a uniform power of $1/Q$ by

over the range $\Lambda\lesssim q_{\text{T}}\lesssim Q$. As one increases $q_{\text{T}}$ from order $\Lambda$, the deviation of $W$ from the cross section increases. But a collinear approximation can be applied to this deviation. As $q_{\text{T}}$ increases the deviation becomes larger, while its collinear approximation becomes better, in proportion.

It is very important that the stated error estimate for TMD factorization applies only when $q_{\text{T}}\lesssim Q$, and can completely fail for higher $q_{\text{T}}$. Similarly the stated error for collinear factorization applies only when $q_{\text{T}}\gtrsim\Lambda$, and can completely fail at lower $q_{\text{T}}$.

Boglione et al. (2015a, b) have found particular difficulties with implementing CSS’s $W+Y$ method in SIDIS at moderately low energies. These are illustrated in Fig. 5, for SIDIS at $Q=10\,\mathrm{GeV}$ at HERA. The solid line is the NLO calculation from collinear factorization. It shows the decrease of the cross section with hadron transverse $P_{\text{T}}$ that QCD predicts at the larger values of $P_{\text{T}}$. But the NLO curve diverges as $P_{\text{T}}\to 0$, where fixed-order calculations in collinear factorization are totally inaccurate.

The dotted curve shows the $W$ term, i.e., the result of TMD factorization, with fits to data determining the non-perturbative part of the TMD functions and their evolution. At small enough $P_{\text{T}}$, it should be a good approximation by itself for the cross section.

The asymptotic low $q_{\text{T}}$ part of the NLO collinear-factorization term is given by the dashed curve labeled “ASY”. This reproduces the NLO calculation at low $P_{\text{T}}$, but deviates from it substantially as $P_{\text{T}}$ increases. The deviation becomes substantial at quite a large factor below $Q$, which shows that there is a substantial numerical degradation of the simplest error estimates that were given in Sec. III. The ASY term goes through zero at around $1.7\,\mathrm{GeV}$ and then becomes negative.

Since the basic low $P_{\text{T}}$ asymptote has a logarithm: $\alpha_{s}\ln(Q/P_{\text{T}})/P_{\text{T}}^{2}$, the negative values are expected, and are in a region where the asymptotic calculation is inapplicable. The bothersome issue is that the inapplicability happens at what appears to be a surprisingly low $P_{\text{T}}$ compared with $Q$. Similarly $W$ goes through zero; this is also expected. The principle of the CSS method is that there should be a region of intermediate $P_{\text{T}}$ or $q_{\text{T}}$ where neither TMD nor collinear factorization is completely degraded in accuracy, at least for high $Q$. In this case, the large $P_{\text{T}}$ part of the TMD factorization and the low $P_{\text{T}}$ part of collinear factorization should approximately match. But this does not happen in Fig. 5. Furthermore the zeros in $W$ and ASY happen at quite different $P_{\text{T}}$.

The $Y$ term is the difference between the fixed-order collinear calculation, in this case NLO, and its small-$P_{\text{T}}$ asymptotic ASY. CSS intended this to correct TMD factorization to collinear factorization at large $q_{\text{T}}$ or $P_{\text{T}}$. It is calculated to be small when $P_{\text{T}}$ is small, as it should be. But it quickly becomes much larger than the presumably approximately correct NLO estimate for the cross section, which rather invalidates its use as a correction.

These plots suggest some ideas for improving the $W+Y$ method.

Perhaps the most important practical problem is that the TMD term, $W$, goes negative at large $P_{\text{T}}$. This indicates that at large $P_{\text{T}}$, $W+Y$ is the difference between substantially larger terms, and therefore shows a strong magnification of the relative effects of truncation errors in the predicted perturbative parts of the cross section.

Associated with this is that the integral over $W$ is exactly zero. This is because in $b_{\text{T}}$-space, the evolution equations show that the integrand is zero: $\tilde{W}(b_{\text{T}})=0$.

From these properties arises a severe problem in getting the integral over $q_{\text{T}}$ of the $W+Y$ formula in Eq. (2) to agree with the collinear factorization results

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  On the left-hand side, the integral $\int\mathop{\mathrm{d}^{2}\mskip-0.59999mu\boldsymbol{q}_{\text{T}}}\nolimits\ifrac{\mathop{\mathrm{d}\mskip-0.59999mu\sigma}\nolimits}{\mathop{\mathrm{d}^{4}\mskip-0.59999muq}\nolimits}$ is given by collinear factorization starting at LO, i.e., $\alpha_{s}^{0}$, up to a power-suppressed error. Fixed-order calculations of the hard scattering are appropriate.

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  On the right-hand side, the integral of $W$ is zero. So the integral of the right-hand side is the integral of $Y$ plus the error term.

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  But $Y$ is obtained from collinear factorization starting at NLO, i.e., $\alpha_{s}^{1}$.

If we used a fixed order of collinear factorization, there is a mismatch of orders of $\alpha_{s}$, and this results in complete mismatch of sizes even at the highest $Q$. Recall that the elementary derivation of the $W+Y$ formula concerned the errors at intermediate $q_{\text{T}}$ and did not concern itself explicitly with the integral over $q_{\text{T}}$.

As regards the integral over all $q_{\text{T}}$, we deduce that either we need resummation of $Y$ to handle the problem, or the error term is intrinsically very large, or both. The knowledge that nevertheless fixed-order calculations with collinear factorization are completely appropriate for the integrated cross section motivates part of our method for improving the $W+Y$ method.

We modify $W$ in two ways, as exhibited in our formula for a modified $W$ term:

First, to avoid problems in $\int\mathop{\mathrm{d}^{2}\mskip-0.59999mu\boldsymbol{q}_{\text{T}}}\nolimits W_{\rm New}$, we provide the $\tilde{W}$ function with a smooth cutoff at small $b_{\text{T}}$:

The integral of (8) over $q_{\text{T}}$ is now given by $\tilde{W}(b_{\text{T}})$ at $b_{\text{T}}$ of order $1/Q$. This is correctly predicted by fixed-order collinear factorization, and agrees with collinear factorization for the integrated cross section at leading order. Higher-order terms bring in the integral of our correspondingly modified $Y$ term, with well-behaved collinear expansions. This prescription is close to that of Bozzi et al. (2006). Their prescription was formulated purely in terms of resummation calculations in massless QCD. Our solution applies to full TMD factorization. The function $\tilde{W}(b,Q)$ has the same functional form as before, and involves exactly the same TMD pdfs and evolution equations; the modification consists in changing the value used for the transverse position argument in the Fourier transform, from $b_{\text{T}}$ to $b_{c}(b_{\text{T}})$. At low $q_{\text{T}}$, larger values of $b_{\text{T}}$ than $1/Q$ dominate, and then the cutoff at small $b_{\text{T}}$ is unimportant; thus the validity of TMD factorization in its target region of $q_{\text{T}}\ll Q$ is unaffected.

The second change is to impose an explicit cutoff at large $q_{\text{T}}$ cutoff, by a factor

This keeps the modified $W$ term from being significantly nonzero at such large $q_{\text{T}}$ that TMD factorization is totally inapplicable.

Correspondingly, to implement a correct matching, we modify $Y$ to

The second factor is the basic implementation of matching of TMD and collinear factorization. But we impose an extra $q_{\text{T}}$ cutoff factor, for example

The reason for the extra cutoffs on $W$ at large $q_{\text{T}}$ and on $Y$ at small $q_{\text{T}}$ is found in the derivation of the errors in Sec. III. That error calculation is only valid when $\Lambda\lesssim q_{\text{T}}\lesssim Q$. Below that range, we should use only the TMD factorization term $W$, i.e., $Y$ should then be close to zero. Above that range, we should only use collinear factorization, so $W$ should then be close to zero.

We modified the $W+Y$ formalism, so that

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  The error in $W+Y$ is suppressed by a power of $Q$ for all $q_{\text{T}}$.

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  The integral over all $q_{\text{T}}$ is now properly behaved with respect to collinear factorization for the integrated cross section.

Further improvements are undoubtedly possible. We have tried to formulate some relevant issues. Generally, to do better, one needs to “look under hood”, to ask questions like:

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  What is the nature of the approximations giving factorization (TMD and collinear)?

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  How much do they fail, with proper account of non-perturbative properties?

These issues are important for subject of this conference, i.e., spin physics. This is because we often want to use TMD factorization at moderate $Q$, notably in the measurement of transverse-spin-dependent TMD pdfs and fragmentation functions.

One important question for SIDIS, is to determine the appropriate criteria for what is large and small transverse momentum relative to $Q$. Is the appropriate variable $q_{\text{T}}$ or $P_{\text{T}}$? I.e., is it transverse momentum of the virtual photon relative to the detected hadrons, or is it the transverse-momentum of the detected final-state hadron relative to the $\gamma^{*}$ and target? These two variables differ by a factor of the fragmentation variable $z$. One can also ask whether that was even the right question.

It would also be useful to have the $Y$-term for SIDIS at NNLO, i.e., $O(\alpha_{s}^{2})$. This could be obtained from the results for collinear factorization for SIDIS at the same order, as reported by Daleo et al. Daleo et al. (2005), but we are not aware that this has been done yet.