Model independent evolution of transverse momentum dependent distribution functions (TMDs) at NNLL

Abstract

We discuss the evolution of the eight leading-twist transverse momentum dependent parton distribution functions, which turns out to be universal and spin independent. By using the highest order perturbatively calculable ingredients at our disposal, we perform the resummation of the large logarithms that appear in the evolution kernel of transverse momentum distributions up to next-to-next-to-leading logarithms (NNLL), thus obtaining an expression for the kernel with highly reduced model dependence. Our results can also be obtained using the standard CSS approach when a particular choice of the b ^∗ prescription is used. In this sense, and while restricted to the perturbative domain of applicability, we consider our results as a “prediction” of the correct value of b _max which is very close to 1.5 GeV⁻¹. We explore under which kinematical conditions the effects of the non-perturbative region are negligible, and hence the evolution of transverse momentum distributions can be applied in a model independent way. The application of the kernel is illustrated by considering the unpolarized transverse momentum dependent parton distribution function and the Sivers function.

Appendices

Appendix A: CSS approach to the evolution of TMDs

In various works following Collins’ approach to TMDs [6, 9, 15], large L _⊥ logarithms in the D term of the evolution kernel in Eq. (7) were resummed using the CSS approach [19], which, for the sake of completeness and comparison, we explain below.

First, the D term is resummed using its RG-evolution in Eq. (8),

$$\begin{aligned} D (b;Q_i ) &= D (b;\mu_b ) + \int _{\mu_b}^{Q_i}\frac{d\bar{\mu}}{\bar{\mu}} \varGamma_{\mathrm{cusp}} , \end{aligned}$$

(A.1)

where large L _⊥ logarithms in the D term on the right hand side are canceled by choosing $\mu_{b} = 2e^{-\gamma_{E}}/b$. Thus, they are resummed by evolving D from μ _b to Q _i . However, since we need to Fourier transform back to momentum space, at some value of b the effective coupling α _s (μ _b ) will hit the Landau pole. In Fig. 7(a) we can see the evolution kernel $\tilde {R}(b;Q_{i}=\sqrt{2.4}~\mathrm{GeV},Q_{f}=5~\mathrm{GeV})$ where we have used Eq. (A.1) to resum the L _⊥ logarithms in D and the appearance of the Landau pole is manifest. The breakdown of the perturbative series is driven by the running coupling α _s (μ _b ), when μ _b is sufficiently small.

Fig. 7

Evolution kernel from $Q_{i}=\sqrt{2.4}~\mathrm{GeV}$ up to Q _f =5 GeV using RG-evolution in Eq. (A.1) to resum the D term. The resummation accuracy is given in Table 2. (a) With $\mu_{b} = 2e^{-\gamma_{E}}/b$ the Landau pole appears clearly. (b) With $\mu_{b^{*}} = 2e^{-\gamma_{E}}/{b^{*}}$ and b _max=1.5 GeV⁻¹ to avoid hitting the Landau pole. (c) Adding the BLNY non-perturbative model to recover the information at large b

In order to avoid this issue, CSS did not actually introduce a sharp cut-off but a smoothed one defined as $b^{*}=b/\sqrt{1+(b/b_{\max})^{2}}$. Obviously b ^∗ cannot exceed b _max and the effective coupling $\alpha_{s}(\mu_{b^{*}})$ does not hit the Landau pole. As is shown in Fig. 7(b), the kernel saturates and does not present any uncontrolled behavior. It is also worth noticing that comparing Fig. 7(a) with Fig. 7(b), we see that the implementation of the b ^∗ prescription has some appreciable effect in the perturbative region, which now depends on this parameter.

The lost information due to the cut-off is recovered by adding a non-perturbative model that has to be extracted from experimental data of a measured cross section. This model not only gives the proper information in the non-perturbative region, but also restores the correct shape of the kernel within the perturbative domain, which was affected by the b ^∗ prescription. When implementing, for example, the Brock–Landry–Nadolsky–Yuan (BLNY )model the evolution kernel can be written as

$$\begin{aligned} & \tilde{R}^{\mathrm{CSS}}(b;Q_i,Q_f) \\ &\quad = \exp \biggl\{ \int_{Q_i}^{Q_f} \frac{d\bar{\mu}}{\bar{\mu}} \gamma_F \biggl(\alpha _s(\bar{\mu}),\ln \frac{Q_f^2}{\bar{\mu}^2} \biggr) \biggr\} \\ &\qquad{}\times \biggl(\frac{Q_f^2}{Q_i^2} \biggr)^ {- [D (b^*;Q_i )+\frac{1}{4}g_2 b^2 ]} , \end{aligned}$$

(A.2)

where D(b ^∗;Q _i ) is resummed using Eq. (A.1). In this model g ₂=0.68 GeV² for b _max=0.5 GeV⁻¹ [42] and g ₂=0.184 GeV² for b _max=1.5 GeV⁻¹ [20]. From the theoretical point of view these two choices are legitimate and they can be used to define the model dependence of the final result. However, considering b _max as a fitting parameter the choice of b _max=1.5 GeV⁻¹ should be preferred according to Ref. [20]. Figure 7(c) shows the complete evolution kernel with the CSS approach, Eq. (A.2), while implementing the BLNY model.

Appendix B: Derivation of D ^R up to NNNLL

Below we provide the details of the derivation of D ^R within the CSS formalism, i.e., solving Eq. (24). Using the perturbative expansion of Γ _cusp(α _s ) and β(α _s ) one can write,

$$\begin{aligned} &\int_{\mu_b}^{Q_i} d(\ln \mu) \varGamma_{\mathrm{cusp}} \\ &\quad = \int_{\alpha_s(\mu_b)}^{\alpha_s(Q_i)} d\alpha\; \frac{\varGamma _{\mathrm{cusp}}(\alpha)}{\beta(\alpha)} \\ &\quad =\int_{\alpha_s(\mu_b)}^{\alpha_s(Q_i)} d\alpha\; \biggl\{ \frac{-\varGamma_0}{2 \alpha\beta_0}+ \frac{\varGamma_0\beta_1-\varGamma_1 \beta_0}{8 \pi\beta_0^2}+ \frac{\alpha (-\beta_0^2 \varGamma_2+\beta_0 \beta_1 \varGamma _1+\beta_0 \beta_2 \varGamma_0-\beta_1^2 \varGamma_0 )}{32 \pi^2 \beta_0^3} \\ &\qquad {} + \frac{\alpha^2 ( -\beta_0^3 \varGamma_3+\beta_0^2 \beta_1 \varGamma_2+\beta_0^2 \beta_2\varGamma_1+\beta_0^2 \beta _3\varGamma_0-\beta_0 \beta_1^2 \varGamma_1-2 \beta_0 \beta_1\beta_2 \varGamma_0+\beta_1^3 \varGamma_0 )}{128 \pi^3\beta_0^4} \biggr\} \\ &\quad = \frac{-\varGamma_0}{2 \beta_0}\ln \frac{\alpha_s(Q_i)}{\alpha _s(\mu_b)}+ \bigl[ \alpha_s(Q_i)-\alpha_s(\mu_b) \bigr]\frac{\varGamma_0\beta _1-\varGamma _1 \beta_0}{8 \pi\beta_0^2}+ \bigl[\alpha_s^2(Q_i)- \alpha_s^2(\mu_b) \bigr]\frac{ (-\beta _0^2 \varGamma_2+\beta_0 \beta_1 \varGamma_1+\beta_0 \beta_2 \varGamma _0-\beta_1^2 \varGamma_0 )}{64 \pi^2 \beta_0^3} \\ &\qquad {}+ \bigl[\alpha_s^3(Q_i)- \alpha_s^3(\mu_b) \bigr]\frac{ ( -\beta _0^3 \varGamma_3+\beta_0^2 \beta_1 \varGamma_2+\beta_0^2 \beta_2\varGamma_1+\beta_0^2 \beta _3\varGamma_0-\beta_0 \beta_1^2 \varGamma_1-2 \beta_0 \beta_1\beta_2 \varGamma_0+\beta_1^3 \varGamma_0 )}{384 \pi^3\beta_0^4} . \end{aligned}$$

(B.1)

Then in the equation above we use the running of the strong coupling to expand α _s (μ _b ) in terms of α _s (Q _i ),

$$\begin{aligned} \alpha_s(\mu_b)&= \alpha_s(Q_i) \frac{1}{1-X} - \alpha_s^2(Q_i) \frac{\beta_1 \ln (1-X)}{4\pi(1-X)^2 \beta_0}- \alpha_s^3(Q_i) \frac{ (-X \beta_0 \beta_2+ \beta_1^2 (X-\ln ^2(1-X)+ \ln (1-X)) )}{16 \pi^2 (1-X)^3 \beta_0^2} \\ &\quad {} -\alpha_s^4(Q_i) \frac{ (\beta_1^3 (X^2+2 \ln ^3(1-X)-5 \ln ^2(1-X)-4 X \ln (1-X) ) )}{128 \pi^3 (X-1)^4 \beta_0^3} \\ &\quad {}-\alpha_s^4(Q_i)\frac{ (+2\beta_0\beta_1\beta_2 ((2 X+1) \ln (1-X)-(X-1) X)+(X-2) X \beta_0^2 \beta_3 )}{128 \pi^3 (X-1)^4\beta_0^3} \end{aligned}$$

(B.2)

and implement it up to the appropriate order in α _s (Q _i ). In order to finally get D ^R at NNLL one should consider also the term D(b;μ _b ) in Eq. (24), which at second order does not vanish due to the presence of the finite d ₂(0) term,

$$\begin{aligned} D^{(2)}(b;\mu_b)= d_2(0) \biggl( \frac{\alpha_s(\mu_b)}{4\pi} \biggr)^2=d_2(0) \frac {a^2}{(1-X)^2} , \end{aligned}$$

(B.3)

with a=α _s (Q _i )/4π. Inserting this result in Eq. (B.1) and the expansion in Eq. (B.2) up to order $\alpha_{s}^{2}(Q_{i})$, one gets our Eq. (16).

Finally, for completeness and future reference, we provide also D ^R at NNNLL,

$$\begin{aligned} D^{R(3)}&= \frac{a^3}{(1-X)^3} \biggl(d_3(0)-2d_2(0) \frac{\beta _1}{\beta_0}\ln (1-X)+D^{R(3)}_\varGamma \biggr) , \\ D^{R(3)}_\varGamma&=-\frac{1}{12\beta_0^4} \bigl[\beta_0^2 \bigl(2 \varGamma_2 \beta_1 \bigl(X \bigl(X^2-3 X+3 \bigr) \\ &\quad {}+3 \ln (1-X) \bigr)+X^2 \bigl(2 \varGamma_1 (X-3) \beta_2 \\ &\quad {}+\varGamma_0 (2 X-3)\beta_3 \bigr) \bigr) \\ &\quad {} -2 \beta_0\beta_1 \bigl(\varGamma_1 \beta_1 \bigl((X-3) X^2+3 \ln ^2(1-X) \bigr) \\ &\quad {}+\varGamma _0 X \beta_2 \bigl(X (2 X-3)-3 \ln (1-X)\bigr) \bigr) \\ &\quad {}-2 \varGamma_3 X \bigl(X^2-3 X+3 \bigr) \beta_0^3 \\ &\quad {} +\varGamma_0 \beta_1^3 \bigl(X^2 (2 X-3)+2 \ln ^3(1-X) \\ &\quad -3 \ln ^2(1-X)-6 X \ln (1-X) \bigr) \bigr] . \end{aligned}$$

(B.4)

Appendix C: Evolution of the hard matching coefficient

The evolution of the hard matching coefficient C _V , where H=|C _V |², is given by

$$\begin{aligned} \begin{array}{l} \displaystyle\frac{d}{d\ln \mu}\ln C_V\bigl(Q^2/ \mu^2\bigr) = \gamma_{C_V} \biggl(\alpha_s( \mu),\ln \displaystyle\frac{Q^2}{\mu^2} \biggr) , \\ \gamma_{C_V} = \varGamma_{\mathrm{cusp}}(\alpha_s)\ln \displaystyle\frac{Q^2}{\mu^2} + \gamma^V(\alpha_s) , \end{array} \end{aligned}$$

(C.1)

where the cusp term is related to the evolution of the Sudakov double logarithms and the remaining term with the evolution of single logarithms. The exact solution of this equation is

$$\begin{aligned} & C_V\bigl(Q^2/\mu_f^2 \bigr) \\ &\quad = C_V\bigl(Q^2/\mu_i^2 \bigr) \exp \biggl[ \int_{\mu_i}^{\mu_f} \frac{d\bar{\mu}}{\bar{\mu}} \gamma_{C_V} \biggl(\alpha_s(\bar{\mu}), \ln \frac{Q^2}{\bar{\mu}^2} \biggr) \biggr] \\ &\quad = C_V\bigl(Q^2/\mu_i^2\bigr) \exp \biggl[ \int_{\alpha_s(\mu_i)}^{\alpha_s(\mu_f)} \frac {d\bar{\alpha}_s}{\beta(\bar{\alpha}_s)} \gamma_{C_V} (\bar{\alpha}_s ) \biggr] , \end{aligned}$$

(C.2)

where we have used d/dlnμ=β(α _s )d/dα _s , where β(α _s )=dα _s /dlnμ is the QCD β-function.

Below we give the expressions for the anomalous dimensions and the QCD β-function, in the $\overline{{\rm MS}}$ renormalization scheme. We use the following expansions:

$$\begin{aligned} &\varGamma_{\mathrm{cusp}} = \sum_{n=1}^{\infty} \varGamma_{n-1} \biggl( \frac{\alpha_s}{4\pi } \biggr)^n , \\ &\gamma^V_{n-1} = \sum_{n=1}^{\infty} \gamma^V_{n} \biggl( \frac{\alpha _s}{4\pi} \biggr)^n , \\ &\beta= -2\alpha_s\sum _{n=1}^\infty\beta_{n-1} \biggl( \frac {\alpha_s}{4\pi} \biggr)^n . \end{aligned}$$

(C.3)

The coefficients for the cusp anomalous dimension Γ _cusp are

$$\begin{aligned} &\varGamma_0 = 4 C_F , \\ &\varGamma_1 = 4 C_F \biggl[ \biggl( \frac{67}{9} - \frac{\pi^2}{3} \biggr) C_A - \frac{20}{9}T_F n_f \biggr] , \\ &\varGamma_2 = 4 C_F \biggl[ C_A^2 \biggl( \frac{245}{6} - \frac{134\pi^2}{27} + \frac{11\pi^4}{45} + \frac{22}{3}\zeta_3 \biggr)\\ &\hphantom{\varGamma_2 =} + C_A T_F n_f \biggl( - \frac{418}{27} + \frac{40\pi^2}{27} - \frac{56}{3}\zeta_3 \biggr) \\ &\hphantom{\varGamma_2 =} {}+ C_F T_F n_f \biggl( - \frac{55}{3} + 16\zeta_3 \biggr) - \frac{16}{27} T_F^2 n_f^2 \biggr] . \end{aligned}$$

(C.4)

The anomalous dimension γ ^V can be determined up to 3-loop order from the partial 3-loop expression for the on-shell quark form factor in QCD. We have

$$\begin{aligned} &\gamma_0^V = -6 C_F , \\ &\gamma_1^V = C_F^2 \bigl( -3 + 4\pi^2 - 48\zeta_3 \bigr) \\ &\hphantom{\gamma_1^V =} {}+C_F C_A \biggl( - \frac{961}{27} - \frac{11\pi^2}{3} + 52\zeta_3 \biggr) \\ &\hphantom{\gamma_1^V =} {}+ C_F T_F n_f \biggl( \frac{260}{27} + \frac{4\pi^2}{3} \biggr) , \\ & \gamma_2^V = C_F^3 \biggl( -29 - 6\pi^2 - \frac{16\pi^4}{5} - 136\zeta_3 \\ &\hphantom{\gamma_2^V =} {}+ \frac{32\pi^2}{3}\zeta_3+ 480\zeta_5 \biggr) \\ &\hphantom{\gamma_2^V =} {}+ C_F^2 C_A \biggl( - \frac{151}{2} + \frac{410\pi^2}{9} + \frac{494\pi^4}{135} - \frac{1688}{3}\zeta_3\\ &\hphantom{\gamma_2^V =} {}- \frac{16\pi^2}{3}\zeta_3 - 240\zeta_5 \biggr) \\ &\hphantom{\gamma_2^V =} {}+ C_F C_A^2 \biggl( - \frac{139345}{1458} - \frac{7163\pi^2}{243} - \frac{83\pi^4}{45} + \frac{7052}{9}\zeta_3 \\ &\hphantom{\gamma_2^V =} {}- \frac{88\pi^2}{9}\zeta_3 - 272\zeta_5 \biggr) \\ &\hphantom{\gamma_2^V =} {}+ C_F^2 T_F n_f \biggl( \frac{5906}{27} - \frac{52\pi^2}{9} - \frac{56\pi^4}{27} + \frac{1024}{9}\zeta_3 \biggr) \\ &\hphantom{\gamma_2^V =} {}+ C_F C_A T_F n_f \biggl( - \frac{34636}{729} + \frac{5188\pi^2}{243} + \frac{44\pi^4}{45} \\ &\hphantom{\gamma_2^V =} {} - \frac{3856}{27} \zeta_3 \biggr) + C_F T_F^2 n_f^2 \biggl( \frac{19336}{729} - \frac{80\pi^2}{27} - \frac{64}{27} \zeta_3 \biggr) . \end{aligned}$$

(C.5)

Finally, the coefficients for the QCD β-function are

$$\begin{aligned} &\beta_0 = \frac{11}{3}C_A - \frac{4}{3}T_F n_f , \\ &\beta_1 = \frac{34}{3}C_A^2 - \frac{20}{3}C_A T_F n_f - 4 C_F T_F n_f , \\ &\beta_2 = \frac{2857}{54}C_A^3 + \biggl( 2 C_F^2 - \frac{205}{9}C_F C_A - \frac{1415}{27}C_A^2 \biggr) T_F n_f\\ &\hphantom{\beta_2 =} {}+ \biggl( \frac{44}{9}C_F + \frac{158}{27}C_A \biggr) T_F^2 n_f^2 , \\ &\beta_3 = \frac{149753}{6} + 3564\zeta_3 - \biggl( \frac{1078361}{162} + \frac{6508}{27}\zeta_3 \biggr) n_f \\ &\hphantom{\beta_3 =} {}+ \biggl( \frac{50065}{162} + \frac{6472}{81}\zeta_3 \biggr) n_f^2 + \frac{1093}{729}n_f^3 , \end{aligned}$$

(C.6)

where for β ₃ we have used N _c =3 and $T_{F}=\frac{1}{2}$.