Electroweak Logarithms in Inclusive Cross Sections

TL;DR

This paper develops a comprehensive framework for resumming electroweak logarithms in high-energy inclusive scattering processes, including detailed calculations at NLL order and implications for PDFs, polarization, and experimental measurements.

Contribution

It introduces a method for all-orders electroweak logarithm resummation at NLL, including the effects of polarization and fragmentation, and extends factorization to non-singlet objects.

Findings

Electroweak logarithms can be resummed via PDF evolution at LL.

Polarization effects are significant for gauge bosons due to chiral SU(2) interactions.

Electroweak effects induce polarization in gluon distributions at high scales.

Abstract

We develop the framework to perform all-orders resummation of electroweak logarithms of Q/M for inclusive scattering processes at energies Q much above the electroweak scale M. We calculate all ingredients needed at next-to-leading logarithmic (NLL) order and provide an explicit recipe to implement this for 2 $\to$ 2 processes. PDF evolution including electroweak corrections, which lead to Sudakov double logarithms, is computed. If only the invariant mass of the final state is measured, all electroweak logarithms can be resummed by the PDF evolution, at least to LL. However, simply identifying a lepton in the final state requires the corresponding fragmentation function and introduces angular dependence through the exchange of soft gauge bosons. Furthermore, we show the importance of polarization effects for gauge bosons, due to the chiral nature of SU(2) - even the gluon distribution in an unpolarized proton becomes polarized at high scales due to electroweak effects. We justify our approach with a factorization analysis, finding that the objects entering the factorization theorem do not need to be SU(2) $\times$ U(1) gauge singlets, even though we perform the factorization and resummation in the symmetric phase. We also discuss a range of extensions, including jets and how to calculate the EW logarithms when you are fully exclusive in the central (detector) region and fully inclusive in the forward (beam) regions.

Figures

Tables

Table 1. Table 1: One-loop diagrams for the renormalization of fermion collinear operators. The columns show the graph and contribution to the μ 𝜇 \mu and ν 𝜈 \nu anomalous dimension. The combinatoric factor of 2 2 2 for the first and third graphs, and − 1 1 -1 for the wavefunction graph has been included. Subsets of the graphs have been summed to give Total 1 subscript Total 1 \text{Total}_{1} and Total 2 subscript Total 2 \text{Total}_{2} . For the singlet fermion PDF, Total 1 subscript Total 1 \text{Total}_{1} and Total 2 subscript Total 2 \text{Total}_{2} have group theory factor c F subscript 𝑐 𝐹 c_{F} . For the adjoint PDF, Total 1 subscript Total 1 \text{Total}_{1} has group theory factor c F − c A / 2 subscript 𝑐 𝐹 subscript 𝑐 𝐴 2 c_{F}-c_{A}/2 and Total 2 subscript Total 2 \text{Total}_{2} has group theory factor c F subscript 𝑐 𝐹 c_{F} .

Table 2. Table 2: One-loop diagrams for the renormalization of scalar collinear operators. Subsets of the graphs have been summed to give Total 1 subscript Total 1 \text{Total}_{1} and Total 2 subscript Total 2 \text{Total}_{2} . For the singlet scalar PDF, Total 1 subscript Total 1 \text{Total}_{1} and Total 2 subscript Total 2 \text{Total}_{2} have group theory factor c F subscript 𝑐 𝐹 c_{F} . For the adjoint PDF, Total 1 subscript Total 1 \text{Total}_{1} has group theory factor c F − c A / 2 subscript 𝑐 𝐹 subscript 𝑐 𝐴 2 c_{F}-c_{A}/2 and Total 2 subscript Total 2 \text{Total}_{2} has group theory factor c F subscript 𝑐 𝐹 c_{F} .

Table 3. Table 3: One-loop Yukawa diagrams for the renormalization of fermion collinear operators. The Yukawa factors multiplying the graphs are given in eqs. ( 3.2 ) and ( 61 ).

Table 4. Table 4: One-loop diagrams for the renormalization of collinear gauge boson operators. The columns show the graph and contribution to the μ 𝜇 \mu and ν 𝜈 \nu anomalous dimension for P G + ​ G + subscript 𝑃 subscript 𝐺 subscript 𝐺 P_{G_{+}G_{+}} , P G + ​ G − subscript 𝑃 subscript 𝐺 subscript 𝐺 P_{G_{+}G_{-}} . At one-loop, P G + ​ G + = P G − ​ G − subscript 𝑃 subscript 𝐺 subscript 𝐺 subscript 𝑃 subscript 𝐺 subscript 𝐺 P_{G_{+}G_{+}}=P_{G_{-}G_{-}} and P G + ​ G − = P G − ​ G + subscript 𝑃 subscript 𝐺 subscript 𝐺 subscript 𝑃 subscript 𝐺 subscript 𝐺 P_{G_{+}G_{-}}=P_{G_{-}G_{+}} . Combinatoric factors have been included. Subsets of the graphs have been summed to give Total 1 subscript Total 1 \text{Total}_{1} and Total 2 subscript Total 2 \text{Total}_{2} . Total 2 subscript Total 2 \text{Total}_{2} has group theory factor 1 1 1 in all cases, since its group theory factors are already included in the table. Total 1 subscript Total 1 \text{Total}_{1} has group theory factors given in eq. ( 3.3 ).

Table 5. Table 5: One-loop diagrams contributing to Q ​ G 𝑄 𝐺 QG mixing. The columns show the graph and contribution to the μ 𝜇 \mu and ν 𝜈 \nu anomalous dimension. The group theory factors are given in eqs. ( 69 ) and ( 70 ).

Table 6. Table 6: The one-loop diagrams contributing to H ​ G 𝐻 𝐺 HG and G ​ H 𝐺 𝐻 GH mixing in the anomalous dimension of collinear operators. The columns show the graph and contribution to the μ 𝜇 \mu and ν 𝜈 \nu anomalous dimension. The group theory factors are given in eqs. ( 69 ), ( 70 ) and ( 3.5 ).

Table 7. Table 7: One-loop diagrams contributing to G ​ Q 𝐺 𝑄 GQ mixing. The columns show the graph and contribution to the μ 𝜇 \mu and ν 𝜈 \nu anomalous dimension. The group theory factors are given in eq. ( 3.5 ).

Table 8. Table 8: The one-loop diagrams contributing to H ​ Q 𝐻 𝑄 HQ and Q ​ H 𝑄 𝐻 QH mixing in the anomalous dimension of collinear operators. The columns show the graph and contribution to the μ 𝜇 \mu and ν 𝜈 \nu anomalous dimension. The Yukawa factors are give in eqs. ( 3.6 ) and ( 3.6 ).

Table 9. Table 9: The one-loop diagram for the soft operator 𝒮 i ​ 𝒮 j subscript 𝒮 𝑖 subscript 𝒮 𝑗 {\mathcal{S}}_{i}{\mathcal{S}}_{j} . The double lines denote the Wilson lines 𝒮 i subscript 𝒮 𝑖 {\mathcal{S}}_{i} and 𝒮 j subscript 𝒮 𝑗 {\mathcal{S}}_{j} . The columns show the graph and its contribution to the μ 𝜇 \mu and ν 𝜈 \nu anomalous dimension, apart from the group theory factors given in the text.

Table 10. Table 10: Perturbative ingredients needed at different orders in resummed perturbation theory. The columns correspond to the loop order of the matching (both at the high scale Q 𝑄 Q and low scale M 𝑀 M ), non-cusp and cusp anomalous dimensions.

	Matching	Non-cusp	Cusp
LL	tree	-	$1$-loop
NLL	tree	$1$-loop	$2$-loop
NLL′	$1$-loop	$1$-loop	$2$-loop
NNLL	$1$-loop	$2$-loop	$3$-loop

Equations

$$\sigma ={\sigma }_0\sum _{m\le 2n}{c}_{nm}{\alpha }_w^n{\ln }^m\frac{Q}{M},$$

$$O_{\ell q}^{(3)}$$

$$O_{\ell u}$$

$$O_{eq}$$

$$O_{eu}$$

$$\displaystyle{\mathcal{L}}_{\rm hard}=\sum_{i}\bigg{(}\prod_{k}\int\!\frac{\mathrm{d}^{4}p_{k}}{(2\pi)^{4}}\bigg{)}\,(2\pi)^{4}\delta^{4}\Big{(}\sum_{m}p_{m}\Big{)}\,\mathcal{H}_{i}(\{p_{k}\})\,O_{i}(\{p_{k}\})\,.$$

$${\mathcal{H}}_{\ell q}^{(3)}=\frac{\mathrm{i}{g}_2^2}{2p_1\cdot p_2},{\mathcal{H}}_{AB}=\frac{\mathrm{i}{g}_1^2{\mathsf{y}}_A{\mathsf{y}}_B}{2p_1\cdot p_2},$$

$$\sum _i{\mathcal{H}}_i{O}_i$$

$$n_i=(1,{\mathbf{n}}_i),{\bar{n}}_i=(1,-{\mathbf{n}}_i),$$

$$n_j=(-1,-{\mathbf{n}}_j),{\bar{n}}_j=(-1,{\mathbf{n}}_j),$$

$$q_4(x)$$

$${\mathcal{W}}_4(x)$$

$${\mathcal{S}}_4$$

$$\sigma \sim \sum _X⟨\nu p⟩{\mathcal{L}}_{\text{hard}}{e}^{-}X⟨e^{-}X⟩{\mathcal{L}}_{\text{hard}}\nu p.$$

$$\displaystyle(\gamma^{\mu}_{\beta_{1}\gamma_{1}}\gamma_{\mu,\beta_{2}\gamma_{2}}\gamma^{\nu}_{\beta_{3}\gamma_{3}}\gamma_{\nu,\beta_{4}\gamma_{4}})\Bigl{[}\sum_{X_{1}}\langle 0|\ell_{1^{\prime},\delta_{3}}|X_{1}\rangle\langle X_{1}|\bar{\ell}_{1,\alpha_{1}}|0\rangle\Bigr{]}\Bigl{[}\langle\nu|\bar{\ell}_{2^{\prime},\alpha_{3}}\ell_{2,\delta_{1}}|\nu\rangle\Bigr{]}$$

$$\displaystyle\Bigl{[}\sum_{X_{3}}\langle 0|q_{3^{\prime},\delta_{4}}|X_{3}\rangle\langle X_{3}|\bar{q}_{3,\alpha_{2}}|0\rangle\Bigr{]}\Bigl{[}\langle p|\bar{q}_{4^{\prime},\alpha_{4}}q_{4,\delta_{2}}|p\rangle\Bigr{]}$$

$$\displaystyle\Bigl{[}\sum_{X_{s}}\langle 0|{\mathcal{S}}_{1,\gamma_{3},\delta_{3}}{\mathcal{S}}_{2,\alpha_{3}\beta_{3}}^{\dagger}{\mathcal{S}}_{3,\gamma_{4},\delta_{4}}{\mathcal{S}}_{4,\alpha_{4}\beta_{4}}^{\dagger}|X_{s}\rangle\langle X_{s}|{\mathcal{S}}_{1,\alpha_{1},\beta_{1}}^{\dagger}{\mathcal{S}}_{2,\gamma_{1}\delta_{1}}{\mathcal{S}}_{3,\alpha_{2}\beta_{2}}^{\dagger}{\mathcal{S}}_{4,\gamma_{2}\delta_{2}}|0\rangle\Bigr{]}\,.$$

$$⟨0|{\mathcal{S}}_{1,{\gamma }_3,{\delta }_3}{\mathcal{S}}_{2,{\alpha }_3{\beta }_3}^{†}{\mathcal{S}}_{3,{\gamma }_4,{\delta }_4}{\mathcal{S}}_{4,{\alpha }_4{\beta }_4}^{†}{\mathcal{S}}_{1,{\alpha }_1,{\beta }_1}^{†}{\mathcal{S}}_{2,{\gamma }_1{\delta }_1}{\mathcal{S}}_{3,{\alpha }_2{\beta }_2}^{†}{\mathcal{S}}_{4,{\gamma }_2{\delta }_2}|0⟩.$$

$${\bar{\ell }}_{i\alpha }{\ell }_{\beta }^j$$

$${\bar{e}}_{\alpha }{e}_{\beta }$$

$$O_{\ell q}^{(3)†}{O}_{\ell q}^{(3)}$$

$$+\frac{4}{N_c{N}_w^2}{\mathcal{C}}_{{\ell }_1}^c{\mathcal{C}}_{{\ell }_2}^d{\mathcal{C}}_{q_3}{\mathcal{C}}_{q_4}\mathrm{tr}({\mathcal{S}}_1{t}^c{\mathcal{S}}_1^{†}{t}^a{\mathcal{S}}_2{t}^d{\mathcal{S}}_2^{†}{t}^b)\mathrm{tr}(t^a{t}^b)$$

$$+\frac{4}{N_c{N}_w^2}{\mathcal{C}}_{{\ell }_1}{\mathcal{C}}_{{\ell }_2}{\mathcal{C}}_{q_3}^c{\mathcal{C}}_{q_4}^d\mathrm{tr}({\mathcal{S}}_3{t}^c{\mathcal{S}}_3^{†}{t}^a{\mathcal{S}}_4{t}^d{\mathcal{S}}_4^{†}{t}^b)\mathrm{tr}(t^a{t}^b)$$

$$+\frac{4}{N_c{N}_w^2}{\mathcal{C}}_{{\ell }_1}^c{\mathcal{C}}_{{\ell }_2}{\mathcal{C}}_{q_3}^d{\mathcal{C}}_{q_4}\mathrm{tr}({\mathcal{S}}_1{t}^c{\mathcal{S}}_1^{†}{t}^a{t}^b)\mathrm{tr}({\mathcal{S}}_3{t}^d{\mathcal{S}}_3^{†}{t}^a{t}^b)+(3\text{ more})$$

$$+\frac{8}{N_c{N}_w}{\mathcal{C}}_{{\ell }_1}^c{\mathcal{C}}_{{\ell }_2}^d{\mathcal{C}}_{q_3}^e{\mathcal{C}}_{q_4}\mathrm{tr}({\mathcal{S}}_1{t}^c{\mathcal{S}}_1^{†}{t}^a{\mathcal{S}}_2{t}^d{\mathcal{S}}_2^{†}{t}^b)\mathrm{tr}({\mathcal{S}}_3{t}^e{\mathcal{S}}_3^{†}{t}^a{t}^b)+(3\text{ more})$$

$$\displaystyle\quad+\frac{16}{N_{c}}\,\mathcal{C}_{\ell_{1}}^{c}\,\mathcal{C}_{\ell_{2}}^{d}\,\mathcal{C}_{q_{3}}^{e}\,\mathcal{C}_{q_{4}}^{f}\,\mathrm{tr}({\mathcal{S}}_{1}t^{c}{\mathcal{S}}_{1}^{\dagger}t^{a}{\mathcal{S}}_{2}t^{d}{\mathcal{S}}_{2}^{\dagger}t^{b})\,\mathrm{tr}({\mathcal{S}}_{3}t^{e}{\mathcal{S}}_{3}^{\dagger}t^{a}{\mathcal{S}}_{4}t^{f}{\mathcal{S}}_{4}^{\dagger}t^{b})\bigg{]}\,.$$

$${\mathcal{C}}_{{\ell }_1}={\bar{\ell }}_1\frac{\bar{n}{/}_1}{2}{\ell }_1,{\mathcal{C}}_{{\ell }_1}^c={\bar{\ell }}_1\frac{\bar{n}{/}_1}{2}{t}^c{\ell }_1,\dots ,$$

$$(t^a{)}^{\alpha }{}_{\beta }(t^a{)}^{\gamma }{}_{\delta }=\frac{1}{2}{\delta }^{\alpha }{}_{\delta }{\delta }^{\gamma }{}_{\beta }-\frac{1}{2N_w}{\delta }^{\alpha }{}_{\beta }{\delta }^{\gamma }{}_{\delta }.$$

$$\mathrm{tr}(t^a{t}^b)\mathrm{tr}(t^a{t}^b)$$

$$\mathrm{tr}({\mathcal{S}}_1{t}^c{\mathcal{S}}_1^{†}{t}^a{\mathcal{S}}_2{t}^d{\mathcal{S}}_2^{†}{t}^b)\mathrm{tr}(t^a{t}^b)$$

$$\mathrm{tr}({\mathcal{S}}_1{t}^c{\mathcal{S}}_1^{†}{t}^a{t}^b)\mathrm{tr}({\mathcal{S}}_3{t}^d{\mathcal{S}}_3^{†}{t}^a{t}^b)$$

$$\mathrm{tr}({\mathcal{S}}_1{t}^c{\mathcal{S}}_1^{†}{t}^a{\mathcal{S}}_2{t}^d{\mathcal{S}}_2^{†}{t}^b)\mathrm{tr}({\mathcal{S}}_3{t}^e{\mathcal{S}}_3^{†}{t}^a{t}^b)$$

$$\mathrm{tr}({\mathcal{S}}_1{t}^c{\mathcal{S}}_1^{†}{t}^a{\mathcal{S}}_2{t}^d{\mathcal{S}}_2^{†}{t}^b)\mathrm{tr}({\mathcal{S}}_3{t}^e{\mathcal{S}}_3^{†}{t}^a{\mathcal{S}}_4{t}^f{\mathcal{S}}_4^{†}{t}^b)$$

$$\stackrel{N_w=2}{=}-\frac{1}{16}{\mathcal{S}}_{12}^{cd}{\mathcal{S}}_{34}^{ef}+\frac{1}{8}{\mathcal{S}}_{13}^{ce}{\mathcal{S}}_{24}^{df}+\frac{1}{8}{\mathcal{S}}_{14}^{cf}{\mathcal{S}}_{23}^{de},$$

$${\mathcal{S}}_{12}^{cd}$$

$${\mathcal{S}}_{123}^{cde}$$

$${\mathcal{S}}_{1234}^{cdef}$$

$$\displaystyle{\mathcal{S}}_{1234}^{cdef}\stackrel{{\scriptstyle N_{w}=2}}{{=}}\tfrac{1}{2}\big{(}{\mathcal{S}}_{12}^{cd}{\mathcal{S}}_{34}^{ef}-{\mathcal{S}}_{13}^{ce}{\mathcal{S}}_{24}^{df}+{\mathcal{S}}_{14}^{cf}{\mathcal{S}}_{23}^{de}\big{)}\,.$$

$$O_{\ell q}^{(3)†}{O}_{\ell q}^{(3)}$$

Full text

††preprint:

Nikhef 2017-067

1 Introduction

At LHC energies, the effect of electroweak (EW) corrections on the cross section can be significant ($\sim 10$%). These are dominated by EW Sudakov double logarithms,

[TABLE]

where $\sigma_{0}$ is the Born cross section, $\alpha_{w}$ is a weak coupling constant ($\alpha_{2}$ or $\alpha_{1}$), $Q$ denotes the hard scale (typically taken to be the partonic center of mass energy $\sqrt{\hat{s}}$) and $M$ is an electroweak scale such as $M_{W}$, $M_{Z}$, $m_{H}$, $m_{t}$, which we consider to be of the same parametric size.111Of course there are also mixed QCD-EW corrections, and we will consider their interplay in the analysis. The energy dependence of EW corrections makes it important to include them when searching for new physics in tails of distributions. It also highlights that these effects are indispensable for cross section predictions at a FCC, where EW logarithms are truly large Mangano:2016jyj , and make order one corrections to the cross section. EW Sudakov logarithms also play an important role in calculations of WIMP dark matter, see e.g. refs. Ciafaloni:2010ti ; Baumgart:2014vma ; Bauer:2014ula ; Ovanesyan:2014fwa ; Baumgart:2017nsr .

Most studies of EW logarithms focus on virtual effects. The underlying assumption is that one is fully exclusive, i.e. all real EW radiation is resolved by detectors. This is not unreasonable because the $W$ and $Z$ boson are massive, and can be tagged experimentally via their decay products. Electroweak Sudakov logarithms were first studied in refs. Ciafaloni:1998xg ; Kuhn:1999de ; Fadin:1999bq ; Beenakker:2000kb ; melles1 , a recipe for the next-to-leading order (NLO) corrections was presented in refs. Denner:2000jv ; Denner:2001gw ; Denner:2006jr and the two-loop logarithms for a four-fermion process were obtained in refs. Kuhn:2001hz ; Jantzen:2005az . Refs. Chiu:2007yn ; Chiu:2007dg developed a resummation framework using Soft-Collinear Effective Theory (SCET) Bauer:2000ew ; Bauer:2000yr ; Bauer:2001ct ; Bauer:2001yt , obtaining results at next-to-leading-logarithmic (NLL) plus NLO accuracy. The effect of real radiation can be important, and was studied at NLO in e.g. refs. Baur:2006sn ; Bell:2010gi ; Stirling:2012ak ; Manohar:2014vxa .

In this paper we start from the opposite extreme, considering inclusive processes. One example is Drell-Yan, $pp\to\ell\bar{\ell}X$, where $X$ is unconstrained, illustrated in fig. 1. When the lepton pair has a small transverse momentum or has a large invariant mass (threshold limit), the cross section contains large double logarithms. Here we will not focus on these regions of phase space, so the QCD corrections do not involve double logarithms. Nevertheless, because the proton is not an electroweak singlet, EW double logarithms remain present Ciafaloni:2000df ; Ciafaloni:2000rp , which is one of the salient features of our analysis. In this paper, we develop a framework to perform the resummation of EW logarithms using a factorization theorem that is valid to all-orders in perturbation theory. Important ingredients for resummation are the collinear splitting functions, which were determined at leading order in refs. Ciafaloni:2001mu ; Ciafaloni:2005fm . These have been implemented into a parton shower Christiansen:2014kba ; Krauss:2014yaa ; Chen:2016wkt and used to resum initial-state radiation by including them in the evolution of parton distribution functions (PDFs) Bauer:2017isx ; Bauer:2017bnh . Our calculation gives the same result for the Sudakov double logarithms as ref. Bauer:2017isx , but we also consider final-state radiation and extend to NLL. Interestingly, we will see that the splitting functions alone are not enough to account for all the EW logarithms, and soft anomalous dimensions must also be included. We also show the importance of polarization effects for gauge bosons, which are a consequence of the chiral nature of $SU(2)$ and the helicity dependence of splitting functions, and were missed in earlier studies.

We achieve resummation using an effective field theory analysis, in the spirit of refs. Chiu:2007yn ; Chiu:2007dg . First the hard scattering is integrated out at the scale $Q$, matching onto an effective field theory in the symmetric phase of $SU(2)\times U(1)$. We then factorize the cross section and use the renormalization group evolution to evolve to the low scale $M$, thereby resumming EW logarithms. Only at the low scale $M$ do we switch to the broken phase. Anomalous dimensions are related to ultraviolet divergences and do not depend on symmetry breaking, which is an infrared effect. The collinear initial- and final-state radiation will be resummed using the DGLAP evolution Gribov:1972ri ; Altarelli:1977zs ; Dokshitzer:1977sg of the corresponding PDFs and fragmentation functions (FFs). Surprisingly, for the nonsinglet PDFs there is also a sensitivity to soft radiation. This introduces rapidity divergences, and we use the rapidity renormalization group Chiu:2011qc ; Chiu:2012ir to resum the corresponding single logarithms of $Q/M$. We calculate all ingredients necessary for resummation at NLL and provide an explicit recipe on how to implement them for $2\to 2$ processes in the appendix.

We end the paper by discussing a range of generalizations: {itemize*}

Resummation beyond NLL.

Other processes.

Kinematic hierarchies which arise when not all of the Mandelstam invariants are of order $Q$.

Jets identified (inclusively) using a jet algorithm

Less inclusive processes where radiation within the range of the detectors is observed, but radiation near the beam axis is not.

The outline of our paper is as follows. Our factorization analysis, which splits the cross section into collinear and soft parts, is described in sec. 2. The renormalization group equations for the collinear sector are given in sec. 3, and for the soft sector in 4. The matching onto the broken phase of the gauge theory is presented in sec. 5. The evolution from the hard scale $Q$ to the electroweak scale $M$ accomplishes the resummation of electroweak logarithms of $Q/M$, as discussed in sec. 6. In sec. 7 we show how our results compare to electroweak resummation for PDFs in the literature. We discuss the generalizations listed above in sec. 8, and conclude in sec. 9. For readers mostly interested in the final results, we provide a recipe to include electroweak resummation at NLL accuracy in appendix A. In appendix B, we give examples of the possible PDF combinations which enter the production of a heavy particle in quark-antiquark annihilation.

2 Factorization

In this section we present our framework for resumming electroweak logarithms in inclusive cross sections, considering as an example deep-inelastic neutrino scattering $\nu p\to\ell X$, for which a tree-level diagram is shown in fig. 2. We start, in sec. 2.1, by integrating out the short-distance scattering at the hard scale $Q$. Here we can work in the symmetric phase of the gauge theory, since $Q\gg M\sim M_{W},M_{Z},m_{h},m_{t}$. The scattering amplitude can be factored into a coefficient and hard scattering operator. We discuss the factorization of the hard-scattering operator into collinear and soft operators in sec. 2.2, which allows one to sum collinear and soft logarithms using RGEs. The gauge and spin indices are disentangled in sec. 2.3, allowing one to write the scattering amplitude for any process in terms of a standard basis of collinear and soft operators.

In previous work on electroweak resummation using SCET in refs. Chiu:2007dg ; Chiu:2007yn ; Chiu:2008vv ; Chiu:2009ft ; Chiu:2009mg ; Chiu:2009yx ; Chiu:2011qc ; Chiu:2012ir ; Fuhrer:2010eu ; Fuhrer:2010vi , the final state was assumed to consist of particles and jets with masses smaller than the electroweak scale such that only virtual electroweak corrections needed to be taken into account. This allows for the entire analysis from $Q$ down to $M$ to be carried out at the amplitude level. In this paper, we are interested in inclusive cross sections where we sum over final states with masses larger than the electroweak scale (e.g. in semi-inclusive cross sections), so we square the amplitude and factorize the cross section above the electroweak scale.

2.1 Matching at the hard scale

The sample process we study is given by lepton-quark scattering, and the hard scattering operators that contribute are

[TABLE]

at leading power in $M/Q\ll 1$. The electroweak doublet fields $\ell=(e_{L},\nu_{L})$ and $q=(u_{L},d_{L})$ are left-handed, the electroweak singlet fields $e=e_{R}$, $u=u_{R}$ and $d=d_{R}$ are right-handed, and $t^{a}$ are the $SU(2)$ generators. For quark-quark scattering, one can also have operators such as $(\bar{q}_{1}\gamma^{\mu}T^{A}q_{2})\,(\bar{q}_{3}\gamma_{\mu}T^{A}q_{4})$ which involve the color generators $T^{A}$, or $(\bar{q}_{1}\gamma^{\mu}t^{a}T^{A}q_{2})\,(\bar{q}_{3}\gamma_{\mu}t^{a}T^{A}q_{4})$ which involve both weak and color generators.

The subscripts on the fields indicate their momentum, e.g. $\bar{e}_{1}$ has momentum $p_{1}$. This is important because the hard-matching coefficients $\mathcal{H}_{i}$ depend on $p_{i}$,

[TABLE]

We will use the convention that all momenta are incoming. Thus an outgoing particle has momentum $p$ with $p^{0}<0$. This convention avoids certain minus signs between incoming and outgoing particles in subsequent results, and allows us to treat both with a unified notation. The field $\bar{e}_{1}$ contributes to processes with either an outgoing right-handed electron or an incoming left-handed positron, and the two are distinguished by the sign of $p_{1}^{0}$.

At tree-level, $O^{(3)}_{\ell q}$ is generated by $SU(2)$ gauge boson exchange. The other operators in eq. (2.1), which we denote by $O_{AB}$ with $A=\ell,e$ and $B=q,u,d$, are due to the exchange of a $U(1)$ gauge boson. This leads to the matching coefficients

[TABLE]

where $g_{2}$ and $g_{1}$ are the $SU(2)$ and $U(1)$ couplings, and $\mathsf{y}_{A}$ and $\mathsf{y}_{B}$ are the $U(1)$ hypercharge of the fields $A$ and $B$. Since $p_{1}\cdot p_{2}\sim Q^{2}\gg M^{2}$, gauge boson masses in the propagator are power suppressed, and have been dropped. For example, for neutrino-proton scattering via $\nu q\to e^{-}X$, the hard-scattering at tree level is given by

[TABLE]

Because we will carry out the collinear and soft evolution to the scale $Q$ of the hard scattering, we do not have to calculate the matching coefficients $\mathcal{H}_{i}$ at one loop to achieve resummation at NLL accuracy.

After integrating out the hard gauge boson to obtain eq. (3), only collinear and soft fluctuations of the fields remain. These can be described using SCET, where the Lagrangian ${\mathcal{L}}_{\rm SCET}$ encodes the dynamics of the collinear and soft fields. For our analysis, we do not need all the technical details of SCET, so we present the discussion in terms of soft and collinear corrections, which should be accessible to a wider audience.222That is, we use pseudo-SCET analogous to pseudocode in computer science.

We will make frequent use of the following light-like vectors for incoming particles,

[TABLE]

where the unit vector $\mathbf{n}_{i}$ points along the direction of $\mathbf{p}_{i}$. For outgoing particles with energy $E_{j}$ and momentum $\mathbf{p}_{j}$, our convention is that $p_{j}=(-E_{j},-\mathbf{p}_{j})$, and

[TABLE]

where $\mathbf{n}_{j}$ is a unit vector in the direction $\mathbf{p}_{j}$.

The matching in eq. (3) removes fluctuations of virtuality $\gtrsim Q$, and the full gauge invariance of the theory is replaced by collinear gauge invariance for each collinear direction, as well as soft gauge invariance Bauer:2001yt . Each field in eq. (2.1) corresponds to a distinct collinear direction, so it must be (made) collinearly gauge invariant by itself. This is accomplished by including collinear Wilson lines in the definitions of fields Bauer:2001ct . By including soft Wilson lines, the interactions between collinear and soft fields can be removed from the Lagrangian Bauer:2001yt , and included in the hard scattering operator. For example, the incoming field $q_{4}$ in eq. (2.1) is short-hand for a collinear fermion field $\psi_{4}$ (typically denoted by $\xi_{n_{4}}$ in SCET) combined with a collinear Wilson line $\mathcal{W}$ in the ${\bar{n}}_{4}$ direction and a soft Wilson line ${\mathcal{S}}$ in the $n_{4}$ direction, (using the covariant derivative convention $D_{\mu}=\partial_{\mu}+igA_{\mu}$)

[TABLE]

$A_{n_{4}}$, $W_{n_{4}}$ and $B_{n_{4}}$ denote $SU(3)$, $SU(2)$ and $U(1)$ gauge fields whose momenta are collinear to the $n_{4}$ direction, and $A_{s},W_{s},B_{s}$ denote soft gauge fields.333The $SU(2)$ gauge field $W$ should not be confused with $\mathcal{W}$, a collinear Wilson line. The Wilson lines $\mathcal{W}$ and ${\mathcal{S}}$ depend on the gauge representation of the particle. The soft Wilson line integral is over the worldline of the particle. For incoming particles, the soft Wilson line integral is from $t=-\infty$ to $t=0$. With our sign convention, eq. (2.1) also holds for outgoing particles, and the minus sign in eq. (7) converts the integral from $t=-\infty$ to $t=0$ into one from $t=0$ to $t=\infty$. The direction of the soft Wilson line affects the sign of $\mathrm{i}0$ terms in the eikonal propagators, and the sign of scattering phases.

The interaction of the collinear fields $\mathcal{W}_{4}$ and $\psi_{4}$ in eq. (2.1), which is given by the full QCD interaction for particles in the $n_{4}$ direction Goerke:2017ioi , leads to the production of a jet of particles in the $n_{4}$ direction, with invariant mass much smaller than $Q$. The soft Wilson line ${\mathcal{S}}_{4}$ sums the emission of soft radiation from the collinear fields, so the collinear fields no longer interact with soft fields in this picture. To avoid additional notation, in the remainder of the paper $q_{4}$ will denote the collinear part of the right-hand side of eq. (2.1), $\mathcal{W}_{4}^{\dagger}\psi_{4}\to q_{4}$, so that $q_{4}$ in eq. (2.1) is now denoted by $q_{4}\to S_{4}q_{4}$.

2.2 Factorization into collinear and soft

Eq. (2.1) factors the operator describing the hard interaction in eq. (2.1) into different collinear sectors and a soft sector that no longer interact. The cross section is given by taking the matrix element of the hard scattering in eq. (3) with initial and final-state particles, squaring, and including the phase-space integration, flux factor and measurement. This is largely an exercise in bookkeeping, where most of the complications arise from the phase space, and leads to the usual factorization theorems for hard scattering processes in QCD.

Schematically, the cross section for $\nu p\to e^{-}X$ is given by

[TABLE]

The fields in ${\mathcal{L}}_{\text{hard}}$ are the product of soft and collinear terms (see eqs. (3) and (2.1)), and the matrix element in eq. (9) is factored into the product of soft and collinear matrix elements, by writing $\ket{e^{-}X}$ as the product $\ket{X_{s}}\otimes\ket{X_{c,1}}\otimes\ket{X_{c,2}}\ldots$ of soft particles and collinear particles in different collinear sectors in the final state. For $\nu p\to e^{-}X$, there are four sectors given by the directions of $\nu$, $p$, $e^{-}$ and the outgoing jet produced by the struck quark. Using only the $O_{\ell q}^{\dagger}O_{\ell q}$ contribution to eq. (9) as an example, the relevant matrix element is

[TABLE]

Here the indices $\alpha_{1},\dots,\delta_{4}$ include both spin and gauge indices. Since the hard interaction eq. (3) has a sum over the momenta of the colliding partons weighted with a hard coefficient $H(\{p_{k}\})$, the labels on the fields in $O^{\dagger}$ have been distinguished from those in $O$ by a prime. Eventually these will be equal because of momentum conservation in the matrix elements.

Eq. (2.2) has factored the total cross section into collinear sectors corresponding to the incoming proton and neutrino, outgoing lepton (in $X_{1}$) and jet (in $X_{3}$), and the soft sector. This factorization is what enables the resummation of logarithms of $Q/M$, by separating the ingredients at different invariant mass and rapidity scales, as discussed in sec. 6. In the next section we show how to disentangle the gauge/spin indices for all combinations of hard-scattering operators. Since we only probe the hard scattering kinematics, the collinear matrix elements will correspond to parton distributions functions for incoming directions and fragmentation functions for outgoing directions, as is the case in QCD factorization for inclusive cross sections. (We avoid kinematic limits, such as small transverse momentum, which would require a transverse momentum dependent parton distribution or fragmentation function.) What is perhaps surprising is the appearance of a soft function, since it would seem that soft radiation is not directly probed by the measurement, i.e. we are not in a kinematic limit that makes the measurement sensitive to soft radiation. For the QCD corrections, color conservation forces the hadronic matrix elements of quark operators to be diagonal in color, leading to color contractions of indices on the soft Wilson lines. Because the observables we consider do not directly probe the soft radiation, we can perform the sum over $\ket{X_{s}}$ and then find that the soft matrix element is the identity by using ${\mathcal{S}}_{i}^{\dagger}{\mathcal{S}}_{i}=1$. The fundamental difference in the electroweak case is that electroweak symmetry is broken, so the matrix elements do not have to be diagonal in electroweak indices,

[TABLE]

The proton matrix element of $q_{4}$ in eq. (2.2) gives the sum of left-handed $u$ and $d$ quark PDFs in the proton. Similarly, the neutrino matrix element corresponds to a lepton PDF in the neutrino. Because the neutrino does not have QCD or QED interactions, this PDF is a delta function at tree-level at the electroweak scale. The matrix element of $q_{3}$ reduces to a quark jet function, after summing on $X_{3}$. The matrix element involving $\ell_{1}$ would reduce to a lepton (electroweak) jet function if one sums over all $X_{1}$. However, in DIS the energy and direction of the outgoing electron are measured, so one sums over $\ket{X_{1}}=\ket{e(p_{e}),X}$ where $p_{e}$ is measured. This corresponds to a fragmentation function, as it only probes the energy $p_{e}^{0}$ (the electron is collinear to the field $\ell_{1}$). On the other hand, the soft function is sensitive to the direction of $p_{e}$ but not its energy. Thus we have factorized the cross section into collinear and soft pieces which can be studied independently.

There is a subtlety in eq. (2.2). The soft Wilson lines have been written as ${\mathcal{S}}$ or ${\mathcal{S}}^{\dagger}$ depending on whether they arose from the field $\psi$ or $\overline{\psi}$. This keeps track of the gauge indices in the Wilson lines. The Wilson lines in $O_{H}$ give the scattering amplitude, whereas those in $O_{H}^{\dagger}$ give the complex conjugate of the amplitude. Thus the Wilson lines from $O_{H}$ are time-ordered, whereas those from $O_{H}^{\dagger}$ are anti-time-ordered. In eq. (2.2), $S_{1}^{\dagger}$, $S_{2}$, $S_{3}^{\dagger}$ and $S_{4}$ are time-ordered, whereas $S_{1}$, $S_{2}^{\dagger}$, $S_{3}$ and $S_{4}^{\dagger}$ are anti-time-ordered. We will not carefully keep track of this in our notation, because our calculation of the anomalous dimension in sec. 4 shows that this is irrelevant.

2.3 Disentangling gauge and spin indices

The next step is to disentangle the spin and gauge indices on the fermion fields in the product of two operators $O^{\dagger}O$, which enter the factorization formula eq. (2.2). This can be achieved by using the relations

[TABLE]

Here $\alpha,\beta$ are spinor indices, $i,j$ are $SU(2)$ gauge indices, and $N_{w}=2$. The lepton fields are treated as massless and assumed to correspond to the same collinear direction $n$. There are similar relations for the quarks. Eventually, we will take the proton matrix element of the quark operators. Since color is an unbroken gauge symmetry, and the proton is a color singlet state, matrix elements of color non-singlet operators in the proton vanish. We therefore drop these from the outset.

We start with the most complicated case, namely $O^{(3)\dagger}_{\ell q}O^{(3)}_{\ell q}$:

[TABLE]

We can use eq. (2.3) to combine $\bar{\ell}_{1}$ and $\ell_{1}$ into a bilinear, $\bar{\ell}_{2}$ and $\ell_{2}$ into a bilinear, etc., and drop color non-singlet operators to obtain

[TABLE]

Here we introduce the abbreviation

[TABLE]

where the superscript distinguishes the gauge group representations of the collinear operator: $\mathcal{C}_{\ell_{1}}$ is a weak singlet, and $\mathcal{C}_{\ell_{1}}^{c}$ is a weak triplet.

We simplify the soft operators using the $SU(N_{w})$ completeness relation

[TABLE]

The relevant identities are

[TABLE]

where the last one only holds for $SU(2)$. Here we used ${\mathcal{S}}_{i}^{\dagger}{\mathcal{S}}_{i}=1$ and introduced the shorthand notation

[TABLE]

For $N_{w}=2$, the last relation in eq. (17) was simplified using

[TABLE]

Using the above relations gives

[TABLE]

We reiterate that a color-adjoint collinear operator of the form $\mathcal{C}_{q_{4}}^{A}=\bar{q}_{4}\tfrac{\bar{n}\!\!\!/_{4}}{2}T^{A}q_{4}$, where $T^{A}$ is a color generator, would never have been considered in QCD. Although it could in principle be kept in intermediate steps of the calculation, it would be dropped at the end because its proton matrix element vanishes, since the proton is a color-singlet state. However, the proton is not an electroweak singlet and gives a nonzero matrix element for the $SU(2)$ adjoint operator $\mathcal{C}_{q_{4}}^{a}=\bar{q}_{4}\tfrac{\bar{n}\!\!\!/_{4}}{2}t^{a}q_{4}$, where $t^{a}$ is an $SU(2)$ generator, see sec. 5. Related to this, we note that only the $SU(2)$ Wilson lines survive in the soft operators in eq. (2.3), since the colored Wilson lines are paired with colored operators which have vanishing proton matrix elements. The new features in the remaining discussion therefore center on $SU(2)$. For $SU(3)$ and $U(1)$ we have the standard PDF and fragmentation function evolution for the collinear operators and we have no soft operators.

Next we consider the interference contribution $O_{\ell q}^{\dagger}O^{(3)}_{\ell q}$ (and its conjugate $O_{\ell q}^{(3)\dagger}O_{\ell q}$), which can be obtained from eq. (13) by dropping the $t^{b}$’s

[TABLE]

This can be simplified using

[TABLE]

to get

[TABLE]

The expressions for $O_{\ell q}^{\dagger}O_{\ell q}$ can directly be obtained from eq. (13), dropping $t^{a}$ and $t^{b}$,

[TABLE]

For $O_{\ell u}^{\dagger}O_{\ell u}$ there is a further simplification compared to eq. (2.3) because the $SU(2)$ doublet $q$ is replaced by the singlet $u$,

[TABLE]

The expression for $O_{\ell d}^{\dagger}O_{\ell d}$ can directly be obtained from eq. (2.3) by replacing $u\to d$, and the expression for $O_{eq}^{\dagger}O_{eq}$ follows from interchanging $q\leftrightarrow\ell$ and $e\leftrightarrow u$. Finally, for $O_{eu}^{\dagger}O_{eu}$

[TABLE]

and similarly for $O_{ed}^{\dagger}O_{ed}$.

The above identities can be used to write the factorized cross section eq. (2.2) as a product of collinear and soft factors, which we now study separately. The collinear factors are the usual PDFs. The soft factors do not arise in QCD factorization theorems, but are present in electroweak cross sections. In appendix B, we give examples of the possible PDF combinations which enter the production of a heavy particle in quark-antiquark annihilation, and show that they are all given in terms of the singlet and triplet PDFs.

3 Collinear evolution

In this section we determine the renormalization group (RG) evolution of the collinear operators entering the factorized cross section. The splitting functions for $z<1$ agree with those computed in ref. Bauer:2017isx . We begin with the collinear operators corresponding to the incoming particles, discussing the fragmentation case in sec. 3.8. The collinear operators that enter the cross section can be written in terms of the usual PDF operators. In QCD processes, PDF operators are singlets under the $SU(3)$ gauge group, and there are no rapidity divergences, as these cancel between real and virtual graphs. In the electroweak case, the factorization formula has terms involving the product of collinear and soft operators which are not separately gauge singlets, as we have seen in the previous section. There are rapidity divergences in the collinear and soft sectors. One can see this must be true in the collinear sector by noting that real and virtual graphs have different group theory factors for gauge non-singlet PDFs.

Before discussing the electroweak case, we first review QCD, for which the standard definitions for the unpolarized PDF quark operator is Collins:1981uw

[TABLE]

Here the Wilson line $\mathcal{W}$ is defined in eq. (2.1) and the null vectors are

[TABLE]

Note that the operator product in eq. (26) is an ordinary product, not a time-ordered product. One can insert a complete set of states between $\mathcal{W}$ and $\mathcal{W}^{\dagger}$, allowing us to evaluate matrix elements of the PDF operator using cut Feynman rules. The PDF operators in eq. (26) are written using standard QCD notation. In terms of collinear fields introduced in eq. (2.1), the quark PDF operator is given by $[W^{\dagger}Q]\to Q$, since the collinear Wilson line was included in $Q$. The quark PDF is given by the matrix elements of this operators in a target state $T$ of momentum $p$,

[TABLE]

where $p^{-}={\bar{n}}\!\cdot\!p$ and the operators are renormalized in the $\overline{\text{MS}}$ scheme. The polarized quark PDF $f_{\Delta Q}$ is given by replacing $\not{\bar{n}}$ by $\not{\bar{n}}\gamma_{5}$. In terms of $f_{Q_{+}}$ and $f_{Q_{-}}$, the distributions of quarks with helicity $h=+1/2$ and $h=-1/2$, the unpolarized and polarized PDFs are $f_{Q}=f_{Q_{+}}+f_{Q_{-}}$ and $f_{\Delta Q}=f_{Q_{+}}-f_{Q_{-}}$.

We need to generalize eq. (26), eq. (28) to include PDFs which are not gauge singlets. The gauge indices in a fermion bilinear $\bar{Q}\,Q$ can be combined into a gauge singlet or adjoint. We thus define two different fermion operators

[TABLE]

where $i,j,k,l$ are gauge indices in the fundamental representation. Note that it is the gauge indices at the $x=\infty$ end of the Wilson line which are combined into a singlet or adjoint. The anti-fermion and anti-scalar PDFs are given by $CP$ conjugation, where $P$ is implemented by reflection in a plane containing the direction of the proton $n$. This amounts to exchanging $\bar{Q}\leftrightarrow Q$ and letting $t^{a}\to-(t^{a})^{T}$ in the adjoint PDFs,

[TABLE]

which can also be written

[TABLE]

on anticommuting the fermion fields. The $Q$ and $\bar{Q}$ PDFs have been defined in the conventional way, such that for example the deep-inelastic structure functions are proportional to $f_{Q}(x,\mu)+f_{\bar{Q}}(x,\mu)$. If the singlet PDF is treated as the PDF for quark number, then the total baryon number is given by the difference of the $Q$ and $\bar{Q}$ PDFs with the sign convention of eq. (3). The triplet PDFs have an $SU(2)$ charge, and this requires the additional minus sign, so that the total $SU(2)$ charge is given by the sum of $Q$ and $\bar{Q}$ triplet PDFs. The matrix elements in a target $T$ of momentum $p$ define the singlet and adjoint PDFs

[TABLE]

The singlet PDF for QCD is the same as the usual PDF. The adjoint PDF vanishes for QCD, but not in the electroweak case.

The generalization of eqs. (3), (3) and (3) to the electroweak case is straightforward. Since the weak interactions are chiral, it is necessary to use polarized PDFs. One defines fermion and antifermion PDFs using eqs. (3) and (3) with the Wilson line in the appropriate $SU(3)\times SU(2)\times U(1)$ representation, and the field $Q$ replaced by the chiral fields $q,l,u,d,e$. For left-handed fields $q$ and $l$, this gives the distribution of particles with helicity $h=-1/2$, and for right-handed fields, the distribution of particles with helicity $h=+1/2$. In terms of the usual polarized and unpolarized quark distributions, the PDF in eq. (26) with $Q\to u$ corresponds to $(f_{u}+f_{\Delta u})/2=f_{u_{+}}$, and with $Q\to q$ corresponds to $(f_{u}-f_{\Delta u})/2+(f_{d}-f_{\Delta d})/2=f_{u_{-}}+f_{d_{-}}$, etc. where the $\pm$ subscript denotes the helicity (not chirality) of the quark. For antiquarks, the relation between helicity and chirality is reversed, so the antiquark PDFs for $Q\to u$ and $Q\to q$ give $(f_{\bar{u}}-f_{\Delta\bar{u}})/2=f_{\bar{u}_{-}}$ and $(f_{\bar{u}}+f_{\Delta\bar{u}})/2+(f_{\bar{d}}+f_{\Delta\bar{d}})/2=f_{\bar{u}_{+}}+f_{\bar{d}_{+}}$, respectively. In the unbroken theory, the PDFs are defined using chiral SM fields, and we will use the notation $f_{u}$, $f_{q}$, $f_{\bar{u}}$, $f_{\bar{q}}$ to denote these four PDFs, dropping the helicity label. In the electroweak sector, we will also denote singlet and adjoint PDFs using superscripts $(I=0)$ and $(I=1)$.

We also need the unpolarized and polarized gluon PDF operators defined by Collins:1981uw ; Manohar:1990kr ; Manohar:1990jx

[TABLE]

($\widetilde{G}_{\alpha\beta}=\frac{1}{2}\epsilon_{\alpha\beta\gamma\delta}G^{\gamma\delta}$ with $\epsilon_{0123}=+1$) whose matrix elements give the PDFs $f_{G_{+}}+f_{G_{-}}$ and $f_{G_{+}}-f_{G_{-}}$. It is more convenient to use $f_{G_{+}}$ and $f_{G_{-}}$, the distribution of helicity $h=1$ and $h=-1$ gauge bosons, given by the sum and difference of the equations in eq. (3).444 ${\bar{n}}_{\mu}G^{\mu\lambda}(\bar{n}\xi)\,{\bar{n}}_{\nu}{\widetilde{G}}^{\nu}{}_{\lambda}(0)=-{\bar{n}}_{\mu}{\widetilde{G}}^{\mu\lambda}(\bar{n}\xi)\,{\bar{n}}_{\nu}{G}^{\nu}{}_{\lambda}(0)$ and ${\bar{n}}_{\mu}G^{\mu\lambda}(\bar{n}\xi)\,{\bar{n}}_{\nu}G^{\nu}{}_{\lambda}(0)={\bar{n}}_{\mu}{\widetilde{G}}^{\mu\lambda}(\bar{n}\xi)\,{\bar{n}}_{\nu}{\widetilde{G}}^{\nu}{}_{\lambda}(0)$ so the other two possibilities for replacing $G^{\mu\lambda}\to\widetilde{G}^{\mu\lambda}$ do not lead to new PDFs. There are also double helicity-flip gauge PDFs which are leading twist Manohar:1988pp ; Jaffe:1989xy . These correspond to a transition where a helicity $h=\pm 1$ gauge boson is emitted and a helicity $h=\mp 1$ gauge boson is absorbed. Since gauge boson helicity changes by two, there has to be a corresponding change in helicity of the target hadron; as a result these operators only contribute to scattering off targets with spin $\geq 1$, and we neglect them here. They occur in the factorization theorem through box graphs.

The transverse $W$ PDF is given by replacing the gluon field-strength tensor by the $SU(2)$ field-strength, and using a Wilson line in the adjoint of $SU(2)$. The transverse $B$ PDF is given by using $B_{\mu\nu}$, and no Wilson line is required since a $U(1)$ field-strength is a gauge singlet. The gauge operator involves two adjoint fields, and

[TABLE]

where the first four representations are in the symmetric product of the two adjoints, and the last three are in the antisymmetric product. The representation $\bar{a}a$ is a traceless tensor $t^{ab}_{cd}$ antisymmetric in its lower and in its upper indices, $\bar{a}s$ is a traceless tensor $t^{ab}_{cd}$ antisymmetric in its lower and symmetric in its upper indices, etc. For the special case of $SU(3)$, $\bar{s}s=\mathbf{27}$, $\bar{a}s=\mathbf{10}$, $\bar{s}a=\mathbf{\overline{10}}$, and the $\bar{a}a$ does not exist. For the special case of $SU(2)$, $\bar{s}s$ is the isospin $I=2$ representation, $\bar{a}s$, $\bar{s}a$, $\bar{a}a$ do not exist, and $\text{adj}_{S}$ does not exist since the $d$-symbol vanishes. Further details on the group theory can be found in app. A of ref. Dashen:1994qi .

The various gauge operators are given by ($G_{\mu\lambda}$ denotes a generic field-strength tensor)

[TABLE]

where $a,b,c$ are gauge indices in the adjoint representation (upper vs. lower indices do not matter, since the adjoint representation is real). $\mathscr{C}_{ab}^{(R,c)}$ is a Clebsch-Gordan coefficient for combining the two adjoints into state $c$ of representation $R$ given in eq. (34). The Clebsch-Gordan coefficients for the singlet, and the symmetric and antisymmetric adjoints are

[TABLE]

The Clebsch-Gordan coefficients for $\bar{a}a$, $\bar{s}s$, $\bar{a}s$ and $\bar{s}a$ are given in ref. Dashen:1994qi . We also have the corresponding polarized PDFs given by replacing $G^{\nu}{}_{\lambda}(0)$ by $-\mathrm{i}\,{\widetilde{G}}^{\nu}{}_{\lambda}(0)$ in eq. (35).

For the $SU(2)\times U(1)$ case, there are some additional PDFs. There are two more isotriplet gauge PDFs,

[TABLE]

A Wilson line is not needed for the $U(1)$ field-strength tensor, since it is a gauge singlet. Taking the Hermitian conjugate gives $[O_{W\!B}^{(I=1,c)}(r^{-})]^{\dagger}=O_{BW}^{(I=1,c)}(r^{-})$. We also have the polarized versions of eq. (35), $O_{\Delta WB}$ and $O_{\Delta BW}$.

For the massive electroweak gauge bosons, eq. (35) and its polarized version only give the PDFs for $h=\pm 1$. We also need the PDFs for $h=0$ longitudinally polarized gauge bosons. As we now discuss, these PDFs can not be written as light-cone Fourier transforms of operators involving the field-strength tensor. For a massive $W$ moving in the $+z$ direction, i.e. $\hat{n}=(0,0,1)$, its momentum and polarization are

[TABLE]

The field strength tensor annihilates a $W$ with amplitude

[TABLE]

For helicity $h=\pm 1$ with polarization vectors $\epsilon_{\pm}$, $\mathcal{A}^{\mu\nu}$ has components of order $E$. For $h=0$,

[TABLE]

and

[TABLE]

Since the $p^{\mu}/M_{W}$ term does not contribute to $\mathcal{A}^{\mu\nu}$, $\mathcal{A}^{\mu\nu}$ has components of order $M_{W}/E$, which are suppressed by $M_{W}^{2}/E^{2}$ relative to transverse polarization, and so are not leading twist.

The longitudinal gauge boson PDFs are given in terms of scalar PDFs, using the Goldstone boson equivalence theorem chanowitz ; bohmbook . We use the PDF operators

[TABLE]

for the Higgs field, which is given by

[TABLE]

in the unbroken and broken phase, respectively. Here $h$ is the physical Higgs particle, and the unphysical scalars $\varphi^{3},\varphi^{\pm}=(\varphi^{1}\mp\mathrm{i}\varphi^{2})/\sqrt{2}$ in the Higgs multiplet can be related to longitudinal electroweak gauge bosons $Z_{L},W^{\pm}_{L}$ using the Goldstone boson equivalence theorem chanowitz ; bohmbook . The $CP$-conjugate $\bar{H}$ PDFs are given by $H\leftrightarrow\bar{H}$ and $t^{a}\to(-t^{a})^{T}$.

There are two additional Higgs PDFs $O_{\widetilde{H}H}^{(I=0)}$, $O_{\widetilde{H}H}^{(I=1,c)}$ given by replacing $H^{\dagger}$ in eq. (3) by $\widetilde{H}^{\dagger}$, where

[TABLE]

The $CP$-conjugate PDFs are given by swapping $H\leftrightarrow\widetilde{H}$, and $t^{a}\to(-t^{a})^{T}$. The operator $O_{\widetilde{H}H}^{(I=0)}$ breaks electromagnetism and doesn’t contribute to factorization formulae. $O_{\widetilde{H}H}^{(I=1,c)}$, $O_{H\widetilde{H}}^{(I=1,c)}$ do, and are included in our analysis. They have $Y=\pm 1$, and can occur in the factorization theorem in pairs. The $\Delta Q=0$ components have non-zero proton matrix elements. Taking Hermitian conjugates gives $[O_{\widetilde{H}H}^{(I=1,c)}(r^{-})]^{\dagger}=-O_{H\widetilde{H}}^{(I=1,c)}(r^{-})$.

3.1 Anomalous dimensions

We will first briefly review rapidity divergences, and the rapidity regulator of ref. Chiu:2011qc ; Chiu:2012ir , that we use to treat them. Rapidity divergences arise in e.g. transverse momentum dependent factorization theorems, where the emission of a single soft gluon involves an integral over its rapidity with a rapidity-independent (i.e. constant) integrand. To bring this divergence under control, the $\eta$ regulator of ref. Chiu:2011qc ; Chiu:2012ir explicitly breaks boost invariance. The resulting $1/\eta$ poles cancel between the collinear and soft operators. These poles lead to renormalization group equations involving the rapidity renormalization scale $\nu$. Just as the $\mu$-evolution can be used to resum invariant mass logarithms, the $\nu$-evolution can be used to resum rapidity logarithms, which arise because the collinear and soft operators have different natural rapidity scales. This will be discussed in more detail in sec. 6, see in particular fig. 3. In our case we do not measure the transverse momentum of the gauge boson, but instead have to account for the gauge boson mass.

The RG equations of the collinear operators take the form

[TABLE]

The lower limit on $z$ turns into $r^{-}/p^{-}$ when its matrix element is taken in a state with momentum $p$. The anomalous dimension $\gamma_{\mu}$ depends not just on $z$ but also on $r^{-}$ because of rapidity divergences. The collinear operators mix under $\mu$ evolution, but are multiplicatively renormalized under $\nu$ evolution. The convolution in eq. (3.1) will be abbreviated by $\otimes$. Since we limit ourselves to one-loop results, we find it convenient to use the notation

[TABLE]

in intermediate expressions. We now compute the anomalous dimensions $\gamma_{\mu}$ and $\gamma_{\nu}$.

The indices $i$ and $j$ in eq. (3.1) run over all the fermion and gauge boson PDFs. We will use the helicity basis $Q_{+}$, $Q_{-}$, $G_{+}$, $G_{-}$ rather than the more conventional basis $Q$, $\Delta Q$, $G$, $\Delta G$ used in QCD. In QCD, there is no $\nu$-evolution and parity invariance implies that the $\mu$-evolution kernels satisfy

[TABLE]

which allows one to write evolution equations which mix $\{f_{Q},\,f_{G}\}$, and separately mix $\{f_{\Delta Q},f_{\Delta G}\}$ using the kernels

[TABLE]

This simplification is not possible in the electroweak sector, since parity is not a good symmetry. We therefore write our results using the helicity basis $\gamma_{Q_{+}Q_{+}}$, $\gamma_{Q_{+}Q_{-}}$, etc.

3.2 $\gamma_{QQ}$ and $\gamma_{HH}$

The one-loop gauge diagrams and resulting contributions to the $QQ$ anomalous dimensions are shown in table 1. Their calculation was performed in section V of ref. Manohar:2012jr , using dimensional regularization for the UV divergences and the rapidity renormalization group Chiu:2011qc ; Chiu:2012ir to treat the rapidity divergences. The graphs are divided into two sets, with the sum of each set given in the table. The value of individual diagrams depends on the gauge choice, but $\text{Total}_{1}$ and $\text{Total}_{2}$ remain the same. The color factor $c_{QQ}(R)$ for $\text{Total}_{1}$ depends on the PDF representation, whereas the color factor for $\text{Total}_{2}$ is $c_{F}$.

The one-loop graphs in Table 1 conserve fermion helicity, implying that the helicity mixing terms $\gamma_{Q_{+}Q_{-}}$ and ${\gamma }_{{}_{}}$