A Recipe for electroweak resummation in inclusive processes at NLL

# Electroweak Logarithms in Inclusive Cross Sections

## Abstract

We develop the framework to perform all-orders resummation of electroweak logarithms of for inclusive scattering processes at energies much above the electroweak scale . We calculate all ingredients needed at next-to-leading logarithmic (NLL) order and provide an explicit recipe to implement this for 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 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.

\synctex

=1 \preprint Nikhef 2017-067 a]Aneesh V. Manohar, b,c]Wouter J. Waalewijn \affiliation[a]Department of Physics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA \affiliation[b]Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands \affiliation[c]Nikhef, Theory Group, Science Park 105, 1098 XG, Amsterdam, The Netherlands

## 1 Introduction

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

 σ=σ0∑m≤2ncnmαnwlnmQM, (1)

where is the Born cross section, is the weak coupling constant, denotes the hard scale (typically taken to be the partonic center of mass energy ) and is an electroweak scale such as , , , , which we consider to be of the same parametric size.1 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 [1], 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. [2, 3, 4, 5, 6].

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 and boson are massive, and can be tagged experimentally via their decay products. Electroweak Sudakov logarithms were first studied in refs. [7, 8, 9, 10, 11], a recipe for the next-to-leading order (NLO) corrections was presented in refs. [12, 13, 14] and the logarithms for a four-fermion process at NNLO was obtained in ref. [15]. Refs. [16, 17] developed a resummation framework using Soft-Collinear Effective Theory (SCET) [18, 19, 20, 21], 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. [22, 23, 24, 25].

In this paper we start from the opposite extreme, considering inclusive processes. One example is Drell-Yan, , where is unconstrained, illustrated in fig. 1. Because the proton is not an electroweak singlet, EW double logarithms remain present [26, 27], 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. [28, 29]. These have been implemented into a parton shower [30, 31, 32] and used to resum initial-state radiation by including them in the evolution of parton distribution functions (PDFs) [33, 34]. Our calculation gives the same result for the Sudakov double logarithms as ref. [33], 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 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. [16, 17]. First the hard scattering is integrated out at the scale , matching onto an effective field theory in the symmetric phase of . We then factorize the cross section and use the renormalization group evolution to evolve to the low scale , thereby resumming EW logarithms. Only at the low scale 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 [35, 36, 37] 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 [38, 39] to resum the corresponding single logarithms of . We calculate all ingredients necessary for resummation at NLL and provide an explicit recipe on how to implement them for 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 . 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 to the electroweak scale accomplishes the resummation of electroweak logarithms of , 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 . We start, in sec. 2.1, by integrating out the short-distance scattering at the hard scale . Here we can work in the symmetric phase of the gauge theory, since . 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. [17, 16, 40, 41, 42, 43, 38, 39, 44, 45], 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 down to 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

 O(3)ℓq =(¯ℓ1γμtaℓ2)(¯q3γμtaq4), Oℓq =(¯ℓ1γμℓ2)(¯q3γμq4), Oℓu =(¯ℓ1γμℓ2)(¯u3γμu4), Oℓd =(¯ℓ1γμℓ2)(¯d3γμd4), Oeq =(¯e1γμe2)(¯q3γμq4), Oeu =(¯e1γμe2)(¯u3γμu4), Oed =(¯e1γμe2)(¯d3γμd4), (2)

at leading power in . The electroweak doublet fields and are left-handed, the electroweak singlet fields , and are right-handed, and are the generators. For quark-quark scattering, one can also have operators such as which involve the color generators , or which involve both weak and color generators.

The subscripts on the fields indicate their momentum, e.g.  has momentum . This is important because the hard-matching coefficients depend on ,

 Lhard=∑i(∏k∫d4pk(2π)4)(2π)4δ4(∑mpm)Hi({pk})Oi({pk}). (3)

We will use the convention that all momenta are incoming. Thus an outgoing particle has momentum with . 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 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 .

At tree-level, is generated by gauge boson exchange, and by gauge boson exchange. This leads to the matching coefficients

 H(3)ℓq=ig222p1⋅p2,HAB=ig21yAyB2p1⋅p2, (4)

where and are the and couplings, and and are the hypercharge of the fields and . Since , gauge boson masses in the propagator are power suppressed, and have been dropped. For example, for neutrino-proton scattering via , the hard-scattering at tree-level given by

 ∑iHiOi =ig222p1⋅p2O(3)ℓq+ig212p1⋅p2[yℓyqOℓq+yℓyuOℓu+yℓydOℓd]. (5)

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 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.2

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

 ni=(1,ni),¯ni=(1,−ni), (6)

where the unit vector points along the direction of . For outgoing particles with energy and momentum , our convention is that , and

 nj=(−1,−nj),¯nj=(−1,nj), (7)

where is a unit vector in the direction .

The matching in eq. (3) removes fluctuations of virtuality , and the full gauge invariance of the theory is replaced by collinear gauge invariance for each collinear direction, as well as soft gauge invariance [21]. 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 [20]. By including soft Wilson lines, the interactions between collinear and soft fields can be removed from the Lagrangian [21], and included in the hard scattering operator. For example, the incoming field in eq. (2.1) is short-hand for a collinear fermion field (typically denoted by in SCET) combined with a collinear Wilson line in the direction and a soft Wilson line in the direction, (using the covariant derivative convention )

 q4(x) =S4∫d4xeip1⋅xW†4(x)ψ4(x), W4(x) =Pexp{−i∫0∞ds¯n4⋅[g3An4(x+s¯n4)+g2Wn4(x+s¯n4)+g1yqBn4(x+s¯n4)]}, S4 =Pexp{−i∫0−∞dsn4⋅[g3As(sn4)+g2Ws(sn4)+g1yqBs(sn4)]}. (8)

, and denote , and gauge fields whose momenta are collinear to the direction, and denote soft gauge fields.3 The Wilson lines and 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 to . With our sign convention, eq. (2.1) also holds for outgoing particles, and the minus sign in eq. (7) converts the integral from to into one from to . The direction of the soft Wilson line affects the sign of terms in the eikonal propagators, and the sign of scattering phases.

The interaction of the collinear fields and in eq. (2.1), which is given by the full QCD interaction for particles in the direction [46], leads to the production of a jet of particles in the direction, with invariant mass much smaller than . The soft Wilson line 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 will denote the collinear part of the right-hand side of eq. (2.1), , so that in eq. (2.1) is now denoted by .

### 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 is given by

 Extra close brace or missing open brace (9)

The fields in are the product of soft and collinear terms (see eq. (2.1)), and the matrix element in eq. (9) is factored into the product of soft and collinear matrix elements, by writing as the product of soft particles and collinear particles in different collinear sectors in the final state. For , there are four sectors given by the directions of , , and the outgoing jet produced by the struck quark. Using only the contribution to eq. (9) as an example, the relevant matrix element is

 (γμβ1γ1γμ,β2γ2γνβ3γ3γν,β4γ4)[∑X1⟨0|ℓ1′,δ3|X1⟩⟨X1|¯ℓ1,α1|0⟩][⟨ν|¯ℓ2′,α3ℓ2,δ1|ν⟩] [∑X3⟨0|u3′,δ4|X3⟩⟨X3|¯u3,α2|0⟩][⟨p|¯u4′,α4u4,δ2|p⟩] Extra open brace or missing close brace (10)

Here the indices 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 , the labels on the fields in have been distinguished from those in 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 ) and jet (in ), and the soft sector. This factorization is what enables the resummation of logarithms of , 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. 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. In the QCD factorization theorem, color conservation forces the hadronic matrix elements of quark operators to be diagonal in color. This leads to color contractions of indices on the soft Wilson lines, and the soft matrix element becomes the identity using . The fundamental difference in the electroweak case is that electroweak symmetry is broken, so the hadron matrix elements do not have to be diagonal in electroweak indices. The observables we consider do not directly probe the soft radiation, so summing over gives the soft function

 ⟨0|S1,γ3,δ3S†2,α3β3S3,γ4,δ4S†4,α4β4S†1,α1,β1S2,γ1δ1S†3,α2β2S4,γ2δ2|0⟩. (11)

The proton matrix element of in eq. (2.2) gives the right-handed quark PDF 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 reduces to a quark jet function, after summing on . The matrix element involving would reduce to a lepton (electroweak) jet function if one sums over all . However, in DIS the energy and direction of the outgoing electron are measured, so one sums over where is measured. This corresponds to a fragmentation function, as it only probes the energy (the electron is collinear to the field ). On the other hand, the soft function is sensitive to the direction of 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 or depending on whether they arose from the field or . This keeps track of the gauge indices in the Wilson lines. The Wilson lines in give the scattering amplitude, whereas those in give the complex conjugate of the amplitude. Thus the Wilson lines from are time-ordered, whereas those from are anti-time-ordered. In eq. (2.2), , , and are time-ordered, whereas , , and 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 , which enter the factorization formula eq. (2.2). This can be achieved by using the relations

 ¯ℓiαℓjβ =12Nwδji(PLn/)βα¯ℓ¯n/2ℓ+(ta)ji(PLn/)βα¯ℓ¯n/2taℓ, ¯eαeβ =12(PRn/)βα¯e¯n/2e. (12)

Here are spinor indices, are gauge indices, and . The lepton fields are treated as massless and assumed to correspond to the same collinear direction . 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.

 O(3)†ℓqO(3)ℓq =(¯ℓ2S†2γνtbS1ℓ1)(¯q4S†4γνtbS3q3)(¯ℓ1S†1γμtaS2ℓ2)(¯q3S†3γμtaS4q4). (13)

We can use eq. (2.3) to combine and into a bilinear, and into a bilinear, etc., and drop color non-singlet operators to obtain

 O(3)†ℓqO(3)ℓq =(n1⋅n3)(n2⋅n4)[1NcN4wCℓ1Cℓ2Cq3Cq4tr(tatb)tr(tatb) (14) +4NcN2wCcℓ1Cdℓ2Cq3Cq4tr(S1tcS†1taS2tdS†2tb)tr(tatb) +4NcN2wCℓ1Cℓ2Ccq3Cdq4tr(S3tcS†3taS4tdS†4tb)tr(tatb) +4NcN2wCcℓ1Cℓ2Cdq3Cq4tr(S1tcS†1tatb)tr(S3tdS†3tatb)+(3 more) +8NcNwCcℓ1Cdℓ2Ceq3Cq4tr(S1tcS†1taS2tdS†2tb)tr(S3teS†3tatb)+(3 more) +16NcCcℓ1Cdℓ2Ceq3Cfq4tr(S1tcS†1taS2tdS†2tb)tr(S3teS†3taS4tfS†4tb)].

Here we introduce the abbreviation

 Cℓ1=¯ℓ1¯n/12ℓ1,Ccℓ1=¯ℓ1¯n/12tcℓ1,…, (15)

where the superscript distinguishes the gauge group representations of the collinear operator: is a weak singlet, and is a weak triplet.

We simplify the soft operators using the completeness relation

 (ta)αβ(ta)γδ=12δαδδγβ−12Nwδαβδγδ. (16)

The relevant identities are

 tr(tatb)tr(tatb) =14(N2w−1), (17) tr(S1tcS†1taS2tdS†2tb)tr(tatb) =−14NwScd12, tr(S1tcS†1tatb)tr(S3tdS†3tatb) =−12NwScd13, tr(S1tcS†1taS2tdS†2tb)tr(S3teS†3tatb) =−14Nw(Scde123+Sced132), tr(S1tcS†1taS2tdS†2tb)tr(S3teS†3taS4tfS†4tb) =14N2wScd12Sef34+14Scf14Sde23−14Nw(Scdef1234+Scfed1432) \lx@stackrelNw=2=−116Scd12Sef34+18Sce13Sdf24+18Scf14Sde23,

where the last one only holds for . Here we used and introduced the shorthand notation

 Scd12 =tr(S1tcS†1S2tdS†2), Scde123 =tr(S1tcS†1S3teS†3S2tdS†2), Scdef1234 =tr(S1tcS†1S2tdS†2S3teS†3S4tfS†4). (18)

For , the last relation in eq. (17) was simplified using

 Extra open brace or missing close brace

Using the above relations gives

 O(3)†ℓqO(3)ℓq =(n1⋅n3)(n2⋅n4)[N2w−14NcN4wCℓ1Cℓ2Cq3Cq4 (19) −1NcN3wCcℓ1Cdℓ2Cq3Cq4Scd12−1NcN3wCℓ1Cℓ2Ccq3Cdq4Scd34 −2NcN3wCcℓ1Cℓ2Cdq3Cq4Scd13+(3 more) −2NcN2wCcℓ1Cdℓ2Ceq3Cq4(Scde123+Sced132)+(3 more) +1NcCcℓ1Cdℓ2Ceq3Cfq4(−Scd12Sef34+2Sce13Sdf24+2Scf14Sde23)].

We reiterate that a color-adjoint collinear operator of the form , where 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 adjoint operator , where is an generator, see sec. 5. Related to this, we note that only the 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 . For and 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 (and its conjugate ), which can be obtained from eq. (13) by dropping the ’s

 O†ℓqO(3)ℓq Extra open brace or missing close brace =(n1⋅n3)(n2⋅n4)[4NcN2wCcℓ1Cℓ2Cdq3Cq4tr(S1tcS†1ta)tr(S3tdS†3ta)+(3 more) +8NcNwCcℓ1Cdℓ2Ceq3Cq4tr(S1tcS†1taS2tdS†2)tr(S3teS†3ta)+(3 more) +16NcCcℓ1Cdℓ2Ceq3Cfq4tr(S1tcS†1taS2tdS†2)tr(S3teS†3taS4tfS†4)]. (20)

This can be simplified using

 tr(S1tcS†1ta)tr(S3tdS†3ta) =12Scd13, tr(S1tcS†1taS2tdS†2)tr(S3teS†3ta) =12Sced132, tr(S1tcS†1taS2tdS†2)tr(S3teS†3taS4tfS†4) =12Scfed1432−12NScd12Sef34 \lx@stackrelNw=2=−14Sce13Sdf24+14Scf14Sde23, (21)

to get

 O†ℓqO(3)ℓq =(n1⋅n3)(n2⋅n4)[2NcN2wCcℓ1Cℓ2Cdq3Cq4Scd13+(3 more) +4NcNwCcℓ1Cdℓ2Ceq3Cq4Sced132+(3 more) +4NcCcℓ1Cdℓ2Ceq3Cfq4(−Sce13Sdf24+Scf14Sde23)]. (22)

The expressions for can directly be obtained from eq. (13), dropping and ,

 O†ℓqOℓq Extra open brace or missing close brace =(n1⋅n3)(n2⋅n4)[1NcN2wCℓ1Cℓ2