Electroweak gauge boson parton distribution functions

Transverse and longitudinal electroweak gauge boson parton distribution functions (PDFs) are computed in terms of deep-inelastic scattering structure functions, following the recently developed method to determine the photon PDF. The calculation provides initial conditions at the electroweak scale for PDF evolution to higher energies. Numerical results for the W± and Z transverse, longitudinal and polarized PDFs, as well as the γZ transverse and polarized PDFs are presented.

W and Z PDFs. Massive gauge bosons have both transverse and longitudinal polarizations, and a new feature of our analysis is the computation of PDFs for longitudinally polarized gauge bosons. In contrast to the photon PDF, nonperturbative contributions are suppressed, allowing us to calculate the gauge boson PDFs in terms of quark PDFs at the electroweak scale.
Section 2 computes the transverse and polarized W ± , Z and γZ gauge boson PDFs (which are the sum and difference of the helicity h = ±1 PDFs) using operator methods. The W ± and Z longitudinal PDFs, i.e. h = 0, are computed in sec. 3. In sec. 4 we compare our results with previous ones in the literature based on the effective W approximation [9][10][11]. We present an alternative derivation in sec. 5 using factorization methods. Numerical values for the PDFs are presented in sec. 6.

Transverse gauge boson PDFs
We start this section with defining the PDFs of transverse gauge bosons. We then derive how these are related to structure functions in deep-inelastic scattering. Evaluating the structure functions to lowest order in the strong coupling α s , we obtain a formula in terms of the quark PDFs.

Definition
We start by briefly reviewing the PDF definition for quarks and gluons in QCD, before discussing the electroweak gauge boson case. We will frequently use light-cone coordinates, decomposing a four-vector p µ as p µ = p − n µ 2 + p +n where n µ = (1, 0, 0, 1) andn µ = (1, 0, 0, −1) are two null vectors with n ·n = 2 and p ⊥ is transverse to both n µ andn µ . The QCD PDF operators are defined as [12] O Q (r − ) = 1 4π for quarks and gluons, respectively. Here W is a Wilson line, 1 along then direction in the fundamental representation for the quark, and in the adjoint representation for the gluon PDF operator, ensuring gauge invariance. For the anti-quark, Q ↔Q and the Wilson line is in the anti-fundamental representation. The PDF operators involve an ordinary product of fields, not the time-ordered product, so the Feynman rules are those for cut graphs. The quark and gluon PDFs are given by the matrix elements of these operators in a proton state of momentum p, where only connected graphs contribute.
In the Standard Model (SM) at high energies, fermion PDFs are defined in terms of the SU (2)×U (1) fields q, , u, d, e, where q, are left-handed SU (2) doublet fields, and u, d, e are right-handed SU (2) singlet fields. The QCD Wilson line is replaced by a SU (3)×SU (2)×U (1) Wilson line in the representation of the fermion field. The new feature in the electroweak case is that the PDF operator does not have to be a SU (2) × U (1) singlet. In particular, for the quark doublet q there are two operators, where i, j are gauge indices in the fundamental representation of SU (2), t a is an SU (2) generator, and α is a gauge index in the fundamental representation of SU is an SU (3) × U (1) singlet, and transforms as an SU (2) adjoint. The proton matrix elements of the operators give the u L and d L PDFs, Since electroweak symmetry is broken, O (adj,a=3) q can have a non-zero matrix element in the proton, such that f u L = f d L . The evolution above the electroweak scale of O (1) q , O (adj,a) q and of the corresponding gauge and Higgs boson operators was computed in ref. [6].
The quark PDF operators in eq. (2.5) in the unbroken theory can be matched onto PDF operators in the broken theory at the electroweak scale. At tree level this matching is trivial, where The gauge PDF operators in irreducible SU (2) representations were given in ref. [6]. At lowest order in electroweak corrections, the matching onto the broken operators is analogous to eq. (2.7) and given in eq. (5.1) of ref. [6]. The relevant PDF operators in the broken theory are 2 9) in terms of the field-strength tensors. Note that the PDFs f Z T γ and f γZ T are related by complex conjugation. The PDF operators in eq.
There are no Wilson lines for γ and Z, since they are neutral.
As we now show, the operators in eq. (2.9) only capture the transverse polarizations. A gauge boson moving in the n direction has momentum and polarization vectors which satisfy k · λ = 0 and * λ · σ = −δ λσ . By calculating the matrix element of the fieldstrength tensors appearing in eq. (2.9) for a gauge boson state, we conclude that the PDF operators only pick out transversely-polarized gauge bosons.
The transverse PDFs f W + contribution, and will be discussed in sec. 3. In addition, we will also consider the polarized W + PDF, etc. In an unpolarized proton target, the gluon distribution f ∆g vanishes, as can be shown by reflecting in the plane of the incident proton. However, the weak interactions violate parity, so f ∆W ± T and f ∆Z T do not vanish. The polarized photon PDF can be written as [14,15] (see footnote 2), where F αβ = 1 2 αβλσ F λσ with 0123 = +1, and we use the 't Hooft-Veltman convention for the -symbol and γ 5 . Similar expressions hold for the polarized versions of the other PDFs in eq. (2.9).

Evaluation
We now discuss how the transverse gauge boson PDFs can be computed from fig. 1, following the procedure in refs. [7,8]. We start by introducing the hadronic tensor and structure functions, briefly repeat the argument for the photon case, and then generalize to the other gauge boson PDFs in eq. (2.9). Only PDFs for unpolarized proton targets will be considered, but it is straightforward to generalize to polarized protons.
The electroweak PDFs at high energies evolve using anomalous dimensions in the unbroken theory computed in refs. [5,6], which contain Sudakov double logarithms. In this paper, we compute the initial conditions to this evolution at the electroweak scale. Since the electroweak gauge bosons are massive, the logarithmic evolution is not important until energies well above the electroweak scale. In addition, there are radiative corrections for the W PDFs from interactions with the Wilson line from graphs shown in fig. 2, which are absent in the photon case. For this reason, we compute the electroweak PDFs to order α ∼ α 2 ∼ α Z . The lower part of the graph in fig. 1 is the hadronic tensor defined as where p is the proton momentum and q is the incoming gauge-boson momentum at vertex j ν . The standard decomposition of W µν is 15) in terms of the structure functions F 1 , F 2 , F 3 , which depend on the Bjorken variable and Q 2 = −q 2 . It is convenient to replace F 1 in our results by the longitudinal structure function, The currents in eq. (2.14) depend on the process. For the photon PDF, j µ is the electromagnetic current and F 3 vanishes. For W PDFs, j µ is the weak charged current, and for Z PDFs, j µ is the weak neutral current. These currents follow from the interaction Lagrangian, which is given by [16] using the conventional normalization of currents in deep-inelastic scattering. Here g 2 = e/sin θ W and g Z = e/(sin θ W cos θ W ) and The structure functions for electromagnetic scattering are denoted by F , where the superscripts indicate the j µ and j ν current used in eq. (2.14). In QCD, to lowest order in α s , F L = 0 and (2.20) and Here the subscripts L, R denote the parton helicities, not chiralities, Q q is the electric charge, and we have neglected CKM mixing and heavier quark flavors. For an unpolarized proton beam, the expressions can be simplified using f u L = f u R = 1 2 f u , etc. We now briefly review the method that ref. [8] used to compute the photon PDF, before applying the same procedure to the other PDFs. The computation of fig. 1 gives (see sec. 6.1 of ref. [8], and dropping vacuum polarization corrections) is the momentum transfer, and W µν is evaluated at (x bj , Q 2 ). The label D is a reminder that the hadronic tensor (and the couplings) are evaluated in D = 4−2 dimensions, and In eq. (2.23), we included P γ = 1, as it will be replaced by other factors for the electroweak case, see eq. (2.30) through eq. (2.32) below. The W µν terms in eq. (2.23) can be written in terms of the structure functions using eq. (2.15), We retain the F 3 term, even though F (γ) 3 = 0, since we will need it for the other PDFs. The splitting function in eq. (2.26) is and is the piece of Q 2 in fractional dimensions. Since we already averaged over the angular directions in obtaining eq. (2.23), we can simply replace We can now immediately get the other transverse PDFs. The only change is the replacement of the photon coupling and propagator by those for massive gauge bosons, and using the appropriate structure function. The W + PDF uses theν structure functions, and the replacement (2.30) and the W − PDF uses eq. (2.30) with the ν structure functions. The Z PDF has (2.31) The γZ and Zγ PDFs use the γZ and Zγ structure functions, with Proceeding as in ref. [8], the integral in eq. (2.23) over Q 2 is divided into an integral from where we assume µ m p . Following the terminology of ref. [8], the two contributions are called the "physical factorization" term f PF and the MS correction, f MS . The physical factorization integral is finite, so one can set D = 4. The MS integral is divergent, and needs to be evaluated in the MS scheme (hence the name) to get the MS PDF. As an example, we illustrate this for the W + PDF. Using eqs. (2.23), (2.26), and (2.30), Since the integral in eq. (2.34) is for Q 2 m 2 p , we have dropped m 2 p /Q 2 terms. Changing variables to Since µ is large, the dependence of F i (x/z, µ 2 s/(1 − z)) on µ is perturbative. To lowest order in α s and α 2 , we can therefore set the second argument of F i to µ 2 without incurring large logarithms, and drop the F L term since it is order α s . This results in The s integral yields The 1/ term is cancelled by the UV counterterm, and the sum of eqs. (2.33) and (2.37) gives The 1/ counterterm agrees with the anomalous dimensions for PDF evolution computed in refs. [5,6]. Alternatively, one can also directly take the µ derivative of eq. (2.39), for which the contribution from the upper limit of the first integral cancels the contribution from rational function of µ 2 and M 2 W , leaving the usual evolution The largest effect not included is the QCD evolution of F (ν) . Eq. (2.40) agrees with the anomalous dimension in refs. [5,6], using eq. (2.20). In obtaining eq. (2.40) we can neglect F L , the µ-dependence of the structure functions and α 2 (µ), since these give terms that are higher order in α 2 or α s . The diagonal W W term in the PDF evolution, which contains Sudakov double logarithms, is also missing, since f W T only starts at order α 2 .
The W − T PDF is given by eq. (2.39), eq. (2.44) withν structure functions replaced by ν structure functions. The Z T PDF is given by eq. (2.39), eq. (2.44) withν structure functions replaced by Z structure functions, M W → M Z , and α 2 → 2α Z . The γZ PDFs require the s integral since there is only one massive propagator. This gives where we consider both α and α Z of order α 2 in writing O(α 2 2 ). Similarly, and the Zγ PDF is obtained by γZ → Zγ.

Longitudinal gauge boson PDFs
In this section we repeat the analysis of sec. 2 for the PDFs of longitudinal gauge bosons. We start again with defining them, using the equivalence theorem to express them in terms of scalar PDFs, and then calculate them in terms of structure functions.

Definition
The operators in eq. (2.9) give the PDFs for transversely polarized gauge bosons. Longitudinally polarized gauge bosons are not produced at leading power in M/Q by the gauge field-strength tensor. Instead, they have to be computed in terms of Goldstone bosons using the Goldstone-boson equivalence theorem [17,18], as was done for electroweak corrections to scattering amplitudes in refs. [19,20]. The scalar (Higgs) PDFs we need are in the unbroken and broken phase, respectively. Here h is the physical Higgs particle, and the unphysical scalars ϕ 3 , ϕ ± = (ϕ 1 ∓ iϕ 2 )/ √ 2 are related to the longitudinal gauge bosons Z L , W ± L through the Goldstone-boson equivalence theorem. For the incoming gauge bosons this is given by iϕ + → W + L , iϕ 3 → Z L , and for gauge bosons on the other side of the cut in fig. 1, −iϕ + → W + L , −iϕ 3 → Z L . This leads to which are equivalent to eqs. (5.4) and (5.5) of ref. [6]. Note that f hZ L and f Z L h are complex conjugates of each other. It is instructive to rederive eq. (3.3), by expanding the Wilson lines in eq. (3.1) to first order where g 2 = e/sin θ W and g Z = e/(sin θ W cos θ W ). Using integration by parts, we have the identity setting the gauge field at infinity to zero, and similarly for the W − term. As a result, The linear combinations in eq. (3.6) are those required by the equivalence theorem. Exploiting gauge invariance, we can evalute the PDFs in the broken phase in unitary gauge using eq. (3.6) with ϕ i → 0. This does not affect renormalizability, since the PDFs in the broken phase only have radiative corrections due to dynamical gluons and photons, i.e. the W and Z are treated as static fields in the same way as heavy quark fields in heavy quark effective theory. Thus we reproduce eq. (3.3), identifying the longitudinal PDFs as where the Wilson lines W in the W PDFs only contain photon fields.

Evaluation
Before evaluating the longitudinal gauge boson PDFs, we note that the Higgs PDFs in eq. (3.3) are suppressed, This happens because the gauge field couplings to the proton are of order g 2 , g Z , whereas the dominant coupling of the Higgs field to the proton is given by the scale anomaly [21], and is order m p /v (for a pedagogical discussion see ref. [22]). (There are of course also contributions of the order of light fermion Yukawa couplings m u,d /v.) We therefore neglect the Higgs PDFs in eq. (3.8) in this paper, but they can be computed using the same method as the gauge boson PDFs. We now repeat the steps in sec. 2.2 for longitudinal gauge bosons, starting with f W + L . The matrix element of the PDF operator gives which combine to yield Using the same procedure of splitting the integral as before gives and similarly for Comparing eq. (3.12) with eq. (2.39) for the transverse W PDF, we see that W L has an extra M 2 W /Q 2 factor in the Q 2 integral. The W T integral grows as ln µ 2 for large values of µ, whereas the W L integral is finite, and dominated by Q 2 ∼ M 2 W . The W L PDF is thus smaller than the W T PDF by ln µ 2 /M 2 W . The split of the longitudinal PDFs into two pieces in eq. (3.12) is not necessary, and one can instead use eq. (3.12) with µ → ∞, but it is useful when comparing with the other PDFs. Differentiating eq. (3.12) with respect to µ gives as expected. Eq. (3.12) was obtained starting from eq. (3.7) in the broken phase. For µ much larger than M W , we need the PDFs in the unbroken theory, which are related to Higgs PDFs by eq. (3.3). Since there is no quark contribution to the Higgs PDF evolution when fermion Yukawa couplings are neglected, eq. (3.13) is expected.

Comparison with previous results
W and Z PDFs have been computed previously using the effective W, Z approximation [9][10][11], i.e. the Fermi-Weizsäcker-Williams [23][24][25] approximation for electroweak gauge bosons. We will compare these with our results. Considering for concreteness the transverse W PDF, the leading-logarithmic contribution from eq. (2.39) is given by and agrees with the effective W approximation result in refs. [9][10][11]. The subleading terms (the last two lines in eq. (2.39)) are smaller by a factor of ln µ 2 /M 2 W . These differ from the corresponding terms in previous results.
The longitudinal W PDF is smaller by a factor of ln µ 2 /M 2 W , and is obtained by integrating which agrees with refs. [9][10][11]. Again, the subleading terms given by the last line in eq. (3.12) differ from previous results.

PDF computation using factorization methods
In this section we present an alternative derivation for the massive gauge boson PDF using standard factorization methods. We will consider both transverse and longitudinal polarizations, and consider a massive gauge boson in a broken U (1) theory to keep the presentation simple. Our calculation exploits the fact that the cross section for the hypothetical process of electron-proton scattering producing a new heavy lepton or scalar in the final state can be written in two ways: in terms of proton structure functions or using proton PDFs. This approach was used in ref. [7] for the photon case. The new feature in the calculation is broken gauge symmetry, which results in massive gauge bosons that can have a longitudinal polarization.

Transverse polarization
Following refs. [7,8], consider the hypothetical inclusive scattering process shown in Fig. 3, where l is a massless fermion, and L is a fermion with mass M L . We will assume that they interact with the massive U (1) gauge boson (called Z) via a magnetic momentum coupling, Here g is the gauge coupling, and we work to leading order in the scale of the new interaction Λ v. The interaction between Z and the protons is governed by L Zp = −gZ µ j µ . We will now calculate the Z PDF by first factorizing the cross section into the hadronic and leptonic tensor, and then factorizing it in terms of PDFs. In doing so, we assume M L → ∞.
The cross section averaged over initial spins and summed over final states is given by in terms of the hadronic tensor W µν (p, q) and the leptonic tensor L µν (k, q), with q = k − k .
The lepton tensor, averaging over initial spins and summing over final ones is The decomposition of the hadronic tensor in terms of structure functions was given by eq. (2.15). F 3 does not contribute to the spin-averaged cross section, since eq. (5.4) is symmetric in µ ↔ ν. The rest of the derivation of σ lp is then identical to sec. 3 of ref. [8], after accounting for the gauge boson mass in the propagator 1/q 4 → 1/(q 2 − M 2 Z ) 2 . The total cross section in the M L → ∞ limit is thus given by where the kinematic variables are and we have split the Q 2 integral into two parts. If µ is large compared to Λ QCD , we can replace F 2 (x/z, Q 2 ) in the second integral by F 2 (x/z, µ 2 ) since the µ evolution is perturbative, and evaluate the integral to obtain We now factorize the cross section into a convolution of hard-scattering cross sections and parton distributions, Figure 4. Feynman diagram for heavy scalar production via Higgs-proton scattering h + p → s + X.
leading power, for which we consider shown in fig. 4. Here the interaction between the Z boson and scalars is described by the gauge invariant Lagrangian where Φ is a charged scalar whose vacuum expectation value breaks the gauge symmetry, and s is a heavy neutral scalar with mass M s . After spontaneous symmetry breaking, in unitary gauge, where h denotes the Higgs field. Eq. (5.12) then becomes (5.14) Note that to obtain a term in which interactions with longitudinal gauge bosons are not power suppressed requires operators involving the Higgs field Φ, resulting in interactions proportional to the vacuum expectation value v, as shown in eq. (5.14). The first step in obtaining the longitudinal Z PDF is to factorize the cross section for the process in eq. (5.11) in terms of structure functions, where the scalar tensor S µν is The scalar tensor couples only to longitudinally-polarized gauge bosons and not to transverse ones, so factorization directly gives the longitudinal Z PDF. The scattering cross section in

Numerics
In this section we present numerical results for the electroweak gauge boson PDFs, obtained using eqs. (2.39), (2.44), (2.47), (2.48), and (3.12). The equations have corrections of order α 2 2 , arising from e.g. the graphs in fig. 2. All QCD corrections and m 2 p /M 2 W power corrections are included automatically by using the deep-inelastic scattering structure functions.
The expressions for the electroweak gauge boson PDFs involve integrations over Q 2 between m 2 p x 2 /(1 − z) and µ 2 /(1 − z), and thus include the elastic scattering and resonance regions. Compared to the photon PDF, the integrands have factors of Q 2 /(Q 2 + M 2 ), where M = M W , M Z . Thus the low-Q 2 part of the integration region is suppressed by ∼ m 2 p /M 2 W ∼ 10 −4 , the size of low-energy weak interaction corrections and smaller than the missing α 2 2 corrections, so we only need values of the structure functions for Q 2 of order the electroweak scale. The x integral still includes elastic scattering at x/z = 1, but for large Q 2 the elastic form-factors are power suppressed. This justifies using the expressions for the F 2 and F 3 structure functions at lowest order QCD in terms of PDFs in eq. (2.20), eq. (2.21), and setting F L to zero (since it starts at order α s ), to evaluate the PDFs. This method is not as accurate as using the experimentally measured structure functions, because it introduces order α s (M W ) radiative corrections as well as m 2 p /M 2 W power corrections. 3 The numerical integrations are done using the PDF set NNPDF31 nlo as 0118 luxqed PDFs [26] and the LHAPDF [27] and ManeParse [28] interfaces. A detailed numerical analysis including PDF evolution and errors is beyond the scope of this paper. The results for electroweak gauge boson PDFs are shown in fig. 5, 6, and 7 for µ = M Z and µ = 1000 GeV using the NNPDF31 central PDF set. The electroweak PDFs have been renormalized in the MS scheme, so they do not have to be positive. They start at order α 2 , rather than order unity, and NLO corrections can be negative.
The transverse PDFs ( fig. 5) are small at µ = M Z , and rapidly grow with energy to be almost comparable to the photon at µ = 1000 GeV due to the evolution in eq. (2.40). The W + T PDF is larger than W − T , since f u > f d in the proton. The PDFs are negative at µ = M Z , but rapidly become positive as the ln Q 2 part dominates over the MS subtraction term. The polarized PDFs ( fig. 6) are negative, since quark PDFs are larger than antiquark PDFs, and left-handed quarks prefer to emit left-hand circularly polarized W bosons. The longitudinal PDFs ( fig. 7) are comparable to the transverse ones at µ = M Z . Since the longitudinal PDFs are µ-independent, only one plot has been shown. The transverse PDFs rapidly become larger than the longitudinal ones as µ increases.

Conclusions
We have computed the electroweak gauge boson PDFs at a scale µ ∼ M W,Z m p for transversely and longitudinally polarized gauge bosons, by computing the proton matrix element of the PDF operator in terms of proton structure functions for charged and neutral current scattering. The PDFs can be evolved to higher energies using the evolution equations derived recently in refs. [5,6]. The electroweak gauge boson PDFs have been computed previously using the effective W approximation [9][10][11]. The leading logarithmic piece of our result agrees with their expressions, but the full order α results differ. Numerical values for the PDFs at the representative scales µ = M Z and µ = 1000 GeV are given in sec. 6. More detailed numerical results are beyond the scope of this paper, and will be given elsewhere.