Electroweak gauge boson parton distribution functions

Journal of High Energy Physics, May 2018

Abstract 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.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:


Electroweak gauge boson parton distribution functions

HJE Electroweak gauge boson parton distribution functions Bartosz Fornal 0 1 2 4 5 Aneesh V. Manohar 0 1 2 4 5 Wouter J. Waalewijn 0 1 2 3 5 0 Science Park 904 , 1098 XH Amsterdam , The Netherlands 1 University of Amsterdam 2 9500 Gilman Drive , La Jolla, CA 92093 , U.S.A 3 Nikhef, Theory Group 4 Department of Physics, University of California , San Diego , USA 5 Science Park 105 , 1098 XG, Amsterdam , The Netherlands 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 Deep Inelastic Scattering (Phenomenology); QCD Phenomenology - results for the W and Z transverse, longitudinal and polarized PDFs, as well as the Z transverse and polarized PDFs are presented. Conclusions Introduction 1 Introduction Comparison with previous results PDF computation using factorization methods 5.1 5.2 Transverse polarization Longitudinal polarization An essential ingredient in calculations of high energy scattering cross sections are the parton distribution functions (PDFs), which describe the incoming protons. These usually only encode QCD e ects, but at the multi-TeV energies probed by collisions at the Large Hadron Collider, electroweak e ects start becoming important. At Future Circular Collider energies, electroweak e ects are order one [1], because of Sudakov double logarithms in the electroweak PDF evolution [2, 3], which are absent for QCD. This di erence is due to the spontaneous breaking of electroweak symmetry, implying that PDFs only have to be QCD (and QED) singlets, but not necessarily electroweak singlets. Indeed, it is the SU(2) U(1) non-singlet PDFs that have Sudakov double logarithms in their evolution. Electroweak contributions to PDF evolution have been computed recently [4{6], which relates PDFs at di erent scales. However, the PDFs themselves have to be determined from experiment. Recently, the photon PDF was calculated directly in terms of deep-inelastic scattering structure functions [7, 8]. In this paper, we use a similar method to compute the 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. { 1 { Section 2 computes the transverse and polarized W , Z and Z gauge boson PDFs (which are the sum and di erence of the helicity h = 1 PDFs) using operator methods. The W and Z longitudinal PDFs, i.e. h = 0, are computed in section 3. In section 4 we compare our results with previous ones in the literature based on the e ective W approximation [9{11]. We present an alternative derivation in section 5 using factorization methods. Numerical values for the PDFs are presented in section 6. 2 structure functions to lowest order in the strong coupling s, we obtain a formula in terms of the quark PDFs. 2.1 fQ(r =p ; ) hpjOQ(r )jpic ; fG(r =p ; ) hpjOG(r )jpic ; (2.4) where only connected graphs contribute. convention D 1It is conventional to use Wy(x) Q(x) for the eld, so the Wilson line must end at x. We use the sign We start by brie y reviewing the PDF de nition 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 as p = p n 2 + p + n 2 + p ; ? p = n p ; p + = n p ; where n = (1; 0; 0; 1) and n = (1; 0; 0; 1) are two null vectors with n n = 2 and p? is transverse to both n and n . The QCD PDF operators are de ned as [12] OQ(r ) = OG(r ) = d e i r [Q(n ) W(n )] n= [Wy(0) Q(0)] ; d e i r n [G (n ) W(n )] n [Wy(0) G for quarks and gluons, respectively. Here W is a Wilson line,1 W(x) = P exp i g ds n A(x + sn) ; along the n 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 elds, 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, Z 0 1 { 2 { (2.1) (2.2) (2.3) In the Standard Model (SM) at high energies, fermion PDFs are de ned in terms of the SU(2) U(1) elds q, `, u, d, e, where q, ` are left-handed SU(2) doublet elds, and u, d, e are right-handed SU(2) singlet elds. The QCD Wilson line is replaced by a SU(3) SU(2) U(1) Wilson line in the representation of the fermion eld. 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, Oq(1)(r ) = Oq(adj;a)(r ) = 4 4 d e i r [q(n ) W(n )]i n= ji [Wy(0) q(0)]j ; d e i r [q(n ) W(n )]i n= [ta]ij [Wy(0) q(0)]j ; (2.5) HJEP05(218)6 where i; j are gauge indices in the fundamental representation of SU(2), ta is an SU(2) generator, and SU(3) SU(2) is a gauge index in the fundamental representation of SU(3). Oq(1) is an U(1) singlet, but Oq(adj;a) is an SU(3) U(1) singlet, and transforms as an SU(2) adjoint. The proton matrix elements of the operators give the uL and dL PDFs, fuL (r =p ; ) = hpj 21 Oq(1)(r ) + Oq(adj;a=3)(r ) jpi ; fdL (r =p ; ) = hpj 21 Oq(1)(r ) Oq(adj;a=3)(r ) jpi : Since electroweak symmetry is broken, Oq(adj;a=3) can have a non-zero matrix element in the proton, such that fuL 6= fdL . The evolution above the electroweak scale of Oq(1), Oq(adj;a) 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 1 2 Oq(1)(r ) = OuL (r ) + OdL (r ) ; Oq(adj;a=3)(r ) = OuL (r ) OdL (r ) ; 1 2 OuL (r ) = d e i r [uL(n ) W(n )] n= [Wy(0) uL(0)] ; and similarly for dL. Essentially all we have done is replace the SU(3) SU(2) U(1) Wilson lines by SU(3) U(1)em Wilson lines, so W in eq. (2.8) only contains gluons and photons. 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 { 3 { (2.6) (2.7) (2.8) broken theory are2 OWT+ (r ) = OWT (r ) = O (r ) = OZT (r ) = OZT (r ) = O ZT (r ) = 1 1 1 1 1 1 2 r 2 r 2 r 2 r 2 r 2 r in terms of the eld-strength tensors. Note that the PDFs fZT and f ZT are related by complex conjugation. The PDF operators in eq. (2.9) are invariant under SU(3) U(1)em gauge transformations. For OWT+ this involves a U(1)em Wilson line W with Q = 1, and has Q = 1. There are no Wilson lines for 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 k = (Ek; 0; 0; k); + = p (0; 1; i; 0); = p (0; 1; i; 0); 0 = (k; 0; 0; Ek); 1 2 1 M which satisfy k = 0 and = . By calculating the matrix element of the eld-strength tensors appearing in eq. (2.9) for a gauge boson state, hk; jn F (n ) n F (0)jk; i = (n k)2( ) + (n )(n )k2 ei(n k) = (n k)2ei(n k) (1; 0; = +; = 0 ; ; (2.9) HJEP05(218)6 (2.10) (2.11) (2.12) we conclude that the PDF operators only pick out transversely-polarized gauge bosons. The transverse PDFs fWT+ , etc. de ned through the operators in eq. (2.9), sum over the helicity h = 1 contributions. The longitudinal gauge boson PDF encodes the h = 0 contribution, and will be discussed in section 3. In addition, we will also consider the polarized W + PDF, f WT+ = fW +(h=1) fW +(h= 1) ; 2Note that W is the SU(2) gauge eld, and W is the Wilson line. We have switched conventions relative to ref. [8], n $ n, p + $ p . The photon PDF operator in ref. [8] was written as the sum of two terms, such that it has manifest antisymmetry under x ! x. However, the commutator of light-cone operators term shown in eq. (2.9). The two terms in O , etc. can be similarly combined. { 4 { (a) (b) eld-strength tensor and the bottom part of the graphs is the hadronic tensor W (p; q). late parity, so f WT as [14, 15] (see footnote 2), etc. In an unpolarized proton target, the gluon distribution f g vanishes, as can be shown by re ecting in the plane of the incident proton. However, the weak interactions vioand f ZT do not vanish. The polarized photon PDF can be written O (r ) = i 2 r d e i r n F (n ) n Fe (0) ; (2.13) where Fe = 12 in eq. (2.9). 2.2 Evaluation F with 0123 = +1, and we use the 't Hooft-Veltman convention for the -symbol and 5. Similar expressions hold for the polarized versions of the other PDFs We now discuss how the transverse gauge boson PDFs can be computed from gure 1, following the procedure in refs. [7, 8]. We start by introducing the hadronic tensor and structure functions, brie y 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 gure 2, which are absent in the photon case. For this reason, we compute the electroweak PDFs to order 2 The lower part of the graph in gure 1 is the hadronic tensor de ned as d4z eiq zhpj jy (z); j (0) jpi ; factorization in terms of structure functions, as it involves a three-point correlator in the proton. where p is the proton momentum and q is the incoming gauge-boson momentum at vertex HJEP05(218)6 j . The standard decomposition of W q2. It is convenient to replace F1 in our results by the longitudinal structure FL(xbj; Q2) 1 + F2(xbj; Q2) 2xbjF1(xbj; Q2) : The currents in eq. (2.14) depend on the process. For the photon PDF, j is the electromagnetic current and F3 vanishes. For W PDFs, j is the weak charged current, and for Z PDFs, j 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. The structure functions for electromagnetic scattering are denoted by Fi scattering p ! e X by Fi( ), for anti-neutrino scattering p ! e+X by Fi current scattering by F (Z), and for i Z interference by F i i ( Z) and F (Z ), where the superscripts indicate the j and j current used in eq. (2.14). In QCD, to lowest order in ( ), for neutrino ( ), for neutral { 6 { (2.15) (2.16) (2.17) (2.18) and F2( )(x; Q2) = 4x[fdL(x; Q)+fuR(x; Q)]; Here the subscripts L; R denote the parton helicities, not chiralities, Qq is the electric charge, gLu = gLd = 1 2 1 2 2 3 + 1 3 sin2 W ; sin2 W ; gRu = gRd = 1 3 2 3 sin2 W ; sin2 W ; (2.22) and we have neglected CKM mixing and heavier quark avors. For an unpolarized proton beam, the expressions can be simpli ed using fuL = fuR = 12 fu, etc. We now brie y review the method that ref. [8] used to compute the photon PDF, before applying the same procedure to the other PDFs. The computation of gure 1 gives (see section 6.1 of ref. [8], and dropping vacuum polarization corrections) f (x; ) = f (x; ) = 8 8 ( ) (S ) Q2(1 z) ( ) (S ) x x 2 2 1 1 { 7 { z ; Q2 = q2 is the momentum transfer, and W is evaluated at (xbj; Q2). The label D is a reminder that the hadronic tensor (and the couplings) are evaluated in D = 4 2 dimensions, and S In eq. (2.23), we included P = 1, as it will be replaced by other factors for the electroweak case, see eq. (2.30) through eq. (2.32) below. The W terms in eq. (2.23) can be written in terms of the structure functions using eq. (2.15), is the piece of Q2 in fractional dimensions. Since we already averaged over the angular directions in obtaining eq. (2.23), we can simply replace 2 Q 2 ! D D 4 2 Q 2? = D D 4 2 Q2(1 z) x2m2p : 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 the structure functions, and the replacement 1 { 8 { (2.25) (2.27) (2.28) (2.29) (2.30) (2.31) (2.32) x(pz )2 h(n q)2 W (D) +q2 n n W (D)i = z ix(p )2 (n q)n q We retain the F3 term, even though F The splitting function in eq. (2.26) is 3 and p q(z) = 1 + (1 z z)2 ; ( ) = 0, since we will need it for the other PDFs. and the W PDF uses eq. (2.30) with the structure functions. The Z PDF has The Z and Z PDFs use the Z and Z structure functions, with where 2 = =sin2 W and Z = =(sin2 W cos2 W ). P ! PW = (Q2 + M W2 )2 P ! PZ = (Q2 + MZ2 )2 P ! P Z = Q2 (Q2 + MZ2 ) ; Q4 Q4 ; : Proceeding as in ref. [8], the integral in eq. (2.23) over Q2 is divided into an integral from m2px2=(1 z) to 2=(1 z), and from 2=(1 z) to 1, where we assume mp. 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 nite, 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), 16 " 2 2( ) Z 1 dz Z 1 z dQ2 Q4 x z Since the integral in eq. (2.34) is for Q2 m2p, we have dropped m2p=Q2 terms. Changing s = Q2(1 2 z) ; xfWPFT+ (x; ) = and xfWMTS+ (x; ) = xfWMTS+ (x; ) = (2.33) (2.34) (2.35) (2.36) (2.37) +O ( ) : (2.38) 16 h ) x z 1 s1+ 2, we can therefore set the second argument of Fi to 2 without incurring large logarithms, and drop the FL term since it is order s. This results in xfWMTS+ (x; ) = The s integral yields The 1= term is cancelled by the UV counterterm, and the sum of eqs. (2.33) and (2.37) gives xfWT+ (x; ) = 82 x z 5 F 2( )(x=z; 2 ) The 1= counterterm agrees with the anomalous dimensions for PDF evolution computed in refs. [5, 6]. Alternatively, one can also directly take the derivative of eq. (2.39), for which the contribution from the upper limit of the rst integral cancels the contribution from rational function of 2 and M W2 , leaving the usual evolution d d fWT+ (x; ) = x z p q(z) F2( )(x=z; Q2) x=z + O( 2 s) : (2.40) The largest e ect not included is the QCD evolution of F ( ). Eq. (2.40) agrees with the anomalous dimension in refs. [5, 6], d d fWT+ (x; ) = x z p q(z) fuL (x=z; Q2) + fdR (x=z; Q2) + : : : (2.41) using eq. (2.20). In obtaining eq. (2.40) we can neglect FL, the -dependence of the structure functions and 2( ), since these give terms that are higher order in 2 or s . The diagonal W W term in the PDF evolution, which contains Sudakov double logarithms, is also missing, since fWT only starts at order 2 . Similarly, for f WT , using eqs. (2.23) and (2.26), f PFWT+ (x; ) = 2 2( ) Z 1 dz Z 1 z dQ2 Q4 16 x z m2px2 Q2 (Q2 + M W2 )2 (2 1 z z) F3(;D) (x=z; Q2) (2.42) and f MWS T+ (x; ) = s1+ s Replacing the second argument of F3 by 2, as before, and using eq. (2.38) gives f WT+ (x; ) = x z x z which agrees with ref. [6]. z)F3( )(x=z; Q2) + O( 2 s) ; (2.44) (2.45) (2.46) (2.47) 3 5 9 ; (2.48) where we consider both and Z of order 2 in writing O( 22). Similarly, f ZT (x; ) = p ( ) Z ( ) Z 1 dz < Z 1 z dQ2 Q2 82 2 4 + (2 z) ln +O( 22) ; x z : 2 functions replaced by Z structure functions, MW ! MZ , and 2 ! 2 Z . The Z PDFs require the s integral e E since there is only one massive propagator. This gives xf ZT (x; ) = p ( ) Z ( ) Z 1 dz < Z 1 z dQ2 Q2 82 4 x z : 4 m2px2 Q2 Q2 +MZ2 1 z and the Z PDF is obtained by Z ! Z . z F 2 ( Z)(x=z; Q2)+ zp q(z)+ L F 2 ( Z)(x=z; Q2) Q2 MZ2 (1 z)+ 2 F 2 ( Z)(x=z; 2) z F 2 ( Z)(x=z; 2)=+O( 22) ; 2 5 9 ; In this section we repeat the analysis of section 2 for the PDFs of longitudinal gauge bosons. We start again with de ning them, using the equivalence theorem to express them in terms of scalar PDFs, and then calculate them in terms of structure functions. 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 eld-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 HJEP05(218)6 OH+ (r ) = OH (r ) = OH0 (r ) = OH0 (r ) = OH0H0 (r ) = OH0H0 (r ) = 2 2 2 2 2 2 r Z 1 r Z 1 r Z 1 r Z 1 r Z 1 r Z 1 1 1 1 1 1 1 d e i r [Hy(n ) W(n )]1 [Wy(0) H(0)]1 ; d e i r [Wy(n ) H(n )]1 [Hy(0) W(0)]1 ; d e i r [Hy(n ) W(n )]2 [Wy(0) H(0)]2 ; d e i r [Wy(n ) H(n )]2 [Hy(0) W(0)]2 ; d e i r [Wy(n ) H(n )]2 [Wy(0) H(0)]2 ; d e i r [Hy(n ) W(n )]2 [Hy(0) W(0)]2 ; with SU(2) U(1) Wilson lines W. The indices 1; 2 in eq. (3.1) pick out the charged and neutral components of the Higgs multiplet, H = H+! H0 = p 1 p i 2 '+ 2 v + h i'3 ! ; in the unbroken and broken phase, respectively. Here h is the physical Higgs particle, and the unphysical scalars '3; ' = ('1 i'2)=p2 are related to the longitudinal gauge bosons ZL; WL through the Goldstone-boson equivalence theorem. For the incoming gauge bosons this is given by i'+ ! WL+, i'3 ! ZL, and for gauge bosons on the other side of the cut (3.1) (3.2) in gure 1, i'+ ! WL+, i'3 ! ZL. This leads to fWL+(x; ) = hpjOH+(xp )jpi ; conjugates of each other. order i 2 Z 0 1 which are equivalent to eqs. (5.4) and (5.5) of ref. [6]. Note that fhZL and fZLh are complex It is instructive to rederive eq. (3.3), by expanding the Wilson lines in eq. (3.1) to rst Hy(x)W(x) = Hy(x) P exp ds n [g2W (x + ns) + g1B(x + ns)] = i ' (x) i p v Z 0 setting the gauge eld at in nity to zero, and similarly for the W term. As a result, Hy(n )W(n ) = i' (n ) Mr W n W (n ) p12 hv+h(n )+i'3(n )+ MrZ n Z(n ) i inside an integral of the form as in eq. (3.1), where we used MW = g2v=2 and MZ = gZ v=2. One can make a similar substitution for the Wy(0)H(0) term. The argument 0 does not depend on the integration variable . However, we can use translation invariance of eq. (3.1) to switch the eld arguments in eq. (3.1) from n and 0 to 0 and n . Eq. (3.5) can be applied again, with an additional minus sign because of the switch n n . Then Wy(0)H(0) is given by the conjugate of eq. (3.6) with r ! r and ! ! 0. 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 where g2 = e=sin W and gZ = e=(sin W cos W ). Using integration by parts, we have the identity ds n Z(n + ns) = e i r ds n Z(n + ns) ds n Z(n + ns) = d e i r n Z(n ); Z 1 1 d Z 1 d d 1 Z 0 1 i r = Z 1 1 Z 1 1 d e i r Z 0 d e i r d Z 0 1 d 1 +: : : (3.5) (3.6) using eq. (3.6) with 'i ! 0. This does not a ect 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 elds in the same way as heavy quark elds in heavy quark e ective theory. Thus we reproduce eq. (3.3), identifying the longitudinal PDFs as d e i r d e i r d e i r hpj[n W (n ) W(n )] [Wy(0) n W +(0)]jpi ; hpj[n W +(n ) W(n )] [Wy(0) n W (0)]jpi ; hpjn Z(n ) n Z(0)jpi ; where the Wilson lines W in the W PDFs only contain photon elds. Before evaluating the longitudinal gauge boson PDFs, we note that the Higgs PDFs in mp MZ fh(x; ) = O fhZL (x; ) = O ; fZLh(x; ) = O : (3.8) This happens because the gauge eld couplings to the proton are of order g2; gZ , whereas the dominant coupling of the Higgs eld to the proton is given by the scale anomaly [21], and is order mp=v (for a pedagogical discussion see ref. [22]). (There are of course also contributions of the order of light fermion Yukawa couplings mu;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 section 2.2 for longitudinal gauge bosons, starting with fWL+ . The matrix element of the PDF operator gives mp MZ which combine to yield Using the same procedure of splitting the integral as before gives xfWL+ (x; ) = 8 2 2 2( ) <Z 1 dz Z 1 z dQ2 8 : x z 4 m2px2 Q2 (Q2 + M W2 )2 1 z M W2 Q2 + 1 z Comparing eq. (3.12) with eq. (2.39) for the transverse W PDF, we see that WL has an extra M W2 =Q2 factor in the Q2 integral. The WT integral grows as ln 2 for large values M W2 . The WL PDF is thus smaller than the WT PDF by ln 2=M W2 . 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 it is useful when comparing with the other PDFs. Di erentiating eq. (3.12) with respect ! 1, but to gives ( ) ! F i( ). For fZL this requires replacing d d xfWL+ (x; ) = 0 + O( 2 s) ; as expected. Eq. (3.12) was obtained starting from eq. (3.7) in the broken phase. For much larger than MW , 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. 4 Comparison with previous results W and Z PDFs have been computed previously using the e ective W; Z approximation [9{11], i.e. the Fermi-Weizsacker-Williams [23{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 xfWT+ (x; ) 2( ) Z 1 dz Z 2 dQ2 Q4 16 16 2( ) ln x z M W2 2 Z 1 dz x z Q2 (Q2 + M W2 )2 zp q(z)F2( )(x=z; Q2) zp q(z)F2( )(x=z; 2) ; and agrees with the e ective W approximation result in refs. [9{11]. The subleading terms (the last two lines in eq. (2.39)) are smaller by a factor of ln 2=M W2 . These di er from the corresponding terms in previous results. The longitudinal W PDF is smaller by a factor of ln 2=M W2 , and is obtained by integrating xfWL+ (x; ) 2( ) Z 1 dz Z 1 dQ2 8 8 2( 2) Z 1 dz x z x z 0 (1 M W2 Q2 Q2 (Q2 + M W2 )2 (1 z) F2( )(x=z; 2) ; z) F2( )(x=z; Q2) which agrees with refs. [9{11]. Again, the subleading terms given by the last line in eq. (3.12) di er from previous results. 5 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 nal 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. 5.1 Transverse polarization Following refs. [7, 8], consider the hypothetical inclusive scattering process (4.1) (4.2) (5.1) (5.2) shown in gure 3, where l is a massless fermion, and L is a fermion with mass ML. We will assume that they interact with the massive U(1) gauge boson (called Z) via a magnetic momentum coupling, l(k) + p(p) ! L(k0) + X; Lint = g L Z l + h.c. : Here g is the gauge coupling, and we work to leading order in the scale of the new interaction v. The interaction between Z and the protons is governed by LZp = gZ j . We will now calculate the Z PDF by rst factorizing the cross section into the hadronic and leptonic tensor, and then factorizing it in terms of PDFs. In doing so, we assume ML ! 1. The cross section averaged over initial spins and summed over nal states is given by HJEP05(218)6 lp = 1 4p k Z d4q g 2 4g2 h ML2 + Q 2 2 ](k=0 + ML)[ ; =q] 2k q)g q q lp = 2 + 2 2 1 z g Q4 2 x z " Q4 2 2 x z Q2 (Q2 +MZ2 )2 where the kinematic variables are xbj = Q2 2p q = x z ; in terms of the hadronic tensor W (p; q) and the leptonic tensor L (k; q), with q = k k0. The lepton tensor, averaging over initial spins and summing over nal ones is ref. [8], after accounting for the gauge boson mass in the propagator 1=q4 The total cross section in the ML ! 1 limit is thus given by The decomposition of the hadronic tensor in terms of structure functions was given by eq. (2.15). F3 does not contribute to the spin-averaged cross section, since eq. (5.4) is symmetric in . The rest of the derivation of lp is then identical to section 3 of ! 1=(q2 MZ2 )2. (p; q) L (k; q) 2 [(k q)2 ML2] (k0 q0) (5.3) 2q2k k + (k q) (2q k + 2q k q q ) i ML2g + 4Q2k k 2(ML2 + Q2) (k q + k q ) : (5.4) i p k = M 2 2x ; F2(x=z; Q2) # 2zQ2 ML2 + z2Q2 ML2 +zp q(z) F2(x=z; Q2) ; (5.5) s = (p + k)2; (5.6) We now factorize the cross section into a convolution of hard-scattering cross sections and parton distributions, These hard-scattering cross sections are the same as in ref. [8], lp(xs) = X a=Z;q;::: x Z 1 dz z ^la(zs; ) z fa=p x x z ; 0 = with z = ML2=s^. It should not come as a surprise that this only describes transverse polarizations, since it is the same as for photons. Indeed, the contribution to the cross section of this process is power suppressed by MZ2 =ML2 for longitudinally polarized gauge bosons. Thus the factorization formula in eq. (5.8) gives the Z PDF summed over the two transverse polarizations only. We can then extract fZT by combining eqs. (5.7) and (5.8), Q2 2 and we have split the Q2 integral into two parts. If is large compared to QCD, we can replace F2(x=z; Q2) in the second integral by F2(x=z; 2) since the evolution is perturbative, and evaluate the integral to obtain h(k) + p(p) ! s(k0) + X ; 2 xfZT (x) = g2 Z 1 dz Z 1 z dQ2 8 2 x z for the W is g=(2p2) rather than g. Longitudinal polarization which agrees with eq. (2.39). The overall normalization di ers by 8 because the coupling Longitudinal gauge boson PDFs present novel features, because they only exist after spontaneous symmetry breaking. The rst step is to identify a process to which they contribute at leading power, for which we consider lp = 2 2 x z 2 2 g 4 Z 1 dz 2 x z 2 x z shown in gure 4. Here the interaction between the Z boson and scalars is described by the gauge invariant Lagrangian y) ; where is a charged scalar whose vacuum expectation value breaks the gauge symmetry, and s is a heavy neutral scalar with mass Ms. After spontaneous symmetry breaking, in unitary gauge, where h denotes the Higgs eld. Eq. (5.12) then becomes = v + h p 2 p(p) Z s(k′) X (5.12) (5.13) (5.14) (5.15) (5.16) (5.17) Note that to obtain a term in which interactions with longitudinal gauge bosons are not power suppressed requires operators involving the Higgs eld , resulting in interactions proportional to the vacuum expectation value v, as shown in eq. (5.14). The rst 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, hp = 1 4 p k Z d4q g 2 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 the limit Ms ! 1 is given by hp = 16 g4v2 Z 1 dz Z Ms2(1 z) z dQ2 1 Ms2 2 " x z z + (z x12mz2p 1) MQ2s2 + x2m2p Ms2 Q4 1 + ( z2Q2 4Ms4 Ms2 2# Q2 Q2 (Q2 + MZ2 )2 Splitting the Q2 integral into two parts at Q2 = 2=(1 z), and neglecting power corrections gives hp = 16 + g4v2 Z 1 dz <Z 1 z 8 2 dQ2 Again, this cross section can also be written in terms of proton PDFs using eq. (5.8). In the limit MZ =Ms ! 0, the Z boson cross section ^hZ is ^hZ (x; ) = g for longitudinally polarized Z bosons, and power suppressed for transversely polarized Z bosons. Thus eq. (5.8) picks out the longitudinal Z PDF. The contribution from quarks is calculated from tree-level Higgs-quark scattering via Z exchange, and is ^hq(z; ) = g4v2z 16 2Ms2 2 ln Ms2(1 z)2 z MZ2 ; where z = Ms2=s^, and is power suppressed relative to eq. (5.19). From eq. (5.18) and eq. (5.19), we get the longitudinal Z boson PDF # (5.18) (5.19) (5.20) # (5.21) xfZL (x; ) = g2MZ2 Z 1 dz Z 1 z 2 + x z g2MZ2 Z 1 dz x z dQ2 " x12mz2p (Q2 +MZ2 )2 F2 x=z; 2 1 z (1 z)2 MZ2 (1 z)+ 2 : This agrees with eq. (3.12) taking into account the overall factor of 1=8, as in eq. (5.10). To simplify the presentation, the calculations in this section have been done for a spontaneously broken U(1) gauge theory. However, it should be clear how these can be extended to the case of a spontaneously broken SU(2) U(1) in the Standard Model. 6 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 2, arising from e.g. the graphs in 2 gure 2. All QCD corrections and m2p=M W2 power corrections are included automatically by using the deep-inelastic scattering structure functions. The expressions for the electroweak gauge boson PDFs involve integrations over Q2 between m2px2=(1 z) and 2=(1 z), and thus include the elastic scattering and resonance regions. Compared to the photon PDF, the integrands have factors of Q2=(Q2 + M 2), where M = MW ; MZ . Thus the low-Q2 part of the integration region is suppressed by Z T γZT γ * 0.5 4-)x 0.5 4 . 0 -110-5 10-4 + WT W-T Z T γZT γ * 0.1 4)-x 4 1 )• 2 µ , x f(x 1 0 -110-5 = 1000 GeV. The unpolarizeWdT+ph(roetdo)n, fPWDTF (dashed, brown) has also been shown (blue), fZT (green) and f ZT (purple) for comparison, multipled by 0:1 at = MZ and by 0:5 at = 1000 GeV, so it ts on the same plot. power corrections.3 m2p=M W2 the missing 10 4, the size of low-energy weak interaction corrections and smaller than 22 corrections, so we only need values of the structure functions for Q2 of order the electroweak scale. The x integral still includes elastic scattering at x=z = 1, but for large Q2 the elastic form-factors are power suppressed. This justi es using the expressions for the F2 and F3 structure functions at lowest order QCD in terms of PDFs in eq. (2.20), eq. (2.21), and setting FL to zero (since it starts at order s), to evaluate the PDFs. This method is not as accurate as using the experimentally measured structure functions, because it introduces order s(MW ) radiative corrections as well as m2p=M W2 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 gure 5, 6, and 7 for = MZ and = 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 2 rather than order unity, and NLO corrections can be negative. The transverse PDFs ( gure 5) are small at = MZ , and rapidly grow with energy to be almost comparable to the photon at = 1000 GeV due to the evolution in eq. (2.40). at The WT+ PDF is larger than WT , since fu > fd in the proton. The PDFs are negative = MZ , but rapidly become positive as the ln Q2 part dominates over the MS subtraction term. The polarized PDFs ( gure 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 ( gure 7) are comparable to the transverse ones at = MZ . Since the longitudinal PDFs are -independent, only one plot has been shown. The transverse PDFs rapidly become larger than the longitudinal ones as increases. 3The radiative corrections depend on s and 2 evaluated at Q2 scales that contribute to the integral. One can minimize these corrections by including higher-order terms in the expressions for Fi. If, instead, the experimentally measured structure functions are used, the corrections depend on 2( ), where > MZ is the scale at which the PDF is evaluated. 4 . 4 0 0 3 3 x01 -1 ΔW+T ΔW-T ΔZT -2 ΔγZT -2.510-5 10-5 10-4 10-3 µ , Z L 10-5 10-4 10-3 We have computed the electroweak gauge boson PDFs at a scale MW;Z mp 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 e ective W approximation [9{11]. The leading logarithmic piece of our result agrees with their expressions, but the full order ical values for the PDFs at the representative scales = MZ and results di er. Numer= 1000 GeV are given in section 6. More detailed numerical results are beyond the scope of this paper, and will be given elsewhere. Acknowledgments This work is supported by the DOE grant DE-SC0009919, the ERC grant ERC-STG-2015677323, the D-ITP consortium, a program of the Netherlands Organization for Scienti c Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW), and the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excellence \Origin and Structure of the Universe." Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] M.L. Mangano et al., Physics at a 100 TeV pp Collider: Standard Model Processes, CERN Yellow Report (2017) 1 [arXiv:1607.01831] [INSPIRE]. [2] M. Ciafaloni, P. Ciafaloni and D. Comelli, Bloch-Nordsieck violating electroweak corrections [3] M. Ciafaloni, P. Ciafaloni and D. Comelli, Electroweak Bloch-Nordsieck violation at the TeV scale: `Strong' weak interactions?, Nucl. Phys. B 589 (2000) 359 [hep-ph/0004071] [4] C.W. Bauer, N. Ferland and B.R. Webber, Combining initial-state resummation with xed-order calculations of electroweak corrections, JHEP 04 (2018) 125 [arXiv:1712.07147] [INSPIRE]. [5] C.W. Bauer, N. Ferland and B.R. Webber, Standard Model Parton Distributions at Very High Energies, JHEP 08 (2017) 036 [arXiv:1703.08562] [INSPIRE]. [6] A.V. Manohar and W.J. Waalewijn, Electroweak Logarithms in Inclusive Cross Sections, arXiv:1802.08687 [INSPIRE]. [arXiv:1607.04266] [INSPIRE]. [7] A. Manohar, P. Nason, G.P. Salam and G. Zanderighi, How bright is the proton? A precise determination of the photon parton distribution function, Phys. Rev. Lett. 117 (2016) 242002 [8] A.V. Manohar, P. Nason, G.P. Salam and G. Zanderighi, The Photon Content of the Proton, JHEP 12 (2017) 046 [arXiv:1708.01256] [INSPIRE]. [9] M.S. Chanowitz and M.K. Gaillard, Multiple Production of W and Z as a Signal of New Strong Interactions, Phys. Lett. B 142 (1984) 85 [INSPIRE]. [10] S. Dawson, The E ective W Approximation, Nucl. Phys. B 249 (1985) 42 [INSPIRE]. [11] G.L. Kane, W.W. Repko and W.B. Rolnick, The E ective W , Z0 Approximation for High-Energy Collisions, Phys. Lett. B 148 (1984) 367 [INSPIRE]. [12] J.C. Collins and D.E. Soper, Parton Distribution and Decay Functions, Nucl. Phys. B 194 [13] R.L. Ja e, Parton Distribution Functions for Twist Four, Nucl. Phys. B 229 (1983) 205 [14] A.V. Manohar, Parton distributions from an operator viewpoint, Phys. Rev. Lett. 65 (1990) [15] A.V. Manohar, Polarized parton distribution functions, Phys. Rev. Lett. 66 (1991) 289 [16] Particle Data Group collaboration, C. Patrignani et al., Review of Particle Physics, Chin. Phys. C 40 (2016) 100001 [INSPIRE]. Amplitudes and Application to Hard Scattering Processes at the LHC, Phys. Rev. D 80 (2009) 094013 [arXiv:0909.0012] [INSPIRE]. Phys. 29 (1924) 315 [INSPIRE]. (1934) 612 [INSPIRE]. ionization and radiation formulae, Phys. Rev. 45 (1934) 729 [INSPIRE]. within a global PDF analysis, arXiv:1712.07053 [INSPIRE]. C 75 (2015) 132 [arXiv:1412.7420] [INSPIRE]. Nucl. Phys. B 261 ( 1985 ) 379 [INSPIRE]. Teubner , Stuttgart, Germany ( 2001 ) [INSPIRE]. [17] M.S. Chanowitz and M.K. Gaillard , The TeV Physics of Strongly Interacting W's and Z's , [18] M. Bohm , A. Denner and H. Joos , Gauge theories of the strong and electroweak interaction , [19] J .-y. Chiu, A. Fuhrer , R. Kelley and A.V. Manohar , Soft and Collinear Functions for the Standard Model , Phys. Rev. D 81 ( 2010 ) 014023 [arXiv: 0909 .0947] [INSPIRE]. [20] J .-y. Chiu, A. Fuhrer , R. Kelley and A.V. Manohar , Factorization Structure of Gauge Theory [21] M.B. Voloshin and V.I. Zakharov , Measuring QCD Anomalies in Hadronic Transitions Between Onium States , Phys. Rev. Lett . 45 ( 1980 ) 688 [INSPIRE]. Goldstone Bosons , Annals Phys . 192 ( 1989 ) 93 [INSPIRE]. [22] R.S. Chivukula , A.G. Cohen , H. Georgi , B. Grinstein and A.V. Manohar , Higgs Decay Into [28] D.B. Clark , E. Godat and F.I. Olness , ManeParse: A Mathematica reader for Parton Distribution Functions, Comput . Phys. Commun . 216 ( 2017 ) 126 [arXiv: 1605 .08012]

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP05%282018%29106.pdf

Bartosz Fornal, Aneesh V. Manohar, Wouter J. Waalewijn. Electroweak gauge boson parton distribution functions, Journal of High Energy Physics, 2018, 106, DOI: 10.1007/JHEP05(2018)106