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 deepinelastic 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 multiTeV 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)
nonsinglet 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 deepinelastic
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 lightcone coordinates,
decomposing a fourvector 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 antiquark,
Q $ Q and the Wilson line is in the antifundamental representation. The PDF operators
involve an ordinary product of elds, not the timeordered 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 lefthanded SU(2) doublet
elds,
and u, d, e are righthanded 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 nonzero 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 eldstrength 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
eldstrength 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 transverselypolarized 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 lightcone operators
term shown in eq. (2.9). The two terms in O
, etc. can be similarly combined.
{ 4 {
(a)
(b)
eldstrength 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 HooftVeltman 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 threepoint correlator in the proton.
where p is the proton momentum and q is the incoming gaugeboson 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 deepinelastic scattering.
The structure functions for electromagnetic scattering are denoted by Fi
scattering p ! e X by Fi( ), for antineutrino 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
eldstrength tensor. Instead, they have to be computed in terms of Goldstone bosons using
the Goldstoneboson 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 Goldstoneboson 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 FermiWeizsackerWilliams [23{25] approximation for electroweak gauge
bosons. We will compare these with our results.
Considering for concreteness the transverse W PDF, the leadinglogarithmic
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 electronproton 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 spinaveraged 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 hardscattering cross sections
and parton distributions,
These hardscattering 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 longitudinallypolarized 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 treelevel Higgsquark 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 deepinelastic 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 lowQ2 part of the integration region is suppressed by
Z
T
γZT
γ * 0.5
4)x 0.5
4
.
0
1105
104
+
WT
WT
Z
T
γZT
γ * 0.1
4)x 4
1
)• 2
µ
,
x
f(x 1
0
1105
= 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 lowenergy 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 formfactors 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 lefthanded quarks prefer to emit lefthand 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 higherorder 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
ΔWT
ΔZT
2 ΔγZT
2.5105
105
104
103
µ
,
Z
L
105
104
103
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 DESC0009919, the ERC grant
ERCSTG2015677323, the DITP 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 (CCBY 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, BlochNordsieck violating electroweak corrections
[3] M. Ciafaloni, P. Ciafaloni and D. Comelli, Electroweak BlochNordsieck violation at the TeV
scale: `Strong' weak interactions?, Nucl. Phys. B 589 (2000) 359 [hepph/0004071]
[4] C.W. Bauer, N. Ferland and B.R. Webber, Combining initialstate resummation with
xedorder 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
HighEnergy 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]