BI-TP 2000/04

DESY 01-121

Edinburgh 2001/11

hep-ph/0109062

Electroweak radiative corrections to

[.5em] W-boson production at hadron colliders

Stefan Dittmaier^{†}^{†}†Heisenberg fellow of the Deutsche Forschungsgemeinschaft DFG.

[1ex] Deutsches Elektronen-Synchrotron DESY,

D-22603 Hamburg, Germany

[2ex] Michael Krämer

[1ex] Department of Physics and Astronomy, University of Edinburgh,

Edinburgh EH9 3JZ, Scotland

[2ex]

Abstract:

The complete set of electroweak corrections to the Drell–Yan-like production of W bosons is calculated and compared to an approximation provided by the leading term of an expansion about the W-resonance pole. All relevant formulae are listed explicitly, and particular attention is paid to issues of gauge invariance and the instability of the W bosons. A detailed discussion of numerical results underlines the phenomenological importance of the electroweak corrections to W-boson production at the Tevatron and at the LHC. While the pole expansion yields a good description of resonance observables, it is not sufficient for the high-energy tail of transverse-momentum distributions, relevant for new-physics searches.

September 2001

## 1 Introduction

The Drell–Yan-like production of W bosons represents one of the cleanest processes with a large cross section at the Tevatron and at the LHC. This reaction is not only well suited for a precise determination of the W-boson mass , it also yields valuable information on the parton structure of the proton. Specifically, the accuracy of – [1] in the measurement envisaged at the LHC will improve upon the precision of to be achieved at LEP2 [2] and Tevatron Run II [3], and thus competes with the precision of the measurement expected at a future collider [4]. Concerning quark distributions, precise measurements of rapidity distributions provide information over a wide range in [5]; a measurement of the d/u ratio would, in particular, be complementary to HERA results. The more direct determination of parton–parton luminosities instead of single parton distributions is even more precise [6]; extracting the corresponding luminosities from Drell–Yan-like processes allows one to predict related processes at the per-cent level.

Owing to the high experimental precision outlined above, the predictions for the processes should match per-cent accuracy; for specific observables the required theoretical accuracy is even higher. To this end, radiative corrections have to be included. In particular, it is important to treat final-state radiation carefully, since photon emission from the final-state lepton significantly changes the lepton momentum, which is used in the determination of the W-boson mass. A first step to include electroweak corrections was already made in Ref. [7], where effects of final-state radiation in the W-boson decay stage were taken into account. Those effects lead to a shift in the value of of the order of 50–150. The approximation of Ref. [7] was improved much later in Ref. [8], where the electroweak corrections to resonant W-boson production [9] were discussed for W production at the Tevatron in detail. The corrections are of the order of the known next-to-next-to-leading order (NNLO) QCD corrections [10] to Drell–Yan-like processes. In only a study of two-photon radiation exists [11], while the virtual counterpart is completely unknown. Discussions of QCD corrections to Drell–Yan-like processes can be found in Refs. [12, 13] and references therein.

In this paper we present the complete calculation of the electroweak corrections, including non-resonant contributions. In particular, we compare the full correction to a pole approximation (similar to the one used in Refs. [8, 9]) that is based on the correction to the production of resonant W bosons. All relevant formulae are listed explicitly. Moreover, a discussion of numerical results is presented for the Tevatron (Run II) and for the LHC. Partial results of this analysis have already been presented in the LHC workshop report [14].

The paper is organized as follows. In Sect. 2 we set our conventions and provide analytical results for the parton-level subprocess. In particular, we describe the calculation of the complete and the “pole-approximated” corrections. Different methods for treating the infrared and collinear singularities are presented and compared with each other. The hadronic cross section is discussed in Sect. 3. In Section 4 we present numerical results for W-boson production at the Tevatron and at the LHC. Our conclusions are given in Sect. 5. Finally, the appendices provide some supplementary formulae.

## 2 The parton process

### 2.1 Conventions and lowest-order cross section

We consider the parton process

(2.1) |

where and generically denote the light up- and down-type quarks, and . The lepton represents . The momenta and helicities of the corresponding particles are given in brackets. The Mandelstam variables are defined by

(2.2) |

We neglect the fermion masses , , whenever possible, i.e. we keep these masses only as regulators in the logarithmic mass singularities originating from collinear photon emission or exchange. Obviously, we have for the non-radiative process . In lowest order only the Feynman diagram shown in Fig. 1 contributes to the scattering amplitude,

and the Born amplitude reads

(2.3) |

with an obvious notation for the Dirac spinors , etc., and the left-handed chirality projector . The electric unit charge is denoted by , the weak mixing angle is fixed by the ratio of the W- and Z-boson masses and , and is the CKM matrix element for the transition.

Strictly speaking, Eq. (2.3) already goes beyond lowest order, since the W-boson width results from the Dyson summation of all insertions of the (imaginary part of the) W self-energy. Defining the mass and the width of the W boson in the on-shell scheme (see e.g. Ref. [15]), the Dyson summation directly leads to a running width, i.e.

(2.4) |

On the other hand, a description of the resonance by an expansion about the complex pole in the complex plane corresponds to a constant width, i.e.

(2.5) |

In lowest order these two parametrizations of the resonance region are fully equivalent, but the corresponding values of the line-shape parameters and differ in higher orders [16, 9, 17],

(2.6) |

Since , it is necessary to state explicitly which parametrization is used in a precision determination of the W-boson mass from the W line-shape.

The differential lowest-order cross section is easily obtained by squaring the lowest-order matrix element of (2.3),

(2.7) |

where the explicit factor results from the average over the quark spins and colours, and is the solid angle of the outgoing in the parton centre-of-mass (CM) frame. The electromagnetic coupling can be set to different values according to different input-parameter schemes. It can be directly identified with the fine-structure constant or the running electromagnetic coupling at a high-energy scale . For instance, it is possible to make use of the value of that is obtained by analyzing [18] the experimental ratio . These choices are called -scheme and -scheme, respectively, in the following. Another value for can be deduced from the Fermi constant , yielding ; this choice is referred to as -scheme. The differences between these schemes will become apparent in the discussion of the corresponding corrections.

### 2.2 Virtual corrections

The virtual one-loop corrections comprise contributions of the transverse part of the self-energy , corrections to the two and vertices, box diagrams, and counterterms. The explicit expression for (in the ‘t Hooft–Feynman gauge) can, e.g., be found in Ref. [15]. The diagrams for the vertex and box corrections, which are shown in Fig. 2, were calculated using standard methods.

The Feynman diagrams and amplitudes were generated with FeynArts [19]. The subsequent algebraic reduction [20] of the one-loop tensor integrals to scalar integrals was performed with FeynCalc [21], and the scalar integrals were evaluated using the methods and results of Ref. [22]. The algebraic part was checked numerically by a completely independent calculation, in which the amplitudes are expressed in terms of tensor coefficients using Mathematica, and the tensor reduction is done numerically. UV divergences are treated in dimensional regularization, and the IR singularity is regularized by an infinitesimal photon mass . The actual calculation is performed in ‘t Hooft–Feynman gauge using the on-shell renormalization scheme described in Ref. [15], where, in particular, all renormalization constants used in this paper can be found. As an additional check we have repeated the calculation within the background-field formalism [23] and found perfect agreement. In the following we sketch the structure of the virtual corrections and emphasize those points that are relevant for the treatment of the resonance and for the change from one input-parameter scheme to another. The complete expressions for the vertex and box corrections are provided in App. A.

While the self-energy and vertex corrections are proportional to the Dirac structure appearing in the lowest-order matrix element , the calculation of the box diagrams leads to additional combinations of Dirac chains. However, since the box diagrams are UV-finite, the four-dimensionality of space-time can be used to reduce all Dirac structures to the one of , see App. A.2. In summary, the complete one-loop amplitude can be expressed in terms of a correction factor times the lowest-order matrix element,

(2.8) |

Thus, in the squared matrix element reads

(2.9) |

so that the Breit–Wigner factors are completely contained in the
lowest-order factor .
Note that the Dyson-summed imaginary
part of the W self-energy, which appears as in ,
is not double-counted, since only the real part of
enters in .^{1}^{1}1Note also that only the imaginary parts
induced by the fermion loops, which lead to the physical decay width in
the resonance propagator, are resummed, while all other parts of
are treated in .
In this way the differences between the on-shell renormalization
of the W-boson mass (see e.g. Ref. [15])
and other variants based on the complex pole
position in the inverse W propagator are of ,
and thus beyond the accuracy of our calculation. In particular,
the problems [24] with the resummation of loops
containing photon exchange are avoided.
The correction factor is decomposed
into four different parts,

(2.10) |

according to the splitting into self-energy, vertex, and box diagrams.

The self-energy correction reads

(2.11) |

where the explicit expression for the unrenormalized self-energy is given in Eq. (B.4) of Ref. [15]. In the on-shell renormalization scheme the renormalization constants for the W-boson mass and field, and , are directly related to .

The vertex corrections are given by

(2.12) |

where the explicit expression for the form factor is
given in App. A.1. The counterterm
for the vertex depends on the input-parameter scheme. In the
-scheme (i.e. the usual on-shell scheme), it is given by^{2}^{2}2We consistently set the CKM matrix to the unit matrix in the
correction factor , since mixing effects in the corrections are negligible. This means that the CKM matrix
appears only in the global factor to the -corrected parton cross section.

(2.13) |

The wave-function renormalization constants and are obtained from the (left-handed part of) the fermion self-energies, and the renormalization of the weak mixing angle, i.e. , is connected to the mass renormalization of the gauge-boson masses. The charge renormalization constant contains logarithms of the light-fermion masses, inducing large corrections proportional to , which are related to the running of the electromagnetic coupling from to a high-energy scale. In order to render these quark-mass logarithms meaningful, it is necessary to adjust these masses to the asymptotic tail of the hadronic contribution to the vacuum polarization of the photon. Using , as defined in Ref. [18], as input this adjustment is implicitly incorporated, and the counterterm reads

(2.14) |

where

(2.15) |

with denoting the photonic vacuum polarization induced by all fermions other than the top quark (see also Ref. [15]). In contrast to the -scheme the counterterm does not involve light quark masses, since all corrections of the form are absorbed in the lowest-order cross section parametrized by . In the -scheme, the transition from to is ruled by the quantity [25, 15], which is deduced from muon decay,

(2.16) |

Therefore, the counterterm reads

(2.17) |

Since is explicitly contained in , the large fermion-mass logarithms are also resummed in the -scheme. Moreover, the lowest-order cross section in -parametrization absorbs large universal corrections induced by the -parameter.

The box correction is the only virtual correction that depends also on the scattering angle, i.e. on the variables and . The explicit expression for is given in App. A.2.

Despite the separation of the resonance pole from the correction factor in (2.8), still contains logarithms that are singular on resonance. Since these singularities would be cured by a Dyson summation of the self-energy inside the loop diagrams, we substitute

(2.18) |

with a fixed width everywhere, independent of the use of a fixed or running width in lowest order. In principle, also a running width could be used on the r.h.s., but the difference to the fixed width is of two-loop order, and thus beyond the accuracy of our calculation. The substitution (2.18) does not disturb the gauge-invariance properties of the one-loop amplitude , i.e. its gauge-parameter independence and the validity of SU(2)U(1) Ward identities. The reason is that the sum of all terms in proportional to separately fulfills the (algebraic) relations arising from gauge invariance, since this logarithm is the only resonant (and thus a unique) term in the relative correction .

### 2.3 Virtual correction in pole approximation

If one is only interested in the production of (nearly) resonant
W bosons, the electroweak corrections can be approximated by an
expansion [26] about the resonance pole, which is
located in the complex plane at up to
higher-order terms. The approximation of taking into account only the
leading term of this expansion is called pole approximation (PA)
and should not be confused with the on-shell approximation for the
W bosons. In contrast to the PA, the on-shell approximation,
where the W bosons are assumed to be stable, does not provide a
description of the W line-shape. In the following we construct a PA
for the virtual correction factor in the same
way as a double-pole approximation was constructed in
Ref. [27] for the more complicated case of
W-pair production in annihilation, fermions.
In this formulation the PA is only applied to the virtual corrections,
while the real corrections are based on the full photon-emission
matrix element, as described in the next section.
In principle, it is also possible to construct a PA for the real
corrections, as for instance described in Ref. [28]
for .^{3}^{3}3More details about the pole expansions that are used in
practice are reviewed in Ref. [29].
However, we prefer to
make use of the full matrix elements, as it was also done in the
variant of the PA used in Ref. [8] for W production in
hadronic collisions.

In the PA the virtual corrections to can be classified into two categories. The first category comprises the corrections to the production and the decay of an on-shell W boson. Owing to the independence of these subprocesses, these corrections are called factorizable. All contributing Feynman graphs are of the generic form shown in Fig. 3.

By definition, the factorizable corrections receive only contributions from the self-energy and the vertex corrections. The corresponding correction factor is obtained from and by setting and . Since we have in the complete on-shell renormalization scheme, we get

(2.19) |

Since the vertex corrections for on-shell W bosons correspond to physical S-matrix elements, both contributions to the factorizable corrections are gauge-invariant. Note that is a constant factor, neither depending on the scattering energy nor on the scattering angle. Moreover, contains IR singularities originating from the logarithms of (2.18); these terms are connected to photon emission from on-shell W bosons and are regularized by the infinitesimal photon mass .

The second category of corrections in the PA are called non-factorizable [30, 31] and comprise all remaining resonant contributions, i.e. all terms in that are non-vanishing for and ,

(2.20) |

Here results from the full off-shell correction upon taking the asymptotic limits and while keeping the ratio fixed. As can be shown by simple power counting [31], only loop diagrams with an internal photon contribute. The relevant diagrams for are shown in Fig. 4.

Since also box diagrams are involved, production and decay do not proceed independently, and the terminology non-factorizable is justified. The limit for the pole expansion has to be defined carefully, because is not the only kinematical variable. The variables and , which are related to the scattering angle of the parton CM frame, range within and are related to by . Therefore, changing while keeping and fixed is inconsistent in general. We circumvent this problem by taking for fixed scattering angle , resulting in the replacements

(2.21) |

The actual calculation of is performed as described in Ref. [31] in detail. The final result is

(2.22) | |||||

where

(2.23) |

is the usual dilogarithm, and denotes the electric charge of the fermion . In (2.22) we made use of . We note that is, by definition, a gauge-invariant quantity. It does not involve mass singularities of the fermions, but it is IR-singular. More precisely, the IR-singular term proportional to exactly compensates the artificially created IR singularity in by setting and there, and these terms of are replaced by the correct logarithms after is added.

In summary, the PA for the virtual correction factor reads

(2.24) |

with and given in (2.19) and (2.22), respectively. The uncertainty induced by omitting non-resonant corrections can be estimated naively by

(2.25) |

where the factor results from the neglect of non-resonant contributions near resonance and the logarithms account for typical enhancements away from resonance.

Finally, we note that we have analytically compared the PA for the
virtual corrections worked out in this section with the results
presented in Ref. [9]. Apart from non-resonant
contributions, which go beyond the validity of the PA, both
results agree.^{4}^{4}4There is a
misprint in Eq. (D.45) in Ref. [9]: the factor 2 in
front of the , and terms needs to be
removed.

### 2.4 Real-photon emission

Real-photonic corrections are induced by the diagrams shown in Fig. 5.

In the calculation of the corresponding amplitudes it is mandatory to respect the Ward identity for the external photon, i.e. electromagnetic current conservation. Writing the amplitudes as with denoting the polarization vector of the outgoing photon, this Ward identity reads . If the W width is zero, this identity is trivially fulfilled. This remains even true for a constant width, since the W-boson mass appears only in the W propagator denominators, i.e. the substitution is a consistent reparametrization of the amplitude in this case. However, if a running W width is introduced naively, i.e. in the W propagators only, the Ward identity is violated. The Ward identity can be restored by taking into account the part of the fermion-loop correction to the vertex that corresponds to the fermion loops in the self-energy leading to the width in the propagator [17, 32]. In Ref. [33] it was shown that this modification simply amounts to the multiplication of the vertex by the factor

(2.26) |

if the photon is on shell (). By construction [17, 32, 33], the width appearing in as well as in the W propagator is the lowest-order width, since it results from the imaginary part of the W self-energy at one loop. Since, however, for the special case of an on-shell photon the relevant imaginary parts are completely parametrized by the ratio , the Ward identity is fulfilled for any numerical value of . Therefore, we are allowed to use a value for that includes also QCD and electroweak radiative corrections. For later convenience, we define

(2.27) |

In addition to the U(1) Ward identity for the external photon, an SU(2) Ward identity [17, 32] becomes relevant for effectively longitudinally polarized W bosons in production at high energies, where the lepton current becomes proportional to the W momentum . As can be checked easily, this Ward identity is maintained in the above treatment of the W width as well.

The helicity amplitudes for the radiative process can be written in a very compact way using the Weyl-van der Waerden spinor formalism. Adopting the conventions of Ref. [34], we obtain

(2.28) | |||||

The amplitudes for the other helicity channels vanish for massless fermions. The spinor products are defined by

(2.29) |

where , are the associated momentum spinors for the light-like momenta

(2.30) |

The contribution of the radiative process to the parton cross section is given by

(2.31) |

where the phase-space integral is defined by

(2.32) |

### 2.5 Treatment of soft and collinear singularities

The phase-space integral (2.31) diverges in the soft () and collinear () regions logarithmically if the photon and fermion masses are set to zero, as done in (2.28). For the treatment of the soft and collinear singularities we applied three different methods, the results of which are in good numerical agreement. In the following we briefly sketch these approaches.

#### 2.5.1 IR phase-space slicing and effective collinear factors

Firstly, we made use of the variant of phase-space slicing that is described in Ref. [35], where the soft-photon region is excluded in the integral (2.31) but the regions of photon emission collinear to the fermions are included.

In the soft-photon region the bremsstrahlung cross section factorizes into the lowest-order cross section and a universal eikonal factor that depends on the photon momentum (see, e.g., Ref. [15]). Integration over in the partonic CM frame yields a simple correction factor to the partonic Born cross section ,

(2.33) | |||||

The factor can be added directly to the virtual correction factor defined in (2.9). It can be checked easily that the photon mass cancels in the sum .

The remaining phase-space integration in (2.31) with still contains the collinear singularities in the regions in which , , or is small. In these regions, however, the asymptotic behaviour of the differential cross section (including its dependence on the fermion masses) has a well-known form. The singular terms are universal and factorize from . A simple approach to include the collinear regions consists in a suitable modification of , which was calculated for vanishing fermion masses. More precisely, is multiplied by an effective collinear factor that is equal to 1 up to terms of () outside the collinear regions, but replaces the poles in , , and by the correctly mass-regularized behaviour. Explicitly, the substitution reads

(2.34) | |||||

The functions and describe collinear photon emission with and without spin flip of the radiating fermion, respectively,

(2.35) |

where is the fermion energy and is the angle of the photon emission from .

#### 2.5.2 IR and collinear phase-space slicing

Instead of using effective collinear factors, alternatively we have also applied phase-space slicing to the collinear singularities, i.e. the collinear regions are now excluded by the angular cuts in the integral (2.31). The IR region is treated as previously, leading to the same correction factor as given in (2.33).

In the collinear cones the photon emission angles can be integrated out by making use of the factorization property of the squared photon-emission matrix elements with the radiator functions , as described in the previous section. The resulting contribution to the bremsstrahlung cross section has the form of a convolution of the lowest-order cross section,

(2.36) | |||||

(2.37) | |||||

(2.38) | |||||

with the splitting function

(2.39) |

For initial-state radiation the respective quark momentum is reduced by the factor so that the partonic CM frame for the hard scattering receives a boost, while this is not the case for final-state radiation. Note that for final-state radiation, i.e. in , the lepton momentum in the final state is , although is relevant for the lowest-order cross section in the hard scattering. Of course, if photons collinear to the lepton are not separated the integration can be carried out explicitly. It can be checked easily that in this case all logarithms of the lepton mass cancel in the sum of virtual and real corrections.

#### 2.5.3 Subtraction method

Finally, we applied the subtraction method presented in Ref. [36], where the so-called “dipole formalism”, originally introduced by Catani and Seymour [37] within massless QCD, was applied to photon radiation and generalized to massive fermions. The general idea of a subtraction method is to subtract and to add a simple auxiliary function from the singular integrand. This auxiliary function has to be chosen such that it cancels all singularities of the original integrand so that the phase-space integration of the difference can be performed numerically. Moreover, the auxiliary function has to be simple enough so that it can be integrated over the singular regions analytically, when the subtracted contribution is added again.

The dipole subtraction function consists of contributions labelled by all ordered pairs of charged external particles, one of which is called emitter, the other one spectator. For we, thus, have six different emitter/spectator cases : , , , , , . The subtraction function that is subtracted from is given by

(2.40) | |||||