Combined Effect of QCD Resummation and QED Radiative
Correction
to boson Observables at the Tevatron
Abstract
A precise determination of the boson mass at the Fermilab Tevatron requires a theoretical calculation in which the effects of the initialstate multiple softgluon emission and the finalstate photonic correction are simultaneously included . Here, we present such a calculation and discuss its prediction on the transverse mass distribution of the boson and the transverse momentum distribution of its decay charged lepton, which are the most relevant observables for measuring the boson mass at hadron colliders.
pacs:
12.38.t;12.15.LkMSUHEP040106
hepph/0401026
As a fundamental parameter of the Standard Model (SM), the mass of the boson () is of particular importance. Aside from being an important test of the SM itself, a precision measurement of , together with an improved measurement of top quark mass (), provides severe indirect bounds on the mass of Higgs boson (). With a precision of 27 MeV for and 2.7 GeV for , which are the target values for Run II of the Fermilab Tevatron collider, in the SM can be predicted with an uncertainty of about 35% Baur:2001yp . Comparison of these indirect constraints on with the results from direct Higgs boson searches, at the LEP2, the Tevatron and the CERN Large Hadron Collider (LHC), will be an important test of the SM. In order to have a precision measurement of , the theoretical uncertainties, dominantly coming from the transverse momentum of the boson (), the uncertainty in parton distribution function (PDF) and the electroweak (EW) radiative corrections to the boson decay, must be controlled to a better accuracy Baur:2000bi .
At the Tevatron, about ninety percent of the production cross section of boson is in the small transverse momentum region, where GeV. When is much smaller than , every softgluon emission will induce a large logarithmic contribution to the distribution so that the orderbyorder perturbative calculation in the theory of Quantum chromodynamics (QCD) cannot accurately describe the spectrum and the contribution from multiple softgluon emission, which contributes to all orders in the expansion of the strong coupling constant , needs to be summed to all orders. It has been shown that by applying the renormalization group analysis, the multiple softgluon radiation effects can be resummed to all orders to predict the distribution that agrees with experimental data Balazs:1995nz ; Balazs:1997xd . RESBOS, a Monte Carlo (MC) program Balazs:1997xd resumming the initialstate softgluon radiations of the hadronically produced lepton pairs through EW vector boson production and decay at hadron colliders , has been used by the CDF and DØ Collaborations at the Tevatron to compare with their data in order to determine . However, RESBOS does not include any higher order EW corrections to describe the vector boson decay. The EW radiative correction, in particular the finalstate QED correction, is crucial for precision measurement of boson mass at the Tevatron, because photon emission from the finalstate charged lepton can significantly modify the lepton momentum which is used in the determination of . In the CDF Run Ib mass measurement, the mass shifts due to radiative effects were estimated to be MeV and MeV for the electron and muon channels, respectively Affolder:2000bp . The full nexttoleading order (NLO) EW corrections have been calculated Dittmaier:2001ay ; Baur:1998kt and resulted in WGRAD Baur:1998kt , a MC program for calculating EW radiative corrections to the process . However, WGRAD does not include the dominant correction originated from the initialstate multiple softgluon emission. To incorporate both the initialstate QCD and and finalstate QED corrections into a parton level MC program is urgently required to reduce the theoretical uncertainties in interpreting the experimental data at the Tevatron. It was shown in Refs. Dittmaier:2001ay ; Baur:1998kt that at the NLO, the EW radiative correction in is dominated by the finalstate QED (FQED) correction. Hence, in this paper we present a consistent calculation which includes both the initialstate multiple softgluon QCD resummation and the finalstate NLO QED corrections, and develop an upgraded version of the RESBOS program, called RESBOSA RESBOSA , to simulate the signal events.
The fully differential cross section for the production and decay of the boson that includes only the effect of the initialstate multiple QCD softgluon emission can be found in Ref. Balazs:1997xd . To include also the finalstate NLO QED contributions, we sum up the following two sets of differential cross sections. One, cf. Eq. (1), contains finalstate QED virtual correction and part of the real photon emission contribution in which photon is either soft or collinear. Another, cf. Eq. (2), includes the hard photon contribution from the real photon emission processes. Denote , , and to be the invariant mass, rapidity, transverse momentum and azimuthal angle of the dilepton pair, respectively. For production and decay, we have
(1)  
and
(2)  
where and represents the twobody and threebody phase space of the vector boson decay products, respectively. In the above equations the parton momentum fractions are defined as and , where is the centerofmass energy of the hadrons and . The renormalization group invariant quantities and , which sum to all orders in all the singular terms that behave as for , are
(3)  
and
(4)  
where is the Born level parton cross section for , and includes the finalstate NLO QED corrections. The notation denotes the convolution Balazs:1997xd
(5) 
and the coefficients are the CabibboKobayashiMaskawa mixing matrix elements. In the above expression represents quark flavors and stands for antiquark flavors. The indices and are meant to sum over quarks and antiquarks or gluons. Summation on these double indices is implied. As compared to the results shown in Refs. Balazs:1995nz ; Balazs:1997xd , and contain additional corrections, which come from the final state QED corrections. The Sudakov exponent in Eqs. (3) and (4) is defined as Balazs:1997xd
(6) 
The explicit forms of the , , and functions and the renormalization constants can be found in Appendix D of Ref. Balazs:1997xd . In our calculation, we have included , , , , and , with canonical choice of ’s. The piece in Eq. (1), which is the difference of the fixed order perturbative result and their singular part, can be found in Appendix E of Ref. Balazs:1997xd .
We follow the prescription in Ref. Wackeroth:1996hz , which decomposes the electroweak contribution to the resonant single production in a general 4fermion process into gauge invariant QEDlike and weak parts, to extract a gauge invariant QEDlike form factor from the photon contribution. The NLO FQED differential cross sections are calculated by using phase space slicing method Baer:1989jg , which introduces two theoretical cutoff parameters, soft cutoff and collinear cutoff , to isolate the soft and collinear singularities associated with the real photon emission subprocesses by partitioning phase space into soft, collinear and hard regions such that
(7) 
The soft region is thus defined by requiring that the photon energy () in the () parton centerofmass frame to satisfy , where is the invariant mass of the () partons. Using the dimensional regularization scheme, we can then evaluate, in ndimensions, the real photon emission diagrams under the soft photon approximation, where the photon momentum is set to be zero in the numerator, and integrate over the soft region. In the soft and collinear regions the cross section is proportional to the Born cross section. The soft singularities originating from the finalstate photon radiation cancel against the corresponding singularities from the finalstate virtual corrections and leave a finite result depending on the soft cutoff parameter . For , the real photon emission diagrams are calculated in four dimensions using the helicity amplitude method. The collinear singularities associated with photon radiation from the finalstate charged lepton is regulated by the finite lepton mass. The end result of the calculation consists of two sets of weighted events corresponding to the and contributions which are included in Eqs. (1) and (2), separately. Each set depends on the soft cutoff parameter . The sum of the two contributions, however, is independent of , as long as the soft cutoff is small enough to validate the softgluon approximation. In our numerical studies, we take which yields a stable numerical result in agreement with Refs. Dittmaier:2001ay ; Baur:1998kt . Through our calculation, we adopt the CERN LEP lineshape prescription of a resonance state and write the boson propagator as
(8) 
where is the width of boson.
To examine how much the combined contributions from the initialstate QCD resummation and the finalstate QED corrections can affect the precision measurement of , we perform Monte Carlo analyses to study a few experimental observables that are most sensitive to the measurement of at the Tevatron (a collider with =1.96 TeV). For the numerical evaluation we chose the following set of SM parameters: , , GeV, GeV, GeV, MeV. Thus, the square of the weak gauge coupling is . Because of the limited space, we focus our attention on the positively charged electron lepton (i.e. positron) only, though our analysis procedure also applies to the lepton. The complete study including both electron and muon leptons will be shown in our forthcoming paper, in which we also extend our study to the LHC.
The events in these analyses are selected by demanding a single isolated high charged lepton in conjunction with large missing transverse energy. To model the acceptance cuts used by the CDF and DØ Collaborations in their mass analyses, we impose the following transverse momentum () and pseudorapidity () cuts on the finalstate leptons:
(9) 
Due to the overwhelming QCD backgrounds, the measurement of boson mass at hadron collider is performed in the leptonic decay channels. Since the longitudinal momentum of the neutrinos produced in the leptonic boson decays () cannot be measured, there is insufficient information to reconstruct the invariant mass of the boson. Instead, the transverse mass distribution of the final state lepton pair, which exhibits a Jacobian edge at , is used to extract out . Transverse mass () of is defined as
(10) 
where is the angle between the charged lepton and the neutrino in the transverse plane. The neutrino transverse momentum () is identified with the missing transverse energy () in the event. In Fig. 1, we show various theory predictions on the distribution. The legend of the figure is defined as follows:

LO : including only the Born level initialstate contribution,

RES : including the initialstate multiple softgluon corrections via QCD resummation,

LO QED : including only the Born level finalstate contribution,

NLO QED : including the finalstate NLO QED corrections.
For example, the solid curve (labelled as RES+NLO QED) in Fig. 1(a) is the prediction from our combined calculation given by Eqs. (1) and (2).
As shown in Fig. 1(a), compared to the lowest order cross section (dotted curve), the initial state QCD resummation effects (dashed curve) increase the cross section at the peak of the distribution by about , and the final state NLO QED corrections (dotdashed curve) decrease it by about , while the combined contributions (solid curve) of the QCD resummation and FQED corrections increase it by . In addition to the change in magnitude, the lineshape of the distribution is significantly modified by the effects of QCD resummation and FQED corrections. To illustrate this point, we plot the ratio of the (RES+NLO QED) differential cross sections to the LO ones as the solid curve in Fig. 1(b). The dashed curve is for the ratio of (LO+NLO QED) to LO. As shown in the figure, the QCD resummation effect dominates the shape of distribution for , while the FQED correction reaches its maximal effect around the Jacobian peak (). Hence, both corrections must be included to accurately predict the distribution of around the Jacobian region to determine . We note that after including the effect due to the finite resolution of the detector (for identifying an isolated electron or muon), the size of the FQED correction is largely reduced Dittmaier:2001ay ; Baur:1998kt .
Although the distribution has been the optimal observable for determining at the Tevatron, it requires an accurate measurement of the missing transverse momentum direction which is in practice difficult to control. On the other hand, the transverse momentum of the decay charged lepton () is less sensitive to the detector resolution, so that it can be used to measure and provide important crosscheck on the result derived from the distribution, for they have different systematic uncertainties. Another important feature of this observable is that distribution is more sensitive to the transverse momentum of boson. Hence, the QCD softgluon resummation effects, the major source of , must be included to reduce the theoretical uncertainty of this method. In Fig. 2(a), we show the distributions predicted by various theory calculations, and in Fig. 2(b), the ratios of the higher order to lowest order cross sections as a function of . The lowest order distribution (dotted curve) shows a clear and sharp Jacobian peak at , and the distribution with the NLO finalstate QED correction (dotdashed curve) also exhibits the similar Jacobian peak with the peak magnitude reduced by about . But the clear and sharp Jacobian peak of the lowest order and NLO FQED distributions (in which ) are strongly smeared by the finite transverse momentum of the boson induced by multiple softgluon radiation, as clearly demonstrated by the QCD resummation distribution (dashed curve) and the combined contributions of the QCD resummation and FQED corrections (solid curve). Similar to the distribution, the QCD resummation effect dominates the whole range, while the FQED correction reaches it maximum around the Jacobian peak (half of ). The combined contribution of the QCD resummation and FQED corrections reaches the order of near the Jacobian peak. Hence, these lead us to conclude that the QCD resummation effects are crucial in the measurement of from fitting the Jacobian kinematical edge of the distribution.
It is also interesting to examine the effect of the finalstate NLO QED correction to the theory predictions with LO (leading order) or RES (resummed) initialstate cross sections, which is described by the observable , defined as
(11) 
where
(12)  
The distributions of as a function of and are shown in the upper part and lower part of Fig. 3, respectively. As expected, the distribution is almost flat all the way from to , but the distribution deviates from one in the Jacobian peak region. This is due to the fact that is more sensitive to than .
In order to study the impact of the presented calculation to the determination of the boson mass, the effect due to the finite resolution of the detector should be included, which will be presented elsewhere.
We thank P. Nadolsky and J.W. Qiu for helpful discussions. This work was supported in part by NSF under grand No. PHY0244919 and PHY0100677.
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