Impact of PDF uncertainties at large x on heavy boson production

Impact of PDF uncertainties at large
on heavy boson production

L. T. Brady, A. Accardi, W. Melnitchouk, J. F. Owens Jefferson Lab, Newport News, Virginia 23606
Harvey Mudd College, Claremont, California 91711
Hampton University, Hampton, Virginia 23668
Florida State University, Tallahassee, Florida 32306
Abstract

We explore the sensitivity of and boson production in hadronic collisions to uncertainties in parton distribution functions (PDFs) at large arising from uncertainties in nuclear corrections when using deuterium data in global QCD fits. The and differential cross sections show increasing influence of nuclear corrections at high boson rapidities, particularly for the quark, which is diluted somewhat in the decay lepton rapidity distributions. The effects of PDF uncertainties on heavy and bosons beyond the Standard Model become progressively more important for larger boson masses or rapidities, both in collisions at the LHC and in scattering at the Tevatron.

preprint: JLAB-THY-11-1454

I Introduction

The discovery and determination of properties of new particles beyond the Standard Model at high-energy colliders depends on accurate knowledge of parton distribution functions (PDFs) of the hadrons involved in the collisions. With the Large Hadron Collider (LHC) at CERN now taking data at unprecedentedly high energies, the effort to control backgrounds in searches for the Higgs boson and other putative particles is taking on paramount urgency. Recently the dependence of the Higgs boson cross section for the dominant gluon–gluon fusion channel on PDFs has been the cause of some debate Alekhin11 (); Thorne11 (); Baglio11 (), highlighting the need for a careful determination of strong interaction inputs such as gluon distributions, the strong coupling constant , and higher-order radiative corrections.

At lower energies, the importance of PDF uncertainties has also been discussed recently in fixed-target experiments, particularly for the quark distribution in the region of large parton momentum fractions () CTEQX (); CJ11 (). Because both proton and deuterium deep-inelastic scattering (DIS) data are required to constrain the quark PDF, uncertainties in the nuclear corrections in the deuteron at large translate into significant and growing uncertainties on the ratio as . Through evolution, this can impact cross section calculations at smaller and larger Kuhlmann00 (), especially in regions where the rapidity is large.

In addition to Higgs boson cross sections, other processes studied at the LHC or the Tevatron at Fermilab that may be sensitive to PDF uncertainties include the production of heavy and bosons associated with additional SU(2)U(1) gauge groups. These are predicted in various extensions of the Standard Model, such as the SO(10) and E grand unified theories, or supersymmetric models, some with and boson masses at the TeV scale (for reviews see, e.g., Refs. Rev:Hewett89 (); Rev:Leike99 (); Rev:Rizzo06 (); Rev:Langacker09 ()). Their production cross sections at large rapidities will involve products of PDFs evaluated with one value of small and the other large, thereby exposing them to uncertainties in PDFs at large , particularly near the kinematic limits.

In this paper we explore the sensitivity of the weak boson production cross sections to uncertainties in PDFs at large . For our numerical estimates we use the PDFs from the recent CTEQ-Jefferson Lab (CJ) next-to-leading order (NLO) global analysis CJ11 () of proton and deuteron data, which quantified the model dependence of the nuclear corrections in the deuteron and the resulting effects on the PDFs. In Sec. II we briefly review the CJ analysis and the origin of the PDF uncertainties, before examining their impact in Sec. III on the physical and boson cross sections, and and lepton charge asymmetries. The effects of large- PDF uncertainties on production rates of heavy and bosons in and collisions at the LHC and Tevatron, respectively, are studied in Sec. IV as a function of the boson mass, and limits placed on the accuracy with which cross sections for bosons of a given mass can currently be determined. (Even though data taking has now been completed at the Tevatron, considerable quantities of data remain to be analyzed.) Finally, some concluding remarks are made in Sec. V.

Ii PDF uncertainties at large

The CJ analysis CJ11 () was a dedicated global NLO fit of proton and deuteron DIS and other high-energy scattering data, which critically examined the effects on PDFs of nuclear corrections in the deuteron structure function. Nuclear corrections were estimated using a smearing function computed within the weak binding approximation KP06 (); Kahn09 (), taking into account nuclear binding and Fermi motion, as well as a range of models describing the possible modification of the nucleon structure function off-shell KP06 (); MSTprd (); MSTplb (). Several deuteron wave functions were considered, based on high-precision nucleon–nucleon potentials, including the nonrelativistic CD-Bonn CDBonn () and AV18 AV18 () wave functions, and the relativistic WJC-1 and WJC-2 wave functions WJC (), as well as the older Paris Paris () wave function for reference. The off-shell corrections were estimated from a relativistic quark spectator model MSTplb (), and from a phenomenological model proposed by Kulagin & Petti KP06 () but modified for the specific case of the deuteron (see Ref. CJ11 () for details).

Figure 1: Ratio of to quark distributions for the CJ PDFs CJ11 () at  GeV. The shaded band illustrates the uncertainty range between maximum (blue dashed) and minimum (red dot-dashed) nuclear corrections in the deuteron.

Combinations of deuteron wave functions and off-shell models giving the smallest and largest nuclear effects were identified, and used to define the range of the nuclear corrections from the minimum (WJC-1 wave function and no off-shell corrections) to the maximum (CD-Bonn wave function and largest off-shell corrections) nuclear corrections. The central values, which are used as a reference, were obtained using the AV18 wave function and an intermediate off-shell correction. The resulting fitted quark distribution ratio is shown in Fig. 1 for the full range of nuclear uncertainties determined in Ref. CJ11 () (see also Refs. HAMP11 (); ARM11 ()). While the quark distribution is relatively well constrained by proton structure function data for all values of , the quark PDF has large uncertainties beyond . Significant uncertainties appear also in the gluon PDF in the region of not directly constrained by data, as well as in the and distributions due to correlations with the quark, induced in the global fits by moderate- jet and dilepton production data, respectively.

Several experiments to directly measure up to are planned at the 12 GeV energy upgraded Jefferson Lab in the near future BoNuS12 (); MARATHON (); SOLID (), which it is hoped will reduce the uncertainties significantly. These include the “MARATHON” MARATHON () experiment, which aims to extract from a measurement of the structure functions of tritium and He with cancellation of nuclear effects to the level A3 (), as well as the “BoNuS” experiment BoNuS12 (), which minimizes nuclear corrections in semi-inclusive DIS from deuterium by tagging slow, backward protons that effectively guarantee scattering from a nearly-free neutron. In addition, parity-violating DIS on a hydrogen target SOLID () will yield a new combination of and PDFs at large , free of any nuclear corrections (see Ref. CJ11 () for further details). In the meantime, however, it is important to establish the limitations that the current uncertainties on the quark PDF at large place on the calculation of observables which may be sensitive to these. In the following we shall illustrate the impact of PDF uncertainties at large arising specifically from the dependence on the model of nuclear corrections in the deuteron. The choice of CJ PDFs for this purpose is merely for convenience, as these are the only PDFs available that explicitly quantify the nuclear model dependence.

Iii and boson production

In this section we discuss the effects of PDF uncertainties at large on the and boson cross sections, and possible constraints on these obtained from measurements at large rapidities. Earlier studies probing the sensitivity of weak boson production to PDF uncertainties were explored in Refs. Berger89 (); MST94 (); CTEQ3 (); Peng97 (); Alekhin10 (); CT10 (); MSTW09 (); NNPDF11 (); NNPDF12 (). The discussion here is not meant to provide an exhaustive account of detailed aspects of boson production, but simply highlight the fact that nuclear corrections in deuterium are an important source of PDF uncertainty at large that has not been addressed in earlier analyses. To begin with we shall review the general formulas for the inclusive weak boson production cross sections in hadronic collisions relevant to current collider experiments.

iii.1 Cross sections

Hadron–hadron collisions involve at least two interacting partons, one from the hadron “beam” and one from the “target”, with momentum fractions and , respectively. At fixed center of mass energy and boson rapidity

(1)

where and are the boson energy and longitudinal momentum in the hadron center of mass frame, the parton momentum fractions are given (at leading order in the strong coupling constant) by

(2)

where is the mass of the produced boson. The absolute value of the rapidity thus ranges from 0 up to . For inclusive production in or collisions, for example, the cross sections (to leading order and neglecting heavy quarks) are given by ColPhy ()

(3a)
(3b)

where is the Fermi constant, and is the Cabibbo mixing angle. The differential cross sections are similar to those in Eqs. (3), but with quark PDFs replaced by the corresponding antiquark PDFs. Consequently the cross sections in collisions are related by , and hence are equivalent when integrated over rapidity. For collisions the individual and cross sections are symmetric with respect to , but otherwise unrelated.

Similarly, the leading order, light quark cross sections for boson production in or collisions are given by ColPhy ()

(4a)
(4b)

where and are the vector and axial-vector couplings of the boson to quark PDG10 (), with and the electromagnetic charge and weak isospin of the quark, respectively, and the weak mixing angle. The symmetry properties of the differential cross sections are such that for both and collisions. Note that in the conventions of Ref. ColPhy () the couplings are two times smaller than the standard ones in Ref. PDG10 (). In the convention used here the couplings in Eqs. (4) are equal to and for and quarks, respectively, where . Since , the effective strengths of the boson couplings to and quarks are therefore similar.

While the expressions in Eqs. (3) and (4) are given at leading order, in practice we compute all cross sections at NLO, including heavy quarks. The leading order expressions give the dominant contributions, however, and are instructive in clearly illustrating that in collisions, for instance, the cross section at large rapidity is mostly dependent on the quark, while the cross section depends mostly on the . Since the nuclear corrections discussed in Sec. II induce the greatest uncertainty into the large- quark PDF, one can immediately deduce that production in scattering will be most affected by these uncertainties, while production will be relatively inert. For collisions, the large- PDF uncertainties will affect cross sections at large positive rapidities, or equivalently cross sections at negative rapidities.

In the following sections we will compute the and boson cross sections numerically to study their sensitivity to PDF uncertainties at large . All calculations will be for collisions at the LHC with  TeV, and for collisions at the Tevatron with  TeV.

iii.2 bosons

The sensitivity of the differential boson cross section to the different PDF behaviors at large is illustrated in Fig. 2 as a function of the boson rapidity , for LHC and Tevatron kinematics. The cross sections are computed from the CJ PDFs CJ11 () with minimal and maximal nuclear corrections, relative to a reference cross section computed from the central PDFs as described in Sec. II.

Figure 2: Differential boson cross section as a function of the rapidity , computed from CJ PDFs with maximum (blue dashed) and minimum (red dot-dashed) nuclear corrections, relative to the reference cross section calculated using the central CJ PDF set CJ11 (). The cross sections are computed for collisions at the LHC with  TeV (left) and for collisions at the Tevatron with  TeV (right).

The behavior of the cross section ratios is qualitatively similar at both the LHC and the Tevatron, with the main difference being the range of rapidities accessible from the respective available energies . At low rapidities the cross sections are relatively insensitive to uncertainties in the large- PDFs, with differences of for at the LHC and at the Tevatron. At larger rapidities, however, there is far greater sensitivity to the large- behavior, particularly of the quark, leading to uncertainty in the differential cross section for at the LHC, and for at the Tevatron, which correspond to parton fractions of . As one approaches the kinematical thresholds of at the LHC and at the Tevatron, these uncertainties increase dramatically, as would be expected from the behavior of the ratio in Fig. 1.

iii.3 cross sections and asymmetries

The behavior of the boson differential cross sections as a function of the rapidity is qualitatively similar to those for the , but with some important differences, as Fig. 3 illustrates for and production at LHC and Tevatron kinematics. Again, there is very little dependence on the large- PDF uncertainties at low rapidity, but increasing sensitivity as the rapidity approaches its kinematic upper limit of at the LHC and at the Tevatron.

Figure 3: Differential (top) and (bottom) boson cross sections as a function of the rapidity , computed from CJ PDFs with maximum (blue dashed) and minimum (red dot-dashed) nuclear corrections, relative to the reference cross sections calculated using the central CJ PDF set CJ11 (). The cross sections are computed for collisions at the LHC with  TeV (left) and for collisions at the Tevatron with  TeV (right).

For bosons, the cross section maintains relatively little dependence on the large- nuclear corrections over the entire rapidity range, barely reaching 4% difference at at the LHC, or at the Tevatron. In contrast, the cross section shows an even stronger dependence on nuclear corrections than the cross section in Fig. 2, deviating significantly from unity for for the LHC and for the Tevatron, and reaching upwards of 40% deviation at and 3 for LHC and Tevatron kinematics, respectively. The greater sensitivity of the production cross section compared with the can be understood from the dominance of the latter by the quark PDF at large , which is relatively insensitive to the nuclear correction uncertainties. The enhancement of the cross section at large for the CJ PDFs with maximum nuclear corrections, and corresponding suppression of the CJ PDFs with minimum nuclear corrections, relative to the central CJ fits essentially follows the trend of the quark PDF in Fig. 1. The slight enhancement of the cross section at large for the CJ PDFs with minimum nuclear corrections reflects the anticorrelation of the quark PDF with respect to the observed in Ref. CJ11 ().

Figure 4: boson asymmetry as a function of the rapidity at LHC (left) and Tevatron (right) kinematics, computed from CJ PDFs CJ11 () with minimum (upper blue solid) and maximum (lower blue solid) nuclear corrections. For comparison, the asymmetries using the ABKM Alekhin10 () (red dot-dashed), CT10 CT10 () (green dashed) and MSTW MSTW09 () (black dotted) PDF sets are also shown.

Taking differences and sums of the and cross sections, one can construct the boson asymmetry,

(5)

where . The asymmetry is shown in Fig. 4 versus the rapidity at the LHC and Tevatron for the CJ PDFs with maximum and minimum nuclear corrections. A clear deviation between the two PDF fits becomes visible at for the LHC and for the Tevatron, corresponding to one of the partons carrying momentum fractions and , respectively. Data on boson asymmetries may therefore provide constraints on the quark distribution ratio already at these moderate values of . In particular, comparison with the CDF data AW:CDF () in Fig. 4 illustrates a preference for larger values at , which corresponds to . As observed in Ref. CJ11 (), since the asymmetry at large can be approximated by

(6)

this would suggest a smaller ratio at large , as would arise for the CJ PDFs with minimum nuclear corrections.

For comparison, the asymmetries calculated from the ABKM Alekhin10 (), CT10 CT10 () and MSTW MSTW09 () PDF sets are also shown in Fig. 4. Each parametrization shows good agreement with the Tevatron CDF data, with the exception of the ABKM fit, which overestimates the asymmetries at intermediate rapidities, . This may be due to the asymmetry data not being fitted directly in the ABKM analysis. At large rapidity the spread in the various PDF sets is comparable to the difference between the CJ PDFs with minimal and maximal nuclear corrections. However, we stress that the origin of the differences between the PDF sets is unrelated to the difference between the two CJ PDF sets. Had the various non-CJ PDFs sets included nuclear uncertainties in their analysis, each one would have a corresponding nuclear uncertainty band similar to the one in Fig. 4, and the combined spread between them would subsequently be significantly larger.

iii.4 Lepton Asymmetries

Experimentally, measurement of bosons asymmetries requires reconstruction of the boson distributions from their leptonic decays, and , with or . On the other hand, lepton charge asymmetries can be constructed directly from the decay products and studied as a function of the lepton pseudorapidity , defined as

(7)

where is the charged lepton momentum, and is the angle between the lepton momentum and the beam axis in the center of mass frame. The lepton asymmetry is then given by

(8)

where is the differential cross section for the production and leptonic decay of the .

Figure 5: Lepton charge asymmetry as a function of lepton pseudorapidity at the LHC (left) and Tevatron (right), computed from CJ PDFs CJ11 () with maximum (blue dotted) and minimum (red solid) nuclear corrections. The calculations are compared with CMS (red squares) and (blue triangles) data for a lepton transverse momentum cut  GeV Ae:CMS (), and preliminary LHCb data (black circles) for  GeV Ae:LHCb (), as well as with D0 data from the Tevatron  GeV Ae:D0 ().

Lepton asymmetry data from the D0 Collaboration Ae:D0 () at Fermilab are shown in Fig. 5 as a function of the pseudorapidity up to . The data are integrated over lepton transverse momenta  GeV, and compared with asymmetries computed from the CJ PDFs with maximum and minimum nuclear corrections using the MCFM (Monte Carlo for FeMtobarn processes) program MCFM (). Good agreement is obtained between the calculated asymmetry and data, although little sensitivity is evident to the large- nuclear uncertainty in the PDFs observed in Fig. 4 until .

Similar behavior is found for the new lepton charge asymmetry data from the LHC. Here lepton asymmetries have been measured by the ATLAS Ae:ATLAS () and CMS Ae:CMS () Collaborations for pseudorapidities , and preliminary results from the LHCb Collaboration Ae:LHCb () extend the coverage to Ae:LHC (). The agreement of the CJ PDFs with the LHC data is good over the entire range of , as Fig. 5 illustrates for CMS and data integrated over lepton transverse momenta  GeV, and LHCb data with  GeV. The dependence on the large- behavior of PDFs, however, becomes visible only for .

The limited sensitivity of the lepton asymmetries to large- PDFs is not surprising, given that the lepton asymmetry is computed by convoluting the boson cross sections with the boson decay distributions, which dilutes the sensitivity to regions where the PDFs are small. Although the lepton asymmetry data are clearly valuable for constraining global PDF fits in general, greater sensitivity to the large- behavior of the ratio may be possible through the reconstruction of the boson asymmetries themselves. Note, however, that the reconstruction of boson asymmetries is limited by theoretical uncertainties such as the modeling of the distributions and higher-order resummation corrections (which also affect the lepton asymmetries), as well as the choice of PDFs used to compute the event reweighting coefficients in the reconstruction.

Iv Heavy and bosons

The production rate of any new heavy boson beyond the Standard Model will naturally depend on its internal properties such as the spin. Many possibilities have been canvassed for how such heavy bosons can arise Rev:Rizzo06 (), including as scalar excitations in -parity violating supersymmetry Z':spin0 (), spin-1 Kaluza-Klein excitations of Standard Model gauge bosons in the presence of extra dimensions Z':spin1 (), or as spin-2 excitations of the graviton Z':spin2 (). On the other hand, if the new bosons are associated with extensions of the Standard Model gauge group, their interactions with fermions will resemble those of the and bosons of the electroweak theory, with different masses and couplings.

In this section we explore this latter possibility, and in particular the sensitivity of the production cross sections to uncertainties in PDFs at large . From Eq. (2) one can see that increasing the mass will directly increase the relevant values, so that higher mass bosons will more readily sample the high- region where the nuclear uncertainties are more prominent. In the calculations discussed here we shall assume that the putative and bosons have the same properties as the Standard Model and bosons, except for their larger masses. The cross sections will of course decrease rapidly with increasing boson mass, so that the effects of the large- PDF uncertainties will become more significant as the mass increases. Of course the details of the predictions will change in more sophisticated models in which the and couplings are different from those in the Standard Model; however, the simplified scenario considered here is sufficient to illustrate the possible impact of large- PDF uncertainties on new physics searches.

Currently the exclusion limit on the mass from the Tevatron, for couplings similar to Standard Model couplings, is  TeV at the 95% confidence level with an integrated luminosity of 5.3 fb W'lim_CDF (). For the neutral boson the limits vary between  GeV and  TeV Z'lim_CDF (), depending on the Standard Model extension considered PDG10 (). The latest results from the LHC place the limits for the mass at  TeV W'lim_LHC () and for the mass at  TeV Z'lim_LHC () in the Sequential Standard Model, with the same couplings to fermions as for the and . A similar constraint on the mass in grand unified theories is also obtained from measurements of atomic parity violation in Cs APV ().

iv.1 Rapidity distributions

The differential cross sections for heavy and bosons will generally have similar behavior as a function of rapidity to those of the physical boson cross sections in Figs. 2 and 3, except for a smaller rapidity range, with the large- region emphasized more strongly for increasing boson mass. According to the study by Erler et al. Erler11 (), collisions at the LHC with  TeV will be sensitive to masses up to  TeV for luminosities between 30 fb and 300 fb, corresponding to the so-called “low-luminosity” and “high-luminosity” LHC scenarios, respectively. (With  TeV the mass limits would vary between and 4.6 TeV.) For collisions at the Tevatron, an energy of  TeV with a luminosity of 10 fb would be expected to allow sensitivity to masses up to  TeV Erler11 (). In this section we therefore consider (and ) masses up to 1 TeV and 3 TeV for Tevatron and LHC kinematics, respectively.

Figure 6: Differential (top), (center) and (bottom) cross sections as a function of the rapidity, computed from CJ PDFs with maximum and minimum nuclear corrections, relative to the reference cross sections calculated using the central CJ PDF set CJ11 (). The LHC cross sections (left) are computed for boson masses  TeV (red dot-dashed), 2 TeV (short-dashed) and 3 TeV (long-dashed) with  TeV, while the Tevatron ratios (right) are shown for  TeV (red dot-dashed), 0.75 TeV (short-dashed) and 1 TeV (long-dashed) with  TeV.

From Eq. (2), larger boson masses naturally restrict the kinematically accessible range of rapidities, so that at the LHC, for example, a 1 TeV (3 TeV) boson can be produced at a maximum rapidity of (0.9), compared with for the Standard Model boson. The kinematic reach and sensitivity to large- PDFs is illustrated in Fig. 6 (top) for the differential cross section ratio as a function of , for masses , 2 and 3 TeV at the LHC, and , 0.75 and 1 TeV at the Tevatron. As the rapidities approach their kinematical thresholds for a given , the uncertainty in the differential cross sections increases significantly, reaching about of the central CJ value for collisions at the LHC, and for collisions at the Tevatron.

Because the couplings of the to and quarks are assumed to be similar (see Sec. III.1), the cross section at the LHC depends on both the combinations and . However, since the and ratios are at large values, which are preferentially sampled for large , the contributions of quarks are suppressed relative to quarks. Consequently the ratios in Fig. 6 are only mildly affected by uncertainties in quark PDFs at large . The production cross section in collisions at the Tevatron, on the other hand, is determined predominantly by the product , which is well constrained and independent of the nuclear model for all , and therefore has an even smaller uncertainty.

For the differential cross sections, the behavior as a function of is qualitatively different for and production, shown in Fig. 6 (center) and (bottom), respectively, with the latter displaying dramatically greater sensitivity to large- PDF uncertainties. This is clear from Eqs. (3), where for collisions the dominant contribution to the cross section depends on the products and . While the quark PDF is insensitive to the nuclear corrections, the distribution varies considerably with the nuclear model, especially at larger values of . At high rapidity the ratio is small, and the cross section is determined by the PDF with large and the PDF with small, both of which are well constrained. For , on the other hand, one has for  TeV and for  TeV, at which the PDF has significantly greater uncertainty than the , yielding up to uncertainties in the cross section. Similarly for collisions at the Tevatron, the cross section is dominated by the combination , which for at high rapidity is relatively well constrained. At central rapidity, with for  TeV, the uncertainties in the cross section remain within the level.

In contrast, the cross section at high rapidity is determined by the products and in collisions, and and in collisions. Consequently, uncertainties in the cross sections at large rapidity, both at the LHC and the Tevatron, arise mainly from the quark at large , and exceed 100% as the kinematic limit in is approached. Qualitatively, the growing uncertainty of the cross section with increasing rapidity resembles the cross section ratio at large in Fig. 3. At central rapidity, the uncertainty in the cross section at the LHC is of the order 10%, arising mainly from the uncertainty in the distribution. At the Tevatron, the cross section is well constrained for small boson mass, but for  TeV, with , becomes increasingly sensitive to uncertainties in the quark PDF, reaching about 20% for  TeV.

iv.2 Integrated cross sections

Integrating over all rapidities, the resulting total cross section computed from CJ PDFs with minimum and maximum nuclear corrections is shown in Fig. 7 as a function of the mass. Relatively little dependence on the PDFs is observed, with effects of observed for  TeV at the LHC and  TeV at the Tevatron. This is not surprising given that total cross sections are dominated by contributions from low values of , where the PDF uncertainties are generally smaller than at high values of , at which the contributions are suppressed by the steeply falling PDFs as . At larger values the uncertainties generally increase, but are subject to greater fluctuations in the antiquark distributions at high , and hence are less reliable.

Figure 7: Integrated boson cross section from Fig. 6 as a function of the mass, computed from CJ PDFs with minimum (red dot-dashed) and maximum (blue dashed) nuclear corrections, relative to the reference cross section calculated using the central CJ PDF set CJ11 (), for LHC (left) and Tevatron (right) kinematics.

The integrated cross sections for bosons in Fig. 8 show somewhat greater sensitivity to large- PDFs as a function of . For bosons produced in collisions at the LHC the uncertainties increase from for  TeV to for  TeV. This behavior stems directly from the increasing uncertainty in the antiquark PDF at large apparent in the rapidity distribution at low in Fig. 6. For boson production in scattering, the uncertainties in the total cross section are smaller than for the at low , remaining for  TeV, but increase to at  TeV due to the uncertainty in the quark. The stronger dependence on the behavior of the quark PDF at large apparent in the differential cross section at high rapidity in Fig. 6 is mostly washed out in the integrated cross section.

Figure 8: As in Fig. 7 but for and cross sections at the LHC (left) and Tevatron (right). Note that the integrated cross sections for and production in collisions at the Tevatron are identical.

Because the cross sections in collisions at the Tevatron are determined by the products and for and , respectively, integrating over rapidity samples all accessible values of and , so that the total and cross sections are equivalent. The dependence of the integrated cross sections on PDFs essentially follows the quark distribution. For masses  TeV there is little sensitivity to the large- behavior of the PDFs, with uncertainty in the cross section ratio, but increasing dependence at larger , with uncertainty at  TeV.

Figure 9: Ratio of boson cross sections , where , versus the boson cross section for varying boson masses, indicated by the numbers (in TeV) next to the points (the points corresponding to the physical and cross sections are labeled by “”). The cross sections are computed from CJ PDFs with minimum (filled red circle) and maximum (open blue circle) nuclear corrections, relative to the reference cross section calculated from the central CJ PDFs CJ11 ().

While the sensitivity of the and cross sections to the large- behavior of PDFs increases with increasing and masses, the absolute values of the cross sections naturally fall with increasing masses, some 3 orders of magnitude from 100 GeV to 3 TeV. This is illustrated in Fig. 9, where the ratio of the integrated cross sections computed from CJ PDFs with minimum and maximum nuclear corrections, relative to the cross section with the central CJ PDFs, is plotted versus the integrated cross section. Here the ratio of the to masses is kept constant in order to study the effect of the increasing mass. For larger boson masses the impact of the large- PDF uncertainties clearly increases, reflecting the trend observed in Figs. 7 and 8. Note that because the integrated cross section is generally larger than the cross section (because of the larger distribution compared with the ), the ratio in Fig. 9 generally follows the ratio of the cross sections in Fig. 8 for increasing boson mass.

V Conclusion

In this paper we have explored the sensitivity of weak boson production in hadronic collisions to parton distributions at large values of . At present there are large uncertainties in the quark distribution, particularly above , arising from the model dependence of nuclear corrections used when analyzing deuteron DIS data in global PDF fits, which can impact cross section measurements at large rapidities. The PDF uncertainties can also affect production cross sections of heavy and bosons beyond the Standard Model at central rapidities.

Using PDFs extracted from the recent CJ global fit CJ11 (), we find increasing sensitivity to the large- region for boson production in collisions at the Tevatron for rapidities , and in collisions at the LHC for . Precision measurements of boson cross sections at these rapidities, at the Tevatron and particularly at the LHC with the LHCb experiment, will be required to impact global fits of the quark and constrain nuclear model uncertainties.

For charged weak bosons, the cross sections are mostly independent of PDF uncertainties due to their preferential coupling to quarks, whereas the cross sections display strong dependence on the quark uncertainties for rapidities at Tevatron and at LHC kinematics. Measurements of boson charge asymmetries at large rapidities, such as those from the CDF Collaboration at Fermilab AW:CDF (), thus provide strong constraints on the behavior of the ratio at large , although such measurements are very challenging given the low rates expected in the relevant regions of kinematics. Direct reconstruction of boson asymmetries in the LHCb experiment for  TeV would also be extremely valuable in providing access to PDFs at . We have also compared charged lepton asymmetries with data from D0 Ae:D0 (), and from the CMS Ae:CMS () and LHCb Collaborations Ae:LHCb () at the LHC, finding good overall agreement with the CJ PDFs, but weak sensitivity to large- PDFs.

The large- PDF uncertainties also affect the production rates of heavy and bosons, and the impact of these was studied as a function of the boson mass for Standard Model couplings. At high rapidity, the cross section in and collisions is mostly sensitive to the quark PDF at large and therefore well constrained. The cross section displays a mild sensitivity to the quark PDF at large , reaching upwards of 40% uncertainty at the kinematic rapidity limit. The effects are even more pronounced for cross sections, where the uncertainties exceed 100% at large .

At central rapidity the cross section is well constrained, with weak sensitivity to and quarks at the LHC for  TeV. The central rapidity cross sections, on the other hand, display sensitivity to large- PDFs of the order for  TeV at the LHC (due to and ) and 0.75 TeV at the Tevatron (due again to quarks). The uncertainties at central rapidity directly propagate to the integrated cross sections, which show effects for production, while somewhat larger for production, amounting to for and for at the LHC for  TeV, and for and at the Tevatron for  TeV.

These considerations place important limits on the ability to accurately measure heavy and cross sections in hadronic collisions, particularly at large rapidities and boson masses near the kinematic thresholds of current colliders. Although our analysis is, for illustration, restricted to heavy vector bosons with Standard Model couplings, and the quantitative effects of the PDF uncertainties would be different in other models, our main point is that caution must be exercised when using PDFs in regions where these are not directly constrained, or their uncertainties underestimated, as is the case at large . The uncertainties in the production cross sections can be reduced by obtaining better constraints on PDFs at large , especially for the quark. Several experiments aimed at determining the quark PDF up to are planned at Jefferson Lab following its 12 GeV energy upgrade BoNuS12 (); MARATHON (); SOLID (). Uncertainties in cross sections at central rapidity, and hence in the integrated cross sections, will also be reduced with improved determinations of antiquark distributions at large , such as the E-906/SeaQuest experiment at Fermilab SeaQuest () which plans to measure up to .

The flavor dependence of weak boson production could also be studied with collisions at the LHC or at the Relativistic Heavy-Ion Collider, with the neutron provided by a beam of deuterons Soffer94 (). Unlike for fixed target experiments, in a collider one can study collisions at large positive and negative rapidities. Therefore, partons in the beam at large can scatter from partons in the target at small , and vice versa. In particular, measurements at large negative rapidity would be sentitive to quarks in the deuteron at large momentum fractions, offering a probe of nuclear corrections complementary to deuterium DIS, and constraining the nuclear uncertainties studied in this paper.

Finally, while we have focussed on the  TeV energy at which the LHC currently operates, in future this is planned to increase to  TeV. The behavior of the cross sections illustrated here will not change qualitatively at the larger energy. However, for physical and bosons, the region in rapidity where sensitivity to nuclear models is greatest will be shifted outside of the acceptance of current experiments, limiting the usefulness of 14 TeV data for large- PDF studies. On the other hand, the higher center of mass energy will increase the accessible mass range (up to  TeV) over a larger range of rapidities.

Acknowledgements

We thank D. del Re and J. Erler for helpful discussions and communications. This work was supported by the DOE contract No. DE-AC05-06OR23177, under which Jefferson Science Associates, LLC operates Jefferson Lab, DOE contract No. DE-FG02-97ER41022, DoD’s ASSURE Program, and the National Science Foundation under NSF Contact Nos. 1062320 and 1002644.

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