Reassessment of the NuTeV determination of the Weinberg angle
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Reassessment of the NuTeV determination of the weak mixing angle
W. Bentz, I. C. Clo¨et, J. T. Londergan, and A. W. Thomas Department of Physics, School of Science, Tokai University, Hiratsuka-shi, Kanagawa 259-1292, Japan Department of Physics, University of Washington, Seattle, WA 98195-1560, USA Department of Physics and Nuclear Theory Center Indiana University, Bloomington, IN 47405, USA CSSM, School of Chemistry and Physics, University of Adelaide, Adelaide SA 5005, Australia,and Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606 USA
In light of the recent discovery of the importance of the isovector EMC effect for the interpretationof the NuTeV determination of sin θ W , it seems timely to reassess the central value and the errors onthis fundamental Standard Model parameter derived from the NuTeV data. We also include earlierwork on charge symmetry violation and the recent limits on a possible asymmetry between s and ¯ s quarks. With these corrections we find a revised NuTeV result of sin θ W = 0 . ± . ± . θ W predicted by the StandardModel. As a further check, we find that the separate ratios of neutral current to charge currentcross sections for neutrinos and for antineutrinos are both in agreement with the Standard Model,at just over one standard deviation, once the corrections described here are applied. PACS numbers: 12.15.+y. 13.15.+g, 24.85.+p
Using a very careful comparison of the charged andneutral current total cross sections for ν and ¯ ν on aniron target, the NuTeV collaboration reported a threestandard deviation discrepancy with the Standard Modelvalue of sin θ W [1]. This was initially taken as an indi-cation of possible new physics, however attempts to un-derstand this anomaly in terms of popular extensions ofthe Standard Model have proven unsuccessful [2, 3]. Atthe same time a number of possible corrections withinthe Standard Model have been suggested [4–10], most ofwhich have a sign likely to reduce this discrepancy.The correction associated with charge symmetry viola-tion (CSV), arising from the u - and d -quark mass differ-ences [11, 12], has been shown to be largely model inde-pendent and to reduce the discrepancy by about 1 σ [4].If the momentum fraction carried by s -quarks in the pro-ton exceeds that carried by ¯ s -quarks, as suggested bychiral physics [13, 14] and recent experimental analy-sis [8], there could be a further reduction, albeit withlarge uncertainties at present [9]. Finally, in Ref. [10] itwas recently pointed out that the excess neutrons in ironlead to an isovector EMC effect that modifies the partondistribution functions (PDFs) of all the nucleons in thenucleus. Qualitatively this has the same sign as the CSVcorrection and a quantitative estimate suggests that it re-duces the NuTeV discrepancy with the Standard Modelby about 1.5 σ [10].These effects are essentially independent and cantherefore be combined in a straightforward manner. Itis then immediately clear that the corrected NuTeV datawill be more consistent with the Standard Model. Ratherthan continuing to report that the data is 3 σ above ex-pectations, we suggest that it is timely to update thederived value of sin θ W .In this Letter we will examine the corrections in turn,assign to each a central value and a conservative error, then combine them to produce a revised value for sin θ W .Our final value issin θ W = 0 . ± . ± . , (1)which is in excellent agreement with the correspondingStandard Model result, namely 0 . ± . θ W = 0 . ± . ± . R ν and R ¯ ν , which arethe ratios of the neutral current (NC) to charged cur-rent (CC) total cross sections for ν and ¯ ν , respectively.Integral to the NuTeV extraction of sin θ W was a de-tailed Monte Carlo simulation of the experiment. How-ever, NuTeV have provided functionals which allow oneto accurately estimate the effect of any proposed correc-tion [16].The NuTeV study was motivated by the observationof Paschos and Wolfenstein [17] that a ratio of cross sec-tions for neutrinos and antineutrinos on an isoscalar tar-get allowed an independent extraction of the weak mixingangle. The so-called Paschos-Wolfenstein (PW) ratio isgiven by [17, 18] R PW = σ ν ANC − σ ¯ ν ANC σ ν ACC − σ ¯ ν ACC ≡ R ν − rR ¯ ν − r . (2)In Eq. (2), R PW is the PW ratio, A represents the nu-clear target, and r = σ ¯ ν ACC /σ ν ACC . Expressing the totalcross-sections in terms of quark distributions, ignoringthe heavy quark flavours and O ( α s ) corrections, the PWratio becomes R PW = (cid:16) − s W (cid:17) h x A u − A i + (cid:16) − s W (cid:17) h x A d − A + x A s − A ih x A d − A + x A s − A i − h x A u − A i , (3)where s W ≡ sin θ W , x A is the Bjorken scaling variablefor the nucleus multiplied by A , h . . . i implies integrationover x A , and q − A ≡ q A − ¯ q A are the non-singlet quarkdistributions of the target.Ignoring quark mass differences, strange quark effectsand electroweak corrections, the u - and d -quark distribu-tions of an isoscalar target will be identical, and in thislimit Eq. (3) becomes R PW N = Z −→ − sin θ W . If cor-rections to this result are small the PW ratio providesan independent determination of the weak mixing angle.Expanding Eq. (3) about the u − A = d − A and s − A ≪ u − A + d − A limits, we obtain the leading PW correction term, namely∆ R PW ≃ (cid:18) − s W (cid:19) h x A u − A − x A d − A − x A s − A ih x A u − A + x A d − A i . (4)Extensive studies of neutrino-nucleus reactions have con-cluded that the most important contributions to Eq. (4)arise from nuclear effects, CSV and strange quarks.These corrections will be discussed in turn below.In discussing the extraction of the weak mixing anglefrom neutrino reactions, it is customary and pedagogi-cally useful to refer to corrections to the PW ratio, and wefollow this practice. However, it is important to remem-ber that in the NuTeV analysis the measured quantitieswere the NC to CC ratios for neutrinos and antineutrinos,and that the weak mixing angle was extracted through aMonte Carlo analysis. For a given effect, the PW ratiowill give only a qualitative estimate of the correction tothe weak mixing angle. Quantitative corrections are ob-tained by using the functionals provided by NuTeV [16].Throughout this work we denote a contribution to Eq. (4)by ∆ R i PW , while the best estimate of the correction tothe NuTeV determination of sin θ W , calculated usinga NuTeV functional, is denoted by ∆ R i ≡ ∆ i sin θ W ,where, in each case, i labels the type of correction. Forcompleteness, at the end of our discussion we also reportthe effect of the corrections which we have considered onthe separate ratios R ν and R ¯ ν . Nuclear corrections — For sufficiently large Q , nu-clear corrections to the PW ratio for an isoscalar nucleusare thought to be negligible. However, the NuTeV ex-periment was performed on a steel target and it is essen-tial to correct for the neutron excess before extractingsin θ W . NuTeV removed the contribution of the excessneutrons to the cross-section by assuming that the targetwas composed of free nucleons.However, the recent results of Clo¨et et al. [10] haveshown that the excess neutrons in iron have an effect on all the nucleons in the nucleus, which is not accountedfor by a subtraction of their naive contribution. In par-ticular, the isovector-vector mean-field generated by thedifference in proton and neutron densities, ρ p ( r ) − ρ n ( r ),acts on every u - and d -quark in the nucleus and re-sults in the break down of the usual assumption that u p ( x ) = d n ( x ) and d p ( x ) = u n ( x ) for the bound nucle- ons. An explicit calculation of this correction was madein Ref. [10], using the approach of Bentz et al. [19–21].The correction associated with the neutron excess canbe evaluated in terms of the consequent contribution to h x A u − A − x A d − A i , using Eq. (4). For nuclei with N > Z the u -quarks feel less vector repulsion than the d -quarks,and in Ref. [10] it was shown that a model indepen-dent consequence of this is that there is a small shiftin quark momentum from the u - to the d -quarks. There-fore, the momentum fraction h x A u − A − x A d − A i in Eq. (4)will be negative [10], even after standard isoscalarity cor-rections are applied. Correcting for the isovector-vectorfield therefore has the model independent effect of reduc-ing the NuTeV result for sin θ W .To estimate the effect on the NuTeV experiment, Clo¨et et al. used a nuclear matter approximation, chose the Z/N ratio to correspond to the NuTeV experimental neu-tron excess and calculated the quark distributions at aneffective density appropriate for Fe, namely 0.89 timesnuclear matter density [22]. Using Eq. (4), this gave anestimate of the isovector correction of ∆ R ρ PW = − . R ρ = − . σ of the NuTeV discrep-ancy with the Standard Model.The sign of this effect is model independent and be-cause it depends only on the difference in the neutronand proton densities in iron and the symmetry energy ofnuclear matter, which are both well known, the magni-tude is expected to be well constrained. As a conservativeestimate of the uncertainty we assign an error twice thatof the difference between the PW correction obtained atnuclear matter density and at 0.89 times that, this gives∆ R ρ = − . ± . . (5)Other studies of nuclear corrections to the PW ratiohave largely been focused on Fermi motion [10, 23] andnuclear shadowing [23–25] effects. Fermi motion correc-tions were found to be small [10, 23] and the NuTeVcollaboration argue that, given their Q -cuts, sizeablecorrections from shadowing would be inconsistent withdata [26]. Therefore we have not included a correctionfrom shadowing here. Charge Symmetry Violation — Before the NuTeV re-sult, two independent studies of the effect of quark massdifferences on proton and neutron PDFs, by Sather [11]and by Rodionov et al. [12], reached very similar conclu-sions. These mass differences violate charge symmetry,the invariance of the QCD Hamiltonian under a rotationby 180 degrees about the 2-axis in isospin space. Theylead to the CSV differences δd − ( x ) = d − p ( x ) − u − n ( x ) , (6) δu − ( x ) = u − p ( x ) − d − n ( x ) , (7)where the subscripts p and n label the proton and neu-tron, respectively. The contribution of CSV in the nu-cleon can be found through Eq. (4) and has the form:∆ R CSVPW = 12 (cid:18) −
73 sin θ W (cid:19) h x δu − − x δd − i (cid:10) x u − p + x d − p (cid:11) . (8)Londergan and Thomas [4] explained the similarity ofthe results obtained by Sather and Rodionov et al. bydemonstrating that the leading contribution to the mo-ment h x δu − − x δd − i is largely model independent andsimply involves the ratio of the up-down mass differ-ence to the nucleon mass. The contribution arising fromthe quark mass differences to Eq. (8) was found to be∆ R δm PW ≃ − . R δm ≃ − . Q evolution of photonemission from the quarks. Because | e u | > | e d | the u -quarks lose momentum to the photon field at a greaterrate than the d -quarks. Therefore a model indepen-dent consequence of QED splitting is that it will re-duce the NuTeV result for sin θ W . Gl¨uck et al. [6]calculated this effect on the NuTeV result and obtained∆ R QEDPW = − . R QED = − . R QEDPW = − . Q = 20 GeV . This cor-rection has the same sign as the CSV term arising fromquark mass differences and the two contributions are al-most independent so we simply add them. The sum of thetwo terms explains roughly half of the NuTeV discrep-ancy with the Standard Model. Assigning a conservative100% error to the QED splitting result and combiningthe errors in quadrature gives a total CSV correction of∆ R CSV = − . ± . . (9)The only experimental information regarding CSV ef-fects on the PDFs is obtained from an MRST study [5].In this case a global analysis was performed on a set ofhigh energy data allowing for explicit CSV in the PDFs.From their global analysis [5] the MRST group found∆ R CSVPW = − . − . < ∆ R CSVPW < . Q dependence of the CSV distributions.Finally, the experiments in the global set of high energydata have different treatments of radiative corrections.It is not clear that these different radiative correctionshave been treated consistently in an analysis of CSV ef-fects. If the CSV effects are as large as are allowed withinthe MRST 90% confidence limit, it should be possible toobserve such effects [27]. However it will be some timebefore experiments can further constrain this result. Strange Quark Asymmetry — A difference in shape be-tween s ( x ) and ¯ s ( x ) in the nucleon was first proposed onthe basis of chiral symmetry in Ref. [13]. However, thesize of s − ( x ) is not constrained by symmetries and badlyneeds further input from experiment or lattice QCD [28].The strange quark correction arises from the term (cid:10) x s − A (cid:11) on the right hand side of Eq. (4). The best direct exper-imental information on (cid:10) x s − A (cid:11) comes from opposite signdimuon production in reactions induced by neutrinos orantineutrinos. Such experiments have been carried outby the CCFR [29] and NuTeV [30] groups. A preciseextraction of (cid:10) x s − A (cid:11) is the NuTeV analysis by Mason et al. [8], which found (cid:10) x s − A (cid:11) = 0 . ± . Q = 16 GeV , where we have added the various errorsin quadrature.Global PDF analyses have also provided estimates of s − ( x ). The recent examination by the NNPDF collabora-tion found h x s − i = 0 . ± . Q = 20 GeV ,which has an error more than six times larger than thatcited by NuTeV. However, unlike the NuTeV dimuon ex-periment, this analysis is not directly sensitive to the s -quark distributions, evidenced by the fact that their up-per limit on s − ( x ) is an order of magnitude larger thanany other sea quark distribution at x ∼ .
5. This largeuncertainty is a consequence of their neural network ap-proach, which was primarily aimed at accurately deter-mining V cd and V cs , not s − ( x ) [9]. Alekhin et al. [31]obtained h x s − i = 0 . ± . ± . B µ from production rates of charmedhadrons in other experiments. The MSTW collabora-tion find a momentum fraction very similar to that ofNuTeV, namely h x s − i = 0 . +0 . − . [32] at Q =10 GeV , while CTEQ report h x s − i = 0 . Q = 1 .
69 GeV , with a 90% confidence interval of − . < h x s − i < . s − A ( x ) must have at least one zero-crossing since its first moment vanishes. For each of theabove analyses the central best fit curve crosses zero atvalues of x less than 0 .
03 for Q > , with the ex-ception of the NNPDF result which has a zero-crossing at x = 0 .
13 for Q = 2 GeV . For example the NuTeV resulthas the zero-crossing at x = 0 . h x s − i ∆ R s ∆ R total sin θ W ± syst.Mason et al. [8] 0 . ± . − . ± . − . ± . . ± . . ± . − . ± . − . ± . . ± large Alekhin et al. [31] 0 . ± . ± . − . ± . ± . − . ± . . ± . . +0 . − . − . − . . − . ± . . ± . . +0 . − . − . +0 . − . − . +0 . − . . +0 . − . This work (Eq. (10)) 0 . ± . . ± . − . ± . . ± . h xs − i , the correction to the Paschos-Wolfensteinratio after applying the NuTeV functional, ∆ R s , and the total correction, ∆ R total , obtained by combining ∆ R ρ , ∆ R CSV and∆ R s , with the errors added in quadrature. The final column shows the value of sin θ W deduced in each case by applying thetotal correction to the published NuTeV result. Note that we show only the systematic error, which is obtained by treating theerror on ∆ R total as a systematic error and combining it in quadrature with the NuTeV systematic error. x value (it is smaller than the lowest x point measuredin the CCFR and NuTeV experiments), and moreover isextremely unlikely on theoretical grounds [13, 34, 35]. Inany quark model calculation the zero-crossing will occurnear x ≃ .
15 (a value similar to that found by NNPDF).In this case the NuTeV strange quark momentum fractionbecomes (cid:10) x s − A (cid:11) = 0 . χ compared to the best value of Mason et al. .Since relatively little is known about the s -quark distri-butions we ignore nuclear effects and therefore assume (cid:10) x s − A (cid:11) ≡ h x s − i throughout this discussion.Clearly the correction to the PW ratio from the strangequark asymmetry has a significant uncertainty. Onthe theoretical grounds explained earlier, we prefer theNuTeV analysis based on a zero-crossing at x ≈ . h x s − i is essentially zero. For the un-certainty we choose the difference between this and theNuTeV determination noted above with the zero-crossingat x = 0 . h x s − i = 0 . ± . .This is a conservative choice for the error since it is sub-stantially larger than the original uncertainty quoted byNuTeV and covers all of the central values of the analy-ses mentioned above. Including the effect of the NuTeVfunctional leads to our preferred value for the strangequark correction to the NuTeV sin θ W result, namely∆ R s = 0 . ± . . (10)The s -quark corrections to the NuTeV result obtainedfrom the other analyses discussion here are summarizedin column three of Table I. Conclusion — The errors associated with the threecorrections given in Eqs. (5), (9) and (10) are systematicand to a very good approximation independent errors.We therefore combine them in quadrature with the orig-inal systematic error quoted by NuTeV. The statisticalerror is, of course, unchanged from the NuTeV analysis.Because of the uncertainty over the strangeness asym-metry, in the last column of Table I we show the effect on sin θ W for each of the recent analyses [8, 9, 31–33] aswell as our own preferred value given in Eq. (10). Everyone of the six results lies within one standard deviationof the Standard Model value for sin θ W . As a best esti-mate of the corrected value we take the average of thesesix values. For the systematic error we note that (apartfrom NNPDF which is unrealistically large) they are allvery similar. Because of the correlations between them,the final quoted systematic error is a simple average of allthe entries in the last column of Table I except NNPDF.This yields the revised value for sin θ W , including all ofthe corrections discussed here, namely:sin θ W = 0 . ± . ± . , (11)which is in excellent agreement with the Standard Modelexpectation of sin θ W = 0 . ± . O ( α s )corrections were also investigated and found to be negli-gible.This updated value for the NuTeV determination ofsin θ W is also shown in Fig. 1, now in the MS-scheme andlabelled as ν -DIS, along with the results of a number ofother completed experiments and the anticipated errorsof several future experiments, which are shown at theappropriate momentum scale Q .In this paper we have summarized various estimatesof the size of both partonic CSV effects and a possiblestrange quark momentum asymmetry. For valence quarkCSV we have relied on well founded theoretical argu-ments to constrain the magnitude of CSV effects arisingfrom quark mass differences. We have also used theo-retical guidance on the zero crossing in s − ( x ), as well asthe most recent analyses of the experimental data to con-strain the strange quark momentum asymmetry. Whenre-evaluated, the NuTeV point is within one standard de-viation of the Standard Model prediction for all analysesof this asymmetry. As the experimental information onthe strange quark asymmetry or charge symmetry viola- APV(Cs) SLAC E158 ν -DIS Z -pole CDFD0Møller [JLab]Qweak [JLab]PV-DIS [JLab]0 . . . . . . s i n θ M S W .
001 0 .
01 0 . Q (GeV) Standard ModelCompleted ExperimentsFuture Experiments
FIG. 1: The curve represents the running of sin θ W in the MSrenormalization scheme [36]. The Z -pole point represents thecombined results of six LEP and SLC experiments [37]. TheCDF [38] and D0 [39] collaboration results (at the Z -pole)and the SLAC E158 [40] result, are labelled accordingly. Theatomic parity violating (APV) result [41] has been shiftedfrom Q → Z -pole, conversion to the MS scheme wasachieved via sin θ eff W = 0 . θ MS W [37]. For the re-sults away from the Z -pole, the discrepancy with the Stan-dard Model curve reflects the disagreement with the StandardModel in the renormalization scheme used in the experimentalanalysis. tion improves it is a simple matter to update the currentanalysis.As a final point, we return to the fact that the NuTeVexperiment actually measured R ν and R ¯ ν , not R PW . Wemight ask how the corrections that we have applied affectthe individual values for these two ratios. For the quan-tity R ν NuTeV measured 0 . ± . R ¯ ν they obtained0 . ± . h x A u − A − x A d − A i : δR ν = 2 (cid:0) g Lu + g Ru (cid:1) (cid:10) x A u − A − x A d − A (cid:11)(cid:10) x A u A + 3 x A d A + x A ¯ u A + x A ¯ d A + 6 x A s A (cid:11) , (12) δR ¯ ν = − (cid:0) g Ld + 3 g Rd (cid:1) (cid:10) x A u − A − x A d − A (cid:11)(cid:10) x A u A + x A d A + 3 x A ¯ u A + 3 x A ¯ d A + 6 x A ¯ s A (cid:11) , (13)where g Lu = 12 −
23 sin θ W g Ru = −
23 sin θ W , (14) g Ld = −
12 + 13 sin θ W g Rd = 13 sin θ W . (15)It is clear from our earlier discussion that h x A u − A − x A d − A i is negative and allowing for the NuTeV functional we find δR ν = − . ± . δR ¯ ν = +0 . ± . h xs − i ). Subtracting δR ν from the NuTeV resultyields a value 0 . ± . ν correction yields R ¯ ν = 0 . ± . χ for R ν and R ¯ ν compared with the Standard Model valuesmoves from 7.19 to 2.58. This represents a very signifi-cant improvement.In conclusion, it should be clear that there is no longerany significant discrepancy between the predictions of theStandard Model evolution and the existing data. How-ever, we look forward to the much higher accuracy in theweak mixing angle which is anticipated in future experi-ments [42–44]. With regard to NuTeV itself, the greatestsingle improvement in the accuracy with which one couldextract sin θ W would come from a more precise determi-nation of h x s − i . 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