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Weak effects in proton beam asymmetries at polarised RHIC and beyond

Weak effects in proton beam asymmetries at polarised RHIC and beyond
Weak effects in proton beam asymmetries at polarised RHIC and beyond

a r X i v :h e p -p h /0509254v 2 20 J u n 2006

SHEP-05-07February 2,2008

Weak e?ects in proton beam asymmetries

at polarised RHIC and beyond

S.Moretti,M.R.Nolten and D.A.Ross

School of Physics and Astronomy,University of Southampton

High?eld,Southampton SO171BJ,UK

Abstract

We report on a calculation of the full one-loop weak corrections through the order α2S αW to parton-parton scattering in all possible channels at the Relativistic Heavy Ion Collider (RHIC)running with polarised pp beams (RHIC-Spin).This study extends the analysis previously carried out for the case of 2→2subprocesses with two external gluons,by including all possible four-quark modes with and without an external gluon.The addi-tional contributions due to the new four-quarks processes are extremely large,of order 50to 100%(of either sign),not only in the case of parity-violating beam asymmetries but also for the parity-conserving ones and (although to a more limited extent)the total cross section.Such O (α2S αW )e?ects on the CP-violating observables would be an as-tounding 5times larger for the case of the LHC with polarised beams –which has been discussed as one of the possible upgrades of the CERN machine –whereas they would be much reduced for the case of the CP-conserving ones as well as the cross section.

Keywords:Electroweak physics,Parity violation,Polarised pp scattering,RHIC

The purely weak component of Electro-Weak(EW)interactions is responsible for inducing parity-violating e?ects in jet observables,detectable through asymmetries in the cross section, which are often regarded as an indication of physics beyond the Standard Model(SM)[1]. These e?ects are further enhanced if polarisation of the incoming beams is exploited,like at the BNL machine mentioned in the abstract[2,3].There have also been some discussions [4,5]on the idea of polarising the proton beams at the Large Hadron Collider(LHC)as one of the possible upgrades of the CERN machine,though no proposal has been put forward yet.At either machine,comparison of theoretical predictions involving parity-violation with experimental data can be used as a powerful tool for con?rming or disproving the existence of some beyond the SM scenarios,such as those involving right-handed weak currents[6], contact interactions[7]and/or new massive gauge bosons[8,9,10].

In view of all this,it becomes of crucial importance to assess the quantitative relevance of weak e?ects entering via O(α2SαW)the?fteen possible2→2partonic subprocesses responsible for jet production in hadronic collisions1,namely:

gg→qˉq(1)

qˉq→gg(2)

qg→qg(3)

ˉq g→ˉq g(4)

qq→qq(5)

ˉqˉq→ˉqˉq(6)

qQ→qQ(same generation)(7)

ˉqˉQ→ˉqˉQ(same generation)(8)

qQ→qQ(di?erent generation)(9)

ˉqˉQ→ˉqˉQ(di?erent generation)(10)

qˉq→qˉq(11)

qˉq→QˉQ(same generation)(12)

qˉq→QˉQ(di?erent generation)(13)

qˉQ→qˉQ(same generation)(14)

qˉQ→qˉQ(di?erent generation),(15) with q and Q referring to quarks of di?erent?avours,limited to u-,d-,s-,c-and b-type(all taken as massless).While the?rst four processes(with external gluons)were already computed in Ref.[3],the eleven four-quark processes are new to this study2.Besides,these four-quark processes can be(soft and collinear)infrared divergent,so that gluon bremsstrahlung e?ects ought to be evaluated to obtain a?nite cross section at the considered order.In addition,for completeness,we have also included the non-divergent2→3subprocesses

qg→qqˉq(16)

ˉq g→ˉqˉq q(17)

qg→qQˉQ(same generation)(18)

ˉq g→ˉqˉQQ(same generation).(19)

By recalling that at the typical RHIC-Spin energies(e.g.,

s=14TeV),gluon-induced processes are largely dominant,particularly at low Bjorken?x. As for what concerns the processes with external gluons,it is worth noticing that no CP-violation occurs at tree-level,so that O(α2SαW)is the?rst non-trivial order at which parity violation is manifest.Regarding four-quark processes,the following should be noted.Parity-violating contributions to channels(5)–(15)are induced already at tree-level,through O(α2EW). Besides,all four-quark channels also exist through the CP-violating O(αSαEW)[11],although subprocesses(9),(10),(13)and(15)only receive Cabibbo-Kobayashi-Maskawa(CKM)sup-pressed contributions at this accuracy(i.e.,they mainly proceed via CP-conserving O(α2S) interactions).Furthermore,notice in general that through O(α2SαW)there are many more diagrams available for channels(1)–(15)than via O(α2S)or indeed O(αSαEW)and O(α2EW). Therefore,in terms of parton luminosity,simple combinatorics and power counting,one should expect the impact of O(α2SαW)terms to be large,certainly in parity-violating observables and possibly in parity-conserving ones as well.This is what we set out to test in this paper3for the case of RHIC and LHC.(See Refs.[12,13]for an account of these e?ects at Tevatron.) Before proceeding further we ought to clarify at this stage that we have only computed purely weak e?ects at one-loop level through O(α2SαW),while in the case of tree-level processes via O(αSαEW)and O(α2EW)also the Electro-Magnetic(EM)contributions are included(and so are the interference e?ects between the two).This is why we are referring in this paper to the purely weak terms by adopting the symbolαW,while reserving the notationαEW for the full EW corrections.Here then,we will haveαW≡αEM/sin2θW(withαEM the Electro-Magnetic(EM)coupling constant andθW the weak mixing angle)whileαEW will refer to the appropriate composition of QED and weak e?ects as dictated by the SM dynamics.

We have not computed one-loop EM e?ects for two reasons.Firstly,their computation would be technically very challenging,because the photon in the loop can become infrared(i.e., soft and collinear)divergent,thus requiring also the inclusion of photon bremsstrahlung e?ects, other than of gluon radiation.Secondly,O(α2SαEM)terms(in the above spirit,αEM signi?es here only the contribution of purely EM interactions)would carry no parity-violating e?ects and their contribution to parity-conserving observables would anyway be overwhelmed by the well known O(α3S)terms[14](see also[15,16]).However,notice that we are not including these next-to-leading order(NLO)QCD corrections either,as we are mainly interested in parity-violating beam asymmetries.

Since we are considering weak corrections that may be identi?ed via their induced parity-violating e?ects and since we wish to apply our results to the case of polarised proton beams,it is convenient to work in terms of helicity Matrix Elements(MEs).Here,we de?ne the helicity

amplitudes by using the formalism discussed in Ref.[17].At one-loop level such helicity amplitudes acquire higher order corrections from:(i)self-energy insertions on the fermions and gauge bosons;(ii)vertex corrections and(iii)box diagrams.The expressions for each of the corresponding one-loop amplitudes have been calculated using FORM[18]and checked by an independent program based on FeynCalc[19].Internal gauge invariance tests have also been performed.The full expressions for the contributions from these graphs are however too lengthy to be reproduced here.

As already mentioned,infrared divergences occur when the virtual or real(bremsstrahlung) gluon is either soft or collinear with the emitting parton and these have been dealt with by using the formalism of Ref.[20],whereby corresponding dipole terms are subtracted from the bremsstrahlung contributions in order to render the phase space integral free of infrared divergences.The integrations over the gluon phase space of these dipole terms were performed analytically in d-dimensions,yielding pole terms which cancelled explicitly against those of the virtual graphs.There remains a divergence from the initial state collinear con?guration, which is absorbed into the scale dependence of the Parton Distribution Functions(PDFs)and must be matched to the scale at which these PDFs are extracted.Recall that the remnant initial state collinear divergence at O(αS)is absorbed by the LO Q2dependence of the PDFs. Therefore,to O(α2SαW),it is su?cient,for the purpose of matching these divergences,to consider the LO PDFs.It is also consistent to use the values of the runningαS obtained form the one-loopβ-function.In order to display the corrections due to genuine weak interactions the same PDFs and strong coupling are used in the LO and NLO observables.

The self-energy and vertex correction graphs contain ultraviolet divergences that have been subtracted here by using the‘modi?ed’Dimensional Reduction(

DR,as opposed to the more usual‘modi?ed’Minimal Subtraction (

DR scheme whereas the EM coupling,αEM,has been taken to be1/128at the above subtraction point.The one exception to this renormalisation scheme has been the case of the self-energy insertions on external fermion lines,which have been subtracted on mass-shell,so that the external fermion?elds create or destroy particle states with the correct normalisation

The top quark entering the loops in reactions with external b’s has been assumed to have mass m t=175GeV and widthΓt=1.55GeV.The Z mass used was M Z=91.19GeV and was related to the W-mass,M W,via the SM formula M W=M Z cosθW,where sin2θW=0.232. (Corresponding widths wereΓZ=2.5GeV andΓW=2.08GeV.)For the strong coupling constant,αS,we have used the one-loop expression withΛ(n f=4)

Figure1:The dependence of the cross section as well as of the beam asymmetries on the jet

transverse energy at tree-level(large frames)and the size of the one-loop weak corrections(small

frames),at the two RHIC-Spin energies

s=200and500GeV are also given.

A P V dσ≡dσ???dσ++.(20) The?rst is parity-conserving while the last two are parity-violating4.

Figs.1shows the size of the O(α2SαW)e?ects relatively to the well known LO results, the latter being de?ned as the sum of all O(α2S),O(αSαEW)and O(α2EW)contributions,for the case of RHIC,for two reference energies.Both the di?erential cross section and the above beam asymmetries are plotted,each as a function of the jet transverse energy.The

O(α2SαW)corrections are already very large at cross section level,by reaching?5(?9)%at √

s=300(600)GeV.In the case of both A L and A P V,in regions away from the threshold at E T≈M W/4(where resonance e?ects emerge),there is no local maximum for positive or negative corrections,as both grow monotonically to the level of+100%(at low E T)and?50 to?70%(at high E T and with increasing collider energy).All such e?ects should comfortably

be observable at RHIC,for the customary values of integrated luminosity,of200and800pb?1,√

in correspondence of

4In the numerical analysis which follows we will assume100%polarisation of the beams.

Figure2:The dependence of the cross section as well as of the beam asymmetries on the jet transverse energy at tree-level(large frames)and the size of the one-loop weak corrections(small

frames),at the LHC energy

s=14TeV),the O(α2SαW) corrections to the total cross section as well as the CP-conserving asymmetry are reasonably under control.In fact,they grow monotonically and reach the≈?3%and≈4%at the kinematic limit of the jet transverse energy(as de?ned by the PDFs),respectively.However, it is debatable as to whether these e?ects can actually be disentangled,as we expect systematic experimental uncertainties to be of the same order.Away from the threshold at E T≈M W/2, O(α2SαW)e?ects onto the parity-violating asymmetries are instead enormous,as they yield a K-factor increasing from?2to?4.5,as E T varies from80to500GeV.Despite the absolute value of the CP-violating asymmetries is rather small in the above interval,the huge LHC luminosity(10fb?1per year should be feasible for,say,a70%polarisation per beam[23]) would render the above higher order corrections manifest.

It is intriguing to understand the di?erent behaviours of the O(α2SαW)e?ects depending on the observable and the collider being considered.To this end,we have presented in Tabs.1–2 the contributions to the E T dependent cross section of subprocesses(1)–(15)through O(α2SαW) separately,at both RHIC-Spin and LHC.The purpose of these tables is to illustrate that the leading partonic composition of the O(α2SαW)corrections is markedly di?erent at the two machines.While at RHIC the key role is played by subprocess(7),at the LHC the conspicuous rise of the gluon-luminosity enhances in turn the yield of channel(3)to a level comparable

to that of mode(7)5.(The hierarchy among the subprocesses seen in Tabs.1–2,for?xed jet transverse energy,is characteristic across most of the available E T range at both colliders.) The O(α2SαW)corrections are particularly large for subprocesses(7)–(8)and(12),mainly in virtue of the large combinatorics involved at loop level(as intimated earlier),with respect to the LO case.

Table1:The contributions of subprocesses(1)–(15)to orderα2SαW with respect to the full LO

result for the total cross section at RHIC-Spin,at E T=70GeV for

s=600GeV.Here,we have paired together the channels with identical Feynman diagram topology.We use GSA as PDFs andμ=E T/2as factorisation/renormalisation scale.Column(a)is the percentage contribution from the O(α2SαW)corrections,column(b)is the percentage correction to the tree-level partonic subprocess and column(c)is the percentage contribution at the tree-level of that partonic subprocess to the di?erential cross section at the relevant E T.

√s=600GeV Subprocess

(a)(c)(a)(c)

gg→gg 1.35 1.17

0.0267–0.179

(2) 4.72E-050.188–0.0003310.184

–0.0589–0.264

(5)–(6)–0.41146.1–1.4247.1

–6.64–14.7

(9)–(10)0.0003930.718–0.006020.711

–0.252–0.714

(12)0.006640.06130.01460.0571

1.23 1.17

(14)0.0406 1.280.0282 1.25

0.0491–0.803

qq′→QQ′orˉqˉq′→ˉQˉQ′0.007100.0148 Total–1.94–5.11

The di?erent behaviours seen in Figs.1–2can easily be interpreted in terms of the LO contributions.In this respect,as already mentioned,Tabs.1–2clearly make the point that the jet phenomenology at RHIC-Spin is dominated by subprocesses initiated by quarks only while at the LHC gluons-induced channels are generally predominant.At RHIC energies,LO pro-duction through orderα2S is dominated by channel(5)whereas at the LHC the overwhelmingly dominantα2S channels are gg→gg(which is not subject to O(α2SαW)corrections,as already mentioned)and subprocess(3).Besides,channel(5)through O(α2S)is mainly concentrated

Table2:The contributions of subprocesses(1)–(15)to orderα2SαW with respect to the full LO

result for the total cross section at LHC,at E T=300GeV for

s=14TeV

(a)(c)

gg→gg41.9

–1.25

(2)–0.003860.228

–0.711

(5)–(6)–0.112 1.70

–17.3

(9)–(10)–0.03300.926

–1.85

(12)0.04660.0540

–1.50

(14)0.0131 1.196

–2.64

qq′→QQ′orˉqˉq′→ˉQˉQ′0.0221

Total–1.075

at low E T while with growing E T the O(αSαEW)and–particularly–O(α2EW)terms gain in relative importance.Furthermore,O(α2S)terms entering channel(5)do not contribute, obviously,to the parity-violating asymmetries.Therefore,it should not be surprising to see at RHIC-Spin that our corrections are very large in the case of the latter,where the LO term is O(αSαEW),respect to which the corrections computed here are suppressed only by one power ofαS.We attribute instead the size of the O(α2SαW)e?ects on the cross section and the parity-conserving asymmetry again to the fact that through O(α2SαW)there are many more diagrams available for such channels than via O(α2S)or indeed O(αSαEW)and O(α2EW).As for the LHC,the fact that gg→gg and subprocess(3)vastly dominates through O(α2S)the dσ/dE T distribution explains why O(α2SαW)corrections are limited to the percent level.In A LL,which has no gg→gg component,O(α2SαW)e?ects become somewhat more visible in comparison.

Furthermore,in the case of the LHC,one should note the monotonic rise of the corrections with increasing jet transverse energy,for all observables studied,which can be attributed to the so-called Sudakov(leading)logarithms[24,25]of the formαW log2(E2T/M2W),which appear

in the presence of higher order weak corrections.These‘double logarithms’are due to a lack

of cancellation of infrared(both soft and collinear)virtual and real emission in higher order contributions due to W-exchange,arising from a violation of the Bloch-Nordsieck theorem

occurring in non-Abelian theories.(In fact,if events with real Z radiation are vetoed in the jet sample,αW log2(E T/M2Z)terms would also a?ect the corrections[12].)Clearly,at LHC energies,E T can be very large,thus probing the kinematic regime of these logarithmic

e?ects,which instead a?ected RHIC only very https://www.doczj.com/doc/1610838381.html,bine then the e?ects of such large logarithms with the fact that A L and A P V receive no pure QCD contributions,and one can

explain the enormous(and increasing with E T)O(α2SαW)corrections to these two observables. In fact,recall that another way of viewing the O(α2SαW)terms computed here is as?rst order QCD corrections to the O(αSαW)terms,which are the leading order contributions to A L and

A P V.From this perspective then,the large results reported here can be understood as large O(αS)corrections.Furthermore,also recall here the following two aspects,already mentioned. Firstly,there are several partonic processes which are CKM suppressed at O(αSαEW)but

which occur without CKM suppression at O(α2SαW).Secondly,processes involving external gluons and weak interactions occur for the?rst time at O(α2SαW).

As one of the purposes of polarised colliders is to measure polarised structure functions, in the ultimate attempt to reconstruct the proton spin,it is of some relevance to see how the O(α2SαW)results obtained so far for GSA compare against GRSV-STN.This is done

in Figs.3–4,where we have also adopted the di?erent choiceμ=E cm(E T)as factorisa-

tion/renormalisation scale,i.e.,the centre-of-mass energy at parton level

s=200and500GeV,is straightforward.As expected,these results do not di?er

dramatically from those obtained for300and600GeV,respectively.Fig.1also shows the size of the corrections at these two additional energies,for our usual observables and default choice of PDFs and factorisation/renormalisation scale.For the purpose of emulating the e?ects of O(αSα2W)terms at whichever collider and energy,we make available our code on request.

Figure3:The dependence of the corrections to the cross section as well as the beam asymmetries on the jet transverse energy for two sets of PDFs,GSA and GRSV-STN,at the two RHIC-Spin energies

Figure4:The dependence of the corrections to the cross section as well as the beam asymme-tries on the jet transverse energy for two sets of PDFs,GSA and GRSV-STN,at the LHC energy

[17]E.Maina,S.Moretti and D.A.Ross,JHEP0304(2003)056.

[18]J.A.M.Vermaseren,math-ph/0010025.

[19]J.K¨u blbeck,M.B¨o hm and A.Denner,https://www.doczj.com/doc/1610838381.html,mun.64(1991)165.

[20]S.Catani and M.H.Seymour,Nucl.Phys.B485(1997)291[Erratum-ibid.B510(1997)

503].

[21]T.Gehrmann and W.J.Stirling,Phys.Rev.D53(1996)6100.

[22]M.Gl¨u ck,E.Reya,M.Stratmann and W.Vogelsang,Phys.Rev.D63(2001)094005.

[23]A.De Roeck,private communication.

[24]M.Melles,Phys.Rept.375(2003)219.

[25]A.Denner,hep-ph/0110155.

[26]S.Moretti and D.A.Ross,in progress.

时代周刊翻译

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