application/xmlPossibility of determining gluon polarization via polarized top pairs in gamma-proton scatteringS. AtağA.A. BillurPhysics Letters B 676 (2009) 155-160. doi:10.1016/j.physletb.2009.04.073journalPhysics Letters BCopyright © 2009 Elsevier B.V. All rights reserved.Elsevier B.V.0370-26936764-58 June 20092009-06-08155-16015516010.1016/j.physletb.2009.04.073http://dx.doi.org/10.1016/j.physletb.2009.04.073doi:10.1016/j.physletb.2009.04.073http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/

Journals

S300.2

PLB25825S0370-2693(09)00515-210.1016/j.physletb.2009.04.073Elsevier B.V.Phenomenology

Fig. 1

Number of events as a function of energy for polarized, unpolarized γp→t¯t cross sections and for unpolarized pp→t¯t cross section with luminosity 20 fb−1. For polarized gluon density LSS2006 with SET=1 is used.

Fig. 2

Top quark spin asymmetry as a function of center of mass energy s for LSS2006 polarized gluon distribution function with SET=1,2,3.

Fig. 3

Differential top quark spin asymmetry A1(cosθ) as a function of scattering angle cosθ for s=800 GeV. LSS2006 polarized gluon distribution function with SET=1,2,3 has been used.

Table 1

Center of mass energy estimations of parent ep and its γp option for a collision of linear electron beam and RHIC polarized proton beams.

Ee (GeV)Ep (GeV)Eep (GeV)Eγpmax (GeV)5001004474075001505484995002006325765002507076447501005484997501506716117502007757067502508667891000100632576100015077570610002008948151000250100091112501007076441250150866789125020010009111250250111810191500100775706150015094986415002001095998150025012251116Table 2

An estimation of Statistical uncertainty for the differential asymmetry at s=1000 GeV with L=20 fb−1 and P=0.7. Here z=cosθ.

zA1(z)δA1(z)−10.430.04−0.80.420.04−0.60.390.05−0.40.340.05−0.20.300.0500.250.050.20.210.050.40.180.050.60.180.050.80.230.0410.430.04Possibility of determining gluon polarization via polarized top pairs in gamma-proton scattering

Atağ

⁎

atag@science.ankara.edu.tr

A.A.

Billur

abillur@science.ankara.edu.tr

Department of Physics, Faculty of Sciences, Ankara University, 06100 Tandogan, Ankara, Turkey⁎Corresponding author.

Editor: G.F. Giudice

Abstract

We study the possibility for directly measuring the polarized gluon distribution in the process γp→t¯t. It is shown that polarization asymmetry of the final top quarks is proportional to the gluon polarization. With available energy and luminosity, the collision of a polarized proton beam and a Compton backscattered photon beam can create polarized top quarks which carry the spin information of the process. Energy dependence and angular distributions of the polarization asymmetry of the top pairs has been discussed including statistical uncertainty.

PACS

14.65.Ha14.70.Dj13.88.+e

Introduction

After the discovery of quarks and gluons as constituent partons inside the nucleons, one of the major question was how the spin of the nucleon is composed by its partons. It is now clear that the spin of the nucleon cannot be shared by valence quarks only. Early information about this feature came from Deep Inelastic Scattering (DIS) of polarized leptons and polarized nuclear targets [1]. Important results were obtained from the data on the spin dependent structure function of the nucleon g1(x,Q2) by European Muon Collaboration (EMC) [2]. Following years, several experiments had been performed at SLAC [3], CERN [4] and DESY [5] to understand the spin structure of the proton in the framework of quantum chromodynamics (QCD). Precise measurements on the g1p,d(x,Q2) were done by the Spin Muon Collaboration (SMC) [4] for several Bjorken-x and Q2 values. These results have reached the precise determination of the quark contribution to the proton spin. Therefore, DIS experiments showed that the quarks carry 1/4 of the proton spin and the remaining parts belongs to the gluon and orbital angular momenta of the quarks and gluons. The spin of the proton can be written in terms of its polarized parton distributions integrated over x-region. Then, based on QCD, the spin sum rule of the proton can be written in terms of parton contributions(1)12=12ΔΣ+ΔG+Lq+Lg, where the right-hand side consists of the contributions of quarks, gluons and their orbital angular momenta. DIS is a fixed target experiment with limited x-region, so it has not enough precision in determining gluon polarization. In addition to the DIS experiments in CERN, DESY and SLAC, Relativistic Heavy Ion Collider (RHIC) has started to operate since 2001 with high energy polarized proton beams. One of the major goals of RHIC covers measuring gluon polarization in the proton [6,7]. DIS and RHIC data have led to the fact that gluon contribution to the proton spin is very small than expected. The fraction of individual contributions of each term in the above sum rule is still an open question. Thus, the determination of the spin carried by gluons has been a challenging task for experimentalists and theorists.Although several asymmetry definitions are sensitive to gluon polarization, both DIS and RHIC experiments have theoretical uncertainties in determining gluon polarization extracted from physical processes involved in scattering. In ep scattering polarized gluon distribution always comes together with quasi-real photon distribution function and quark fragmentation function to hadrons. In pp scattering with hadronic final states, the product of gluon–gluon or quark–gluon distributions and quark fragmentation functions are included by the cross sections. In order to reduce these theoretical uncertainties one needs a direct way to measure ΔG alone. In theoretical point of view, this is possible only through real gamma-proton scattering. At earlier works, the scattering of a polarized real gamma beam on a polarized fixed nuclear target was studied to investigate polarized gluon distribution with J/Ψ, K and π meson production at final states [8]. RHIC experiments with polarized proton beams at 100 GeV energy (or s=200 GeV collider energy) has been done for a few years. RHIC at Brookhaven National Laboratory has the capability of polarized proton beams up to 250 GeV. Therefore, it is reasonable to consider a scattering of a polarized proton beam with a real photon beam where high energy photon beam can be achieved through Compton backscattering of parent linear electron beam [9]. Center of mass energy estimations in parent ep collisions and its γp mode are given in Table 1

at linear electron beam and RHIC proton beam energy regions.

In this work, the possibility of probing gluon polarization with polarized top–antitop quark pair production will be discussed in γp→t¯t process. This process occurs via γg→t¯t subprocess with t and u channels. As will be seen in the next section, polarization of the top–antitop pairs or one of the top quarks is correlated to the gluon polarization in the cross section. Observation of polarized tops in the final state or top-polarization asymmetry is sensitive to gluon polarization. When the gluon polarization is absent final top polarization vanishes. In the next section, we give the details of the top polarization, the scattering of polarized gluon with unpolarized real photons, asymmetry definitions and numerical results at possible energy and luminosity values. Section 3 is devoted to conclusion and discussion.2

Polarized top quark pair production and asymmetry in γp→t¯t

Let us first discuss the features of top quarks and its polarization. The top quark is the most massive fundamental fermion which plays a special role to test QCD and to probe new physics. Because of its large mass, top quark decays immediately through weak interaction after being produced, without hadronization. Therefore, decay products give information directly on top quark properties. In particular, top pair production is quite suitable to draw the spin information of the process which can be determined from the angular distributions of their electroweak decay products [10]. In the Standard Model (SM), the dominant electroweak decay chain of the top quark is(2)t→W+b→bℓ+ν/bd¯u. The correlation among the top spin and its decay product can be shown simply in the top quark rest frame. In this frame, the decay angular distribution is given by(3)1ΓdΓdcosαi=12(1+βicosαi), where αi is the angle between the top quark ith decay product and top quark spin quantization axis. βi is called the correlation degree between the decay products and top spin where βi=1 for i=ℓ+,d¯,s¯ and βi=−0.4 for i=b. More details about the above expression and its implication can be found in Ref. [11]. Thus, the top polarization asymmetry can provide additional observables when initial particle polarizations are taken into account.As a first step to achieve cross section, the t and u channel amplitudes for the subprocess γg→t¯t is written below(4)iM1=(−igsλa2)(−igeQt)[u¯1γμiq̸1−mtγνv2]eν(k1)eμ(k2),(5)iM2=(−igsλa2)(−igeQt)[u¯1γνiq̸2−mtγμv2]eν(k1)eμ(k2), with(6)q1=k1−p2,q2=k2−p2,u1=u(p1,s1),v2=v(p2,s2), where k1, k2, p1 and p2 are momenta of the photon, gluon, top quark and antitop quark. e(k1) and e(k2) are the polarization vectors of the photon and the gluon. gs, ge and Qt are strong coupling, electromagnetic coupling and electric charge number of the top quark. λa are Gell-Mann matrices. If we assume k→1 and k→2 are in the zˆ and -zˆ direction, the gluon polarization vector in the helicity basis can be given by(7)eμ(k2)=12(0,λg,−i,0),λg=∓1.As will be explained below, we do not need polarization of the photon then we sum over photon spins in the squared amplitude. In order to obtain the cross section which depends on the spins of the final top quarks we use the following projection operator(8)∑s1u(p1,s1)u¯(p1,s1)=12(1+γ5s̸1)(p̸1+mt) where the spin of the top quark s1 in the helicity basis is(9)s1μ=λt(|p→1|mt,E1mtp→1|p→1|),λt=±1.Depending on the helicities λg,λt,λt¯ of the gluon, top and antitop quarks, we have calculated the differential cross section for the subprocess γg→t¯t by squaring Feynman amplitudes and using trace method(10)dσˆdz(λg,λt,λt¯)=βNcπααsQt24sˆ(1−β2z2)2[−4βλg{(β2−1)(λt−λt¯)+β(λt+λt¯)(1−z2)z}+2β4(λtλt¯−1)(1+(1−z2)2)+4β2(1+λtλt¯z2)(1−z2)−2(λtλt¯−1)] where sˆ=(k1+k2)2 is the square of the center of mass energy of the photon–gluon or tt¯ pair. It is easy to get the cross section which depends on λg and λt by summing over λt¯ when the polarization of only one of the top quarks is considered(11)dσˆdz(λg,λt)=βNcπααsQt24sˆ(1−β2z2)2[−8βλgλt{(β2−1)+βz(1−z2)}+4{−β4(1+(1−z2)2)+2β2(1−z2)+1}], where(12)z=cosθ,β=1−4mt2sˆ.Nc=1/2 and αs=gs2/(4π) are the color factor and the strong coupling constant. For completeness, let us integrate over scattering angle to obtain total cross section for the subprocess(13)σˆ(λg,λt,λt¯)=βNcπααsQt24sˆ[4λg(λt−λt¯){12(1−β2)log1+β1−β+β}+4β3{(1−λtλt¯)β5−(2−λtλt¯)β3−3λtλt¯β+12log1+β1−β[−(1−λtλt¯)β6+λtλt¯β4+3(1−λtλt¯)β2+3λtλt¯]}],(14)σˆ(λg,λt)=2βNcπααsQt2sˆ[λgλt{12(1−β2)log1+β1−β+β}+{(3β−β3)12log1+β1−β+β2−2}].In both cross sections the gluon helicity appears as a product by the top quark helicity. This feature creates a nonzero asymmetry which vanishes when the gluon or top quark is unpolarized. Another way to get an asymmetry is to have a polarized photon beam if the final quarks are taken unpolarized. But the polarizations of the Compton backscattered photons have the distribution with respect to their energy. This is an additional uncertainty coming from the determination of the photon polarization. This is one of the reason that we consider unpolarized photon beam. As can be seen from both cross sections it is possible to define two different asymmetries(15)A1=σ+(λt=1)−σ+(λt=−1)σ+(λt=1)+σ+(λt=−1)=Δσσ,(16)A2=σ+(λt=1,λt¯=−1)−σ+(λt=−1,λt¯=1)σ+(λt=1,λt¯=−1)+σ+(λt=−1,λt¯=1)=Δσ2σ2, where σ+ denotes the spin dependent cross section with the positive helicity of the proton. Integrated cross section Δσ and σ over momentum fractions can be written by(17)Δσ=∫x1minx1maxdx1∫x2min1dx2fγ(x1)Δg(x2,Q2)Δσˆ(sˆ),(18)σ=∫x1minx1maxdx1∫x2min1dx2fγ(x1)g(x2,Q2)σˆ(sˆ), where Δg, g are the polarized and unpolarized gluon distribution function and fγ is the energy spectrum of the Compton backscattered photons [9]. The gluon-spin dependent part of the cross section Δσˆ and the unpolarized part σˆ for the subprocess is given by(19)Δσˆ=2βNcπααsQt2sˆ[(1−β2)log1+β1−β+2β],(20)σˆ=2NcπααsQt2sˆ[(3−β4)log1+β1−β+2β(β2−2)]. Similar expressions for Δσ2 and σ2 can be extracted from related cross sections.Now let us consider the case where polarization of one of the top quarks will be observed. In order to detect top spin polarization through weak decay products the decay channel bb¯(νℓ)(jj) (golden channel) with their branching ratio 30% is the most favorable. The important thing is to know the reasonable number of events in the final state which makes a sizable detection. In Fig. 1

the number of events are shown as a function of energy for luminosity L=20 fb−1. Polarized and unpolarized cross sections and comparison with pp scattering for the unpolarized case are taken into account. qq¯→tt¯ and gg→tt¯ cross sections are provided in Appendix B. We have used LSS2006 parton distribution functions (PDF) for polarized gluon distribution. LSS2006 consists of positive polarized gluon density SET=1, negative polarized gluon density SET=2 and the one that changes sign as a function of xSET=3[12]. For unpolarized gluon distribution, MRST2006 PDF [13] have been used. Energy spectrum of Compton backscattered photons fγ is given in Appendix A. The number of events has been defined as N=LσBr where L and Br are luminosity and semi-leptonic branching ratio. As can be seen from Fig. 1, number of events for unpolarized γp→t¯t process is higher than polarized one because of final state polarization sum. From comparison, at lower energies number of events from γp→t¯t is substantially larger than the number of events from pp→t¯t process. For example, at s=700 GeV the factor of increase is about 80, at s=1000 GeV it is about 20. This is due to the energy carried by parton which is higher in γp scattering.

To investigate gluon polarization, top quark spin asymmetry A1 can be regarded as the best observable. The plot of this asymmetry as a function of energy is shown in Fig. 2

for LSS2006 polarized gluon distribution function with SET=1,2,3.

From Fig. 2 we see that the asymmetry becomes maximum about at s=800 GeV. This highly sizable asymmetry value comes from the high mass of the top quark.Angular distribution of top quark asymmetry is shown in Fig. 3

for s=800 GeV with the same polarized gluon distributions as in Fig. 2. Higher asymmetry appears in the forward and backward region. As seen from the figures, the asymmetries for SET=1 and SET=3 show almost the same behavior.

We have calculated statistical uncertainty δA1 of the asymmetry A1 by the following formula(21)δA1=1P1−A12P2N, where N is the number of events for the unpolarized final state and P is the proton beam polarization. An estimation has been made for the differential asymmetry as a function of scattering angle of the top quark with P=0.7, L=20 fb−1 and 30% semi-leptonic branching ratio at s=1000 GeV. Table 2

shows this estimation for LSS2006, SET=1. Higher energy and luminosity will make the statistical uncertainty better.

As discussed in the beginning of the section, the asymmetry A2 does the same job. In this case, one should measure polarizations of both of the top quarks. Then the leptonic channels of top pairs give clearer signals with a branching ratio 5%. This branching ratio creates 6 times lower number of events than semi-leptonic channel at the same energy and luminosity. This feature can be compensated with higher center of mass energy and higher luminosity.3

Conclusion

Contribution of the polarized gluon distribution to the polarized proton can be directly probed by Compton backscattered photon beam in the process γp→t¯t. Measuring the polarization of the top quarks in the final state provides the spin information of the process based on the fact that the top quark decays without hadronization. When enough energy and luminosity are available in γ-proton collision to produce top quarks, polarization asymmetry of the final top quarks is the promising observable for determining gluon polarization. However, there are several points to study for designing and establishing such an experiment including detector. Since our work is devoted to the discussion of the theoretical steps only, experimental arguments are out of our scope.Appendix A

Compton backscattered photons

After the development on the research for linear electron-positron colliders, future γe, γγ and γp modes with real photons have been discussed as complementary to basic colliders. In this section we give the spectrum of the real gamma beam obtained by the Compton backscattering of laser photons off linear electron beam(A.1)fγ/e=1g(ζ)[1−y+11−y−4yζ(1−y)+4y2ζ2(1−y)2], where(A.2)g(ζ)=(1−4ζ−8ζ2)log(ζ+1)+12+8ζ−12(ζ+1)2, with(A.3)y=EγEe,ζ=4E0EeMe2,ymax=ζ1+ζ.Here, E0 and Ee are energy of the incoming laser photon and initial energy of the electron beam before Compton backscattering. Eγ is the energy of the backscattered photon. The maximum value of y reaches 0.83 when ζ=4.8.Appendix B

Cross sections for qq¯→tt¯ and gg→tt¯

In proton–proton scattering the differential cross sections contributing to tt¯ final states are given below [14](B.1)dσˆdt(qq¯→tt¯)=4παs29s4[(m2−t)2+(m2−u)2+2m2s],(B.2)dσˆdt(gg→tt¯)=παs28s2[6(m2−t)(m2−u)s2−m2(s−4m2)3(m2−t)(m2−u)+4(m2−t)(m2−u)−8m2(m2+t)3(m2−t)2+4(m2−t)(m2−u)−8m2(m2+u)3(m2−u)2−3(m2−t)(m2−u)+3m2(u−t)s(m2−t)−3(m2−t)(m2−u)−3m2(u−t)s(m2−u)], where m is the top quark mass and s, t and u are the Mandelstam invariants for the subprocess.

References

[1]

M.J.

Alguard

Phys. Rev. Lett.

1978

[2]European Muon Collaboration

Ashman

Phys. Lett. B

206

1988

364

European Muon Collaboration

Ashman

Nucl. Phys. B

328

1989

[3]SLAC-E142 Collaboration

P.L.

Anthony

Phys. Rev. Lett.

1993

959

SLAC-E143 Collaboration

Abe

Phys. Rev. Lett.

1995

SLAC-E143 Collaboration

Abe

Phys. Rev. D

1998

112003

SLAC-E155 Collaboration

P.L.

Anthony

Phys. Lett. B

493

2000

[4]CERN-SM Collaboration

Adeva

Phys. Rev. D

1998

112002

COMPASS Collaboration

E.S.

AgeevPhys. Lett. B

633

2006

[5]DESY-HERMES Collaboration

Ackerstaff

Phys. Lett. B

404

1997

383

DESY-HERMES Collaboration

Airapetian

Phys. Lett. B

442

1998

484

[6]

S.S.

Adler

Phys. Rev. Lett.

2004

202002

S.S.

Adler

Phys. Rev. Lett.

2005

202001

S.S.

Adler

Phys. Rev. D

2006

091102(R)

[7]

B.I.

Abelev

Phys. Rev. Lett.

2006

252001

B.I.

Abelev

Phys. Rev. Lett.

100

2008

232003

[8]

Atag

Europhys. Lett.

1995

273

Atag

Nucl. Instrum. Methods Phys. Res. A

381

1996

[9]

I.F.

Ginzburg

Nucl. Instrum. Methods Phys. Res.

205

1983

I.F.

Ginzburg

Nucl. Instrum. Methods Phys. Res. A

219

1984

V.I.

TelnovNucl. Instrum. Methods Phys. Res. A

294

1990

D.I. Borden, D.A. Bauer, D.O. Caldwel, SLAC Report No. SLAC-PUB-5715, Stanford, 1992

[10]

Jezabek

J.H.

KuhnNucl. Phys. B

320

1989

JezabekNucl. Phys. B (Proc. Suppl.)

1994

197

Mahlon

ParkePhys. Rev. D

1996

4886

Mahlon

ParkePhys. Lett. B

411

1997

173

BernreutherJ. Phys. G

2008

083001

[11]

Mahlon

arXiv:hep-ph/0011349

[12]

Leader

A.V.

Sidorov

D.B.

StamenovPhys. Rev. D

2007

074027

[13]

A.D.

Martin

W.J.

Stirling

R.S.

Thorne

WattPhys. Lett. B

652

2007

292

[14]

Barger

PhillipsCollider Physics1987Addison–Wesley Publ. Co.p. 374