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Spin-Spin Asymmetries in Large Transverse

arXiv:hep-ph/9208250v1 26 Aug 1992PSU/TH/113MAD/PH/709August 1992Spin-Spin Asymmetries in Large TransverseMomentum Higgs Boson ProductionM. A. DoncheskiDepartment of PhysicsUniversity of WisconsinMadison, WI 53706andR.

W. Robinett and L. WeinkaufDepartment of PhysicsThe Pennsylvania State UniversityUniversity Park, PA 16802AbstractWe examine the spin-dependence of standard model Higgs boson productionat large transverse momentum via the processes gg →gH0, qg →qH0, andqq →gH0. The partonic level spin-spin asymmetries (ˆaLL) for these processesare large at SSC/LHC energies.

The prospects for probing the spin-dependence of the standard model of particlephysics at collider and supercollider energies have recently received renewed attention[1], partly because of the successful experimental tests of the Siberian snake concept[2]. Polarization options at collider energies[3, 4] and supercollider energies [5, 6] andthe physics programs possible at such facilities have been discussed extensively.Predictions for the longitudinal spin structure of hard scattering events at such en-ergies require two ingredients: a knowledge of the helicity structure of the contributingmatrix elements and parameterizations of the spin-dependent parton distributions ofthe proton.

Lowest order predictions for the spin structure of many standard colliderprocesses now exist, often quoted as partonic level asymmetries,ˆaLL ≡ˆσ(++) −ˆσ(+−)ˆσ(++) + ˆσ(+−) ,(where ± refers to the helicity of the incident partons parallel or antiparallel to thepolarized incident nucleon. )Polarized lepton-nucleon production experiments haveprovided some knowledge of the spin structure of the valence quark distribution in thenucleon (at relatively large x) but the spin dependence of the sea quark and gluondistributions is yet to be measured directly, and, in fact, a large part of a programof polarized pp collisions at RHIC would be a comprehensive mapping out of thesedistributions[7].As mentioned above, many calculations of the partonic level asymmetries for stan-dard model processes exist (see, e.g., [6], [8]–[11] and references therein).

While likelynot relevant for a first generation program of polarized pp collisions at collider energies,the production of the standard model Higgs boson (H0) at a polarized supercollider(such as SSC or LHC) has also been discussed to some extent [6]. The calculations1

of Ref. [6] have focused on low transverse momentum H0 production via gluon fusion(gg →H0) and weak gauge boson fusion (qqqq →W +W −→H0) and the partoniclevel asymmetries in these cases are known to be quite large, e.g.

ˆaLL(gg →H0) = +1.Higgs boson production at large transverse momentum, via the processes gg →gH0, qg →qH0, and qq →gH0 has also been considered in the unpolarized case byEllis et al. [12] and Baur and Glover [13].

The matrix elements for these processes, de-pending as they do on heavy quark loops, are, in general, quite complicated functionsof the kinematic variables ˆs, ˆt, ˆu, Mt, and MH but simplify dramatically in the limitof Mt >> MH, ˆs, ˆu, ˆt. It was noted in Ref.

[11], that in this limit the partonic levelasymmetries for these processes reduce to those for the production of a 1S0 quarkoniumstate, which are known to be reasonably large[9]. In this note, we extend this prelimi-nary observation concerning the spin dependence of large transverse momentum Higgsboson production in three obvious ways.

We first explore the effects of a finite heavyquark mass on the partonic level asymmetries (ˆaLL) and then calculate the averagespin-spin asymmetry < ˆaLL > in high pT H0 production at SSC and LHC energies todetermine a rough measure of the ‘analyzing power’ of such processes. Finally, we esti-mate the range in observable longitudinal spin-spin asymmetries ALL by using severalparameterizations of the polarized gluon distributions indicating a range in the currentuncertainty about the contribution of gluons to the proton spin.The cross-section for qq →H0g (and the crossed process qg →qH0) depend onlyon a single heavy quark triangle graph and the resulting ggH0 form factor appears asa simple multiplicative factor in the amplitude.

While this factor changes the totalcross-section, it as no effect on the helicity structure of the matrix elements so the2

partonic level spin-spin asymmetries for these two processes are independent of Mtand are given byˆaLL(qg →qH0) = ˆs2 −ˆu2ˆs2 + ˆu2(1)andˆaLL(qq →gH0) = −1. (2)These asymmetries were plotted for several values of√ˆs/MH in Ref.

[11] for illustra-tion. The amplitude for the dominant gg →gH0 process depends on both triangle andbox diagrams and can be expanded in terms of two invariant functions (see, e.g., Ref.

[12], Eqn. (A.6)) so that the helicity structure of the interaction does, in fact, dependon the mass of the quark in the internal loops.

In the notation of Ref. [12], one hasdˆσdˆt =116πˆs214·64Xspins,colors|M|2(3)where the spin and color summed invariant matrix elements are given byX|M|2=αwα3S96ˆsˆtˆuM8HM2W(|A2(ˆs, ˆt, ˆu)|2 + |A2(ˆu, ˆs, ˆt)|2+|A2(ˆt, ˆu, ˆs)|2 + |A4(ˆs, ˆt, ˆu)|2) .

(4)The dimensionless functions A2 and A4 are given in terms of standard loop integralsand thus depend on the quark loop mass and are actually proportional to the gluonhelicity amplitudes which we require for the partonic level spin-spin asymmetry. Infact, we findˆaLL(gg →gH0) = |A4(ˆs, ˆt, ˆu)|2 + |A2(ˆs, ˆt, ˆu)|2 −|A2(ˆu, ˆs, ˆt)|2 −|A2(ˆt, ˆu, ˆs)|2|A4(ˆs, ˆt, ˆu)|2 + |A2(ˆs, ˆt, ˆu)|2 + |A2(ˆu, ˆs, ˆt)|2 + |A2(ˆt, ˆu, ˆs)|2 .

(5)In the Mt →∞limit, one hasA4(ˆs, ˆt, ˆu) →−13,A2(ˆs, ˆt, ˆu) →−ˆs23M4H(6)3

so that the partonic level asymmetry is simplyˆaLL(gg →gH0) = M8H + ˆs4 −ˆt4 −ˆu4M8H + ˆs4 + ˆt4 + ˆu4(7)which is the same result as for gg →g 1S0 quarkonium production as found in Ref. [11].As the only change is in the purely gluon induced processes, we illustrate results forthat sector only.

In Fig. 1 we plot the partonic level asymmetries versus the center-of-mass angle (cos(θ∗)) for several values of ˆs/M2H in the large Mt limit.

(We note thatwhen ˆs →M2H the asymmetry approaches the limit +1, corresponding to gg →H0,independent of angle.) In Fig.

2, we then plot the ratio of the ‘exact’ expressions forthe partonic level ˆaLL (Eqn. 5) to the infinite quark mass limit (Eqn.

7) versus cos(θ∗)for the same values of ˆs/M2H and two values of Mt/MH. In general, the asymmetriesare somewhat reduced for finite values of Mt but not dramatically so.We next plot in Fig.

3 the differential cross-sections for large transverse momentumH0 production, dσ/dpT versus pT for two choices of MH and for SSC and LHC energiesto acquire some feel for the event rates possible using these mechanisms. As a testof our calculation, we are able to reproduce the corresponding figures from Refs.

[12,13]. Then to ensure that the spin-dependence of the matrix elements is large in thekinematic regions for high pT Higgs boson production, in Fig.

4 we plot the averagepartonic level asymmetry, defined via< ˆaLL >≡PijR dxaR dxbfi(xa, Q2)fj(xb, Q2) dˆσij · ˆaijLLPijR dxaR dxbfi(xa, Q2)fj(xb, Q2) dˆσij(8)where fi(x, Q2) are the appropriate parton distributions. We use EHLQ2 distributions[14] for consistency with Ref.

[12] as well as the choice of momentum scale Q2 =4

M2H +p2T, and include all three relevant subprocesses. We see that the average partoniclevel asymmetries are quite reasonable in all of the kinematic regimes relevant forhigh pT Higgs production.

Finally, we can include the effects of the polarized partondistributions by calculating values of the observable spin-spin asymmetry,ALL ≡PijR dxaR dxb∆fi(xa, Q2)∆fj(xb, Q2) dˆσij · ˆaijLLPijR dxaR dxbfi(xa, Q2)fj(xb, Q2) dˆσij(9)where now the ∆fi(x, Q2) ≡f (+)i(x, Q2) −f (−)i(x, Q2) are defined in terms of the spindependent parton distributions. Using the parameterizations of the polarized partondistributions given by Soffer et al.

[6] and Bourrely et al. [15], we then find theasymmetries plotted in Fig.

5. These two choices are representative of a standardlysmall gluon polarization in the nucleon (Ref.

[6]) and EMC-motivated ‘large’ gluoncontribution to the proton spin (Ref. [15]) and therefore give some idea of the possiblerange in observable spin dependence in this reaction.

In neither case is the observableasymmetry larger that 1%; since the average partonic level asymmetry, < ˆaLL >, wasseen to be large (Fig. 4), the small observable asymmetry is due to the smallness ofthe polarized gluon distribution in the kinematic region probed.In conclusion, we have found that the partonic level longitudinal spin-spin asym-metries in the kinematic regions relevant for large pT Higgs boson production are quitelarge but that the observable spin dependence depends critically on the, as yet unmea-sured, polarized gluon distribution in the proton.

Hopefully, measurements of variousstandard model processes (such as jet and direct photon production) with a polar-ized proton-proton colliding beam facility such as at RHIC will provide the first directinformation on the size of the gluon contribution to the proton spin.We are very grateful for conversations with U. Baur and R. Stuart, who also kindlyprovided us with various computer programs. This work was supported in part by the5

National Science Foundation under grant PHY–9001744 (R.R. ), by the Texas NationalResearch Laboratory Commission under an SSC Junior Faculty Fellowship (R.R.

),by the University of Wisconsin Alumni Research Foundation (M.D. ), by the U. S.Department of Energy under contract DE-AC02-76ER00881 (M.D.

), and by the TexasNational Research Laboratory Commission under grant No. RGFY9173 (M.D.

).6

References[1] Proceedings of the Polarized Collider Workshop, Penn State University, 1990,edited by J. Collins, S. Heppelmann, and R. W. Robinett (AIP Conf. Proc.

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[3] C. Bourrely, J. Ph. Guillet, and J. Soffer, Nucl.

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[4] For a review of the prospects for a program of polarized pp collisions at RHIC, see“Polarized Proton at RHIC”, G. Bunce, J. Collins, S. Heppelmann, R. Jaffe, S.Y. Lee, Y. Makdisi, R. W. Robinett, J. Soffer, M. Tannenbaum, D. Underwood,and A. Yokosawa, Particle World 3 (1992) 1.

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Figure CaptionsFig. 1.

Partonic level asymmetries ˆaLL for gg →gH0 versus cos(θ∗) (where θ∗is thecenter-of-mass scattering angle) in the Mt →∞limit. ˆs/M2H = 2 (20, 200) isshown in the solid (dashed, dotdashed) curve.Fig.

2. Ratio of ‘exact’ partonic level asymmetry ˆaLL to that in the Mt →∞limit(ˆaLL(∞)) versus cos(θ∗) for two values of Mt/MH (Mt/MH = 0.2 (0.8) on theleft (right) respectively).

Three values of ˆs/M2H are shown as in Fig. 1.

We usethe fact that the angular distribution is symmetric around y = cos(θ∗) = 0.Fig. 3.

Differential cross-section, dσ/dpT (nb/GeV ) versus pT (GeV ) for Higgs bosonproduction for MH = 100 GeV, (200 GeV ) for √s = 40 TeV dashed (dotdashed)curve, and for √s = 17 TeV solid (dotted) curve. The parton distributions ofRef.

[14] are used (EHLQ2) with the scale choice Q2 = M2H + p2T.Fig. 4.

The average partonic level asymmetry < ˆaLL > (as defined in Eqn. 8) in thequantity dσ/dpT versus pT (GeV ).

Curves are labelled as in Fig. 3.Fig.

5. The observable spin-spin asymmetry, ALL (defined in Eqn.

9) in the quantitydσ/dpT versus pT (GeV ). Asymmetries for √s = 40 TeV , MH = 200 GeV andMt = 130 (∞) GeV for the polarized parton distribution functions of Ref.

[6]solid (dotted) curve, and Ref. [15] dashed (dotdashed) curve.9

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