Parity violation induced by the chromo-anapole form factor of the bottom
quark, generated from weak corrections, is studied in polarized hadron
collisions. The forward-backward asymmetry in the bottom quark pair production
at polarized $pp$ and $p\bar{p}$ colliders is evaluated in the Standard Model
and in a Two Higgs Doublet Model to examine the effects of parity violation. In
the models studied, promising results are found for polarized $p\bar{p}$
colliders.Comment: To be published in Physics Letters B, 3 references adde

Abachi

Amarjit Soni

Bourrely

Bunce

Cheng

Chung Kao

Contougouris

David Atwood

Donoghue

Georgi

Gunion

Hall

Karliner

Keller

Taxil

Physics Letters B

English

arXiv

Parity violation induced by the chromo-anapole form factor of the bottom quark, generated from weak corrections, is studied in polarized hadron collisions. The forward-backward asymmetry in the bottom quark pair production at polarized pp and p{anti p} colliders is evaluated in the Standard Model and in a Two Higgs Doublet Model to examine the effects of parity violation. In the models studied, promising results are found for polarized p{anti p} colliders

Kao, Chung

Atwood, David

Soni, Amarjit

UNT (University of North Texas) Digital Library

arXiv:hep-ph/9608383 v2   13 Jan 1997UR-1384MADPH-96-968Parity Violation in Bottom Quark Pair Productionat Polarized Hadron CollidersChung Kaoa,b, David Atwoodc and Amarjit SonidaDepartment of Physics and Astronomy, University of Rochester,Rochester, NY 14627, USAbDepartment of Physics, University of Wisconsin, Madison, WI 53706, USAcTheory Group, CEBAF, Newport News, VA 23606, USAdDepartment of Physics, Brookhaven National Laboratory, Upton, NY 11973, USAAbstractParity violation induced by the chromo-anapole form factor of the bottom quark,generated from weak corrections, is studied in polarized hadron collisions. Theforward-backward asymmetry in the bottom quark pair production at polarized ppand pp̄ colliders is evaluated in the Standard Model and in a Two Higgs DoubletModel to examine the effects of parity violation. In the models studied, promisingresults are found for polarized pp̄ colliders.1 IntroductionThe large production cross section of bottom quark pairs (bb̄) at polarized hadron col-liders [1, 2] coupled with the relatively long lifetime of the b-quark, implies that detailedexperimental study of the properties of the b-quark at those facilities could also be veryuseful. Since the CKM mixing angle Vtb is close to one, the b quark couples rather readilyto the virtual top quark in loop diagrams. Now the top quark is so heavy (mt ∼ 175 GeV)[3, 4] that the b-quark becomes very sensitive to electroweak radiative corrections as thesecorrections often tend to grow with the virtual quark mass [5]. Precision studies of the b’sare thus very useful in testing the Standard Model (SM) and in searching for new physics.In the hadronic environment, while the production cross sections are high, a quantitativeunderstanding of the effects of QCD can be a very difficult challenge. For this reason, intesting the SM and in searching for clues of new physics, it is perhaps better to focuson observables that tend to be robust to QCD corrections. For that reason, in general,the production cross section is not a good observable. We propose to focus instead onsignatures of parity violation in b-quark production since QCD corrections cannot generateparity violation. Parity violating asymmetries that are ratios of cross-sections should beless sensitive to QCD corrections as well as to the uncertainties in the parton distributionfunctions. Furthermore, in study of parity violation, one may be able to make use ofpolarized incident p(p̄) beams [6]. We will concentrate on one type of parity violatingobservable, that is the forward-backward asymmetry. In its differential form, it is definedas:δA(Mbb̄) ≡dσF /dMbb̄ − dσB/dMbb̄dσF /dMbb̄ + dσB/dMbb̄(1)where the subscripts F and B stand for the forward and the backward hemispheres respec-tively.For the reactions of interest to us, i.e. pp → bb̄ + X or pp̄ → bb̄ + X, we consider1two sources of contributions to such a parity violating observable. These are the chromo-anapole form factor of the gbb̄ vertex (note g ≡ gluon) and the tree level electroweakprocess qq̄ → Z → bb̄. The latter contribution (Z exchange) is significant only when theinvariant mass (Mbb̄) of the bb̄ pair is close to the Z mass (MZ). It can be removed, ifnecessary, by imposing an appropriate cut on the Mbb̄. Thus, the chromo-anapole formfactor is a very important contributor to the parity violating signal and consequently it isour primary focus.Recently, several suggestions have been made to study parity violation asymmetries inpolarized hadron collisions for (1) the production of one jet, two jets, and two jet plusphoton in the SM [7]; (2) the production of W± and Z in the SM [8]; and (3) the inclusiveproduction of one jet with a new handed interaction between subconstituents of quarks[9]. These works primarily deal with interference of electroweak and strong interactions onlight quarks.In this Letter, we present the first study on parity violation generated from one loopweak corrections to bottom quark pair production at polarized hadron colliders. The keydifference with the works in Refs. [7, 8, 9] is, as alluded to in the opening paragraph, thatthe bb̄ pair in the final state in our study is a very sensitive tool of the effects of the topquark which in turn is sensitive to non-standard effects. Specifically, we will evaluate theforward-backward asymmetry in the bottom quark pair production at polarized pp and pp̄colliders in the Standard Model and in a Two Higgs Doublet Model to examine the effectsof parity violation from the interactions of bb̄ with spin-1 and spin-0 fields.2 Form FactorsLet us write the gbb̄ vertex as− igsū(p1)T aΓµv(p2) (2)2where gs = the strong coupling, T a = the SU(3) matrices, u(p1) and v(p2) are the Diracspinors of b and b̄ with outgoing momenta p1 and p2 and k = p1 + p2 is the momentum ofthe gluon. At the tree level Γµ0 = γµ. The 1-loop vertex function can be expressed asΓµ = γµ[A(k2)−B(k2)γ5]+(p1 − p2)µ[C(k2)−D(k2)γ5]+(p1 + p2)µ[E(k2)− F (k2)γ5] (3)Current conservation demands that E = 0 and B = −k2F (k2)/2mb, where k = p1 +p2 andk2 = M2bb̄ = ŝ in the bb̄ center of mass (CM) frame. Applying the Gordon identities we canre-writeΓµ = F1(k2)γµ − F2(k2)iσµνkν+a(k2)γνγ5(k2gµν − kµkν) + d(k2)iσµνkνγ5 (4)where F1(k2) = A(k2) + 2mbC(k2), F1(0) = the chromo-charge; F2(k2) = C(k2), F2(0) =the anomalous chromo-magnetic moment; a(k2) = −B(k2)/k2 = F/(2mb), a(0) = chromo-anapole moment; d(k2) = D(k2), and d(0) = the chromo-electric dipole moment.In the SM, the dominant one loop weak corrections that contribute to the chromo-anapole moment arise from diagrams with the W+ and its Goldstone counterpart, theG+. In many extensions of the SM, e. g. in models which contain more than one doubletsof Higgs, there are charged Higgs bosons (H±) contributing to the chromo-anapole formfactor of fermions at the one-loop order. As is well known the simplest of such extensionsconsists of two Higgs doublets [10] φ1 and φ2 with vacuum expectation values (VEVs) v1and v2. After symmetry breaking, there remain five physical Higgs bosons [11]: a pair ofsingly charged Higgs bosons H±, two neutral CP-even scalars H (heavier) and h (lighter),and a neutral CP-odd pseudoscalar A. The ratio of the two VEVs is usually expressed astan β ≡ v2/v1.3In our analysis, we have considered such a Two Higgs Doublet Model (THDM) withthe Yukawa interactions of model II [12], which is required in the minimal supersymmetricmodel (MSSM)1. [11] In this model, one doublet (φ1) couples to down-type quarks andcharged leptons while the other (φ2) couples to up-type quarks and neutrinos. The oneloop weak corrections from diagrams with the W+, the G+ and the H+, yield dominantcontribution to the b quark anapole moment from the virtual top quark in the loop. In ourcalculations of these corrections we will set Vtb = 1 and also we will ignore CP violatingeffects. We have employed the ’t Hooft–Feynman gauge for loop calculations with MG± =MW .The forward-backward asymmetry in bb̄ production is primarily driven by ReB(k2)which is related to the chromo-anapole form factor (a(k2)) via B(k2) = −k2a(k2). There-fore, in Fig. 1 we present ReB(k2) coming separately from diagrams with the G±(BG), theW±(BW ) and the Z as well as the total contribution in the SM. As an illustration of abeyond the SM effect, we also show ReB(k2) for THDM for MH+ = 200 GeV and vari-ous values of tan β. The Z boson exchange contribution to the b-chromo-anapole momentis very small since the coupling product 2gbV gbA is very small. The form factor ReB(k2) isenhanced near k2 ∼ 4m2t , where k2 = ŝ = M2bb̄, due to the tt̄ threshold which appears in dia-grams with charged boson (i. e. W± or H±) exchanges. Since B(k2) = −k2a(k2) = −ŝa(ŝ),its value is expected to grow with Mbb̄. From Fig. 1 we see that numerically this increasesets in for Mbb̄ ∼> 500 GeV.In the THDM, the form factor B(k2) gets an additional contribution, BH(k2), from thecharged Higgs boson. Clearly BH is a function of MH+ and tan β, being proportional tocot2 β{1− [(mb/mt) tan2 β]2}. Therefore, it has the same sign as BG for tan β <√mt/mb ∼6, but has an opposite sign for tan β >√mt/mb. Thus the effects of parity violation are1 In the MSSM, though,there are additional contributions to the chromo-anapole moment from loopdiagrams with the charginos and the squarks as well.4enhanced in the THDM if tanβ is less than√mt/mb. For tan β =√mt/mb, the BHvanishes and the total B(k2) becomes that of the SM.3 Forward Backward AsymmetryLet us define the cross section for the sub-process, qq̄ → bb̄, in each helicity state of quarksin the initial state asσ̂λ1λ2 ≡ σ̂(qλ1 q̄λ2 → bb̄) (5)where λ1,2 represents a right-handed (R) or a left-handed (L) helicity of the quark and theantiquark. The cross sections in the forward (0 ≤ θ ≤ π) and the backward (−π ≤ θ ≤ 0)directions are defined asσ̂Fλ1λ2 ≡∫ 10dσ̂λ1λ2dzdzσ̂Bλ1λ2 ≡∫ 0−1dσ̂λ1λ2dzdzσ̂λ1λ2 = σ̂Fλ1λ2+ σ̂Bλ1λ2∆σ̂λ1λ2 = σ̂Fλ1λ2− σ̂Bλ1λ2 (6)where z = cos θ, with θ being the scattering angle of the b in the bb̄ center of mass frame, σ̂is the total cross section of the qq̄ subprocess and ∆σ̂ is the difference of the cross sectionsin the forward and the backward directions of the subprocess. (Note that variables in theCM frame of the qq̄ subprocess are denoted with a ˆ on top of them).The tree level expressions for these cross sections areσ̂0RL = σ̂0LR=g4s27πŝ(1 +2m2bs)σ̂0LL = σ̂0RR = 0 (7)At the tree level, the difference of the cross sections in the forward and backward directionsis zero in each helicity state of qq̄ because parity is conserved in QCD.5The one loop weak corrections to the qq̄ cross section isσ̂1RL = σ̂1LR=g4s27πŝ[Re(A)(2 +4m2bŝ) +Re(C)(mb)(−1 +4m2bŝ)] (8)And the difference of qq̄ cross sections in the forward and backward directions is∆σ̂RL = −∆σ̂LR= − g4s18πŝβ2[Re(B)] (9)where β =√1− 4m2b/ŝ and A, B and C are form factors defined in Eq. 3. A nonzero∆σ̂RL is of course a signature of parity violation.In polarized pp and pp̄ collisions, the differential cross section for the production of bb̄takes the formdσdx1dx2= σ̂RL[qR(x1, ŝ)q̄L(x2, ŝ) + qR(x2, ŝ)q̄L(x1, ŝ]+ σ̂LR[qL(x1, ŝ)q̄R(x2, ŝ) + qL(x2, ŝ)q̄R(x1, ŝ]+ σ̂RR[qR(x1, ŝ)q̄R(x2, ŝ) + qR(x2, ŝ)q̄R(x1, ŝ]+ σ̂LL[qL(x1, ŝ)q̄L(x2, ŝ) + qL(x2, ŝ)q̄L(x1, ŝ] (10)where x1 and x2 are momentum fractions of the partons in the initial beam.Parity violation can be studied at hadron colliders by using polarized initial beams,e.g. pRpL or pRp̄L etc. Let us consider two hadron colliders with polarized beams (a) pRpLand (b) pRp̄L. With polarized pp or pp̄ beams, we can define q+i (x, ŝ) [q−i (x, ŝ)] to be theprobability density [13] for a quark of flavor i and momentum fraction x with a helicity ofthe same [opposite] sign as the helicity of the proton. The differential cross section, dσRLin polarized pRpL collisions becomedσRLdx1dx2=dσLRdx1dx26= σ̂RL[q+(x1, ŝ)q̄+(x2, ŝ) + q−(x2, ŝ)q̄−(x1, ŝ)+q−(x1, ŝ)q̄−(x2, ŝ) + q+(x2, ŝ)q̄+(x1, ŝ)]σRL =∫ 1xmin1dx1∫ 1xmin2dx2dσRLdx1dx2(11)For gg → bb̄, we need to sum over all helicity states: gRgL, gLgR gRgR and gLgL.The difference of σ in the forward (0 < θ < π) and the backward (−π < θ < 0)directions at the bb̄ CM frame in polarized pλ1pλ2 collisions areδσRLdx1dx2= − δσLRdx1dx2= ∆σ̂RL[q+(x1, ŝ)q̄+(x2, ŝ)− q−(x2, ŝ)q̄−(x1, ŝ)−q−(x1, ŝ)q̄−(x2, ŝ) + q+(x2, ŝ)q̄+(x1, ŝ)]∆σRL =∫ 1xmin1dx1∫ 1xmin2dx2δσRLdx1dx2(12)At high energy, parity violation from chromo-anapole form factor is expected to increasewith Mbb̄. It is therefore useful to examine the differential forward-backward asymmetrygiven in Eq. 1. The numerator has contribution only from qq̄→ bb̄, while the denominatorhas contributions from both qq̄ → bb̄ and gg → bb̄. Gluon fusion produces a large numberof bb̄ pairs, which tend to contribute significantly to the denominator of Eq. 1 thus reducingthe signal for the parity violating asymmetry from qq̄ in pp collisions. In pp̄ collisions, theantiquark density is greatly enhanced, therefore, parity violation signals from qq̄ → bb̄ ismuch larger in pp̄ than in pp collisions at the same energy. This differential asymmetryis presented in Figures 2 and 3 for polarized pRpL and pRp̄L collisions at several values ofenergy. The parity violation signal peaks at the Mbb̄ = 2mt in the differential of forward-backward asymmetry versus Mbb̄. Also shown is the same asymmetry in qq̄ → Z → bb̄; itdominates if Mbb̄ is close to MZ , but becomes negligible for |Mbb̄ −MZ | > 10 GeV.The integrated forward-backward asymmetry is defined asA ≡ NF −NBNF +NB=σF − σBσF + σB7N = Lσ (13)where NF and NB are the number of bb̄ pairs in the forward and the backward directions;and L is the integrated luminosity. The statistical uncertainty (∆A) and the statisticalsignificance (NS) of the integrated asymmetry are∆A ' 1√NF +NB(14)NS ≡ A/∆A. (15)The difference (∆σ) and the total (σ) of the cross sections σF and σB with one loopweak corrections, as well as the integrated asymmetry (A) and its statistical significance(NS) are presented in Table 1 for an integrated luminosity of 10 fb−1 and 100 fb−1. For√s ≥ 1 TeV, this asymmetry might be visible with L = 100 fb−1 and a cut on Mbb̄. We findthat requiring a higher Mbb̄ can efficiently enhance the asymmetry from weak correctionsin qq̄ → bb̄. A high Mbb̄ cut not only reduces the cross section from gluon fusion it alsoreduces the asymmetry from qq̄ → Z → bb̄ as well.We close with two brief remarks.1) In our calculations we have retained only the parity violating effects originating fromqq̄ → bb̄. In principle, gluon fusion, i.e. gg → bb̄, can also contribute to parity violationdue to electroweak corrections [14]. However, dimensional reasoning indicates that suchcontributions are probably small at lower energy, at least for√ŝ ∼< 2mt. In any case, it isvery unlikely that the contribution to the forward backward asymmetry from such a sourcewill cancel away the qq̄ contribution for all values of Mbb̄.2) It would clearly be extremely interesting to consider these parity violating effectsin other extensions of the SM; in particular in the MSSM. As we noted before, in MSSM,there will be additional loop contributions not contained in a THDM that could enhancethe asymmetry [14].84 ConclusionsTo summarize, at a hadron collider with polarized pp and√s = 500 GeV (e.g. RHIC),parity violation in bb̄ production will be dominated by qq̄ → Z → bb̄. In polarized pRp̄Lcollisions with√s ≥ 1000 GeV and enough integrated luminosity, it might be possible toobserve parity violation signals from SM weak corrections in bb̄ production. In the THDMwith Model II Yukawa interactions, the chromo-anapole form factor of the bottom quarkgenerated from leading weak corrections is enhanced for tan β close to one. It is enhancedby the factor cot2 β for tan β less than one. Therefore, the signal of parity violation in bb̄production is greatly enhanced for tanβ ∼< 1 and MH+ less than about 300 GeV.2 Thestudy of parity violation in bb̄ production might provide a good opportunity to study newinteractions between the third generation quarks and (charged) spin-0 as well as spin-1bosons.AcknowledgementsWe are grateful to Stephane Keller and Jeff Owens for providing polarized parton distri-bution functions and for beneficial discussions. C.K. and D.A. would like to thank BillBardeen and the Theoretical Physics Department of Fermilab for hospitality where partof this research was completed. A.S. would like to thank Mike Tannenbaum for inter-esting discussions. This research was supported in part by the the U.S. Department ofEnergy Grant Nos. DE-FG05-87ER40319 (Rochester), DE-FG02-95ER40896 (Wisconsin),DE-AC05-84ER40150 (CEBAF), and DE-AC-76CH00016 (BNL), and in part by the Uni-versity of Wisconsin Research Committee with funds granted by the Wisconsin AlumniResearch Foundation.2 In the same model, parity violation in tt̄ production [15], is greatly enhanced for tanβ ∼> mt/mb. InRef. [15], parity violation in tt̄ production was studied in unpolarized pp̄ collisions.9References[1] A. P. Contougouris, S. Papadopoulas and B. Kamal, Phys. Lett. B 246 (1990) 523.[2] M. Karliner and R. Robinett, Phys. Lett. B 324 (1994) 209.[3] F. Abe et al., [CDF], Phys. Rev. Lett. 74 (1995) 2626.[4] S. Abachi et al., [D∅], Phys. Rev. Lett. 74 (1995) 2632.[5] See, e.g., W.S. Hou, R.S. Willey and A. Soni, Phys. Rev. Lett. 58 (1987) 1608.[6] G. Bunce et al., Particle World 3, 1 (1992); See also, Proceedings of the PolarizedCollider Workshop, Penn State University 1990, edited by J. Collins, S. Heppelmann,and R. Robinett, AIP conf. Proceedings No. 223 (AIP, New York, 1991).[7] H.-Y. Cheng, M.-L. Huang and C.F. Wai, Phys. Rev. D 49 (1994) 1272.[8] C. Bourrely and J. Soffer, Nucl. Phys. B 423 (1994) 329.[9] P. Taxil and J.M. Virey, Phys. Lett. B 364 (1995) 181.[10] H.M. Georgi, Hadronic J. 1 (1978) 155.[11] J.F. Gunion, H.E. Haber, G.L. Kane and S. Dawson, The Higgs Hunter’s Guide,Addison-Wesley, Redwood City, CA (1990) and references therein.[12] J.F. Donoghue and L.-F. Li, Phys. Rev. D19 (1979) 945; L. Hall and M. Wise, Nucl.Phys. B187 (1981) 397.[13] S. Keller and J.F. Owens Phys. Rev. D49 (1994) 1199.[14] D. Atwood, C. Kao and A. Soni, work in progress.[15] C. Kao, Phys. Lett. B 348 (1995) 155.10TablesTable 1. The difference (∆σ) and the total (σ) of cross sections σF and σB, and theasymmetry of pRp̄L → bb̄ + X as defined in Eq. 13 generated by weak corrections in (a)the SM and (b) the THDM with MH+ = 200 GeV and tanβ = 1, for√s = 500 GeV withMbb̄ > 100 GeV,√s = 1 TeV with Mbb̄ > 200 GeV and√s = 2 TeV with Mbb̄ > 300 GeV.Also shown is the statistical significance (NS) for L = 10 fb−1 and 100 fb−1.√s ∆σ σ(qq̄) σ(gg) A NS NS(GeV) (pb) (pb) (pb) (%) (10 fb−1) (100 fb−1)(a) SM500 -0.130 124 81.2 -0.063 0.91 2.91000 -0.081 22.9 17.2 -0.20 1.3 4.12000 -0.122 16.6 33.3 -0.25 1.7 5.5(b) THDM500 -0.156 124 81.2 -0.076 1.1 3.41000 -0.106 22.9 17.2 -0.27 1.7 5.32000 -0.163 16.7 33.3 -0.33 2.3 7.311FiguresFig. 1. The real part of the chromo-anapole form factor B(ŝ) as a function of Mbb̄ from (a)the diagrams with G+ (dash), the W+ (dot-dash), the Z (dot) and the SM Total; and theTHDM Total for tanβ = 0.5, 1, 3, 10 and 35, with (b) mH+ = 200 GeV and (c) mH+ = 400GeV. Note that ŝ = Mbb̄.Fig. 2. The differential forward-backward asymmetry (δA) defined in Eq. 1 versus Mbb̄, inpolarized pRpL collisions for (a)√s = 500 GeV (b)√s = 1000 GeV and (c)√s = 2000GeV. This asymmetry has been evaluated using the (total) chromo-anapole form factorfrom leading weak corrections in the SM as well as in the THDM for mH+ = 200 GeV andtan β = 0.5, 1, and 35.Fig. 3. The differential forward-backward asymmetry (δA) defined in Eq. 1 versus Mbb̄, inpolarized pRp̄L collisions for (a)√s = 500 GeV (b)√s = 1000 GeV and (c)√s = 2000GeV. This asymmetry has been evaluated using the (total) chromo-anapole form factorfrom leading weak corrections in the SM as well as in the THDM for mH+ = 200 GeV andtan β = 0.5, 1, and 35.12arXiv:hep-ph/9608383 v2   13 Jan 1997FIG 1/Atwood, Kao and SoniarXiv:hep-ph/9608383 v2   13 Jan 1997arXiv:hep-ph/9608383 v2   13 Jan 1997

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