Hard gluon emission from colored scalar pairsbreak in e^+ e^- annihilation

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a r X i v :h e p -p h /9603203v 1 1 M a r 1996TU–497TIT-HEP-308February 2008Hard gluon emission from colored scalar pairsin e +e −annihilationKen-ichi HikasaDepartment of Physics,Tohoku UniversityAoba-ku,Sendai 980-77,JapanJunji HisanoDepartment of Physics,Tokyo Institute of TechnologyOh-okayama,Meguro-ku,Tokyo 152,JapanABSTRACTWe study QCD correction to the pair production of colored scalar particles in electron-positron annihilation with an emphasis on gluon emission in the final state.We discuss the usefulness of working in a “quasi-two-body”frame and present the helicity amplitudes for the process.We compare the final state con-figuration with fermion pair production and find that the three-jet fraction for the scalars shows quantitative difference from that for fermions.1.IntroductionThe known elementary particles are either spin-1/2fermions(quarks and leptons)or spin-1gauge bosons.No elementary spin-0particle has been found to date.Yet,scalar particles appear in mostfield theory models of particles. Higgs bosons are the key ingredient for electroweak symmetry breaking in the standard model and its extensions.Supersymmetry predicts a scalar partner for each fermion degree of freedom,thus the existence of three generations of squarks and sleptons.Grand unified E6model contains colored scalar particles which may be interpreted either as leptoquarks or diquarks.Many of these scalars can be pair produced in e+e−annihilation with a cross section comparable to or somewhat smaller than that for fermions.Experimental searches for these particles have been extensively performed[1].If the scalar particle is colored(like the squark and leptoquark),the cross section is modified by QCD corrections.The calculation of the O(αs)correction is essentially the same as scalar QED one-loop calculation which is known for a long time.1The O(αs)correction for the scalar pair production is numerically quite important.In the high energy limit,the total correction factor for scalar pair production is four times larger than that for fermion pair production[3]:σ(e+e−→ζ¯ζ(g))=σ0 1+3C Rαs1A detailed description can be found in Ref.2.A typographical error is corrected in Ref.3.ζ¯ζg,both of which are infrared divergent.This divergence is cancelled when the two contributions are added.Essential for this cancellation is only the part of the three-body phase space where the gluon is soft(or collinear to one of the scalars when the scalar mass goes to zero).The rest of the three-body phase space corresponds to thefinal state with a hard gluon which may be observed as a separate jet(“three-jet”final state).The gluon emission process in scalar top quark pair production has been studied by Beenakker,H¨o pker,and Zerwas[4],who calculated the fully differ-ential cross section for e+e−→˜t¯˜t g.In the present paper,we derive the crosssection in a different Lorentz frame(“quasi-two-body”frame)in which the he-licity amplitudes have a simple interpretation.We then extend their analysis and compare the three-jet cross sections with those for scalar and fermion pair production processes.The QCD correction we calculate in this paper is just the effects arising from gluon gauge interactions.In the supersymmetric standard model there are additional interactions with the same strength involving gluinos.The gluino coupling contribution to the virtual O(αs)correction is discussed by Arhrib, Capdequi-Peyranere,and Djouadi[5].2.Lowest-Order Cross SectionWe consider the process e+e−→ζ¯ζwhich occurs via s-channelγor Z ex-change.Additional t or u-channel exchanges have to be included for selectron (or electron sneutrino)production or a leptoquark if it couples to electrons with a nonnegligible Yukawa coupling.In this section we list the lowest-order amplitude and cross section for com-pleteness.The Feynman graph is shown in Fig.1.The amplitude is found to be (see Fig.1for the momentum assignments)M0=e2withHµ=−Qζ¯v(¯p e)γµu(p e)+T3ζ−Qζsin2θWs−m2Z¯v(¯p e)γµ(v e−a eγ5)u(p e).(2.2)Here s is the e+e−c.m.energy squared,Qζand T3ζare the electric charge and the third component of weak isospin ofζ,andv e=−14.(2.3)We will neglect the electron mass throughout the paper.For longitudinally polarized electrons with polarization P colliding with un-polarized positrons,the cross section can be writtendσ=1+P4dσ−,(2.4)where dσ±denotes the cross section for the e−R e+L and e−L e+R initial states.We finddσ±2sH2±sin2θ,(2.5) withH±=−Qζ+(v e∓a e)(T3ζ−Qζsin2θW)s−m2Z.(2.6)Here,d R is the dimension of the color SU(3)representation ofζ(3for the fun-damental representation),andβ=3sH2±,(2.8) which we will denote byσ0±.3.O(αs)correction3.1Virtual one-loop correctionFeynman diagrams for the process e+e−→ζ¯ζat O(αs)are depicted in Fig.2.The diagrams(a)–(c)are the one-particle-irreducible vertex corrections. The mixed four-point vertex in(b)and(c)appears because the scalar kinetic term has two covariant derivatives.The diagrams(d)and(e)areζself-energy corrections,and the diagrams(f)–(h)show the counterterm contribution.We regularize the ultraviolet divergence by dimensional reduction2with D= 4−2ǫ,and the infrared singularity by an infinitesimal gluon massλ.This pro-cedure does not cause problems with gauge invariance since the diagrams do not contain non-Abelian gauge vertices.We adopt the on-shell renormalization scheme to determine the counterterms. The mass and wave function renormalization constants forζare found to beδm24π 3 1µ2+7 ,(3.1)Zζ−1=C Rαsǫ−γE+log4π −2logλ2cosθW sinθWΛµ(3.4) with the sameΛµ.At the lowest order we have F(q2)=1.The counterterm for the vertex can be found using the Ward identity.The same result is obtained by requiring no O(αs)correction at q2=0,i.e.,F(0)=1. Expanding the form factor asF(q2)=1+C Rαs2βlog1+βλ2+1+β21+β −log2β1−β−11−β+π2βlog1+β2β log m21−β2−2.(3.6)Here Li2(x)is the dilogarithm(Spence)function3Li2(x)=− x0dt log(1−t)/t. At O(αs),the lowest order cross section is multiplied by the factor1+C Rαs3For a convenient expansion for numerical evaluation of dilogarithm,see Ref.6.3.2Real gluon emissionThe Feynman diagrams for the process e+e−→ζ¯ζg at the lowest order are shown in Fig.3.The amplitudes are found to beM=e2g s T a p·k−(p−¯p−k)µ¯pαdτd cosΘd cosθdφ=d Rα2H2±πv(1−τ)× A(1+cos2Θ)+B(1−3cos2Θ)+C sinΘcosΘcosφ+D sin2Θcos2φ ,(3.9)whereA=1+2β2τv2sin2θ(1−τ)2(1−v2cos2θ)2,(3.10b)C=−4√(1−τ)2(1−v2cos2θ)2,(3.10c)D=−2τv2sin2θ(β2−v2cos2θ)1−4m2/τs(3.11) is the velocity ofζin theζ¯ζc.m.frame.If one integrates the cross section over the whole3-body phase space,one encounters infrared divergence coming from soft-gluon region.We separate this region from the rest by the condition that the gluon energy k0<ω,where the cutoffωis infinitesimally small.In this region,we regularize the divergence by giving an infinitesimal mass λto the gluon,which satisfiesλ≪ω.We recalculate the amplitude withfinite mass.The amplitude factorizes into the lowest-order part and the soft gluon factor.Since a very soft gluon does not alter the kinematics of theζ¯ζstate,we can also integrate over the soft gluon phase space ignoring momentum conservation. Integrate over this soft-gluon region,wefind the cross sectionσsoft=σ0C Rαs2βlog1+βλ2+1+β21+β −log2β1−β−11−β−π2+11−β ,(3.12)whereσ0is the lowest-order cross section.For the rest of the phase space,we can set the gluon mass to zero.Wefirst integrate overΘ,φtofinddσ±πv(1−τ)(1−τ)2(1−v2cos2θ)2 ,(3.13)which we will use to study the event configuration in the next section.The integrated cross section isσhard=σ0C Rαs2βlog1+β4ω2+1+β21+β +2Li2−1−β1+βlog1+β2log21+β6 −3log44β3(3+β2)(1−β2)log1+β2β2(3+7β2) .(3.14)Theωdependence cancels out when these two cross sections are summed. The infrared divergence is cancelled by adding the virtual one-loop correction discussed in the previous subsection.3.3Total correctionThe O(αs)corrected cross section for e+e−→ζ¯ζ(g)can thus be written asσ=σ0 1+C RαsβA(β)+11−β+31+β +2Li2−1−β1+βlog1+β1−β −3βlog4βA(β)+33+22β2−7β41−β+3(5−3β2)βA(β)+11−β+3axial current is equal to that for vector current at the high energy limit(which reflects the chiral symmetry in this limit),but approaches the scalar curve at smallβ.This latter behavior reflects the fact that spin does not play an important role in the nonrelativistic region.The difference of the vector current result is due to its S-wave dynamics contrast to the P wave behavior of the others.At the high energy limitβ→1,onefinds∆=3,∆V=∆A=32β−2,∆V≃π23s H2±C Rαs β2+11−exp(−πC Rαs/β) ,(3.22)which gives at the threshold∆=π2C2Rα2sIf one approaches closer to the threshold,one will encounter P-waveζ¯ζbound states,similar to charmonium and bottomonium resonances.These states are expected to be less pronounced than the fermion bound states,since the resonance formation cross section is proportional to the derivative of the wave function at the origin and is suppressed by a factor ofα2s compared to the fermion bound states.In the Coulomb potential approximation,this subthreshold cross section is given by[7]σ±=π2d Rα2n5δ(s−M2n),(3.24)where M n=2m−C2Rα2s m/4n2.This Coulombic approximation ignores the effect of confinement and is not adequate especially for higher bound states.One needs to solve Schr¨o dinger equation with a realistic color-force potential to obtain the cross section in the resonance region.Ifζhas a short lifetime withΓ>∼α2s m,it does not live long enough to form a bound state.There will be only a smooth bump or shoulder instead of the resonances at the threshold.The situation will be similar to the case for the top quark pair production[8]and the cross section can be calculated quite reliably within perturbative QCD.4.Jet configurationIn this section we study the shape of the three-bodyfinal stateζ¯ζg when the gluon is‘visible’.We compare various distributions with those for the more familiarfinal state of a fermion-pair plus a gluon.To define thefinal state configuration,it is convenient to use Lorentz-invariant kinematic variables.We use the following Dalitz variablesx=2p·ks,(4.1)for which the phase space density is constant.Theζ¯ζg cross section is(we dropthe subscript±from here on)dσπ1β2−1¯x +1+β2x¯x−1−β2x2+1π 2Li2(x c)+log2x c−π2x c+(3−2x c)(1−2x c) ,(4.4) whereas for fermions(both vector and axial)we haveσ3j/σ0=C Rαs6−1xc+5to the lowest-order cross section.)The three-jet fraction for the scalar is larger than that for the fermion at large x c but becomes smaller at small x c<∼0.1.Theapproximate form for small x c isσ3j/σ0≃C Rαs6 (4.6) for scalars,andσ3j/σ0≃C Rαs2log x c+56 (4.7)for fermions.These predictions are not reliable at very small x c since higher order contributions become important.The three-jet fraction forfinite m has a very complicated expression and we do not write down the analytic expression here.Instead,we show the numerical result forσ3j/σfor several values ofβin Fig.6.For afixed value of x c,the three-jet fraction rapidly decreases asβbecomes smaller.Hard gluon emission is much suppressed when one approaches to the threshold.This suppression comes from the available phase space and from the structure of the gluon-emission vertex. The large QCD enhancement near the threshold can be attributed to two-jet configuration and one can assume that most events in the threshold region have no extra jets in experimental searches for new colored scalar particles.Toward the threshold region,the three-jet fraction in Fig.6for scalar pairs becomes similar to that for fermion pairs from axial current.The fermion pair from vector current shows a different behavior.This again reflects the fact that spin of the particle is not important in the nonrelativistic region.In jet analyses in e+e−physics,one normally uses a jet-defining algorithm by combining momentum of hadrons and compares the result with the corresponding quantity in the quark-gluon system.A commonly adopted prescription uses the invariant mass of the hadron system to define a jet.To conform with thisalgorithm,we may use the invariant mass variablesy=(p+k)2s,(4.8)instead of x and¯x.These variables are related byy=x+m2s.(4.9)The definition of three-jet events in terms of y isy>y c and¯y>y c.(4.10)The three-jet fraction as a function of y c is shown in Fig.7.So far we have assumed that we can somehow distinguish aζ-jet and a gluon jet.If this is not the case,we need to treat the threefinal particles in a democratic way and define the three-jet region asy>y c and¯y>y c and y g>y c,(4.11)where y g=(p+¯p)/s.Since the region y g→0(or y g→4m2/s)contains no singularity,this definition does not lead to significant modification of the result for small y c.In the high energy limitβ→1,the three-jet fraction is found to beσ3j/σ0=C Rαs1−y c +log21−y c6−2(1−2y c)log1−2y cπ 2Li2y c y c−π22(1−2y c)log1−2y c4(1−3y c)(5+3y c) .(4.13)5.ConclusionsWe have studied O(αs)QCD correction to colored scalar pair production in e+e−annihilation.Emphasis is put on gluon emission in thefinal state.We compared the configuration of the three-jetfinal state with those in fermion pair production and found that the two cross sections are quantitatively quite different.This has to be taken into account in the search and study of production of scalar particles such as squarks and leptoquarks at LEP-2and future linear colliders.In calculating the helicity amplitudes e+e−→ζ¯ζg,we found that it is conve-nient to decompose the amplitude into two subprocesses e+e−→V∗(V∗=γ,Z) and V∗→ζ¯ζg.The amplitudes take a simple form if one works in two different Lorentz frames,the e+e−c.m.frame for the former and theζ¯ζc.m.frame in the latter.In particular,working in the“quasi-two-body”frame in the latter1→3 subprocess allows one to take advantage of the strong constraints of rotational invariance on two-body states.AcknowledgmentsOne of us(J.H.)is a fellow of the Japan Society for the Promotion of Sci-ence.This work is partly supported by Grant-in-Aid for Scientific Research (no.06640369)from the Japan Ministry of Education,Science,Sports,and Cul-ture.Appendix A.Helicity amplitudes for e+e−→ζ¯ζgA number of formulations[9]to calculate helicity amplitudes[10]are found in the literature.Most of them are particularly suited for many-bodyfinal states and numerical evaluation of the amplitudes.The technique we employ here is a method[11]based on spherical-vector basis,which is suited for manual analytic calculations and gives amplitudes with clear physical interpretation.The method is most useful for2→2body processes(also1→2,2→1).In these processes, angular momentum conservation strongly constrains the form of the amplitudes. In our technique,this constraint can be explicitly extracted in the form of a Wigner D function which depends on the scattering angles.This part represents the variation of the amplitude which is imposed by kinematics.The rest of the amplitude contains purely dynamical information.In general,the helicity amplitude for the processa(p a,λa)+b(p b,λb)→c(p c,λc)+d(p d,λd)(λa is the helicity of the particle a,etc.)in the c.m.frame can be written asM= M(E c.m.,cosθ;{λ})d J0λi,λf(θ)e i(λi−λf)φ,(A1)whereλi=λa−λb,λf=λc−λd,J0=max(λi,λf),and(θ,φ)are the scattering angles(in the frame p a is along the z axis, p c is in the direction(θ,φ)).The d Jλλ′is the Wigner d function.The last two factors in this formula represent the“minimal”angular distribution imposed by the kinematics.The physical meaning of J0is the smallest allowed angular momentum for the process.The cosθdependence of M comes from higher partial waves with J>J0.If the process has only one partial wave J(e.g.for two-body decays or e+e−→µ+µ−), the whole angular dependence can be extracted in the form of d J and theφ-dependent phase factor.This technique can be applied to the process we consider if we note the following two points:1)The amplitude can be decomposed into two parts:(A)production of avirtual vector boson V∗,e+e−→V∗(V=γor Z);(B)the virtual vector boson decaying toζ¯ζg.The helicity amplitude for the whole process is the product of the two amplitudes with afixed V∗helicity(λV=±1,0),which are then summed over:M=− V1λn≡λV−λ=±2M=2τv2sin2θ2τv2sinθcosθ(1−τ)(1−v2cos2θ)−2.(A7) The helicity amplitude for e−e+→V∗is simpler.We take the V∗direction as the z axis and the e−direction lying on the xz plane(with a positive x component).The angle between the two directions is denoted byΘ.This choice is made to relate the two frames by the boost along the z axis,under which the helicity of V∗does not change.The amplitude conserves electron chirality so that onlyλi=λ(e−)−λ(e+)=±1is allowed.M=(−1)λV g i√(1+λµcosΘ)|λ|=|µ|=1,2−λ2sinΘ|λ|=1,µ=0.(A9)Appendix B.Three-body phase spaceThe differential cross section can be calculated from the helicity amplitudes with the Lorentz-invariant three-body phase spacedΦ3=(2π)4δ4(q−p−¯p−k)d3p(2π)32¯p0d3k1024π4dτd cosΘd cosθdφ,(B2)where1−β2<τ<1,and v=(1−4m2/τs)1/2is theζvelocity in theζ¯ζc.m. frame.(We have integrated over one azimuthal angle on which the cross section has trivial dependence.)Two of the kinematic variables,τandθ,specify thefinal state configuration and the other two determine the orientation of the configuration with respect to the initial beam axis.Integrating overΘandφone hasdΦ3=sv(1−τ)128π3v2(β2−v2)constant(Dalitz variables).sdΦ3=.(B6)x+¯xThe physical phase space region is bounded by the inequalitym2x¯x(1−x−¯x)−γµ(v F−a Fγ5),(C1)cosθW sinθWwith1v F=2 T3(F L)−T3(F R) .(C3) At O(αs),the correction factor is common for the photon and the vector part of the Z current.The axial part of the Z current is modified differently if m F=0.We write the general Lorentz structure of the vector and axial vertices for on-shell fermionsΓµ=F V 1(q 2)γµ+F V 2(q 2)iσµνq ν/m F(C4)for the vector current andΓ5µ=F A 1(q 2)γµγ5+F A 2(q 2)q µγ5/m F(C5)for the axial current.We have retained only terms appearing at O (αs ),i.e.,terms allowed by CP invariance and vector current conservation.We normalizethe vertices such that F V,A 1=1at the lowest order.F V,A2do not appear in thisorder.We write the O (αs )corrected vertices asF V,A 1(q 2)=1+C R αs2πf V,A 2(q 2).(C7)We find the renormalized vector form factors for q 2>4m 2Ff V1=−1+β21−β+1logm 2FβLi 21−β2βlog1+β4log 21+β3+1+2β21−β−2+iπ1+β2λ2+log 4β22β,(C8)f V2=1−β21−β+iπ,(C9)where β=(1−4m 2F /q 2)1/2.We have used on-mass-shell renormalization con-dition such that F V 1(0)=1.Once the prescription for the vector vertex is fixed,there is no freedom to choose a renormalization condition for the axial vertex be-cause of electroweak gauge symmetry.In particular,F A 1(0)deviates from unity.The axial form factors are found to bef A1=−1+β21−β+1logm 2FβLi 21−β2βlog1+β4log 21+β3+2+β21−β−2+iπ1+β2λ2+log4β22β,(C10)f A 2=1−β21−β+iπ +2β .(C11)The second form factor F A 2represents the pseudoscalar part of the axial current and does not contribute to the reaction we are interested in as long as the electron mass is neglected.The differential cross section for the 3-body F ¯F g final state is for the vector currentdσV ±πv (1−τ)1−v 2cos 2θ+2(3−β2)τv 2sin 2θdτd cos θ=σA0±C R αs2β3×2−β2+v 2cos 2θ(1−τ)2(1−v 2cos 2θ)2,(C13)with the lowest order cross sectionsσV0±=8πd Rα22,(C14)σA0±=8πd Rα2cos2θW sin2θWs cos2θW sin2θWsdx d¯x =σV0±C Rαsβ 1x+xx+1214 1¯x2,(C18)and for the axial vector currentdσA±π14β2 ¯x¯x+1−β2x+1214 1¯x2.(C19)The total O(αs)correction can be written as the sum of the three contribu-tions,virtual,soft,and hard corrections.We list each correction term for the scalar-pair production for comparison∆virtual=1βlog1+β∆soft =1βlog 1+ββA h (β)−11−β+12log 1+βλ2+(1+β2)Li 2 1−β1+βlog1+β4log 21+β3 ,(C23)A s (β)=1+β21−β−βlog 4ω21+β−log 2β1−β−11−β−π22log 1+β4ω2+(1+β2)2Li 21−β1+β−log21−β+11−β−π21−β2−4βlog β.(C25)The function A (β)in (3.17)is the sum of these threeA (β)=A v (β)+A s (β)+A h (β).(C26)Turning to the fermion-pair production,it is found that the dilogarithmic terms are exactly the same as for the scalar-pair production.For the vector partwefind∆V virtual=ℜe f V1+6βA v(β)+β(4−β2)1−β−2,(C27)∆V soft=1βlog1+ββA h(β)+9−2β2+β41−β+39−17β2βA v(β)+2+β21−β−2,(C30)∆A soft=1βlog1+ββA h(β)+11−β+1References[1]Particle Data Group,L.Montanet et al.,Phys.Rev.D50,1173(1994).[2]J.Schwinger,Particles,Sources,and Fields,Vol.II(Addison-Wesley,Mass.,1973),Chapter5-4.[3]M.Drees and K.Hikasa,Phys.Lett.B252,127(1990).[4]W.Beenakker,R.H¨o pker,and P.M.Zerwas,Phys.Lett.B349,463(1995).[5]A.Arhrib,M.Capdequi-Peyranere,and A.Djouadi,Phys.Rev.D52,1404(1995).[6]G.’t Hooft and M.Veltman,Nucl.Phys.B153,365(1979).[7]I.I.Bigi,V.S.Fadin,and V.Khoze,Nucl.Phys.B377,461(1992).[8]V.S.Fadin and 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order.Fig.2.Feynman diagrams for e+e−→ζ¯ζat O(αs).Fig.3.Feynman diagrams for e+e−→ζ¯ζg.Fig.4.Total O(αs)correction to the pair production cross section of scalar pair (solid);fermion pair via vector current(dash);fermion pair via axialcurrent(dotted).All curves are for particles in the fundamental colorrepresentation(C R=4/3)and the strong coupling constantαs=0.12. Fig.5.Three-jet fraction for scalar(solid)and fermion(dashed)at the high energy limit,forαs=0.12and C R=4/3.Fig.6.Three-jet fraction for scalar(solid),fermion-vector(dashed),and fermion-axial(dotted)forβ=0.4,0.6,0.8,and1.0,αs=0.12and C R=4/3. Fig.7.Same as Fig.6,but as a function of a jet-defining variable y c.-e ( )p e+e ( )p _e ζ ( )p _-ζ ( )p Fig. 1-e ( )p e+e ( )p _e p _p b)c)d)f)g)Fig. 2-e ( )pe +e ( )p _e p _p g (k)b)c)Fig. 31.01.21.41.61.82.00.00.20.40.60.8 1.0βσσ0Fig. 4scalarfermion (vector)fermion (axial)0.00010.0010.010.110.010.1x c Fig. 5scalar fermionσ3jσ0.00010.0010.010.1100.10.20.30.40.5x cFig. 6σ3j σscalarfermion (vector).fermion (axial)β=1.0β=0.8β=0.6β=0.40.00010.0010.010.1100.10.20.30.40.5y CFig. 7σ3j σscalarfermion (vector)fermion (axial)β=1.0β=0.8β=0.6β=0.4。