Angular distribution coefficients of Z(W) boson produced in e + e − collisions at s √ =240 GeV
AAngular distribution coefficients of Z(W) boson produced in e + e − collisions at √ s = 240 GeV Yu-Dong Wang Jian-Xiong Wang Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, ChinaSchool of Physics, University of Chinese Academy of Sciences, Beijing 100049, China
Abstract:
At designed CEPC, similar to hadron collider, the angular distribution coefficients of decay lepton pairfrom produced Z(W) boson in e + e − collisions are predicted. Their dependence on cosθ Z ( cosθ W ) are presented in fourdifferent polarization frame. Furthermore, the value of angular coefficients in different bins of cosθ Z are presentedin the C-S frame. In comparison to the case at hadron collider, better accurate measurement for Z ( W ) is expectedsince there exists less background and W could be reconstructed from it’s leptonic decay channel. This work supplya way to precise test the electroweak production mechanism or some effect induced from new physics in the futuremeasurements at the CEPC. Key words:
Drell-Yan process, Z boson, CEPC, angular distribution coefficients.
PACS:
The Drell-Yan process, firstly proposed in ref.[1],which study the angular distribution of lepton pair fromvector boson decay which produced in the hadron-hadroncollisions. It is predicted in the simplest case that thedifferential cross section is proportional to 1 + cos θ atlowest order when the vector boson is virtual photon.With the emission and absorption of partons with largetransverse momentum, there is a factor of 2 enhancementto the total cross section[2], and the angular distributionbecome more general[3–5]. Through measurements ofthe angular distribution coefficients of final-state lepton,many theoretical works such as the violation of Lam-Tung relation[4], the forward-backward asymmetry oflepton pair productions[7] were studied.The Drell-Yan type processes provide a powerfulmethod to understand the production mechanism ofgauge boson and explore the new physics. In 1983, W and Z boson are discovered [8, 9], and some followingmeasurements was found to be consistent with the pre-dictions of the V-A Standard Model[10–12]. The mea-surement of angular distribution coefficients of leptonpair in Z/γ ∗ production was first reported for p ¯ p colli-sions at 1.96 TeV by CDF Collaboration[13], and the re-sults were found in good agreement with the predictionsof QCD fixed-order perturbation theory. The measure-ments were also done in the CMS and ATLAS collabora-tions at √ s =8 TeV[14–18]. Meanwhile, many theoreticalwork on predictions of the inclusive Z boson productionwhich involving emission of partons of large transverse momenta were done [19, 20].The circular electron positron collider(CEPC) is pro-posed to build in future. It is designed that the center-of-mass (CM) energy would be has maximum energy 240GeV and a higher luminosity than the linear collider [21],there will be less background compared with hadron col-lider. The CEPC project aims to precise test the prop-erties of the higgs, Z and W boson and search for newphysics. Compared to the process at hadron collider, thesimilar one e + + e − → Z/γ ∗ ( W ) + X → l + l − ( l − ¯ ν l ) + X isof interest and deserved to study.In this paper we study the angular distribution coef-ficients of Z boson inclusive production. In comparisonto Z boson hadroproduction, the energy of Z boson isfixed at Leading-order (LO) at e + e − collider. Thus fordetailed study, we present these angular distribution co-efficients dependence of cosθ Z (cos θ W ), θ Z ( θ W ) is polarangle of Z(W) boson in laboratory frame.The angular differential cross section can be writtenas dσdcosθ Z d Ω = (cid:88) λλ (cid:48) dσ λλ (cid:48) dcosθ Z f λλ (cid:48) ( θ, ϕ ) (1)Where θ and ϕ are polar and azimuthal angles of thelepton in Z(W) rest frame, and d Ω = dcosθdϕ . dσ λλ (cid:48) and f λλ (cid:48) are production density matrix of process e + + e − → Z/γ ∗ + X and decay density matrix of Z/γ ∗ → l + + l − respectively.The content of this paper is divided into followingparts. In Sec.2 we present the general expression ofthe lepton angular distribution of this process, and also
1) E-mail: [email protected]) E-mail: [email protected] a r X i v : . [ h e p - ph ] D ec epresent the angular coefficients by the gauge bosonproduction density matrix elements. In Sec.3 we cal-culate angular distribution coefficients numerically fortotal and the differential cross section in different polar-ization frame. In Sec.4 we draw the figures of angularcoefficients of Z( W − ) production dependence of cosθ Z ( cosθ W ). Also the calculation of coefficients of Z produc-tion processes at different bins of cosθ Z are done. Thenthe summary and conclusion is given at last Sec. For simplicity, following discussion focus on the Z bo-son production and the situation is same for W boson.In the process that e + ( p ) + e − ( p ) → Z ( p Z ) + X ( p X ) → l + ( k )+ l − ( k )+ X ( p X )( l is µ or e ), there are two planesneed to define which named production plane and decayplane. In the lab frame, the first one is formed by beamdirection and (cid:126)p Z , between which the angle is θ Z . Theother one is formed by (cid:126)p Z and (cid:126)k , the corresponding an-gle in Z boson rest frame is θ . Finally the angle betweenthe production and decay plane is ϕ , which is invariantunder lorentz transformation from the lab frame to theZ-rest frame.The dilepton angular distribution is defined in the Zboson rest frame, and we use this frame for all follow-ing discussion. The invariant mass window of Z bosonis chosen around 91.19 GeV, the contribution from γ ∗ issuppressed by a large factor and lead to a error less than1%. The momentum of Z boson, l − and l + are expressedas p Z = ( E, , , ,k = E , sinθcosϕ, sinθsinϕ, cosθ ) ,k = E , − sinθcosϕ, − sinθsinϕ, − cosθ ) , (2)Where the mass of fermion (mass of e, µ, u, ... ) is set tozero approximately. There are four commonly used po-larization frames[23] correspond different choice Z -axis,which are recoil(helicity) frame( (cid:126)Z = − (cid:126)p + (cid:126)p | (cid:126)p + (cid:126)p | ), Gottfried-Jackson frame( (cid:126)Z = (cid:126)p | (cid:126)p | ), target frame( (cid:126)Z = − (cid:126)p | (cid:126)p | ) and theCollins-Soper(C-S)[22]( (cid:126)Z ∝ (cid:126)p | (cid:126)p | + (cid:126)p | (cid:126)p | ). The three-vector (cid:126)p and (cid:126)p used here refer to the Z boson-rest frame. Thelast frame was frequently used in the measurements inhadron collision. For example, the polarization vector inhelicity frame are expressed as (cid:15) ± = (0 , ∓ √ , − i √ , , (cid:15) = (0 , , , . (3)The amplitude of each channel( X i ) in inclusive Z bo-son production can be written as M i = M µe + e − → ZX i − i ( g µν − p zµ p zν m z + im z Γ ) p z − m z − im z Γ M νZ → l + l − = (cid:88) λ M µe + e − → ZX i (cid:15) λµ (cid:15) ∗ λν m z Γ M νZ → l + l − = (cid:88) λ a λ ( X i ) 1 m z Γ b λ (4)Where Γ is the decay width of Z boson, a λ ( X i ) = M µe + + e − → Z + X i (cid:15) λµ and b λ = (cid:15) ∗ λν M νZ → l + l − . Both a λ ( X i )and b λ are Lorentz invariant. Therefore they can be cal-culated in different frame. a λ ( X i ) and b λ are calculatedin the lab frame and the Z boson rest frame respectively.The production and decay density matrix are defined as σ λλ (cid:48) = (cid:88) i a λ ( X i ) a ∗ λ (cid:48) ( X i ) ,D λλ (cid:48) = (cid:88) s ,s b λ b ∗ λ (cid:48) ,b λ =¯ µ ( k , s )( ig v γ µ + ig a γ µ γ ) ν ( k , s ) (cid:15) ∗ λ , (5)Where the decay density matrix D λλ (cid:48) is easy to obtainedand the production matrix σ λλ (cid:48) is discussed in appendixA. By applying D λλ (cid:48) and σ λλ (cid:48) , the differential cross sec-tion is expressed as [4] dσd Ω ∝ ( g v + g a ) Sp (1 + λ θ cos θ + λ ϕ sin θcos (2 ϕ ) + λ θϕ sin (2 θ ) cosϕ + λ ⊥ ϕ sin θsin (2 ϕ ) + λ ⊥ θϕ sin (2 θ ) sinϕ )+ α θ cosθ + α θϕ sinθcosϕ + α ⊥ θϕ sinθsinϕ ) (6)Where θ and ϕ are the polar and azimuthal angles ofdilepton in the rest frame of Z boson. The coefficients ofeach term is given as follows, S = σ ++ + σ −− + 2 σ ,λ θ = σ ++ + σ −− − σ S , λ ϕ = 2 Re ( σ − + ) S ,λ ⊥ ϕ = − Im ( σ − + ) S , λ θϕ = √ Re ( σ +0 − σ − ) S ,λ ⊥ θϕ = √ Im ( σ +0 + σ − ) S , α θ = − A l ( σ ++ − σ −− ) S ,α θϕ = 2 √ A l Re ( σ +0 + σ − ) S , α ⊥ θϕ = 2 √ A l Im ( σ +0 − σ − ) S . (7)The asymmetry parameter of fermion f is A f = g v g a g v + g a ,given in PDG[24], its value are 0 . ± . . ± .
015 for electron and muon respectively. In thispaper, A l = 0 . α θ , α θϕ and α ⊥ θϕ comparedto the case in which J/ Ψ production decay to lepton2air[25] due to the presence of parity-violation coupling g a . When g a = 0, these terms disappear. From above ex-pressions, terms of α θ , α θϕ and α ⊥ θϕ which proportionalto sin ϕ or sin2 ϕ come from contributions of imaginarypart of density matrix elements.As presented in appendix A, there are relationsEq.(15) for the real part in σ λλ (cid:48) which are from thecoupling g v . According to Eq.(7), this part does notcontribute to the value of α θ , α θϕ . Meanwhile g a g v = − √ − A f A f = 0 . g a is small. It is expected that the valueof α θ and α θϕ are much smaller than other coefficients.In particular, the following relations are obtained, − ≤ λ θ ≤ , − ≤ λ ϕ ≤ , − √ ≤ λ θϕ ≤ √ , − A l ≤ α θ ≤ A l , −√ A l ≤ α θϕ ≤ √ A l , − ≤ λ ⊥ ϕ ≤ , − √ ≤ λ ⊥ θϕ ≤ √ , −√ A l ≤ α ⊥ θϕ ≤ √ A l , (8) Fig. 1. The differential cross section of Z( W − )boson production. At leading-order(LO), the production of Z bosoncome from following three processes e + + e − → Z + γ → l + + l − + γ,e + + e − → Z + Z → l + + l − + Z,e + + e − → Z + H → l + + l − + H. (9)The production density matrices of Z boson in theseprocesses are calculated by using package FDC[26], andobtain the value of angular distribution coefficients.In these three processes, the Zγ production have alarger contribution on the cross section compared theothers. The cross section are 46.64 pb, 0.96 pb and 0.20pb for Zγ , ZZ and ZH production channels respective.Summed up all above production channels, the value ofdilepton angular distribution coefficients at Z pole areshown in the table 1. The total cross section refer to theinclusive Z boson production e + + e − → Z + X .The cross section is much smaller compared with theDrell-Yan type process in hadron collisions[27] at LO,to obtain accurate measurements, larger integrated lu-minosity is in need. From the CEPC design report [21],at the CM energy √ s = 240 GeV, the luminosity of theCEPC is about 3 × cm − s − . The integrated valueis about 0 . ab − / year(it operate about 8 months eachyear), the total number of Z boson produced is about3 . × each year. It is expected to run 7 years at thisenergy, according to the fraction of Z decay modes[24],the total events of electron and muon pair should be1 . × . Also, the events of jets if we use the jets toreconstruct Z boson is about 1 . × . In ATLAS[18],the total events of lepton pair is about 1 . × , it isclosed to the value at the CEPC. Moreover, there is lessbackground at the CEPC, it is expected a good accu-racy in the future CEPC experiment to measure angulardistribution coefficients of inclusive Z boson production. Table 1. The value of angular distribution coefficients and total cross section for Z boson productions at √ s = 240GeV in the Recoil frame. cosθ Z λ θ λ ϕ λ θϕ α θ α θϕ λ ⊥ ϕ λ ⊥ θϕ α ⊥ θϕ total cross section(pb) cosθ Z > cosθ Z < Table 2. The value of angular distribution coefficients and total cross section for W − boson productions at √ s =240 GeV in Recoil frame. cosθ W λ θ λ ϕ λ θϕ α θ α θϕ λ ⊥ ϕ λ ⊥ θϕ α ⊥ θϕ total cross section(pb) cosθ W > cosθ W < λ ⊥ ϕ , λ ⊥ θϕ and α ⊥ θϕ are all 0. All theseterm are proportional of imaginary part of the densitymatrix elements in Eq.(7). In the calculation, the imag-inary part of density matrix is zero at LO, and theseterms will not be discussed hereafter. However, for Zboson production at hadron collider[6], these coefficientsare nonzero at Next-to-Leading order (NLO) correctionsin QCD.From table 1, the value of λ θϕ and α θ are 0 whentake full range of cosθ Z , since they are antisymmetricin value for cosθ Z > cosθ Z <
0. Usually, there isa Lam-Tung[3] relation for the coefficients λ θ and λ ϕ . Itis presented in the Drell-Yan process that 1 − λ θ = 4 λ ϕ with the vector boson being of gauge invariance condi-tion (i.e. virtual photon) for full final state phase space(except the dilepton) integration. For the electroweakinteraction at this situation we found that λ θ + 4 λ ϕ isabout equal to 0.97, of course, it does not obey the re-lation as it should be. From the calculation, the valueof off-diagonal density matrix elements which defined inEq.(5) are about 100 times smaller compare to the diag-onal matrix elements. The conclusion is obtained fromthe expression of coefficients Eq.(7) that the value of λ θ is far larger than the others, just as shown in the table 1.The process that e + + e − → W − + X → u + ¯ d + X onW boson pole is also calculated(In the W − productionprocess of LO, X can only be W + ). For W + productionprocess, the results should be symmetric with W − . In ta-ble 2, the value of coefficient for cosθ Z < cosθ Z > cosθ Z > W − boson produced at the CEPC peryear at the CM energy 240 GeV is about 8 . × , thisoffer a high accuracy for W boson detection. However,it is need to rebuild jets then to reconstruct W boson.Comparing to hadron collider where there are a lot ofsource of jets due to the complicated background, at theCEPC there is a advantage to rebuild the jets from W bo-son decay with less background. In the W decay modesfrom PDG, the hadrons fraction is about 67.4% and thelepton fraction except τ is about 21.3%. By four mo-mentum conversation, the momentum of anti-neutrinofrom W − decay could be obtained and then W − is re-constructed as: p ν = ( p + p − p X − p e − ) = 0 , ( p ¯ ν + p e − ) = ( p + p − p X ) = m W (10)where p X is the sum of momentum of all the final stateparticles except the lepton. If p ν = 0 then it is terrifiedthat this is the momentum of anti-neutrino and can beused to reconstruct the W − boson too. Finally, the total proportion of W − which can be reconstructed throughlepton and jets are about 67.4%+67.4% × .
3% = 81 . We present the differential cross section and angulardistribution coefficients dependence of the θ Z , which isthe polar angle of Z boson in the lab frame. In the ex-periment, usually the measurements are done in limitedregion. The following bins in cosθ Z which have enoughevents are selected in the table 3 to test the value ofangular distribution coefficients. Since the differentialcross section is symmetric in cosθ Z , the bins in range − < cosθ Z < < cosθ Z <
1. For the thedetector at the CEPC, the conical space with an openingangle is about 6.78 ∼ cosθ Z is about 0.99, i.e. particles in therange | cosθ Z | < .
99 can be measured well.
Fig. 2. Angular distribution coefficients of the in-clusive Z boson production dependence on cosθ Z .Table 3. The cross section for inclusive Z boson production at different bins of cosθ Z and corresponding number f events estimated for the designed CEPC experiments. cos θ Z − .
45 0 . − . . − . . − .
94 0 . − .
99 0 . − . σ ( pb ) 0.83 0.71 1.20 0.54 1.80 20.00N(1 year) 6.6 × × × × × × N(7 year) 4.7 × × × × × × In the figure 2, we show the angular coefficients de-pendence of cosθ Z in four different polarization frame,which are recoil frame, Collins-Soper frame, target frameand Gottfried-Jackson frame. In the first two frame, theangular coefficients have apparent symmetry. For recoilframe, λ θ , λ ϕ , α θϕ are even and λ θϕ , α θ are odd under cosθ Z ↔ − cosθ Z . For Collins-Soper frame, λ θ , λ ϕ , α θ are even and λ θϕ , α θϕ are odd under cosθ Z ↔ − cosθ Z .Since these symmetry, as seen in the table 1 that thetotal value of λ θϕ and α θ are 0. The coefficients α θ and α θϕ which come from the parity-violation coupling part,are much smaller than value of λ θ , λ ϕ and λ θϕ just asthe expectation discussed in Sec.2. Fig. 3. Angular distribution coefficients of theinclusive W − boson production dependence on cosθ W . Then in the figure 3, the same plots for W − produc- tions is given. For more discussion about W − produc-tions, it could be refer to the process in hadron colli-sion [28]. Table 4. The angular distribution coefficients forZ boson production at each bin in cosθ Z at C-Sframe. cosθ Z λ θ λ ϕ λ θϕ α θ α θϕ -1 . − -0 .
99 0.996 0 0.015 -0.028 0-0 . − -0 .
94 0.843 0.017 0.196 -0.054 0.012-0 . − -0 . . − -0 . . − -0 .
45 0.335 0.110 0.223 -0.040 0.019-0 . − − .
45 0.216 0.135 -0.100 -0.036 -0.0090 . − . . − . . − .
94 0.723 0.024 -0.228 -0.050 -0.0150 . − .
99 0.843 0.017 0.196 -0.054 0.0120 . − . Table 5. The angular distribution coefficients for W − boson production at each bin in cosθ Z at C-Sframe. cosθ W λ θ λ ϕ λ θϕ α θ α θϕ − .
454 0.328 -0.105 0.295 0.652 -0.7760 . − .
707 0.635 -0.160 0.209 1.288 -0.5970 . − .
891 0.772 -0.185 0.019 1.598 -0.2260 . − .
987 0.688 -0.171 -0.180 1.592 0.1360 . − We present the detailed definition for the theoreticalcalculation and experimental measurement on the leptonangular distribution coefficients of inclusive Z(W) bosonproduction needed for the designed CEPC. The generalexpression of the cosθ dependence for lepton angular dis-tribution coefficients in the Z boson rest frame are rep-resented by the production density matrix elements of e + + e − → Z + X , and their range is given.From the numerical results, it is clear that the eventnumber of lepton pair estimated at the CEPC is at sameorder of magnitude compared to that in the ATLAS. Thebetter accurate measurements is expected since there ex-ists less background. In comparison to case at hadroncollider, the measurement for W is of a advantage thatthe momentum of the missing anti-neutrino from W − W − is reconstructed.The two jets decay channels of Z ( W ) can also be mea-sured with less background. The angular distributioncoefficients of Z( W − ) boson production dependence of cosθ Z ( cosθ W ) is calculated in the four different polariza-tion frame. Furthermore, the value of angular coefficientsin different bins of cosθ Z are presented in the C-S frame.The calculation and results in this paper supply a wayto precise test the electroweak production mechanism orsome effect induced from new physics in the future mea-surements at the CEPC.There should be study on this subject to includeMonte Carlo simulation with detector and background,to include NLO electroweak correction for the productionand Z ( W ) boson decay, NLO QCD correction to Z ( W )boson decay, the correction to narrow width approxima-tion and initial-state-radiation effect. all these parts isout of the scope of this work and should be addressed atfuture study. We thank Dr. Bin Gong for the discussions. Thiswork was supported by the National Natural ScienceFoundation of China with Grant No. 11475183.
A The relations of the density matrix el-ements
The production density matrix of the vector boson iswritten as σ λλ (cid:48) =( M µ (cid:15) µλ )( M ν (cid:15) νλ ) ∗ , (11)where (cid:15) λ is the polarization of the vector boson and λ = + , , − , which is defined by (cid:15) = (cid:15) z ,(cid:15) ± = 1 √ ∓ (cid:15) x − i(cid:15) y ) , (12)By the definition of Eq.(12) and rewrite M µ as M µ = M µ + iM µ that M µ and M µ is real(for the situationwhen there is no weak interaction in M µ , M = 0) . σ ++ =( M µ (cid:15) µ + )( − M ∗ ν (cid:15) ν − )= 12 [( M µ M ν + M µ M ν )( (cid:15) µx (cid:15) νx + (cid:15) µy (cid:15) νy ) − ( M µ M ν − M µ M ν )( (cid:15) µy (cid:15) νx − (cid:15) µx (cid:15) νy )] ,σ −− =( M µ (cid:15) µ + )( − M ∗ ν (cid:15) ν − )= 12 [( M µ M ν + M µ M ν )( (cid:15) µx (cid:15) νx + (cid:15) µy (cid:15) νy )+( M µ M ν − M µ M ν )( (cid:15) µy (cid:15) νx − (cid:15) µx (cid:15) νy )] ,σ + − =( M µ (cid:15) µ + )( − M ∗ ν (cid:15) ν + )= −
12 ( M µ M ν + M µ M ν )[( (cid:15) µx (cid:15) νx − (cid:15) µy (cid:15) νy )+ i ( (cid:15) µx (cid:15) νy + (cid:15) µy (cid:15) νx )] ,σ − + =( M µ (cid:15) µ − )( − M ∗ ν (cid:15) ν − )= −
12 ( M µ M ν + M µ M ν )[( (cid:15) µx (cid:15) νx − (cid:15) µy (cid:15) νy ) − i ( (cid:15) µx (cid:15) νy + (cid:15) µy (cid:15) νx )] ,σ =( M µ (cid:15) µ )( − M ∗ ν (cid:15) ν − )= − √ (cid:15) µz { [( M µ M ν + M µ M ν ) (cid:15) νx +( M µ M ν − M µ M ν ) (cid:15) νy ] − i [( M µ M ν + M µ M ν ) (cid:15) νy − ( M µ M ν − M µ M ν ) (cid:15) νx ] } ,σ +0 =( M µ (cid:15) µ + )( M ∗ ν (cid:15) ν )= − √ (cid:15) µz { [( M µ M ν + M µ M ν ) (cid:15) νx +( M µ M ν − M µ M ν ) (cid:15) νy ]+ i [( M µ M ν + M µ M ν ) (cid:15) νy − ( M µ M ν − M µ M ν ) (cid:15) νx ] } ,σ − =( M µ (cid:15) µ )( − M ∗ ν (cid:15) ν + )= 1 √ (cid:15) µz { [( M µ M ν + M µ M ν ) (cid:15) νx − ( M µ M ν − M µ M ν ) (cid:15) νy ]+ i [( M µ M ν + M µ M ν ) (cid:15) νy +( M µ M ν − M µ M ν ) (cid:15) νx ] } ,σ − =( M µ (cid:15) µ − )( M ∗ ν (cid:15) ν )= 1 √ (cid:15) µz { [( M µ M ν + M µ M ν ) (cid:15) νx − ( M µ M ν − M µ M ν ) (cid:15) νy ] − i [( M µ M ν + M µ M ν ) (cid:15) νy +( M µ M ν − M µ M ν ) (cid:15) νx ] } . (13) From above calculation, we see that σ ++ is not equalto σ −− , unless the situation that M µ is real. Besides, weobtain following relations, σ + − =( σ − + ) ∗ ,σ +0 =( σ ) ∗ ,σ − =( σ − ) ∗ , (14)If M µ is real, from Eq.(14) we could obtain relationsas σ ++ = σ −− ,σ + − =( σ − + ) ∗ ,σ +0 =( σ ) ∗ = − σ − = − ( σ − ) ∗ , (15) References
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