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SIMULATION OF T-VIOLATION IN THE

arXiv:hep-ph/9207271v1 30 Jul 1992UPR-0517TJuly, 1992FINAL-STATE-INTERACTIONSIMULATION OF T-VIOLATION IN THETOP-QUARK SEMILEPTONIC DECAYJiang LiuDepartment of Physics, University of Pennsylvania, Philadelphia, PA 19104ABSTRACTThe standard electroweak final-state interaction induces a false T-odd correla-tion in the top-quark semileptonic decay. The correlation parameter is calculatedin the standard model and found to be considerably larger than those that couldbe produced by genuine T-violation effects in a large class of theoretical models.PACS#: 11.30.Er, 13.30.Ce.1

1. IntroductionFinal-state interactions play an important role in the determination of CP andT violation.

A test for CP violation is to compare the partial decay rates of a par-ticle and its antiparticle. In this case final-state interactions are necessary since intheir absence the partial decay rates are equal from CPT invariance even if CP isviolated.

General formalism for calculating such partial rate differences based onCPT invariance and unitarity has recently been developed,1 and its applicationsto B meson decays (Ref. 1) and to t-quark decays2 have revealed some interest-ing relations between final-state interaction and CP violation observables in weakdecays.A test3 for T violation is to observe a “T-odd correlation”, such as those ofthe form ⃗σ · (⃗p1 × ⃗p2) where ⃗σ is a spin and ⃗p1 and ⃗p2 are momenta.

In contrastto the partial decay difference, a T-odd correlation can be produced by final-stateinteractions even if T invariance holds. Thus, to use such correlations as a test ofT violation the final-state-interaction effect must be negligible or calculable.This paper will be concerned with the t-quark semileptonic decay t →bW →bνℓ¯ℓin the standard model.

Copious production of t-quarks at future high-energycolliders such as the SSC and the LHC have aroused considerable interest in ex-ploring the origin of CP and T violation via t-quark interactions.4 In particular,a recent study5 of the possibility of using the T-odd correlation has shown thatit has a reasonable sensitivity to some non-standard sources of T Violation. Sincesuch correlations can be produced by standard model physics alone, it is timelyto undertake a computation of the final-state-interaction effect due entirely to thestandard electroweak interaction, which, up to the one-loop level, respects T andCP invariance in Cabibbo-allowed weak decays such as t →bW + →bνℓ¯ℓ.2

2. Final-State-Interaction EffectThe computation of final-state-interaction effects on the T-odd correlation haslong been of interest.

Early examples of the calculation involved nuclear β decay,6hyperon semileptonic decay,7 and K±,0ℓ3decays.8 The parameter of interest is thecoefficient of the T-odd correlation term in the decay spectrum, which in nuclearβ decay, for instance, has the following form in the leading approximationdΓdΩedΩνedEe∼1 + a⃗pe · ⃗pνeEeEνe+ ⃗σ ·hA ⃗peEe+ B ⃗pνeEνe+ D⃗pe × ⃗pνeEeEνei,(1)where ⃗σ is the polarization of the parent nucleus and ⃗pe(Ee) and ⃗pνe(Eνe) are theelectron and neutrino momentum (energy), respectively. In this example, the dom-inant contribution arises from electromagnetic final-state interaction.

The effectdepends, among other things, on the recoil of the decaying particle, and thus thesize of the T-odd correlation parameter D is of order D ∼αEe/M (ZαEe/M) inneutron (nuclear) β decay, where M is a nucleon mass. Since Ee is typically oforder 1 MeV , the recoil effect, which is characterized by the ratio Ee/M, is rathertiny.

Hence D is highly suppressed in neutron β decay with D typically of theorder of 10−5 −10−6. A considerably larger result (10−3 −10−4) can be obtainedin some nuclear β decays due to the enhancement Z ≫1 (Ref.

6). The typicalvalue of the T-odd correlation is between 10−3 and 10−4 in a neutral K0ℓ3 decay.The result in a charged K±ℓ3 decay is still smaller (10−5 −10−6), because there thefinal-state pion is neutral and the effect can only arise from two-loop graphs.In terms of weak-current interactions the t-quark semileptonic decay is analo-gous in many respects to nuclear β decay.

However, the disparity between mt andmb implies that the T-odd correlation in the decay t →bνℓ¯ℓdoes not have a recoil3

suppression. Indeed, compared to nuclear β decay, where the recoil effect is of order10−3, in the t semileptonic decay such effects are given by E¯e/mt, which is of orderunity.

As a consequence, we expect that the final-state-interaction contribution tothe T-odd correlation parameter is roughlyD(t →bW →bνe¯e) ∼α|Qd|E¯emt∼α9 ∼10−3,(2)where Qd = −1/3 is the b-quark charge, and we have taken E¯e/mt ∼1/3.In what follows we will concentrate on the decay t →bW →bνe¯e. Insofar asthe lepton mass can be ignored, our result holds for the other t-quark semileptonicdecays as well.A large mt implies that the decay t →bνe¯e proceeds dominantly through theW resonance.The smallness of the W width (ΓW /MW ≈0.026) then makesthe calculation of the leading final-state-interaction effect very simple.

Neglectingthe b-quark and lepton masses, the leading contributions are generated by graphsdisplayed in Fig. 1 withM(t →bνe¯e) = ig√22[¯uνe(pνe)γλLv¯e(p¯e)][¯ub(p′)Γλut(p)]k2 −M2W + iΓW MW,(3)where k = p −p′ is the momentum transfer carried by the W, L and R are thehelicity projection operators, and the effective vertex Γλ, which includes one-loopinteraction corrections from (Fig.

1b) and (Fig. 1c), can be parameterized asΓλ = F1(k2)γλL −iF2(k2)mtσλµkµR,(4)where σλµ = i2[γλ, γµ].

Terms of the form γλR and σλµkµL vanish in the limitmb = 0. Also, the kλ term drops out for me = mνe = 0.

While the form factor4

F1 = 1 + Oαπintroduces a correction to the weak interaction charge g, F2 givesan anomalous moment to the ¯btW vertex.In analogous to nuclear β decay one may define a T-odd correlation parameterD:dΓdΩ=g4(2π)5mtEνeE¯e|k2 −M2W + iΓW MW|2h1−k22mtEνe+D1−2E¯emt⃗σt· ⃗p¯e × ⃗pνeE¯eEνei+...(5)withD = m2t ImF2(M2W )(6a)evaluated at k2 = M2W. The ellipses in Eq.

(5) refer to the other terms of nointerest to us and dΩ= (d3⃗p¯e/2E¯e)(d3⃗pνe/2Eνe)(d3⃗p′/2p′0). In reaching (6a) wehave taken F1 = 1.The final-state interaction in nuclear β decay takes place between the daugh-ter nucleus and the electron.By contrast, the dominant effect in the decayt →bW + →b¯eνe arises from bW →bW rescattering.

By employing the uni-tarity formula given by Wolfenstein (Ref.1 ) one can show that the relevantinteractions are those which scatter a bW + state to other bW + states with differ-ent spin configurations. As a result, the T-odd correlation parameter is directlyproportional to the absorptive part of the form factor F2 which connects hadronstates with different helicities.

We find (the detail of the calculation is summarizedin the Appendix)ImF2(M2W ) = −αQd2m2t1 −12M2Wm2t+ α(1 + 2Qds2)8(m2t −M2W )h 1c2 −1s2I1 + 2s2I2i,(6b)5

where s2 = sin2 θW , c2 = cos2 θW andI1 =2 +h1 +2m2t M2Z(m2t −M2W )2ilnM2Zm2tM2Zm2t + (m2t −M2W)2,I2 =1 −M2Wm2th1 −12M2Wm2t+ 2M2Zm2t −M2W+ 3M2W M2Z(m2t −M2W)2i+M2Zm2t −M2Wh2 + 2M2Zm2t −M2W+ 3M2W M2Z(m2t −M2W)2ilnM2Zm2tM2Zm2t + (m2t −M2W )2. (6c)In Eq.

(6b) the first term comes from the photon graphs and the second from theZ. For a very heavy top the result is dominated by the Z exchange diagram andhas a logarithmic dependence on mt.

Asymptoticaly it approacheslimmt→∞D = α6h1 + 341 −2s23h 2c2 + 1c2 −1s2ln M2Zm2tii. (7)The numerical results for D from Eqs.

(6a) to (6c) are summarized9 in Table1 for mt between 100 GeV and 200 GeV . One sees that D is between 1 × 10−3and 5 × 10−3, as we expected from the simple dimensional argument Eq.

(2). Theresult shows a slow increase with larger values of mt in this region.The T-odd correlation may be reparameterized in terms of an asymmetry pa-rameter A, which is related to the difference of the decay W + →¯eνe occuring inthe opposite sides of the ⃗σt × ⃗p′ plane (Ref.

5)A = −3(m2t −M2W )4(m2t + 2M2W)mtMW Im(F1F ∗2 )|F1|2≈−3(m2t −M2W )4(m2t + 2M2W )mtMWImF ∗2 ,(9)where ImF ∗2 is given by Eqs. (6b) and (6c) with an additional over all minussign.

The results for A are summarized in the last column of Table 1. They varyfrom 1 × 10−4 to 1 × 10−3 for mt = 100 −200 GeV .

In comparison with the6

maximal-allowed T violation effect in the models considered in Ref. 5 in whichA < 5 × 10−5 ∼5 × 10−4, the standard model final-state interaction produces amuch larger false effect.It is difficult to calculate the T-odd parameter to an accuracy of ∼30%.

Themajor theoretical uncertainties of the present calculation come from neglectingQCD corrections, which introduce a sizable interference between the absorptivepart of F1 from electroweak interactions and the real part of F2 from QCD. Anorder of ∼(1 ∼10)% correction due to this effect alone is possible.

A still morecomplicated contribution arises from the interference between ImF2 calculatedabove and the real part of F1 due to QCD. Other uncertainties arise from neglecting(1) the WZ threshold effect (relevant if mt > MW + MZ + mb) and (2) all thebox-diagrams.

The contribution of the latter also depends on the angle between⃗p¯e and ⃗p′ in a rather complicated way. All of these contributions are suppressedby the ratio ΓW/MW , however.

The calculation of these next-leading terms wouldbe crutial should future experiments approach the precision of D ∼10−3.T-odd correlations of the form ⃗σ¯e ·(⃗pνe ×⃗p¯e), ⃗σb ·(⃗pνe ×⃗p¯e) and P- and CP-oddcorrelation of the form ⃗σt · (⃗σb × ⃗p′) are much more difficult to measure experimen-tally, and thus will not be considered in this paper.7

3. ConclusionWe have calculated the T-odd correlation ⃗σt·(⃗p¯e×⃗pνe) induced by the standardelectroweak final-state interactions in the decay t →bW + →b¯ℓνℓ, and found thatthe result has a logarithmic dependence on the t-quark mass and is dominatedby the bW →bW rescattering due to a Z exchange in the heavy top limit.

Formt in the range 100 GeV to 200 GeV the correlation parameter D defined in Eq. (5) is between 1 × 10−3 to 5 × 10−3, and the asymmetry parameter A given byEq.

(9) is between 1 × 10−4 to 1 × 10−3. It is shown that the standard modelphysics can simulate a false T-odd signal, with its magnitude exceeding genuineT-violation effects of a size that could possibly be produced in a large class oftheoretical models.

To get rid of this pure final-state interaction effect one mayconsider comparing the asymmetry parameter for both t →bW + and ¯t →¯bW −,as in the study of CP-violating parameters α + ¯α and β + ¯β in the Λ decays.10Acknowledgements: I wish to thank Paul Langacker and Lincoln Wolfenstein forvaluable discussions and comments. I would also like to thank Aspen Center forPhysics for hospitality during the completion of this work.This research wassupported in part by the U.S. Department of Energy under contract DE-ACO2-76-ERO-3071 and the SSC National Fellowship.8

FIGURE CAPTIONFig. 1.

Feynman graphs generating the dominant contributions to the T-oddcorrelation. The calculation is carried out in the Feynman-’t Hooft gauge.

φ is theHiggs-Goldstone-boson.TABLE CAPTIONTable 1. The result for the T-odd correlation in the t-quark semileptonic decay.The parameters D and A are defind in Eq.

(5) and Eq. (9), respectively.9

APPENDIXWe give some details of the calculation in this appendix. The technique isstandard11 except that we use the Minkowskian metric gλβ = diag(1, −1, −1, −1).The one- and two-point functions are defined asA(m) = −iµ(n−4)0ZdnK(2π)n1[K2 −m2 + iǫ],B(m1, m2; k) = −iµ(n−4)0ZdnK(2π)n1[K2 −m21 + iǫ][(K + k)2 −m22 + iǫ],(A.1)where ǫ →0+, and we use dimensional regularization to isolate the ultra-violetdivergences.

The only relevant three-point function isC0 = −iZd4K(2π)41[K2 −M2Z + iǫ][(K −k)2 −M2W + iǫ][(K + p′)2 −m2b + iǫ]. (A.2)We findImA(m) = 0,ImB(m1, m2; k) =116πk2qλ(k2, m21, m22)θ[k2 −(m1 + m2)2],ImC0 =116πqλ(m2t, M2W , m2b)lnM2Zm2tM2Zm2t + λ(m2t, M2W , m2b)θ[m2t −(MW + mb)2],(A.3)where λ(x, y, z) = x2 + y2 + z2 −2xy −2xz −2yz.

In evaluating ImC0 we haveput all the external lines on their mass-shell.Neglecting the b-quark and lepton masses, the final-state-interaction effect dueto a photon exchange isΓλ(γ) = −e2Qd4mtp′λR(a1 + b1),(A.4)10

where a1 and b1 are the coefficients defined in the following integrals:−iZd4K(2π)4KλK2[(K −k)2 −M2W ][(K + p′)2 −m2b] = a1kλ + a2p′λ,(A.5)−iµ(n−4)oZdnK(2π)nKλKβK2[(K −k)2 −M2W ][(K + p′)2 −m2b]= b1(kλp′β + kβp′λ) + b2gλβ + b3kλkβ + b4p′λp′β. (A.6)We finda1 = B(MW , 0; k) −B(MW , 0; p)m2t −M2W,b1 = −12A(MW) −M2W B(0, MW; p)m2t (m2t −M2W).

(A.7)It then follows from Eqs. (A.3), (A.4) and (A.7) that the absorptive part of Γλ(γ)isΓλabs(γ) = αQdmt1 −M2W2m2tp′λR.

(A.8)It can be written in a more conventional form by applying the Gordon identity[¯ub(p′)p′λRut(p)] = i2[¯ub(p′)σλµkµRut(p)] + ....(A.9)The result due to Z exchange isΓλ(Z) = −e2(1+2Qds2)mtp′λRh 1c2 −1s2(−a′1 +a′2 +C0)−2s2(a′1 +b′1)i, (A.10)where the coefficients a′1, a′2 and b′1 are defined analogously−iZd4K(2π)4Kλ[K2 −M2Z][(K −k)2 −M2W ][(K + p′)2 −m2b] = a′1kλ + a′2p′λ,−iµ(n−4)0ZdnK(2π)nKλKβ[K2 −M2Z][(K −k)2 −M2W ][(K + p′)2 −m2b]= b′1(kλp′β + kβp′λ) + b′2gλβ + b′3kλkβ + b′4p′λp′β. (A.11)11

We finda′1 =1m2t −M2WhB(MZ, MW ; k) −B(MW , 0; p) −M2ZC0i,(A.12)a′2 = −1m2t −M2WhB(MZ, 0; p′) −B(MW , 0; p) −M2ZC0i−2M2W(m2t −M2W )2hB(MZ, MW; k) −B(MW , 0; p) −M2ZC0i,(A.13)b′1 = −1(m2t −M2W )2h121 −M2Wm2thA(MW ) −M2W B(MW, 0; p)i+ 12hA(MZ) −M2ZB(MW , MZ; k)i+ 2M2ZhB(MW, 0; p) −B(MZ, 0; p′)i−3 M2WM2Zm2t −M2WhB(MW , MZ; k) −B(MW , 0; p)i+h2 + 3M2Wm2t −M2W+ m2t −M2WM2ZiM4ZC0i. (A.14)One can check that in the limit MZ = 0 a1 and a′1 become identical and so do b1and b′1.

The logarithmic dependence on mt in the limit mt →∞arises becauseΓλ(Z) has a term which is directly proportional to C0 (see (A.10)).It then follows that the absorptive part of Γλ(Z) isΓλabs(Z) = −α(1 + 2Qds2)4(m2t −M2W) mtp′λRh 1c2 −1s2I1 + 2s2I2i,(A.15)where I1,2 are given by Eq. (6c).

Adding (A.8) and (A.15) we obtain the resultsgiven by Eqs. (6a) to (6c) of the text.12

mt GeV :DA100:1.0 × 10−31.0 × 10−4110:1.4 × 10−31.7 × 10−4120:1.8 × 10−32.6 × 10−4130:2.2 × 10−33.6 × 10−4140:2.7 × 10−34.6 × 10−4150:3.1 × 10−35.7 × 10−4160:3.6 × 10−36.7 × 10−4170:4.0 × 10−37.6 × 10−4180:4.4 × 10−38.5 × 10−4190:4.8 × 10−39.2 × 10−4200:5.2 × 10−31.0 × 10−3Table 113

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