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QCD CORRECTIONS TO THE CHARGED-HIGGS-BOSON

arXiv:hep-ph/9205245v1 29 May 1992UPR-0508TUM-TH-92-13May 1992QCD CORRECTIONS TO THE CHARGED-HIGGS-BOSONDECAY OF A HEAVY TOP QUARKJiang LiuDepartment of Physics, University of Pennsylvania, Philadelphia, PA 19104York-Peng YaoRandall Laboratory of Physics, University of Michigan, Ann Arbor, MI 48109ABSTRACTIt is shown that up to an over all scale the lowest-order QCD corrections tot →H+b and to t →W +b are the same in the heavy top limit. Asymptotically,they are given by −4αs3π [π23 −54], resulting in a reduction in the decay rate by about9%, rather than 6% reported previously in the literature.

This is verified explicitlyby an analytic calculation.The application of the equivalence theorem to thisprocess is also discussed.PACS# 12.38.Bx1

1. IntroductionIf the top quark t decays according to the standard model, the CDF experimenthas set a lower limit1 of 91 GeV .

A stringent constraint can also be obtainedfrom high-precision measurements: the result of a global fit to all available dataconcluded2 mt = 149+26−31 GeV . With such a heavy mass, the discovery of the topquark may result in a rich harvest of physics results,3 and the study of top decay mayeven provide a window to some new physics.

One particularly interesting examplewould be the decay t →H+b if kinematically allowed, where H+ is a charged-Higgs-boson that occurs when more than one (non-singlet) Higgs representationare included in the theory. H+ must exist in a supersymmetric model, and in manyothers.4 In this paper we wish to study the lowest-order QCD corrections to thisdecay mode.The lowest-order QCD corrections to the decay t →H+b have been calculatedbefore.5 However, the result of that calculation is erroneous.

This, for reasons tobe explained below, can be most easily seen by examining its asymptotic behavior.Given the potential interest of observing the decay at high-energy colliders, it isnecessary to have a more careful calculation of its order O(αs) QCD corrections.We find that for a heavy top the QCD corrections introduce a reduction in thedecay rate by about 9%, which differs substantially from the earlier result of 6%of Ref. 5.The rest of this paper is orgainized as follows.

In the next section we discuss thelowest-order QCD corrections to the decay t →bH+ in the heavy top quark limit.It is shown that up to an over all scale the result is the same as for t →bW +.The lowest-order QCD corrections to the decay t →bW + have been calculatedindependently by several groups,6−10 and all agree with each other.Thus, by2

employing the equivalence theorem11 the heavy-top-limit result for t →bH+ canbe obtained from the existing result of Refs. 6-10.

The justification for applyingthe equivalence theorem to this particular calculation is also discussed.In section 3 we provide an explicit calculation for arbitrary mt and mH+,where mt and mH+ are the masses of the top quark and the charged-Higgs-bosonrespectively, but taking mb = 0 for simplicity.The result shows the expectedasymptotic behavior. Our conclusion is given in section 4, and a few technicaldetails are summarized in the Appendix.2.

Asymptotic ResultThe lowest-order QCD corrections to t →bH+ and to t →bW + are relatedfor the following reasons.Consider the heavy top quark limit.The tree-levelinteraction Lagrangian for the decay t →bH+ is given byLH+ = −η mtmH+¯bRtH−+ h.c.,(2.1)where η is a dimensionless constant determined by the specific theoretical model,and R = 12(1 + γ5). (2.1) is a simplification of the interactions of a large class oftheoretical models by neglecting terms directly proportional to mb.

The importantfeature here is that to the lowest-order interactions η, mH+ and the Higgs fieldH+ do not receive QCD corrections, because to this order the gluons only interactwith the quarks.This feature is also shared by the interaction LagrangianLφ+ = g√2 mtMW¯bRtφ−+ h.c.(2.2)for the Higgs goldstone boson φ+ in the limit mt ≫mb. As a consequence, up3

to an over all scale determined by their interaction strength difference, the lowest-order QCD corrections to t →bH+ are the same as to t →bφ+ if the calculationof the latter is carried out (to be specific) in the Feynman-’t Hooft gauge and MWis replaced by mH+. In addition, the amplitude for t →bφ+ is related to that fort →bW + by a Ward identity, which in the Feynman-’t Hooft gauge isMWA(t →bφ+) = kµAµ(t →bW +),(2.3)where A(t →bφ+) and ǫµ(k)Aµ(t →bW +) are the amplitudes for t →bφ+ andt →bW +, respectively, and ǫµ(k) is the W polarization vector.

Thus, knowingthe QCD corrections to the Green’s function Aµ for the decay t →bW + oneimmediately obtains the result A for t →bφ+ from (2.3).However, (2.3) does not necessarily imply that such relations hold also for thedecay rate. In fact, in the absence of CP violation the one-loop QCD correctedinteraction LagrangianLeff = ¯b(p′)Γµt(p)W µ(k)(2.4)with an on-shell b- and t-quark and an arbitrary W has three independent formfactors which may be parameterized as F1, F2 and F3Γµ = g√2nF1(k2)γµL −Ncαs2πhF2(k2)iσµνkν + F3(k2)kµimtRo,(2.5)where Nc = 4/3 and σµν =i2[γµ, γν].

The F3 term in Aµ(t →bW +) providesa nonzero contribution to A(t →bφ+) via (2.3) but not to the t →bW + decayrate because ǫµkµ = 0. By contrast, the anomalous moment term F2 contributesto t →bW + but not to A(t →bφ+) because σµνkµkν = 0.

Thus, the complete4

lowest-order QCD corrections to the decay rate for t →bW + are not the same asto that for t →bφ+.Nevertheless, to the leading order in mt the aforementioned difference disap-pears in the limit mt →∞. This follows because mt is the only heavy scale in ques-tion, and on dimensional grounds one has from (2.5) that limmt→∞F2,3/F1 = m−2t .Indeed, explicit calculations show that both F2 and F3 vanish to the leading orderin mt.

F1 and F2 have been given explicitly in Refs. 7, 8F1(M2W ) = 1 −Ncαs2π∆0,F2(M2W ) = −12M2Wln1 −M2Wm2t,(2.6)where∆0 =2 +32 + ln µ2M2Wlnmbmtm2t −M2W+ 12ln µ2m2t+ ln µ2m2b+ 12ln2M2Wm2t −M2W+ ln2m2tm2t −M2W−14ln2 m2tM2W+ ln2 m2bM2W+ SpM2Wm2t,(2.7)µ →0 is a fictitious gluon mass and Sp(x) =R 10 dy ln y/(y −x−1) is the Spencefunction.

These results are valid when mbMW /(m2t −M2W) ≪1. F3 has not beengiven explicitly before.

From a straightforward calculation we findF3(M2W) =1M2Wh m2tM2W−32lnm2tm2t −M2W−1i. (2.8)From (2.6) and (2.8) one seeslimmt→∞F2(M2W) =limmt→∞F3(M2W ) = 0,(2.9)in accordance with the dimensional argument.

As a result, in this limit the ratesfor t →bW + and t →bφ+ plus their QCD corrections are in fact the same, andthe former can be calculated from the equivalence theorem.5

The result for t →bW + is known (Refs. 6 - 10 )Γ(t →bW +) = Γ0(t →bW +)n1 −NcαsπhSpM2Wm2t−Sp1 −M2Wm2t+ π22io−GF m3t8√2π Ncαsπn1 −M2Wm2t−2M4Wm4tM2Wm2tln M2Wm2t+ 121 −M2Wm2t25 + 4M2Wm2tln1 −M2Wm2t−141 −M2Wm2t5 + 9M2Wm2t−6M4Wm4to,(2.10)where Γ0(t →bW +) = GF (m2t −M2W )2(1 + 2M2W /m2t)/8√2πmt is the tree-levelrate.

In (2.10) the contribution from a virtual gluon exchange isΓ(t →bW +)virt = Γ0(t →bW +)n1−Ncαsπh∆0−321+2M2Wm2t−1lnm2tm2t −M2Wio,(2.11)where the ∆0 term is due to F1, and the last term is due to the anomalous momentF2. The real gluon emission contribution to (2.10) isΓ(t →bW +g)real = GFM2W8√2πmtαsπNcn(m2t −M2W)2M2W1 + 2M2Wm2th∆0 −ln1 −M2Wm2t+ Sp1 −M2Wm2t−SpM2Wm2t−π22i+ M2W m2tM2W−2M2Wm2t−1ln m2tM2W+ (m2t −M2W )2712 + 54m2tM2W−32M2Wm2to.

(2.12)The condition for the validity of (2.12) is mbmt/(m2t −M2W ) ≪1.Turning back to t →bH+, the tree-level decay rate from (2.1) isΓ0(t →bH+) = η232π m2tm2H+−12mH+mt2mt. (2.13)6

We already know that up to an over all scale its lowest-order QCD correction is thesame as for t →bφ+ (with the exchange of MW by mH+), and the latter is identicalto t →bW + in the heavy top limit. In fact, one can show that such relations alsohold for the virtual- and real-gluon emission contributions seperately.

From (2.10),we havelimmt→∞Γ(t →bW +)Γ0(t →bW +) = 1 −Ncαsππ23 −54,(2.14)where we have made use of Sp(0) = 0 and Sp(1) = π2/6, and hencelimmt→∞Γ(t →bH+)Γ0(t →bH+) = 1 −Ncαsππ23 −54. (2.15)Compared with the result given by Ref.

5 in the limit mt →∞, Eq. (2.15) has adifferent constant, 5/4.

In Ref. 5 that constant is 9/4.3.

Explicit CalculationIn this section we verify (2.15) by an explicit calculation. For simplicity, wewill ignore the bottom quark mass mb, but allow mt and mH+ to be arbitrary.QCD corrections from a virtual gluon exchange introduce a correction to theinteraction vertex, wave function renormalizations to t and b, and a mass renor-malization to mt.

They have been calculated explicitly in Ref. 7.

The result isΓ(t →bH+)virt = Γ0(t →bH+)n1−Ncαsπh∆0+ m2tm2H+−32lnm2tm2t −m2H+−1io. (3.1)The last term of (2.11), which arises from the anomalous moment F2, is nowreplaced in (3.1) by the F3 term (with the exchange of MW ↔mH+).

As we7

expected, in the limit mt →∞both F2 and F3 vanish and the QCD correctionsin (2.11) and (3.1) are the same.The calculation for the decay t →bH+g with a real gluon emission is alsostraightforward. We find (details can be found in the Appendix)Γ(t →bH+g)real = Γ0(t →bH+)Ncαsπh∆0 −ln1 −m2H+m2t+ Sp1 −m2H+m2t−Spm2H+m2t+m2H+m2t −m2H+ln m2tm2H++ 54 −π22i.

(3.2)Again, comparing (3.2) and (2.12) we see that in the heavy top limit their QCDcorrections are indentical. It then follows from (3.1) and (3.2) that the final resultfor t →bH+ including its lowest-order QCD corrections isΓ(t →bH+) = Γ0(t →bH+)n1 −Ncαsπh52 −m2tm2H+ln1 −m2H+m2t−m2H+m2t −m2H+ln m2tm2H++ Spm2H+m2t−Sp1 −m2H+m2t+ π22 −94io,(3.3)which is free from infrared and collinear divergences.

It reduces to (2.15) in thelimit mt →∞. Although (3.3) is obtained by taking mb = 0, one can show that itremains as a good approximation as long asmbmtm2t −m2H+≪1.(3.4)Eq.

(3.3) differs from the result of Ref.5 by a term −(m2t /m2H+) ln(1 −m2H+/m2t) in the squared brackets.This missing term approaches to 1 in theheavy top limit. Numerically, the QCD corrections given by (3.3) turn out to beabout −9% (for αs = 0.1) if the top quark is very heavy, rather than −6% reportedin Ref.

5.8

4. ConclusionWe have calculated the lowest-order QCD corrections to the decay t →bH+.

Asimple analytic result is obtained for mbmt/(m2t −m2H+) ≪1. It is shown that fora heavy top quark, the order O(αs) QCD corrections reduce the tree-level rate ofthe decay t →bH+ by about 9% (for αs = 0.1) rather than 6% reported previouslyin the literature.Following an observation that the lowest-order QCD corrections to the interac-tions ¯btH−and ¯btφ−(in the Feynman-’t Hooft gauge) are identical in the heavy toplimit up to an over all scale, we have shown that asymptotically the lowest-orderQCD corrections to t →bH+ and to t →bW + are the same again up to an overall scale.

We also verified explicitly that the anomalous form factors F2 and F3vanish in leading order, and as a result the leading term of the QCD corrections tot →bW + in the heavy top limit can be calculated from the equivalence theorem.Acknowledgements: We wish to thank Paul Langacker for valuable discussions andcomments. This work was supported in part by the U. S. Department of Energy,contract DE-AC02-76-ERO-3071 (J. L.) and DE-AC02-76-ERO-1112 (Y. P.

Y. ).9

APPENDIXIn this Appendix we give some details for the calculation of Γ(t →bH+g)real.The result for this decay with only a soft gluon emission is known (Ref. 7).

Therate for t →bH+g with a hard gluon emission has also been calculated numericallybefore.12 These result are sensitive to the infrared- and the collinear-cut determinedby the experiment apparatus. Here we will present an analytic calculation thattakes both the soft and the hard gluon emission into account.

The calculation willbe carried out in the limit mb = 0. The condition for the validity of its result canbe extended to that given by (3.4) of the text.The matrix element of the decay t(p) →b(p′)H+(k)g(q) isM(t →bH+g) = ηgsǫν(q) mtmH+¯ub(p′)hR1p/ −q/ −mtγν + γν1p/′ + q/ −mbRiut(p),(A.1)where for simplicity we have not displayed the color matrix λ/2 explicitly.

Thespin-summed matrix square is−Ncη2g28mtmb mtmH+2hI1 + I2 + I3i,(A.2)whereI1 =1[(p −q)2 −m2t ]2h−8[p′ · (p −q)][p · (p −q)] + 4(p −q)2(p · p′)+ 16m2t [p′ · (p −q)] −4m2t (p · p′)i,I2 =1[(p′ + q)2 −m2b]2h−8[p′ · (p′ + q)][p · (p′ + q)] + 4(p′ + q)2(p · p′)+ 16m2b[p · (p′ + q)] −4m2b(p · p′)i,I3 =1[(p −q)2 −m2t ][(p′ + q)2 −m2b]h16[p · (p′ + q)][p′ · (p −q)] −8m2t [p′ · (p′ + q)]i. (A.3)10

It then follows thatΓ(t →bH+g)real = −Ncη2g2s4(2π)5mt mtmH+2hΓ1 + Γ2 + Γ3i,(A.4)whereΓ1,2,3 =Z d3⃗p′2p′0d3⃗q2q0d3⃗k2k0δ4(p −p′ −q −k)I1,2,3. (A.5)To evaluate (A.5), we employ the standard method of decomposing a three-body phase space integral into products of two-body phase space integrals.

Intro-ducing a fictitious gluon mass µ to regularize the infrared singularity , we findΓ1 = π24m2t(mt−µ)2Z(mH++mb)2da2λ1/2(m2t , a2, µ2)λ1/2(a2, m2H+, m2b)a2(a2 −m2t )2×h−2(a2 −m2H+)(a2 + m2t ) + 8m2t (a2 −m2H+) + (a2 −m2H+)(a4 −m4t )a2i= −π2m2th(m2t −m2H+)21 + lnµmtm2t −m2H+−m2t m2H+21 + 52m2H+m2tln m2tm2H++ m2t −m2H+452m2H+ + 92m2ti,(A.6)where λ(x, y, z) = x2 + y2 + z2 −2(xy + xz + yz). Also,Γ2 = π24m2t(mt−mH+)2Z(mb+µ)2db2λ1/2(m2t , b2, m2H+)λ1/2(b2, m2b, µ2)b2(b2 −m2b)2× (m2t + b2 −m2H+)h−2b2 + 6m2b + b4 −m4bb2i= −π2m2th(m2t −m2H+)21 + 12 ln µ2(m2t −m2H+)mtm3b−14(2m2t m2H+ −m4H+) ln m2tm2H+−38(m2t −m2H+)m2t −53m2H+i.

(A.7)11

The calculation of Γ3 is most complicated, we findΓ3 = π2m2t(mt−mH+)2Z(mb+µ)2db2λ1/2(m2t , b2, m2H+)b2 −m2bnλ1/2(b2, m2b, µ2)b2(m2t −m2H+ + b2)+h(m2t −m2H+ + b2)(m2t −m2H+ + m2b) −m2t (b2 + m2b)iJ0o,(A.8)whereJ0 =1λ1/2(m2t , b2, m2H+)× ln (b2 −m2b + µ2)(m2t + b2 −m2H+) −λ1/2(m2t , b2, m2H+)λ1/2(b2, µ2, m2b)(b2 −m2b + µ2)(m2t + b2 −m2H+) + λ1/2(m2t , b2, m2H+)λ1/2(b2, µ2, m2b). (A.9)Neglecting terms of the order of and smaller than mbmt/(m2t −m2H+), we find from(A.8) and (A.9)Γ3 = −π2m2tn(m2t −m2H+)2h2 + lnµ2m2H+lnmbmtm2t −m2H+−14ln2 m2bm2H++ ln2 m2tm2H++ 12ln2m2tm2t −m2H++ ln2m2H+m2t −m2H++ Sp1 −m2H+m2t−π22 + 1i+ 2m2tm2H+ ln m2tm2H+−12m4t1 −m2H+m2t1 + 3m2H+m2to.

(A.10)Substituting (A.6), (A.7) and (A.10) into (A.4), we obtain the result given by (3.2)of the text.12

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