On the QCD corrections to the charged Higgs decay
영어 요약 시작:
We studied the QCD corrections to the decay of a heavy quark. We considered the case where the mass of the decaying quark is sufficiently large that the masses of the up and down type quarks are neglected. In this limit, the first-order QCD corrections to the process t →H+b are equal to those for the process t →W +b.
We used dimensional regularization to regularize both ultraviolet and infrared divergences. This greatly simplifies the calculation, especially for the real radiation part. We calculated the virtual gluon correction to the vertex tH+b and found that it amounts to a multiplication of the tree-level rate by a factor Λ. The counterterm for this vertex involves the wave function and mass renormalization constants.
We also considered the effect of real gluon radiation from the initial or final quark. We parametrized the three-body phase space integration using the variables x = 2t · G and z = 1 −2t · b. After summing over the polarizations of the b quark and the gluon, and averaging over the polarizations of the t quark, the squares of the amplitudes become |A1|2 , |A2|2 , and A1A∗2 + A2A∗1.
We found that although the respective phase space integrations give different results from the analogous amplitudes with a W+ replacing the charged scalar, their sum nevertheless gives the same total contribution in both processes. This is in agreement with the general argument based on the equivalence theorem.
Finally, we added the effects of the virtual and real gluons to obtain the first-order QCD correction, so the decay rate becomes Γ(t →H+b) = GF√2πm3t cot^2 β |Vtb|^2 (1 + αs/3π(5 − 4π^2/3)).
This result is identical to that obtained in ref. [3] and also to the analogous correction to the decay t →W +b [5–8]. If we take αs = 0.1, the first-order correction in the limit mb = mH = 0 is approximately equal −0.87%. This disagrees with the value reported in refs. [1,2].
On the QCD corrections to the charged Higgs decay
arXiv:hep-ph/9208240v1 21 Aug 1992Alberta Thy-25-92August 1992On the QCD corrections to the charged Higgs decayof a heavy quark.Andrzej Czarnecki∗and Sacha Davidson∗∗Department of Physics, University of Alberta, Edmonton, Canada T6G 2J1Using dimensional regularization for both infrared and ultraviolet divergences,we confirm that the QCD corrections to the decay width Γ(t →H+b) are equal tothose to Γ(t →W +b) in the limit of a large t quark mass.
Many extensions of the Standard Model contain more than the one necessary Higgsdoublet, and the new degrees of freedom appear as extra Higgs scalars, some of which arecharged (see [1] and references therein). We consider here the decay t →H+b in a modelwith two Higgs doublets.
The first order QCD correction to this process has been calculatedby two groups [1–3] who disagree with each other. In this letter we present a calculationof this correction in the limit of a very heavy top quark, i.e., neglecting the masses of thebottom quark and the Higgs boson.
The effect of finite mH and mb will be addressed in aforthcoming paper. We use dimensional regularization to cope with both the ultraviolet andinfrared divergences, which greatly simplifies the calculation, especially the real radiationpart.We take H1 and H2 to be the doublets whose vacuum expectation values respectivelygive masses to the down and up type quarks.
The physical charged Higgs H+ is a linearcombination of the charged components of H1 and H2, so if we neglect all the Yukawacouplings except that of H2 to the third generation = h(2)tt , the top only couples to the H2component of H+. The interaction Lagrangian relevant to the decay t →H+b is then:L = h(2)tt cos βVtbH+¯t1 −γ52b + h.c.=g2√2mWVtb cot βmtH+¯t (1 −γ5) b + h.c.(1)where H+ = cos βH+2 −sin βH+1 and cot β = ⟨H1⟩/ ⟨H2⟩is the ratio of vacuum expectationvalues of the two Higgs doublets.In the following calculations we take the space-time dimension to be D = 4 −2ǫ.
Themass of the decaying quark is taken to be the renormalization mass scale, and we also useit as a unit of energy: mt = 1. In the limit of a very heavy top quark the above interactionleads to the tree-level decay rate:Γ(0) t →H+b= GF√2Γ (1 −ǫ)23−2ǫπ1−ǫΓ (2 −2ǫ) cot2 β |Vtb|2 →GF8√2π cot2 β |Vtb|2 .
(2)The first order QCD corrections to this formula arise due tovirtual gluon exchange and radiation of a real gluon. We first deal with the virtual gluoncorrection to the vertex tH+b.
In the limit mb = mH = 0 the spinor structure of this vertex2
remains unchanged and the unrenormalized correction amounts to the multiplication of thetree level rate by a factorΛ = CFg2sΓ (1 + ǫ)(4π)D2−1ǫ2 + 2ǫ. (3)where the colour factor CF is 4/3 for SU(3).
The counterterm for this vertex involves thewave function and mass renormalization constants [4,1]:Λc.t. = 12Zt2 −1+ 12Zb2 −1−δmtmt−δmbmb.
(4)If we use the same ǫ to regularize both UV and IR divergencies we obtain for the renor-malization constants:Zt2 −1 = −δmtmt= CFg2sΓ (1 + ǫ)(4π)D2−3ǫ −4,Zb2 −1 = δmbmb= 0. (5)The contribution of the virtual correction to the decay rate is:Γ(1)virtt →H+b= 2 (Λ + Λc.t.) Γ(0) t →H+b= GF√2 cot2 β |Vtb|2 αs2−3+4ǫπ−2+2ǫ3Γ (2 −2ǫ) −2ǫ2 −5ǫ −12 −π23!.
(6)We now turn our attention to the effect of real gluon radiation from the initial or finalquark. If we denote the amplitudes for these processes byA1 and A2 respectively, the contribution of the real radiation to the decay width is:Γ(1)realt →H+bG= 12GF√2 cot2 β |Vtb|2 4παsZdR3 (t; b, H, G) |A1 + A2|2(7)where the coupling constants have been factored out and t, b, H and G denote the four-momenta of the initial and final quarks, charged Higgs boson and the gluon.The advantage of using dimensional regularization for the infrared and colinear diver-gences is that we need not introduce a mass for the gluon and the integration over three bodymassless phase space is very simple.
We choose to parametrise it by the variables x = 2t · Gand z = 1 −2t · b in terms of which the three body phase space integration becomes:3
ZdR3 (t; b, H, G) = 24ǫ−7π2ǫ−3Γ (2 −2ǫ)Z 10dx(1 −x)ǫZ x0dzzǫ (x −z)ǫ. (8)After summing over the polarizations of the b quark and the gluon, and averaging over thepolarizations of the t quark, the squares of the amplitudes become:|A1|2 = −4x2 [2(1 −x) + x(1 −ǫ)(z −x)]|A2|2 =4xx −z(1 −ǫ)A1A∗2 + A2A∗1 =8x(x −z) [1 −x + x(1 −ǫ)(z −x)](9)The integration over the phase space can be done exactly in any dimension.
In the limitǫ →0 the contribution of the real radiation becomes:Γ(1)realt →H+b= GF√2 cot2 β |Vtb|2 αs2−3+4ǫπ−2+2ǫ3Γ (2 −2ǫ) 2ǫ2 + 5ǫ + 17 −π2. (10)Although the respective phase space integrations of|A1|2 , |A1|2 andA1A∗2 + A2A∗1 give different results from the analogous amplitudes with a W + replacingthe charged scalar, their sum nevertheless gives the same total contribution in both processes.This is in agreement with the general argument based on the equivalence theorem in ref.
[3].Finally we add the effects of the virtual and real gluons to obtain the first order QCDcorrection, so the decay rate (with mt reinstated) becomes:Γt →H+b=GF8√2πm3t cot2 β |Vtb|2"1 + αs3π 5 −4π23!#. (11)This is identical to the result obtained in ref.
[3] and also to the analogous correctionto the decay t →W +b [5–8]. If we take αs = 0.1 the first order correction in the limitmb = mH = 0 is approximately equal −0.87%.
This is in disagreement with the valuereported in ref. [1,2].ACKNOWLEDGMENTSWe gratefully acknowledge support for our doctoral scholarships: A.C. from the KillamFoundation and S.D.
from NSERC. This research was also partially supported by a grant4
to A. N. Kamal from the Natural Sciences and Engineering Research Council (NSERC) ofCanada.This preprint was typeset using REV TEX.5
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