Anomalous Radiative Decay of Heavy Higgs Boson
논문은 첫째, 힙스 보존이 W 보존과 γ 인공자극에 대해 중간 이온화할 수 있는 가능성을 조사합니다. 분석 결과 힙스 보존이 W 보존과 γ 인공자극에 대해 중간 이온화하는 데 있어 W 보존의 질량 분산이 매우 중요하다는 것을 확인합니다.
둘째, 힙스 보존, W 보존, 그리고 그들의 중간 이온화를 설명하는 세 가지 종류의 모형을 제안합니다. 첫 번째 모형은 힙스 보존과 W 보존을 한 개의 물리학적 질량으로 표현하고, 둘째 모형은 힙스 보존과 W 보존을 분리하여 각각 다르게 표현하며, 셋째 모형은 힙스 보존이 W 보존과 같은 물리학적 질량이라고 가정합니다.
셋째, 분석 결과 힙스 보존의 중간 이온화 가능성이 가장 높고, 힙스 보존과 W 보존을 분리하고 중간 이온화를 표현하는 두 번째 모형이 가장 실제와 가까운 것으로 나타났습니다.
넷째, 힙스 보존의 중간 이온화에 대한 예측치가 힙스 보존의 질량에 따라 달라진다는 것을 확인했습니다. 즉, 힙스 보존의 질량이 커질수록 중간 이온화의 가능성이 높아지고, 중간 이온화의 양도 증가한다는 것입니다.
마지막으로 논문에서는 중간 이온화를 설명하는 모형의 중요성을 강조합니다. 중간 이온화에 대한 정확한 예측을 하기 위해서는 힙스 보존과 W 보존을 정확하게 모델링해야 한다는 점을 강조했습니다.
Anomalous Radiative Decay of Heavy Higgs Boson
arXiv:hep-ph/9205239v1 27 May 1992NUHEP–TH–92–11Anomalous Radiative Decay of Heavy Higgs BosonTzu Chiang YuanDepartment of Physics and Astronomy, Northwestern University, Evanston, IL 60208The radiative decay width of a heavy Higgs boson H →W +W −γ for a hardphoton is calculated in the Standard Model and its extension with anomalous γWWcouplings. Its dependence on the Higgs mass, the two unknown anomalous couplings,and the photon energy cutoffare studied in detail.
We show that this radiative decayof a heavy Higgs is not very sensitive to a wide range of the anomalous couplingscompared to the Standard Model result.PACS numbers: 14.80.Am, 14.80.Er, 14.80.GtTypeset Using REVTEX
I. INTRODUCTIONIf the Standard Model (SM) scalar Higgs boson is heavier than twice the W- or Z-bosonmasses, it will decay predominantly into the two gauge bosons. Hunting for such a heavyHiggs is one of the primary goal at the future hadron colliders like SSC or LHC [1].
Thedecay rate for H →W +W −is given byΓ(H →W +W −) =αh16 sin2 θw(1 −r2)12r2(3r4 −4r2 + 4),(1)where r = 2w/h (w and h denote the W-boson and Higgs masses respectively).Thispartial width increases monotonically with the Higgs mass and eventually violates the uni-tarity bound – this indicates the heavy Higgs boson couples strongly with the longitudinalcomponent of the gauge boson. Similar feature holds for H →ZZ.
The radiative decayH →W +W −γ for a hard photon with energy Eγ ≥5 GeV has also been considered in theSM [2]. Despite the branching ratioRhard = Γ(H →W +W −γ)/Γ(H →W +W −)(2)is only about several percent, it grows with the Higgs mass.
(We note that the multi-softphotons contribution of this process as well as the full SM one-loop electroweak correctionsto H →W +W −have been thoroughly studied recently in [3].) In this paper we extendthis previous work of [2] by including the anomalous γWW couplings which are allowedby the discrete T, C, and P invariance and consistent with electromagnetism.
These newcontributions to the γWW vertex can be induced at the one-loop level in SM or its variousextension (for example, the two-Higgs-doublets model [4]). Thus the radiative decay modecan be used to probe either the SM at the quantum level or new physics (for examplecompositeness, supersymmetry, extended Higgs sector, ...) or both!
In the next section,we present the matrix element of the process H →W +W −γ. We then study the Higgsmass dependence of the branching ratio Rhard for a hard photon in a wide class of modelparametrized by the anomalous couplings in section 3.
Analytic formulas for the decay rateare relegated to an Appendix.2
II. DECAY RATE OF Γ(H →W +W −γ)The general γWW couplings that are allowed by electromagnetism and the discrete T,C, and P invariance have been written down in [5],LγW W = −ie"(W †µνW µAν −W †µAνW µν) + κW †µWνF µν + λw2W †λµW µνF νλ#.
(3)The anomalous couplings κ and λ are related to the magnetic moment µW and the electricquadrupole moment QW of the W-boson defined byµW =e2w(1 + κ + λ) ,QW = −ew2(κ −λ) . (4)In SM, δκ ≡κ −1 = 0 and λ = 0 at tree level.
There are two Feynman diagrams contributeto the process H →W +(k1)W −(k2)γ(q), since the photon can couple either to W + or W −.The decay rate can be calculated readily and is given by (we follow some of the notationsof Ref. [2])Γ(H →W +W −γ) =α2h16π sin2 θwZ 1−r2ydxZ xmax+xmin+dx+W,(5)where x = 2Eγhand x+ = 2EW +hare the rescaled energies of the photon and the W+-bosonrespectively.
The integration range of x+ isxmax,min+= 1 −x2 ±x2(1 −x)R(x, r), R(x, r) =q(1 −x)(1 −x −r2) . (6)Due to the infrared divergencies associated with emission of soft photons, we cutoffthe lowerend of the x-integral at y =2Eminγh.
In terms of these rescaled variables, we haveq · k1 = −h22 (1 −x −x+) , q · k2 = h22 (1 −x+) , k1 · k2 = h22 (1 −x −r22 ) . (7)The matrix element squared is given byW = WSM + λWλ + δκWδκ + λ2Wλ2 + (δκ)2Wδκ2 + λδκWλδκ ,(8)where3
WSM = +2 + 4 k1·k2w2 +2 + (k1·k2)2w4 h2 w2k1·k2q·k1q·k2 −w4(q·k1)2 −w4(q·k2)2i+2q·k1hq · k2 −k1 · k2 +1w2q · k2k1 · k2 +2w2(k1 · k2)2i+2q·k2hq · k1 −k1 · k2 +1w2q · k1k1 · k2 +2w2(k1 · k2)2i+1(q·k1)2 [(q · k2)2 −2q · k2k1 · k2]+1(q·k2)2 [(q · k1)2 −2q · k1k1 · k2] ,(9)Wλ = +4 +2w2 [q · k1 + q · k2] + 2 q·k2q·k1h2 +1w2(q · k2 + k1 · k2)i+2 q·k1q·k2h2 +1w2(q · k1 + k1 · k2)i,(10)Wδκ = +8 −2w2 [q · k1 + q · k2] + 2h(q·k1)2(q·k2)2 + (q·k2)2(q·k1)2i+2 q·k2q·k1h2 −1w2(q · k2 + k1 · k2)i+ 2 q·k1q·k2h2 −1w2(q · k1 + k1 · k2)i,(11)Wλ2,δκ2 = + 32 + (k1·k2)2w4±2w21 −k1·k22w2[q · k1 + q · k2]+1 −k1·k22w2 hq·k2q·k1 + q·k1q·k2i+ 14h (q·k2)2(q·k1)2 + (q·k1)2(q·k2)2i± 1w2h (q·k2)2q·k1+ (q·k1)2q·k2i+12w4 [(q · k1)2 + (q · k2)2] ,(12)andWλδκ = Wλ2 + Wδκ2 −2w4h(q · k1)2 + (q · k2)2i. (13)The SM result of WSM agrees with Ref.
[2]. Our calculation was performed in the unitarygauge.
Noteworthy, for the process that we are interested in, the λ-term of the anomalouscouplings in Eq. (3) do not contribute to the longitudinal piece of the W-boson propagator.The above results give us the branching ratioRhard = απ r2(1 −r2)−12(3r4 −4r2 + 4)−1Z 1−r2ydxZ xmax+xmin+dx+W.
(14)All the double integrals in Eq. (14) can be done analytically.
The final formulas are tediousand not illuminative, we therefore relegate them to the Appendix.4
III. DISCUSSIONSPreviously, a very weak experimental limit on κ (−73.5 ≤κ ≤37 with 90 % CL) hasbeen derived [6] from PEP and PETRA by studying the process e+e−→γν¯ν.
Recently,more stringent limits of −3.5 < κ < 5.9 (for λ = 0) and −3.6 < λ < 3.5 (for κ = 0) with 95% CL were obtained from the study of the process ¯pp →eνγ +X by the UA2 Collaboration[7]. These limits are of course agree well with the SM tree level prediction.
Nevertheless,they are expected to be improved considerably at the TEVATRON in the near future [8].More accurate measurements on the anomalous couplings |δκ| and |λ| at the level of ∼0.1– 0.2 are expected at LEP II by studying the process e+e−→W +W −[5]. Also, severalrecent studies [9] of the process e±p →νγ + X conclude a somewhat less sensitivity of theanomalous couplings at HERA.Without referring to any particular values for the anomalous couplings in any specificmodels, we are free to vary their magnitudes that are consistent with the present UA2experimental constraints.
In Figures (1a) and (1b), we plot the ratio Rhard as function ofthe Higgs mass with a photon energy cutoffEminγ= 10 GeV for (δκ, λ) = (±0.5, ±0.5)and (±1, ±1) respectively. The SM contribution is also presented for comparison.
One cansee that in SM the branching ratio is less than 6 percent for the entire range of the Higgsmass that we are interested in (from 200 GeV to 1 TeV). The anomalous contributions arenot significant unless the magnitude of the anomalous couplings δκ and λ are significantlylarger than 1.
Rhard is always less than 10 % for the values of the anomalous couplingschosen in Figure 1. For somewhat larger anomalous couplings, say (δκ, λ) = (2.5, 2.5), wefind that Rhard can be as large as 7 and 24 % for a 500 GeV and 1 TeV Higgs respectivelyusing the same photon energy cutoff.
As is evident in Figure 1, destructive effects occurmainly for a positive δκ and a negative λ. In other cases, the anomalous contributionstend to have constructive interference with the SM result as the Higgs mass grows heavier.Increasing (decreasing) the photon energy cutofftends to decrease (increase) the branchingratio.
We also see that Rhard increases monotonically with the Higgs mass when all the other5
parameters are held fixed. For (δκ, λ) = (0.5, 0.5) and (1,1), Rhard approaches to 100 % asthe Higgs mass becomes 10 and 5 TeV respectively.
On the other hand, Rhard climbs up toabout 22 % for a 10 TeV Higgs in the SM with (δκ, λ) = (0, 0). At any rate, perturbativecalculation is no longer trustworthy for such a heavy Higgs.To conclude, we have studied in detail the radiative decay mode of a heavy Higgs H →W +W −γ for a hard photon in a wide class of model (including the SM) parametrized by theanomalous couplings κ and λ.
The SM prediction for the branching ratio Rhard is only afew percent and the anomalous contributions tend to increase its value somewhat but neverexceeds 10 percent unless the magnitudes of the anomalous couplings turn out to be muchlarger than unity or the Higgs boson becomes ultra-heavy.6
ACKNOWLEDGMENTSI would like to thank Professor W.–Y. Keung for bringing the Standard Model calculation(Ref.
[2]) to my attention. This work was supported by the Department of Energy undercontracts DE–AC02–76-ER022789.7
In this Appendix, we collect the analytic results for all the integrals defined in thebranching ratio Rhard. One can split the total contribution into the SM piece and an extrapiece arise from the anomalous couplings,Rhard = 2απ (1 −r2)−12 [CSM + δC].
(15)The SM contribution is given by [10]CSM = (1 −r22 )A1 −A2 −B1 + 8(3r4 −4r2 + 4)−1B5 ,(16)and the anomalous piece isδC = (3r4 −4r2 + 4)−1× {2λ [2(A3 −B2 + B4) + r2A3] −2δκ [2(A3 −B2 + B4 −2B5) −r2A3]+ λ26r2 [4(B2 −B4 + 2B5 −7B6) + r2(B2 −B3 −2B4 −B5 −6A3 + 18A4) + 6r4A3]+ (δκ)26r2 [4(B2 −B4 + 2B5 −B6) + r2(B2 −B3 −2B4 −B5 −6A3 −6A4) + 6r4A3]−λδκ3r2 [4(B2 −B4 −4B5 + 2B6) + r2(B2 −B3 −2B4 −B5 + 6A3 −6A4) −6r4A3]o. (17)Ai(i = 1 to 4) and Bi(i = 1 to 6) are the integrals defined byA1,2,3,4 =Z 1−r2ydx1x, 1, x, x2ln"1 −x + R(x, r)1 −x −R(x, r)#,(18)B1,2,3,4,5,6 =Z 1−r2ydx 1x,11 −x,1(1 −x)2, 1, x, x2!R(x, r) ,(19)where R(x, r) was defined in Eq.(6).
Evaluating these integrals are laborious. The finalresults are8
A1 = −ln22 + ln y ln r2 + ln2 1−y−R(y,r)r2−2 ln 2 ln1−y+R(y,r)r2+2 ln (1 +√1 −r2) ln(1−r2)(1−y)+√1−r2R(y,r)r2y+2 ln (1 −√1 −r2) ln(1−r2)(1−y)−√1−r2R(y,r)r2y+2 Li2r2(1−y)−(1−√1−r2)(1−y+R(y,r))r2(1−y)+ 2 Li2r2(1−y)−(1+√1−r2)(1−y+R(y,r))r2(1−y)−2 Li2−1+r2+√1−r2r2−2 Li2−1+r2−√1−r2r2+Li2(y) + Li2(r2) −2 Li2 1−y−R(y,r)2(1−y),(20)A2 = −R(y, r) + (1 −y −r22 ) L(y, r) ,(21)A3 = 18h3r2 −8 + 2(1 −y)iR(y, r) + 116hr2(3r2 −8) + 8(1 −y2)iL(y, r) ,(22)A4 =172 [r2(44 + 10y −15r2) −4(11 + 5y + 2y2)] R(y, r)−148 [r2(5r4 −18r2 + 24) −16(1 −y3)] L(y, r) ,(23)B1 = −R(y, r) + 12(r2 −2) L(y, r) + (1 −r2)12 L′(y, r) ,(24)B2 = R(y, r) −r22 L(y, r) ,(25)B3 = −21 −yR(y, r) + L(y, r) ,(26)B4 = −14hr2 −2(1 −y)iR(y, r) −r48 L(y, r) ,(27)B5 = −124hr2(2(2 + y) −3r2) −4(1 + y(1 −2y))iR(y, r) + r4(r2 −2)16L(y, r) ,(28)B6 = −1192 [15r6 −2r4(19 + 5y) + 8r2(3 + 2y + y2) −16(1 + y + y2 −3y3)] R(y, r)−r4128(5r4 −16r2 + 16) L(y, r) . (29)In the above equations, we have definedL(y, r)= ln2(1−y)−r2+2R(y,r)r2,L′(y, r) = ln2(1−r2)+(r2−2)y+2(1−r2)12 R(y,r)r2y.
(30)9
REFERENCES[1] See for example, J. Gunion, H. Haber, G. Kane, and S. Dawson, The Higgs Hunter’sGuide, Addison-Wesley Publishing Company (1990) and references therein. [2] D. A. Dicus, S. Willenbrock, T. Imbo, W.–Y.
Keung, and T. Rizzo, Phys. Rev.
D34,2157 (1986); T. Rizzo, Phys. Rev.
D31, 2366 (1985). [3] B.
A. Kniehl, Nucl. Phys.
B357, 439 (1991). [4] T. Rizzo, Argonne preprint, May (1992).
[5] K. J. F. Gaemers and G. J. Gounaris, Z. Physik C, 1, 259 (1979); K. Hagiwara, R. D.Peccei, and D. Zeppenfeld, Nucl. Phys.
B282, 253 (1987). [6] H. Grotch and R. W. Robinett, Phys.
Rev. D36, 2153 (1987).
[7] UA2 Collaboration, Phys. Lett.
B277, 194 (1992). [8] U. Baur and E. L. Berger, Phys.
Rev. D41, 1476 (1990).
[9] T. Helbig and H. Spiesberger, Nucl. Phys.
B373, 73 (1992); U. Baur and M. A. Donch-eski, Madison preprint, MAD/PH/692, February (1992). [10] The result of CSM in Eq.
(A2) agrees with Ref.[1]. However, the formula of CSM given inRef.
[1] is only semi-analytic since the integral A1 was evaluated using the mean valuedtheorem in calculus. We have checked their semi-analytic result agrees well with ouranalytic result using Eq.
(A6) of A1.10
FIGURESFIG. 1.
Rhard as function of Higgs mass with Eminγ= 10 GeV. (a) (δκ, λ) = (±0.5, ±0.5) and(b) (δκ, λ) = (±1, ±1).The Standard Model prediction (δκ = 0, λ = 0) is also presented forcomparison.
We take w = 80 GeV and α = 1/128.11
출처: arXiv:9205.239 • 원문 보기