W-«rr \ъ~гя, Studying Triple Higgs Vertex in the Process 77 > HE at TeV Energies INSTITUTE FOR HJGH ENERGY PHYSICS ШЕР ОТФ

Size: px

Start display at page:

Download "W*-«*rr \ъ~гя, Studying Triple Higgs Vertex in the Process 77 > HE at TeV Energies INSTITUTE FOR HJGH ENERGY PHYSICS ШЕР ОТФ"

Collin Brett Norton
5 years ago
Views:

1 / / /. INSTITUTE FOR HJGH ENERGY PHYSICS W*-«*rr \ъ~гя, ШЕР ОТФ G.V. Jikia 1 and Yu.F. Pirogov 3 Studying Triple Higgs Vertex in the Process 77 > HE at TeV Energies Submitted to Physics Letters В 'Б П1лЦ address: Jikialsms dccnce.iftep.su ''E mail acidrcb*: PiixisoVgmx.decnet.ihep.su Protvino 1992

2 IDK M 24 Abstract G.V. Jikia, Yu.F. Pirogov Studying Triple Higgs Vertex in the Process r Encrg es. JHEP Preprint Protvino, p.12, refs.: 14. НИ at TeV In the framework of the equivalence theorem the one loop helicity amp. i. ides and cross srrtmn lor the double Higgs production process 77 HH are calculated. It is,hown that the cross section is measurable at TeV 77 colliders and is marginally sensitive to the triple Higgs coupling variation. Аннотаяия Г.В.Цжикия, Ю.Ф.Пирогов Изучение тройной хиггеовской вершины в процессе 77 ИМ при тэвных энергиях: Препринт ИФВЭ Протвино, с, библиогр.: приближении теоремы эквивалентности вычислены однопетлевые спиральные амплп гуды и сечения парного рождения хиггеовских бозонов в реакции 77» ИИ. Показано, что указанные сечения измеримы на 77 коллайдерах тэвкых энергий. Величина сечения находится на пределе чувствительности к вариации треххиггеовской константы связи. Institute for High Energy Physics, 1992.

1. A central issue for experimental particle physics of the nearer future is looking for the neutral Higgs boson Я, which is a remnant of the Higgs electroweak symmetry breaking mechanism in the

3 1. A central issue for experimental particle physics of the nearer future is looking for the neutral Higgs boson Я, which is a remnant of the Higgs electroweak symmetry breaking mechanism in the standard model (SM). Once (if any) the Я boson is found (alongside with the t quark), the particle content of the minimal SM will be completed. After that it will be necessary to study experimentally as thoroughly as possible all the SM vertices, related to these heavy particles. Fine tuning of all the parameters is required for renormalizability and unitarity of the SM. Any deviation from this fine tuning would lead to loosing these precious properties and signal appearance of a new physics responsible for these discrepancies. Among the ingredients of the SM is the triple Higgs vertex Я 3 which could be studied in the double Higgs production processes [1 9]. 2. At the e + e~ colliders the following processes of the double Higgs production are considered. 1) The process e + e" > HHZ which proceeds via double Higgs bremsstrahlung from the s channel Z boson [1]. The appropriate cross section a decreases with the cm. energy squared as о ~ s 1. So for the Higgs mass тц > 50 GeV its value is < 1 fb [2]. For intermediate mass Higgs (m H = 0(100 GeV)) one has a < 0(1O~' fb) which makes the process hardly observable. 2) The process e + e~ > HH is due to the W and Z loop diagrams [3]. The cross section decreases with s a little slower than s" 1, but is numerically small (in R units it is O(10 4 ) at v^ = 0(1 TeV). I n addition, in the limit m e = 0 the relevant Feynman diagrams do not contain the Я 3 vertex. This makes the process of no use for the present purposes. 3) The competing WW fusion process e + e~ vvhh [4] has the cross section increasing as log(s/m^), but still < O(10 _1 fb) at -/s = 2 TeV for intermediate and heavy Higgs bosons. As for the {ip colliders they provide both analogous mechanisms and some new ones. 1

4 1) PP» ЯЯ via WW/ZZ fusion [5]. For intermediate mass Higgs one has a < O(10 fb) at sfs = 40 TeV. So, in the hadron collider environment the process can hardly be studied. 2; The same process via gg fusion [6,7j. This mechanism dominates that of WW/ZZ fusion for a heavy t quark (ra, > 100 GeV) and not too heavy Н{тц < 600 GeV), that is in the whole region attainable. The best possibility is for the light and intermediate mass Higgs % < О (100 GeV) where a > ОМП' 'fh). Nevertheless the process is inappropriate for the Higgs self interactions study because the t quark box diagram dominates that containing the (quark triangle and H 3 vertex. The reason is that the ggh loop form factor containing the internal fermiou decreases as s~' with s growing. The possible exception is the light Higgs (т.ц < loogev), but uncertainties in the gluon structure functions and the scale of a, would render the triple Higgs coupling measurement inconclusive. 3) A subtler mechanism, of the light quark annihilation qq» Я Я through weak boson loops [8] is ineffective as in the e + e" case. The cross section is much smaller even than that for WW/ZZ fusion. The physical reason is that both for transverse and longitudinal virtual bosons some of the couplings either with the light quarks or with Я boson are suppressed or not enhanced. 4) The ti fusion mechanism [9] is operative only for heavy Higgs (mj > 400 GeV) where it dominates both gg and WW/ZZ fusion. But still, the cross section is at the level of 0(1 fb) at s/s = 40 TeV. a) Double Higgs boson bremsstrahlung from W and Z bosons at pp > HHW/Z process [2]. For the intermediate mass Higgs (m H = О (100 GeV)) the cross section is 0(1 fb). But using the additional W/Z boson as a trigger one hopes to suppress the background. So, it follows from the preceding that only light and intermediate Higgs self coupling has some chances to be studied in the foreseeable future at the e + e~ or pp supercolliders. The heavy Higgs self interactions are out of reach for these machines. 3. We propose to study the double Higgs production at the 77 colliders of TeV energies. This provides in a sense a complimentary possibility to study in the best way the properties of the heavy Higgs in comparison with the light one. The reason is that the 77Я form factor contrary to that of ggh tends to a constant with s increasing due to longitudinal W boson contributions. This in turn allows one to overcome in some cases the t loop "background" and discriminate the triple Higgs coupling. Out of four helicity amplitudes Ai\ 1: x,< Aj, Aa = ±1, of the process 77» 2

5 ЯЯ, СР invariance leaves only two independent ones: A<++ = M = MQ, M + -=M- + = M 2. (1) Here the Lorentz and gauge invariant amplitudes Mj, J = 0,2, are labeled by the minimum total angular momentum J, contributing to the helicity amplitudes. The normalizations are such that the helicity differential cross section is -*Г~лъ* [м^- The diagrams of the process are presented in Fig. 1. Here gauge invariant subset (a) both for t loops and W loops containing the usual box is generically labeled a "box" and that containing the usual triangle is called a "triangle". Each of the gauge invariant subsets is finite separately. It is evident that the triangle gives contribution only to Mo, while the box contributes both to Mo and Mi. We have calculated the W loop diagrams using the equivalence theorem approximation [10] in the R(( = 1) gauge. The result is as follows: Alo(triangle) = _ \ + 2 <U + 2m? v C(l,2)l, (3) 2m\y s nifj i J (2) Mo(box) = ^f 4, 4mU.D(l,2,3) + 17(2,1,3) + D(l,3,2)1 477ipy L - J + [(m 4 ff tu)2?(l,3,2) + 2(t - т г )С(1,3) + 2{u m 2 H)C(2,3)]j, (4) 4, Л4 2 (Ьох) = ^ 2m^[D(l,2,3)+ 17(2,1,3)+ D(1,3,2)] + r \s(s 2mj,)C(l,2) 2t(t m^)c(l,3) 2u(u т г н)с(2,3) +(4 2 + u 2 2m 2 v)c(3,4) + W! (l,2,3) + sr :!.D(2,l,3)]}. In these formulae s,t,u are the usual Mandelstam variables. (5) I 3

The algorithm [11] has been used to reduce the tensor loop integrals to scalar loop throe and four point functions [12] C(i,j) = C(p t,pj) and D(i,j,k) = D{pi,Pj,Pk) (see also ref.

6 The algorithm [11] has been used to reduce the tensor loop integrals to scalar loop throe and four point functions [12] C(i,j) = C(p t,pj) and D(i,j,k) = D{pi,Pj,Pk) (see also ref. [7]), which are given by C(pu Pi ) = -2 / (?2 _ ras) ( (q + p.f _ m,) ( (e + p. +p.f _ m,y (6) >()),,P.,Pi) = J_ /, A.V У (,! _ mj) ((, + p,f _ m2) ((, + j,, + Tjf _ m2) ((, + p, + P j + p k f - m») Indices i,j,k = 1, 2 refer to the initial photons and i,j,fc = 3, 4 to the final Higgs bosons. All the momenta are taken to be incoming, m is the mass of the internal loop particle (t or W, respectively). Due to the symmetry properties C(pi,Pj) = C(p.j,Pi) and D(p i: pj,p t ) = D{pt,Pj,Pi) formulae (3) (5) are explicitly symmetric w.r.t. the initial photon permutations. Using the momentum conservation pi +Рг + Рз +РА 0 one can show that (3) (5) are also symmetric under the interchange of the two Higgs bosons. The expressions for the t loops coincide with those of ref. [7] divided by im%r (and an additional factor of 1 for Mi) and are not given here. 4. The results of the numerical calculations are presented in Figs Fig. 2 gives the dominant cross section for the photon helicities Ai = Аг = ±1 versus Higgs mass at three cm. energies y/s = 0.5,1 and 2 TeV being urgent ill the foreseeable future. For each of the energies the total cross sections for the most probable range of values m ( = 100, 150 and 200 GeV are presented. The separate contributions of the W and t loops (for the same m ( ) are presented too. It is seen that the results are in a sense counter intuitive: the heavier the Higgs boson the larger the cross section of its double production (almost up to the kinematical boundary). This is a dramatic manifestation of the decoupling violation for the virtual longitudinal Ws and heavy Higgs boson interactions. So, the process considered is best tailored to the study of a relatively heavy Higgs boson. A somewhat irregular m t dependence in Fig. 2a can be traced back to the threshold behaviour of the amplitude at -/s ~ 2m t. Now, the equivalence theorem we are using to simplify the calculations, accounts only for the enhanced electroweak contributions. That is, it leaves only terms of the order of 0(gmnlmw) and 0(gy/s/mw) neglecting ones of (7) 4

7 the usual order 0(g). So, the results for the intermediate Higgs can change somewhat in exact calculations. But one may hope they are sufficiently correct for, say, тн > 3mw (so that дтц/mw > 2) and could be representative for somewhat lighter mj. Fig. 3 presents the same results for the suppressed helicity configurations Ai = A2 = ±1. We maintain the results only for y/s = 2 TeV because for lower energies the cross section is too small (<r + _ < 10~'fb and 10" 2 fb for /s = 1 and 0.5 TeV, respectively). 5. The main interest of the process under consideration is the feasibility it provides to study some new Higgs interactions. Say, if the Higgs boson is composite [13] its interactions with other fields and with itself could deviate from that of the SM. Considering e.g. the simplest composite Higgs model Sf/(3)x{7(l)/S{/(2)x(/(l) [14] one can find the corresponding anomalous Higgs interactions. In particular, this is true for the modified Higgs self interaction potential. In what follows we do not stick precisely to any of such models and consider as a representative example some general Higgs potential variation of the lowest dimension, such that the minimum position is left unshifted: 6У = (ф<ф-^/2)\ (8) Here ф is the Higgs doublet field, Л is the Higgs compositeness mass scale and A3 some dimensionless constant. In terms of the Goldstone w*, z and Higgs components expression (8) takes the form 6 V = ^К»" + г 2 /НЯ! /2 +»Я) 1, (9) so that 6V = ^ [1/Я 3 + 3v 2 {w + W- + z 2 /2) Я 2 ] +..., (10) where dots mean higher terms not contributing to the process of interest. In the unitary gauge 6V reduces to the triple Higgs vertex variation alone. This allows us to consider 5V as a gauge invariant variation of that vertex. In terms of the SM three Higgs coupling A = m 2 H/2v one has 0К = ^ [ я 3 + ^+ш-я , (и) 2 v [ v 1 where к is the dimensionless anomalous Higgs self coupling (к = О corresponds to the SM limit). As a function of the Higgs compositeness scale the parameter 5

8 -(C)' For гпн v ~ 250 GeV and Л/\/Аз > 1 TeV one should expect к < 0.1, so only rather small deviations from the SM value may be urgent. In the general fij gauge, required for the equivalence theorem approximation we are working in, the w + w~h 2 anomalous term in SV should be properly taken into account as well. The maintenance of the gauge invariance is crucial to render the calculations unambiguous and, hence, reliable. Fig. 4 presents the dominant helicity Ai = A2 = ±1 cross section versus the total cm. energy \/s for different typical values of the Higgs mass (m# = 250, 500 and 800 GeV, respectively) and m t = 150 GeV. The variation of the cross section with the anomalous coupling к is shown by the dashed lines. The value к = 1 corresponds to the limit of the triple-higgs interaction being ''switched off". The solid line is the SM value (к = 0). For comparison the helicity Ai = Aj = ±1 cross section, which is insensitive to the к parameter, is presented too. Its value restricts the regions of sensitivity to small /c. To elucidate somewhat the cross section sensitivity to the к parameter we present in Fig. 5 the cross section versus к for some typical values of other variables: ^/s = 1 TeV, m, = 150 GeV and m H = 400, 450 and 480 GeV (such that the cross section is relatively large). Though the process 77 > HH may be the best one among those considered up to now to study triple-higgs vertex, only rather large deviations (к = 0(1)) from the SM can be hoped to be seen, at least in the foreseeable future. 6. In conclusion, the double Higgs boson production. ; n 77 collisions at TeV energies is feasible to be studied at future high luminosity 77 colliders (! Cdt = fb _1 ). But the direct test of the trilinear Higgs self-coupling is rather marginal. Thus, studying the triple-higgs vertex is a challenge for the high energy precision experiments. I 6

9 WA w,t vww* * WA J>< (a) W W W VWWf "^ (b) Fig. 1. The generic one-loop Feynman diagrams for the process 77» HH: (a) "box"; (b) "triangle". The graphs contributing to the top and W loops are marked by the col responding symbols. The wavy lines are the initial photons, dashed lines - final Higgs bosons. The graphs with the photon and Higgs permutations should be added when required. I 7

10 OO M.CGeVI 0) Fig. 2. The 77 t HH total cross section for the equal photon helicities Ai = A 2 = ±1 versus the Higgs mass: (a) ^s = 0.5 TeV, (b),/j = 1 TeV and (c) s/s =2 TeV. The dashed lines present the W loop contributions, dotted lines - that of the t loops, solid lines - the total results. 8

11 fc -I V 10 : v = 2TeV A,- m, : 2oo X 2 = + 1 Y M / ^ r ^""4 10 г -.. : ioo/ / ' -... rrz^j 1 " '-... Ю. niiiii id,!' 1, i, i Ln LJI,, i in N,i, M [GeVl Fig. 3. The same as in Fig. 2 but for the unequal photon helicities Ai = A 2 : ±1 and v/s =2 TeV only. 9

12 Fig. 4. The 77 > HH total cross section for the equal photon helicities Ai = A2 = ±1 versus cm. energy for the different anomalous triple Higgs coupling as a multiple к times the SM value: (а) т.ц = 250 GeV, (b) m«= 500 GeV and (c) m H = 800 GeV. The SM corresponds to к = 0. The unequal photon helicities cross section (dotted lines) is insensitive to к. The top mass is m, = 150 GeV. 10

13 = 6. v^ = 1 TeV 1 г л, = л 2 = + 1 / 2 - ь / / -\ / 7 : 4 \ / = 480 ^Л ' / 4 ч v м и = 450, У /. м и = 400 / / /' Fig, 5. The 77 > HH total cross section for equal photon helicities Ai = A2 = ±1 and л/s = 1 TeV versus the anomalous triple-higgs coupling к at different values of the Higgs mass: m,, = 400, 450 and 480 GeV. The top mass is m t =150 GeV. 11

14 References [1] G.J. Gounaris, D. Schildknecht and F.M. Renard, Phys. Lett. 83B (1979) 191. [2] V. Barger, T. Han and R.J.N. Phillips, Pbys. Rev. D38 (1988) [3] K.J.F. Gaemers and F. Hoogeveen, Z. Phys. С - Part and Fields 26 (1984) 249. [4] V. Barger and T. Han, Mod. Phys. Lett. A5 (1990) 667. [5] W.-Y. Kevmg, Mod. Phys. Lett. A2 (1987) 765; O.J.P. Eboli et al., Phys. Lett. 197B (1987) 269; D.A. Dicus, K.J. Kallianpur and S.S.D. Willenbrock, Phys. Lett. 200B (1988) 187. [6] D.A. Dicus, С. Као and S.S.D. WiUenbrock, Phys. Lett. 203B (1988) 457. [7] E.W.N. Glover and J.J. van der Bij, Nucl. Phys. B309 (1988) 282. [8] D.A. Dicus, Z. Phys. С - Part, and Fields 39 (1988) 583. [9] K.J. Kallianpur, Phys. Lett. 215B (1988) 392. (10) D.A. Dicus and V. Mathur, Phys. Rev. D7 (1973) 3111; J.M. Cornwall, D.N. Levin and G. Tiktopoulos, Phys. Rev. D10 (1974) 1145; B.W. Lee, C. Quigg and H. Thacker, Phys. Rev. D16 (1977) 1519; M. Veltman, Acta Phys. Polon. B8 (1977) 475; R.N. Cahn and S. Dawson, Phys. Lett. 136B (1984) 196; 138B(E) (1984) 464; M.S. Chanowitz and M.K. Gaillard, Nucl. Phys. B261 (1985) 379. [11] G.J. van Oldenborgh and J.A.M. Vermaseren, Z. Phys. С - Part, and Fields 46 (1990) 425. [12] G. 4 Hooft and M. Veltman, Nucl. Phys. B153 (1979) 365 [13] D.B. Kaplan and H. Georgi, Phys. Lett. 136B (1984) 183. (14] H. Georgi, Pbys. Lett. 151B (1985) 57; Yu.F. Pirogov, Inst, for High Energy Physics preprint, IHEP (1991). Received 20 February,

Джикия Г.З., Пирогов Ю.Ф. Изучение тройной хигтсовской вершины в процессе ^ -Ш при тзвных энергиях. Ответственный эа вгшусж Г.З.Джикия. Подписано к печати 24.02.S2 г.

15 Джикия Г.З., Пирогов Ю.Ф. Изучение тройной хигтсовской вершины в процессе ^ -Ш при тзвных энергиях. Ответственный эа вгшусж Г.З.Джикия. Подписано к печати S2 г. Формат 60x90Дб. Офсетная печать. Печ.л.0,75. Уч-нзд.л.0,93. Тара» 270. Заказ 101. индекс Цена I р.40 к. Институт фвашш высоких энергии, , Московской out. Протвино,

16 j p. 40 к. Лвдекс ПРЕПРИНТ 92-29, И Ф В Э. 1992

Longitudinal/Goldstone Boson Equivalence and Phenomenology of Probing the Electroweak Symmetry Breaking

July, 1995 VPI-IHEP-95-1 TUIMP-TH-95/67 MSUHEP-5071 Longitudinal/Goldstone Boson Equivalence and Phenomenology of Probing the Electroweak Symmetry Breaking Hong-Jian He Department of Physics and Institute