Production of heavy leptons from gluon fusion

Volume 156B, number 5,6 PHYSICS LETTERS 27 June 1985

P R O D U C T I O N O F H E A V Y L E P T O N S F R O M G L U O N F U S I O N

Scott S.D. W I L L E N B R O C K and D u a n e A. D I C U S

Theory Group and Center for Particle Theory, University of Texas, Austin, TX 78712, USA

Received 12 February 1985

We show the production of heavy lepton pairs in pp and pp collisions at multi-TeV energies is dominated by gluon fusion for lepton masses above 100-200 GeV. We also find that the decay of the Higgs boson into heavy lepton pairs may provide a clear signal of Higgs production.

Since the discovery of the muon, physicists have been puzzling over the curious mass spectrum of the elementary fermions. The most striking qualitative feature o f this spectrum is the occurrence of generations or families o f fermions, each of which contains two leptons and two flavors of quarks. At present the number of known generations is three. Despite con- siderable theoretical effort and many interesting spe- culations, no one theory has emerged which accounts for the observed mass spectrum or even provides an explanation of the number of generations.

In light of this situation, we should take seriously the possible existence o f a fourth generation of quarks and leptons. Extrapolating from the three known generations, we might expect these particles to have masses on the order o f 1 0 0 - 1 0 0 0 GeV. Pro- duct ion of these particles will require very high energy and luminosity accelerators. Furthermore, their large masses indicate that they will most likely be very unstable, requiring the use of indirect means to detect them.

In this let ter we will concentrate on the product ion of heavy lepton pairs in pp and ~p colliders. For lepton masses below 100 -200 GeV the dominant contri- but ion to di lepton product ion is the familiar Dre l i - Yan process [1 ], shown in fig. l a ,1 . The cross sections for di lepton product ion via the DreU-Yan mechanism in pp collisions are shown in fig. 2 for various collider energies and lepton masses ,2 . The rapid decrease in the cross section with increasing lepton mass is largely due to the l[g behavior o f the

0370-2693/85•$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Dre l l -Yan process, where x/~--is the total CM energy o f the qua rk -an t iqua rk subprocess. Requiring at least 150 events in the rapidi ty interval - 1 . 5 < y < 1.5 with an integrated luminosity o f 1039 cm - 2 allows a minimum cross section (do/dy)ly=o of about 5 × 10 -5 nb, corresponding to a maximum detectable lepton mass of 140, 180 and 240 GeV at x/~ -= 10, 20 and 40 TeV, respectively ,3 . The reach of a ~p collider is essentially the same [2].

We have found, however, that there is another con- t r ibut ion to dflepton product ion in pp a ~p col- sions at multi-TeV energies which dominates the Drell- Yan process for lepton masses above 100 -200 GeV. In contrast to the Dre l l -Yan process, the lepton pair is produced by the fusion o f two gluons, as shown in figs. l b and lc . The gluon fusion cross sections are shown in figs. 2 and 3 superimposed on the DreU-

All cross sections are calculated in the patton model, using set 2 of the distribution functions (A = 290 MeV) given in the appendix of ref. [2]. We also use the gauge boson masses and weak mixing angle given in section IV.A. of ref. [2], and describe the sealing of the strong coupling constant a s with the usual renormalization group equation, (2.42) in ref. [2].

.2 This graph may be compared with fig. 166 of ref. [2], which we have found to be in error. The kinematical sup- pression of the cross section due to the mass of the produced leptons is not included correctly. This amounts to the cross sections being too large by about 30%.

,3 There are additional contributions to dilepton production, but they do not qualitatively affect out conclusions. See ref. [3].

429


(Q)

q E

(b)

(c)

E g

H

Q E g

Fig. 1. Feynman diagrams for dilepton production in pp and ~p collisions: (a) Drell-Yan process via an intermediate photon or Z; (b), (c) gluon fusion process via a virtual quark loop and an intermediate Z or Higgs boson.

Yah cross sections. The two figures differ only in the I-Iiggs mass assumed. We have drawn bands rather than curves to allow for our ignorance of the masses of the quarks of the fourth generation, which contribute via the virtual quark loop that connects the gluons to the intermediate Z and I-Iiggs bosons. As these graphs show, the cross sections at x/s -= 40 TeV are large enough to allow the possible detection of leptons of masses up to 1 TeV and beyond! The gluon fusion mechanism may also extend the reach of colliders at ~s '= 10 and 20 TeV, depending on the mass of the Higgs boson and the luminosities attained.

At first sight it may seem surprising that the contribution to dllepton production from gluon fusion can be larger than the Drell-Yan contribution. After all, the gluon fusion process is second order in the strong interaction (~a2) while the Drell-Yan process is zeroth order ,4. This naive reasoning neglects three important enhancements to the gluon fusion mechanism for the production of heavy leptons at high energies:

(i) The gluon fusion cross section is independent of energy to first approximation, while the Drell-Yan

¢4 At the energies we are considering, '~s ~ 0.1.

process decreases like 1/g. This is an enhancement if x/~--> M w, since the gluon fusion process is proportional to I[M2w .

(ii) The coupling of the Z and Higgs bosonss to the leptons in the gluon fusion process is not just the usual weak coupling g but rather g(mL/MW). This is an enhancement i fm t > M w. There are terms due to Z exchange in the Drell-Yan process which are simi- larly enhanced, but they are also proportional to mq/ M w where mq is the mass of the light partons, and hence small.

(iii) The rate of gluon-gluon interactions is much greater than the rate of quark-antiquark interactions in both pp and ~p collisions. This effect is present only if the coUider energy is much greater than the mass of the produced leptons, and is increasingly more important at higher energies for a given lepton ITlaSS.

The enhancements (i) and (ii) serve to make the subprocess cross section for gluon fusion at most comparable to that of the Drell-Yan process for the lepton masses we are considering. It is (iii) which allows the gluon fusion mechanism to dominate the DreU-Yan process for heavy dilepton production in pp and ~p collisions at high energies.

We now undertake a more detailed description of the gluon fusion mechanism. In order to produce elec- troweakly interacting leptons from strongly interacting gluons we must have a virtual quark loop, since quarks are the only particles with both strong and electroweak properties. The quark loop must be con- nected to the leptons by a neutral boson, which sug- gests the photon, Z and I-Iiggs particles. However, the photon exchange diagram vanishes due to Furry's theorem; this also shows that only the axial vector coupling of the Z contributes.

The total cross section for the gluon fusion subprocess is calculated from the coherent sum of figs. lb and lc and the corresponding diagrams with the gluon lines crossed. However, the Z exchange diagram is an antisymmetric two-index tensor due to the axial vector coupling of the Z, while the I-Iiggs exchange diagram is a symmetric two-index tensor; the interfer- ence term therefore vanishes, and the diagrams add incoherently. This allows us to study the contributions from Z and I-Iiggs exchange separately.

The spin and color averaged cross section for gluon fusion dilepton production via Z exchange is

430


6Z(g ) = (1/20487rXt~2t~2/sin40wXmL/Mw)2Mw2~[II 2, (1)

where x/~'is the total CM subprocess energy, M w and m L are the masses o f the W boson and the produced lepton, 0 w is the weak mixing angle, and ~ is the ve- locity of the lepton in the CM frame. Note the ab- sence of a resonance at x/~-= M z; this is due to Yang's theorem [4], which forbids the production of a state of total angular momentum d = 1 from two massless spin one states. The subprocess energy x/~-appears only implicitly in the integral [5]

1 1-x

f f dy xy (2) Q 0 0 xy - mb/g"

The cross section is calculated from a coherent sum of diagrams of the form shown in fig. lb, each with a different flavor of quark in the loop. The mass of the quark mQ appears only in the integral 1, via the ratio m-~/g. Since it is only the axial-vector coupling o f the quark to the Z which contributes to the diagram, the quarks with SU(2)L quantum number T3L = +1/2 contribute positively to the sum while those with T3L = - 1 / 2 contribute negatively. We have de-

noted this with a (+) in (2). Hence the contributions from quarks o f a given generation tend to cancel.

To be more precise, we must consider the amount that a given quark flavor contributes to the sum in (2). Roughly speaking, this contribution is unity if mQ ~< x/~-]2 and zero otherwise. Hence a given generation makes an appreciable contribution only if one quark's mass is less than x/~2 and the other's is greater than this. Since the minimum value of V~'is 2m L, Ill 2 counts the number of generations which have one quark with a mass less than or about equal to m L and the other quark somewhat more massive than m L.

Since we are considering the production o f leptons of mass m L t> 100 GeV, it is clear that the three known generations o f quarks make a negligible contribution to the Z exchange cross section , s . Hence it is only the fourth generation of quarks which can give a large contribution to the Z exchange process, provid- ed their masses satisfy the criteria above.

The spin and color averaged cross section for gluon

*s We are assuming a top quark mass of 30-50 GeV as report- ed in ref. [6].

fusion dilepton production via Higgs exchange is

bH(g ) = (1/4608rr)(tx2a2/sin40wXmL/Mw)2M~2f33[N] 2

2 2 × g2/[(g _ m2H)2 + FHmH], (3)

where m H and F n are the Higgs mass and width. Al- though similar to the Z exchange cross section (1), there are several differences. There is a resonance at x/T= m H, i.e. for the production o f a real Higgs. Re- call that there is no such resonance in the Z exchange process since the production o f a real Z is forbidden by Yang's theorem. The I-Iiggs exchange cross section also has a smaller numerical coefficient as well as two extra powers of/3. Lastly, the integral N is quite different from the integral I, as we will now discuss.

The integral N is well known in the literature on Higgs boson production from gluon fusion [7]. It is given by

1 1-x g= 3 f dy 1 - 4 x y

Q o o 1 - xyg/m2Q (4)

Roughly speaking, N receives a contribution o f unity from every quark with mass mQ > 0.2 V~. Since >1 2 m L , N counts the number of quarks with mass greater than or about equal to m L. For m L/> 100 GeV, this implies that N will count only the quarks in the fourth generation and any generations beyond this.

We have seen that the cross sections for gluon fusion dilepton production via Z and I-Iiggs exchange depend on the masses of the quarks of the fourth generation. We do not know the masses o f these quarks, which we will refer to as m U and m D for the quark with T3L = +1/2 and T3L = --1/2, respectively. However, based on our experience with the second and third generations, we might expect m D ~ m L and m U >> m D. If this is the case then Ill 2 ~ 1 while INI 2 "~ 4, assuming there are no generations beyond the fourth, which would further contribute to N. This factor of 4 compensates the smaller numerical coefficient and the ex t r a ~2 factor of the Higgs exchange cross section (3). The result is that the contributions to dilepton production from Z and Higgs exchange are comparable at subprocess energies away from the Higgs resonance. At subprocess energies near the Higgs resonance the Higgs exchange process is enhanced by a factor of approximately (mH/FH) 2.

431


We would like to emphasize that the total cross section for di lepton product ion via gluon fusion is ac- tually rather insensitive to the masses o f the fourth generation quarks. As long as these masses are greater than or equal to m L the I-Iiggs contr ibution is large. The Z contr ibut ion is sensitive to the quark masses, but since it is at most comparable to the I-Iiggs con- t r ibut ion it can only affect the total cross section by a factor of 2. One should also keep in mind that the I-Iiggs contr ibution will be considerably larger i f there are addit ional generations beyond the fourth.

Strictly speaking, our calculation is only legitimate for lepton masses below 1.2 TeV and quark masses below 5 5 0 - 7 0 0 GeV. These are the bounds imposed by requiring that the ampli tude for F F ~ F F satisfy the condit ion o f tree unitari ty, where F is the appro- priate fermion [8]. These bounds signal the l~oint at which the theory becomes strongly coupled and no longer permits a perturbative analysis. Nevertheless, we should expect large cross sections even if some or all o f the fermion masses involved are greater than the unltari ty bounds. With this in mind we now proceed to discuss the cross sections shown in figs. 2 and 3.

In figs. 2 and 3 we have graphed the contr ibution from gluon fusion to the rapidi ty distr ibution at zero rapidi ty of pp -~ LL as a function o f the mass o f the

P ' - - - ' - Dreh - Yon I0 "~ ii II~ff'X~ / / / / . Glu0n fusion

~ o 40

i0 -~

\ ,, \ ' , ~ Y ' ~ x 4 0 "V'g(TeV)

I0 "~ \, " , . ' ~ ' - . . ',,, 2"o~>>>--...

, ,

0 0.2 0.4 0.6 0.8 I.O 1.2 mL(TeV)

Fig. 3. Same as fig. 3 except that the Higgs mass is chosen to be 500 GeV.

produced lepton ,6 . The rapidi ty distr ibution is obtained from the subprocess cross sections (2) and (4) via

da e-21Yl

"d-yy = !m~L/S d'r FG(~/~ eY)FG(VC~ e-Y)

I I i i i i i i

I~ ' -Dre l l -Yon 10"3 ~ 7 / / / . G l u o n fusion

If! ~ mH= I00 GeV

~= I0 "4 ~

- - o 40

\

I0"~ \ \ 2 0 ~ , A "~ ..

0 0.2 0.4 0.6 0.8 1.0 1.2 mL(TeV)

Fig. 2. (da/dy)lv=o versus the mass of the produced lepton for pp ~ LL at ~ollider energies x/s "= 10, 20, and 40 TeV. The dashed lines are from the Drell-Yan process and the shaded bands from gluon fusion. The l-liggs mass is chosen to be 100 GeV.

X [OZ(7"s)+ (rH('rs)] , (5)

where r = g/s, FG(X) is the number distr ibution of gluons carrying momentum fraction x of the total pro- ton momentum [2], and y is the rapidi ty of the subprocess CM frame with respect to the lab frame. Our graphs were obtained by setting y = 0 in (5).

In fig. 2 we have assumed a relatively light Higgs o f mass 100 GeV. For the fourth generation quark masses we have used m D = m L and m U = m D + Am where we have varied Am from 0 to 600 GeV, producing bands rather than curves. For Am = 0 the cross sec- t ion is due entirely to Higgs exchange. This corresponds to the lower edge of the band. The upper edge corresponds to Am = 600 GeV, in which case the contr ibutions from Z and Higgs exchange are comparable. Increasing Am even further does not increase the cross sections. Since the Higgs mass is below the thresh-

,6 The result for ~p ~ LL is identical since the gluon content of p and ~ is the same.

432


old energy for production of lepton pairs of mass m L/> 100 GeV, the bands do not display the effects of the I-Iiggs resonance.

Fig. 3 is identical to fig. 2 except we have raised the Higgs mass to 500 GeV. The result is an appreciable increase in the cross section for leptons of mass between about 50 and 250 GeV, due to the Higgs resonance. The maximum enhancement occurs at m L = 180 GeV and is an increase of nearly an order of magnitude. In general the maximum enhancement is roughly proportional to (mH/FH) 2. Since F a increas- es rapidly with increasing mH, the resonance effect is bigger for smaller Higgs masses. The maximum cross section occurs at m L ~ mill3 since this is the lepton mass which maximizes the branching ratio o f the I-Iiggs into lepton pairs.

The graphs show that the cross sections due to gluon fusion are not very sensitive to the masses o f the fourth generation quarks, as we have anticipated. Indeed, they are much more sensitive to the I-Iiggs mass. This raises the possibility that the I-Iiggs boson could be detected by observing an excess in the production rate of heavy lepton pairs. As we have mentioned, if there are additional generations beyond the fourth the Higgs contribution will be even greater due to the factor INI 2 in (3), where N counts the number of heavy (mQ f> mL) quark flavors.

The total cross section is obtained by integrating (5) over y from lnx/rmi n to -in%/rmin, where rmi n = 4m2/s. We have found that one may estimate the total cross section from our graphs by multiplying (do/dy)l~y_~__._by the range of the y integration ( - 2 lnx/rmin) and by a factor of 1/3 (1/2 near the Higgs resonance) to allow for the shape o f the rapidi- ty distribution do[dy.

We have discussed only the production of heavy lepton pairs in pp and ~p collisions; one must also consider the detection of these particles. For m L /> 100 GeV the produced leptons will decay very rapidly via L ~ Wv L if v L, the neutrino associated with L, is sufficiently light ~7. Since this neutrino will go un- detected, the signal for heavy lepton production is W+W - plus missing transverse momentum. Whether this may be observed above the background is un- known.

Although we have in mind charged leptons, the

,7 Of course in the standard model neutrinos are assumed to be massless.

gluon fusion mechanism may also be useful for producing heavy neutrinos. The Z exchange cross section (3) is the same since only the axial vector coupling contributes. The I-Iiggs contribution depends on the coupling of the Higgs to massive neutrinos, which varies according to the mechanism used to generate a neutrino mass.

One might also consider the production of heavy quark pairs via our mechanism. However, gluon fusion already contributes to quark production at tree level, and we would be surprised to find that our one-loop process was in fact bigger. The cross section for quark pair production in pp and ~p collisions is shown in fig. 163 of ref. [2]. Comparing with our figures (mul- tiplied by 3 for color) shows that our process is larger only for quark masses which exceed the unitarity bounds, in which case our calculation cannot be trust- ed.

In conclusion, we have shown that lepton pair production via gluon fusion in pp and ~p collisions at multi-TeV energies may allow the detection of leptons of masses up to 1 TeV and beyond in the next generation of accelerators. The same process may be useful for detecting the Higgs boson by observing its decay into heavy lepton pairs.

We would like to thank C. Burgess, S. Eubank, S. Nandi, J. Polchinski, and S. Weinberg for valuable discussions. The work of S.S.D.W. was supported by the National Science Foundation under Grant No. PHY 83-04629 and in part by the Robert A. Welch Founda- tion; the work of D.A.D. was supported in part by the US Department o f Energy.

[1 ] S.D. DreU and T.M. Yan, Phys. Rev. Lett. 25 (1970) 316; Ann. Phys. (NY) 66 (1971) 578.

[2] E. Eichten, I. Hinchliffe, K. Lane and C. Quigg, Rev. Mod. I "s ¢6 (1984) 579.

[3] H.D. tolitzer, Phys. Lett. 70B (1977) 430; H. Georgi, Phys. Rev. D17 (1978) 3010.

[4] C.N. Yang, Phys. Rev. 77 (1950) 242. [5 ] J.S. Bell and R. Jackiw, Nuovo Cimento 60 (1969) 47. [6] UA1Collab., G. Arnison et al., Phys. Lett. 147B (1984)

493. [7] H.M. Georgi, S.L. Glashow, M.E. Machacek and D.V.

Nanopoulos, Phys. Rev. Lett. 40 (1978) 692; see also H.A. Gordon et al., in: Proc. 1982 DPF Summer Study on Elementary particle physics and future facili- ties, eds. R. Donaldson et al. (Snowmass, CO, June 28- July 16, 1982) p. 161.

[8] M.S. Chanowitz, M.A. Furman and I. Hinchliffe, Phys. Lett. 78B (1978) 285; Nucl. Phys. B153 (1979) 402.

433

Documents

Production of heavy leptons from gluon fusion