---

Dilepton Production in e−p and e+e−Colliders

arXiv:hep-ph/9206244v1 23 Jun 1992IFP-428-UNCJune 1992Dilepton Production in e−p and e+e−CollidersJyoti Agrawal, Paul H. Frampton and Daniel NgInstitute of Field physicsDepartment of Physics and AstronomyUniversity of North CarolinaChapel Hill, North Carolina 27599-3255In an e−p collider, a striking signature for a dilepton gauge boson is e−p →e+µ−µ−+ anything ; this cross-section is calculated by using the helicity amplitudetechnique. At HERA, with center-of-mass energy √s = 314GeV , a dilepton massabove 150GeV is inaccessible but at LEPII-LHC, with a center-of-mass energy √s =1790GeV , masses up to 650 GeV can be discovered.

In an e+e−collider, the signatureis e+e−→2e+2µ−or 2e−2µ+ . The cross-sections of this process are also calculatedfor the center-of-mass energies √s = 200, 500 and 1000GeV .PACS numbers : 14.80.Er, 12.15.Cc, 12.15.Ji, 13.10.+qTypeset Using REVTEX1

I. INTRODUCTIONIn a recent paper [1], two of the present authors (P.H.F. and D.N.) have studied the phe-nomenology of dilepton gauge bosons predicted by certain simple extensions of the standardmodel of strong and electroweak interactions.

In particular, the existence of a SU(2)L dou-blet (X−−, X−) of vector gauge bosons with lepton number L = 2 is a plausible predictionof a general class of theories in which the electroweak SU(2)⊗U(1) gauge group is expandedto SU(3) ⊗U(1) (e.g. Ref [2]).

The crucial theoretical and practical question is then whatis the mass scale MX ?In Ref. [1], a lower bound of MX > 120GeV was established by studying e+e−→e+e−aswell as the ”wrong” muon decay µ−→e−νe¯νµ ; the s-channel resonance in e−e−→X−−→µ−µ−was also computed.

For polarized muons, a stronger limit [3] of MX > 230 GeV wasestimated. Since certain assumption about the couplings of the dilepton were made in Refs.

[1,2] we shall here entertain a more general range 100GeV < MX < 1TeV , although itshould be borne in mind that the lower end is probably already excluded by existing data.In the present paper, we shall focus on some striking signatures of lepton number violatingprocesses in electron-proton and electron-positron colliders. A light dilepton gauge boson asanticipated in Ref.

[2] couples democratically to the three lepton family associated with e,µand τ. Total lepton number L = Le +Lµ +Lτ is conserved but the separate flavors of leptonLe, Lµ, Lτ are violated.

This is different from the minimal standard model where Le, Lµ, Lτare necessarily separately conserved. This in turn means that there exist dramatic signaturesfor light (below 1TeV ) dileptons which violate Le, Lµ and hence have no background eventsfrom standard model processes; such evidence for a dilepton gauge boson will be accessibleto the next generation of e−p and e+e−colliders as we shall show by explicit estimates ofthe relevant cross sections.In an electron-proton collider one may see the process e−p →e+µ−µ−+ anything withzero background from the Standard Model.

This is relevant to the HERA collider at DESYin Hamburg, Germany; presently beginning operation with 30GeV electrons colliding on2

820GeV protons (√s = 314GeV ) with luminosity [4] 1.6 × 1031cm−2s−1(0.5fb−1yr−1). Inthe future an e−p collider is planned at CERN with 100 GeV electrons on 8000 GeV protons(LEPII-LHC) (√s = 1790GeV ) and luminosity 2 × 1032cm−2s−1(6fb−1yr−1).In an e+e−collider, one may see the background-free process e+e−→2e+2µ−or 2e−2µ+.

This is relevant to LEPII and the Next Linear Collider (NLC) with the center-of-massenergies √s = 200, 500 and 1000GeV and luminosities [4] 1.7 × 1031, 1 × 1032 and 1 ×1033cm−2s−1 (0.5, 3 and 30fb−1yr−1).The outline of this paper is as follows. In Sect.

II, we compute the amplitude of theFeynman diagrams of the above processes. In Sect.

III, the cross-sections are calculatedfor e−p and e+e−colliders. In Sect.

IV, there are some concluding remarks. Appendix Acontains the analysis of production of a real on-shell dilepton; we used this to check ourcomputations.II.

FEYNMAN DIAGRAMS AND HELICITY AMPLITUDESA. PreliminaryFor the processes we consider in this paper, it is most convenient to calculate Feynmandiagrams using the method of helicity amplitudes, particularly when the external particlesare taken to be massless, which is a sensible approximation in the present case.

The formal-ism can be found in Ref. [5].

The outer product of a massless spinor with momentum p andhelicity λ(= ±1) isuλ(p)¯uλ(p) = ωλ ̸p , ωλ = 12(1 + λγ5) . (2.1)Let us define two four-vectors kµ0 and kµ1 with the following properties:k0 · k0 = 0 , k1 · k1 = −1 , k0 · k1 = 0 .

(2.2)Hence any massless spinors with momentum p and helicity λ can be constructed from u−(k0)by the following relations,3

u+(k0) ≠k1u−(k0) , uλ(p) ≠pu−λ(k0)/q2p · k0 . (2.3)The expressions in Eq.

(2.3) can be verified by substituting into Eq.(2.1). From the secondequation, we have uλ(−p) = iuλ(p).

Therefore, there is an (unobservable) overall phasewhen we replace an antifermion spinor by a fermion spinor.For massless spinors, there are only two non-zero invariant products which are definedas follows,s(p, q) = ¯u+(p)u−(q) = −s(q, p) , t(p, q) = ¯u−(p)u+(q) = [s(q, p)]∗. (2.4)In fact, it is enough to derive the expression of s by using Eqs.(2.1)-(2.3).

We obtains(p, q) = ¯u−(k0) ̸p ̸q ̸k1 u−(k0)/q4(k0 · p)(k0 · q)= Tr[̸p ̸q ̸k1 ̸k0 ω+]/q4(k0 · p)(k0 · q) . (2.5)The expression for t(p, q) can be obtained from the second equation in Eq.(2.4).

To calcu-lation the invariant quantity s(p, q), we can choose k0 and k1 to be, for example,k0 = (1, 1, 0, 0) , k1 = (0, 0, 1, 0) . (2.6)With the help of Eqs.

(2.5) and (2.6), s(p, q) are given bys(p, q) = (py + ipz)"q0 −qxp0 −px#1/2−(qy + iqz)"p0 −pxq0 −qx#1/2. (2.7)Using Eqs.

(2.1) and (2.4), we can derive the following useful formulae:γµu±(p)¯u±(q)γµ = −2u∓(q)¯u∓(p) ,(2.8a)γµu+(p)¯u−(q)γµ = 2ω−t(q, p) ,(2.8b)γµu−(p)¯u+(q)γµ = 2ω+s(q, p) . (2.8c)Therefore, we can express any amplitude with external massless fermions in terms of theinvariant quantities s and t. For more general applications of the helicity amplitude techniqueinvolving massive particles, the reader is recommended to read Ref.

[5]. For the purpose ofthis paper, however, the above preliminary introduction is sufficient.4

B. The amplitudes of e−q →e+2µ−qIn this section, we will compute the helicity amplitudes for the process e−q →e+2µ−q .The Feynman diagrams are shown in Fig.

1. Using the Feynman Rules given in Ref [1], thecorresponding amplitudes are given byAmp(a) = g3l√2!2e2Qq−1(p2 −p4)2−1(p5 + p6)2 −M2X + iMXΓX×1(p3 + p5 + p6)2M(a) ,(2.9a)Amp(b) = g3l√2!2e2Qq−1(p2 −p4)2−1(p5 + p6)2 −M2X + iMXΓX×1(−p1 + p5 + p6)2M(b) ,(2.9b)Amp(c) = 2 g3l√2!2e2Qq−1(p2 −p4)2−1(p1 −p3)2 −M2X + iMXΓX×−1(p5 −p6)2 −M2X + iMXΓXM(c) ,(2.9c)Amp(d) = g3l√2!2e2Qq−1(p2 −p4)2−1(p1 −p3)2 −M2X + iMXΓX×1(p1 −p3 −p5)2M(d) ,(2.9d)Amp(e) = g3l√2!2e2Qq−1(p2 −p4)2−1(p1 −p3)2 −M2X + iMXΓX×1(p1 −p3 −p6)2M(e) ,(2.9e)whereM(a) = ¯u(p4)γαu(p2)¯u(p6)γµγ5C¯uT(p5)vT(p3)Cγµγ5(̸p3+ ̸p5+ ̸p6)γau(p1) ,(2.10a)M(b) = ¯u(p4)γαu(p2)¯u(p6)γµγ5C¯uT(p5) vT(p3)Cγα(−̸p1+ ̸p5+ ̸p6)γµγ5u(p1) ,(2.10b)5

M(c)= ¯u(p4)γαu(p2)¯u(p6)γµγ5C¯uT(p5)vT(p3)Cγβγ5u(p1)× [(p2 −p4 + p5 + p6)βgµα + (−p5 −p6 −p1 + p3)αgµβ+(p1 −p3 −p2 + p4)µgαβ] ,(2.10c)M(d) = ¯u(p4)γαu(p2)¯u(p6)γα(̸p1−̸p3−̸p5)γµγ5C¯uT(p5)vT(p3)Cγµγ5u(p1) ,(2.10d)M(e) = ¯u(p4)γαu(p2)¯u(p6)γµγ5(̸p1−̸p3−̸p6)γαC¯uT(p5)vT(p3)Cγµγ5u(p1) . (2.10e)ΓX is the total width of X−−which decays into e−e−, µ−µ−and τ −τ −democratically.

Aftersome Dirac matrix manipulation, Eq. (2.10c) can be rewritten asM(c) = −M(a) −M(c) .

(2.11)For massless spinors, we can replace v(p) by u(p). Therefore, we can decompose M(a) −(e) into various helicities as follows:M±±±(a) =¯u+(p4)γαu+(p2)¯u−(p4)γαu−(p2)¯u+(p6)γµu+(p5)−¯u−(p6)γαu−(p5)×¯u+(p3)γµ(̸p3+ ̸p5+ ̸p6)γαu+(p1)−¯u−(p3)γµ(̸p3+ ̸p5+ ̸p6)γαu−(p1),(2.12a)M±±±(b) =¯u+(p4)γαu+(p2)¯u−(p4)γαu−(p2)¯u+(p6)γµu+(p5)−¯u−(p6)γµu−(p5)×¯u+(p3)γα(−̸p1+ ̸p5+ ̸p6)γµu+(p1)−¯u−(p3)γα(−̸p1+ ̸p5+ ̸p6)γµu−(p1),(2.12b)M±±±(c) = −M±±±(a) −M±±±(b) ,(2.12c)M±±±(d) =¯u+(p4)γαu+(p2)¯u−(p4)γαu−(p2)¯u+(p6)γα(̸p1−̸p3−̸p5)γµu+(p5)−¯u−(p6)γα(̸p1−̸p3−̸p5)γµu−(p5)×¯u+(p3)γµu+(p1)−¯u−(p3)γµu−(p1),(2.12d)6

M±±±(e) =¯u+(p4)γαu+(p2)¯u−(p4)γαu−(p2)¯u+(p6)γα(̸p1−̸p3−̸p6)γµu+(p5)−¯u−(p6)γα(̸p1−̸p3−̸p6)γµu−(p5)×¯u+(p3)γµu+(p1)−¯u−(p3)γµu−(p1). (2.12e)Since Mλ1λ2λ3 = M∗−λ1−λ2−λ3, we need calculate only the helicity amplitudes,M+±±(a) −(e),explicitly in terms of s and t. The results are given in Table 1.III.

CROSS-SECTIONSSince the violation of Le and Lµ conservation is clearly evidenced by the processes e−q →e+2µ−q and e+e−→2e+2µ−or 2e−2µ+ , it is totally free of the minimal standard modelbackground. Before we proceed, let us justify neglecting the Feynman diagrams in whichthe photon of Figs.

1(a)-(e) is replaced by a Z-boson. Aside from the suppression dueto the mass in the Z propagator, the axial vector couplings of electron and Z-boson donot contribute in this process because of the Fermi statistics, see Ref.

[1]. Only the vectorcoupling of Z contributes, but it is proportional to gv = ( 14 −sin2θW) ≃0.02.

The threeboson coupling of X−−−X++−Z is also proportional to gv from the group theory. Therefore,the Z-boson contributes at most 0.5% to the processes and it can be safely neglected.A.

e−p collidersTo evaluate the production cross-section for the process e−p →e+µ−µ−+ anything inthe electron-proton colliders, we use EHLQ [6] parton structure functions (set 1), Fq(x) forquark q. Hence the production cross-section for the process is given byσ(MX) =Z 10 dxXqFq(x, Q2)ˆσ(√ˆs = xs, MX) ,(3.1)where ˆσ is the elementary cross-section of the process e−q →e+2µ−q ; x is the fractionalmomentum of the proton carried by the quark q, hence√ˆs is the center of mass energy7

available for e−q →e+2µ−q . Q2, defined to be −(p2 −p4)2, is the scale for the structurefunctions for quarks.

The result for σ(MX) are shown in Fig. 2 for the cases √s = 314GeV(HERA) and 1790GeV (LEPII-LHC).For HERA, the planned luminosity is 1.6 × 1031cm−2s−1 giving an annual integratedluminosity of 0.5fb−1yr−1.

From Fig. 2, we see that there will be less than one event peryear if the mass of the dilepton is heavier than 120GeV .

The situation become hopeless forMX > 150GeV without an upgrade in energy and/or luminosity. For example, an up-gradein center of mass energy up to 400GeV will allow, for the same luminoisity, discovery ofdileptons up to about 200GeV .

We thus conclude, given the mass bounds mentioned in theintroduction, that the chance of HERA discovering such a dilepton state is very marginal.At LEPII-LHC with √s = 1790GeV the prospects for dilepton discovery are far better.The expected luminosity is about 2 × 1032cm−2s−1 and hence annual integrated luminosity6fb−1yr−1. Requiring at least 2 events per year for e−p →e+µ−µ−+ anything , we candetect MX up to 650GeV .B.

e+e−collidersAt an e+e−collider, dilepton signatures include e+e−→2e+2µ−or 2e−2µ+ .Thiscalculation is quite similar to e−q →e+2µ−q described above and we include also thecharge-conjugation of the corresponding Feynman diagrams. We have computed the resultfor the center-of-mass energies √s = 200GeV (LEPII), 500GeV and 1000GeV (possible NLCenergies).

The results are displayed in Fig. 3.

Requiring at least 2 events per year, we candetect MX up to 180,450 and 950 GeV in e+e−colliders with energies √s = 200, 500 and1000GeV assuming the integrated luminosities [4] to be 0.5, 3 and 30 fb−1yr−1 respectively.The amplitude-squared for the real production of the dilepton is given in the AppendixA. The production cross-sections are also calculated and compared with the curves in Figs.

(2) and (3). We find that the contribution from the Feynman diagrams Figs.

1 (d) and(e) are at most 10% relative to that of other diagrams in a wide range of dilepton mass8

MX except at the high values of MX in which the curves have longer tails. Therefore, itis important to include Figs.

1 (d) and (e) in our calculation in order to provide a betterestimation for the maximum MX being probed in high energy colliders.IV. CONCLUSIONWe have considered a direct search for doubly-charged dilepton X−−(X++) by lepton-number violating processes in e−p and e+e−colliders.

The mass of X−−ranging from 100to 1000GeV is expected from the theory of SU(15). The striking signature for a dileptongauge boson is e−p →e+µ−µ−+anything in an e−p collider and e+e−→2e+2µ−or 2e−2µ+in an e+e−collider.

The chance of discovering a dilepton at HERA is very marginal unlessMX is less than 150GeV . The direct discovery of such a dilepton state depends on futurecolliders such as LEPII-LHC and NLC at which interesting mass ranges will be explored.ACKNOWLEDGMENTSWe thank S. L. Glashow for a useful suggestion.

This work was supported in part by theU. S. Department of Energy under Grant No.

DE-FG05-85ER-40219.9

REFERENCES[1] P. H. Frampton and D. Ng, Phys. Rev.

D 45, 4240 (1992)[2] P. H. Frampton and B. H. Lee, Phys. Rev.

Lett. 64, 619 (1992)[3] E. D. Carlson and P. H. Frampton Phys.

Lett. B, 1992 (to be published)[4] Particle Data Group, Phys.

Lett. B 239, 1 (1990)[5] R. Kleiss and W. J. Stirling, Nucl.

Phys. B 262, 235 (1985)[6] E. Eichten et al., Rev.

Mod. Phys.

56, 579 (1984)10

FIGURESFIG. 1.

Feynman diagrams for e−q →e+µ−µ−qFIG. 2.

Cross-sections for the process e−p →e+µ−µ−+ anything with Q2 > 25GeV 2 at thecenter-of-mass energies √s = 314GeV (solid line) and √s = 1790GeV (dashed line)FIG. 3.

Cross-sections for the process e+e−→2e−2µ+ or 2e+2µ−with Q2 > 25GeV 2 at thecenter-of-mass energies √s = 200GeV (solid line), √s = 500GeV (dashed line) and √s = 1000GeV(dotted line)11

TABLESTABLE I. Helicity amplitudes for the Feynman diagrams shown in Fig. 1M+ + ++ + −+ −++ −−M(a)4t(p2, p1)s(p6, p3)−4t(p5, p3)s(p4, p1)−4t(p2, p1)s(p5, p3)4t(p6, p3)s(p4, p1)×[t(p5, p3)s(p4, p3)×[t(p2, p3)s(p6, p3)×[t(p6, p3)s(p4, p3)×[t(p2, p3)s(p5, p3)+t(p5, p6)s(p4, p6)]+t(p2, p5)s(p6, p5)]+t(p6, p5)s(p4, p5)]+t(p2, p6)s(p5, p6)]M(b)4t(p5, p1)s(p4, p3)−4t(p2, p3)s(p6, p1)−4t(p6, p1)s(p4, p3)4t(p2, p3)s(p5, p1)×[t(p2, p5)s(p6, p5)×[t(p5, p6)s(p4, p6)×[t(p2, p6)s(p5, p6)×[t(p6, p5)s(p4, p5)−t(p2, p1)s(p6, p1)]−t(p5, p1)s(p4, p1)]−t(p2, p1)s(p5, p1)]−t(p6, p1)s(p4, p1)]M(c)−M(a) −M(b)M(d)4t(p1, p5)s(p4, p6)−4t(p3, p5)s(p4, p6)−4t(p2, p6)s(p3, p5)4t(p2, p6)s(p1, p5)×[t(p2, p1)s(p3, p1)×[−t(p2, p3)s(p1, p3)×[−t(p1, p3)s(p4, p3)×[t(p3, p1)s(p4, p1)−t(p2, p5)s(p3, p5)]−t(p2, p5)s(p1, p5)]−t(p1, p5)s(p4, p5)]−t(p3, p5)s(p4, p5)]M(e)4t(p2, p5)s(p3, p6)−4t(p2, p5)s(p1, p6)−4t(p1, p6)s(p4, p5)4t(p3, p6)s(p4, p5)×[−t(p1, p3)s(p4, p3)×[t(p3, p1)s(p4, p1)×[t(p2, p1)s(p3, p1)×[−t(p2, p3)s(p1, p3)−t(p1, p6)s(p4, p6)]−t(p3, p6)s(p4, p6)]−t(p2, p6)s(p3, p6)]−t(p2, p6)s(p1, p6)]12

APPENDIX A:In this appendix, we will present the calculation of the real production of dilepton X−−.If there is sufficient center-of-mass energy and the dilepton is light enough, it will be possibleto produce a real dilepton in the final state. This limit of light dilepton mass provides, inany case, a useful check on all the calculations given in the main text.

Clearly, for a lightdilepton, the calculation based on the Feynman diagrams given in Figure 1 with a Breit-Wigner form of the dilepton propagator should agree with a real dilepton calculation usingthree-body (rather than four-body) phase space. It is because of the fact that Figs.

1(a)-(c)are dominant over Figs. 1 (d) and (e).

The success of this comparison gives us confidencethat the four-body phase space calculation in the main text is reliable. Only the first threediagrams in Fig.

1 are relevant. The amplitude isAmp = Qqe2 g3l√2ǫµ(p)aµαbα1(p2 −p4)2 ,(A1)whereaµα = vT(p3)Cγµγ5̸p3+ ̸p(p3 + p)2γau(p1) + vT(p3)Cγα−̸p1+ ̸p(−p1 + p)2γµγ5u(p1)+vT(p3)Cγβγ5u(p1)−2(p1 −p3)2 −M2X[(p2 −p4 + p)βgµα+(−p −p1 + p3)αgµβ + (p1 −p3 −p2 + p4)µgαβ] ,(A2)andbα = ¯u(p4)γαu(p2) ,(A3)where p1 and p3 are the momenta for the electron and positron; p2 and p4 are the momentafor the initial and final quarks; p and ǫµ(p) are the momenta and polarization vector for thedilepton X−−.

Here eQq is the quark electric charge and g3l/√2 is the coupling constantfor the X++ −e −e interaction. We have neglected the unimportant width of X−−in thepropagator.

Notice that (p2 −p4)µaµα = 0 because of electromagnetic gauge invariance.Using momentum conservation and Dirac algebra (see Eq. (2.11) in the text), aµα in theEq.

(2) can be rewritten as13

aµα ="1(p3 + p)2 +2(p1 −p3)2 −M2X#vT(p3)Cγµ(̸p3+ ̸p)γαγ5u(p1)+"1(−p1 + p)2 +2(p1 −p3)2 −M2X#vT(p3)Cγα(−̸p1+ ̸p)γµγ5u(p1) . (A4)pµaµα is not zero because the dilepton is not coupled to a conserved current; in fact it isgiven explicitly bypµaµα = 2(p2 −p4)2(p1 −p3)2 −M2XvT(p3)Cγαγ5u(p1) .

(A5)Using the polarization sum P ǫµ(p)ǫν(p) = −gµν +pµpν/M2X, the amplitude-squared is givenby|Amp|2 = Qqe4 g3l√2!21(p2 −p4)4 −gµν + pµpνM2X!aµαa∗νβbαbβ∗. (A6)Therefore |Amp|2, with the help of Eq.

(5), can be calculated to be|Amp|2 = Q2qe4 g3l√2!264(p2 −p4)4 1(p3 + p)2 +2(p1 −p3)2 −M2X!2×h2 p3.p (p1.p2 p.p4 + p1.p4 p.p2) −M2X (p1.p2 p3.p4 + p1.p4 p3.p2)i+ 1(−p1 + p)2 +2(p1 −p3)2 −M2X!2×h2 p1.p (p3.p2 p.p4 + p3.p4 p.p2) −M2X (p1.p2 p3.p4 + p1.p4 p3.p2)i+2 1(p3 + p)2 +2(p1 −p3)2 −M2X! 1(−p1 + p)2 +2(p1 −p3)2 −M2X!×−M2X p1.p3 p2.p4 + 2 p1.p p3.p2 p3.p4 −2 p3.p p1.p2 p1.p4+(2 p1.p3 + p1.p −p3.p)(p1.p2 p3.p4 + p1.p4 p3.p2)+p1.p3 (p1.p2 p.p4 + p1.p4 p.p2 −p.p2 p3.p4 −p.p4 p3.p2)+2 1(p1 −p3)2 −M2X!2 (p2 −p4)4M2X(p1.p2 p3.p4 + p1.p4 p3.p2).

(A7)The above equation is used to calculate the production of a real dilepton in the e−p ande+e−collders. We then compared this result with Figs.

(2) and (3). We find agreementfor light dilepton mass with the curves in Figs.

(2) and (3) and that, as expected, the fullcalculation allowing a virtual dilepton gives an extra contribution in the tail of high MXvalues.14

---

출처: arXiv:9206.244 • 원문 보기