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Multiple photon effects in fermion-(anti)fermion

arXiv:hep-ph/9203217v1 18 Mar 1992UTHEP-92-0101January 1992Multiple photon effects in fermion-(anti)fermionscattering at SSC energies∗D. B. DeLaney, S. Jadach,† Ch.

Shio, G. Siopsis, B. F. L. WardDepartment of Physics and AstronomyThe University of TennesseeKnoxville, TN 37996–1200ABSTRACTWe use the theory of Yennie, Frautschi and Suura to realize, via Monte Carlomethods, the process f(¯f) →f′ (¯f)′ + nγ at SSC and LHC energies, where f andf′ are quarks or leptons. QED infrared divergences are canceled to all orders inperturbation theory.

The resulting Monte Carlo event generator, SSC-YFS2, isused to study the effects of initial-state photon radiation on these processes in theSSC environment. Sample Monte Carlo data are presented and discussed.

We findthat the respective multiple-photon effects must be taken into account in discussingprecise predictions for SSC physics processes.∗Supported in part by the Texas National Research Laboratory Commission for the Super-conducting Super Collider Laboratory via grant RCFY9101, and by the Department ofEnergy Contract No. DOE-AC05-76ER03956.† Permanent address: Institute of Physics, Jagellonian University, Cracow, Poland.

1. IntroductionNow that the SSC is under construction, it is extremely important to preparefor the maximal physics utilization and exploration of the new frontier which itwill probe.

In particular, it is of some import to determine the effects of higher-order radiative corrections on the SSC physics processes of interest so that optimaldiscrimination between signal and background can be realized. In this paper, weexplore the first step in the determination of such corrections by computing themfor the basic QED-related effects in f(¯f) →f′ (¯f)′) + nγ at SSC energies usingthe methods which two of us (S.J.

and B.F.L.W.) introduced for the analogousprocesses e+e−→f ¯f + nγ for high precision Z0 physics at SLC and LEP.

Thus,here, we extend the SLC/LEP Monte Carlo event generator YFS2 Fortran in thefirst paper in Ref. [1] to the SSC physics environment.

An analogous study ofthe multiple-gluon radiation in SSC processes such as qq →q′q′ + nG (where q, q′represent quarks and G a gluon) will appear elsewhere [2].In the SSC environment, multiple-photon and multiple-gluon radiative effectsare expected to be important, in partial analogy with the significance of multiple-photon radiation in the SLC and LEP environments. The analogy is partial becausethe would-be resonance at the SSC, the Higgs, is actually quite broad comparedto the Z0 at SLC and LEP.

Nonetheless, if ¯k0 ≈0.001√s/2 is a typical infraredresolution factor for an SSC detector, then the probability of an incoming u-quarkto radiate at the SSC is, e.g.,P(¯k0 ≤k ≤√s/2) ≈2αQ2uπ(log(√s/6mu)2 −1) log(√s/¯k0)= 0.44 ,(1.1)where √s = 40 TeV and mu ∼5.1 MeV. Hence, such radiation and its attendanteffect on the respective SSC event structure must be computed to the standardSLC/LEP precision to assess its interplay with detector cuts, physics signals andphysics backgrounds .

This would then leave only the multiple-gluon radiative ef-2

fects to be taken into account to gain a complete view of higher-order radiative ef-fects to SSC physics processes. Such gluon radiation will be taken up elsewhere [2].Our strategy is to treat the incoming quark radiation via the YFS theory [3]so that we realize it on an event-by-event basis using the methods in Ref.

[1].The full multiple-photon character of the final state, including the physical four-momentum vectors of the photons, is then made available to the users of theattendant new multiple-photon event generator SSC-YFS2. The implementation ofarbitrary detector cuts on the respective simulated cross sections is straightforward.A logical next step is to include the effects of the final-state multiple-photonradiation via the extension of the Monte Carlo event generators BHLUMI2.0 For-tran [1,4] and YFS3 Fortran [3] to the SSC environment in analogy with our ex-tension of YFS2 Fortran in the current work.

We shall discuss these extensions ina future publication [2].Our work is organized as follows in the paper: in the next Section, we reviewthe YFS methods as they are implemented in YFS2 in the first paper in Ref. [1], sothat this paper is self-contained; in Section 3, we discuss how we extend YFS2 toSSC processes and energies to get the Fortran Monte Carlo event generator SSC-YFS2; in Section 4, we present some sample Monte Carlo data for SSC physicsprocesses and comment on their implications; finally, in Section 5, we present ouroutlook and summary remarks.2.

Review of YFS methodsIn this Section, we review the methods used in Ref. [1] to realize the YFStheory via the Monte Carlo event generator YFS2 Fortran for e+e−→f ¯f + nγ,f ̸= e, in the Z0 energy regime.

We begin by recalling the key ingredients of thesemethods.The YFS Monte Carlo methods in Ref. [1] take advantage of the expansion of3

the total cross section for the process illustrated in Fig. 1,e+(p1) + e−(p2) −→f(q1) + ¯f(q2) + γ(k1) + .

. .

+ γ(kn) ,(2.1)in terms of the YFS hard-photon residuals [3] ¯βn(k1, . .

. , kn), which are free of allvirtual and real infrared divergences to all orders in the QED coupling constant α,and of the products of the YFS infrared emission factors [3]eS(k) = −α4π2 p1p1 · k −p2p2 · k2+ · · · ,(2.2)where .

. .

represents the remaining terms obtained from that shown by the appro-priate substitutions of {(e, p1) , (−e, p2)} with {(Qfe, q1) , (−Qfe, q2)}, with dueattention to signs associated with the direction of the flow of charge. The mostinfrared-singular contribution to the cross section involves n factors of eS, as we seefrom the following expression for the respective cross section:dσ(n) =eS(k1) · · · eS(kn)β0(p1, p2, q1, q2) + .

. .

+ βn(k1, . .

. , kn)× 1n!δ4 p1 + p2 −q1 −q2 −nXi=1ki!d3q1q01d3q2q02d3k1k01· · · d3knk0ne2αReB,(2.3)where B is the YFS virtual infrared function and is given in Refs.

[1] and [3].Basing ourselves on Eq. (2.3), in YFS2 Fortran we proceed as follows.We use the YFS form factor,FY FS(p1, p2, ǫ) = exp2αReB +Z d3kk0 eS(p1, p2, k) (1 −θ(k0 −ǫ√s/2))= expαπ (2(ln(s/m2e) −1) ln ǫ + 12 ln(s/m2e) −1 + π2/3,(2.4)to compensate for the omission of small-energy photons with k0 < ǫ√s/2 forǫ ≪1 from the phase space in (2.3) to all orders in α, as is effected by insertingQni=1 θ(2k0i /√s −ǫ) into (2.3) and summing over all dσ(n).

Here we presume we4

are in the e+e−center-of-mass frame. For YFS2, the hard photon residuals β0,1,2are used, as two of us have explained in Ref.

[1]. This means that, in the YFS2Monte Carlo itself, some choice must be made for the reduction of the n-photon+f ¯f phase space to the j-photon +f ¯f phase space (n = 0, 1, 2, .

. ., j = 0, 1, 2,n ≥j), which is involved in the definition of the residuals βi (i = 0, 1, 2).

Wecall this choice the reduction procedure R and the exact YFS result Pn dσ(n)is independent of it, if it is done according to the rigorous YFS theory.Ourchoice for R is explained in [1]. Finally, we should emphasize that, for efficientevent generation, it is always desirable to generate a background population ofevents according to a set of distributions dσ′(n) which embody all of the generalfeatures of Eq.

(2.3), but which remove unnecessary details, and to restore theexact distributions dσ(n) in (2.3) by rejection methods. In Refs.

[1] we follow thisstrategy in constructing YFS2; in addition, several changes of variables are usedto make the background generation simpler and more efficient from the standpointof CPU time. In this way we have realized the YFS theory for e+e−→f ¯f + nγwith β0, β1, and β2, where we should emphasize that β2 has only been includedin the second-order leading-log approximation [6].In the next section, we discuss how we extend YFS2 to more general incomingf ¯f and ff initial and final states, as well as the modifications needed to make theprogram applicable in the SSC energy regime.3.

YFS2 at SSC energiesIn this section we describe how one extends the YFS2 Monte Carlo program inRef. [1] to realize particle interaction at SSC energies.

Such an extension involvesthe introduction of new physics (mainly through a modification of the Born crosssection), numerical problems (due to the very high energies involved, care is neededfor the accuracy of the formulas), and certain technical problems associated withthe Monte Carlo weight rejection method.5

We begin by discussing the modifications made to introduce the new physicsat SSC energies. The new program, SSC-YFS2, computes the cross section for theinteractionf(p1) +(¯f)(p2) −→f′(q1) +(¯f)′(q2) + γ(k1) + .

. .

+ γ(kn) ,(3.1)where f is any lepton or quark.♮This is still not the most general form of SSCinteractions, because the incoming fermions are of the same type (identical orparticle-antiparticle).We are currently generalizing the program to include allinteractions, and shall report on the results shortly [2].The mass parameters mq used for the quarks are the Lagrangian quark masses.We have in mind that the overall momentum transfer in the interactions will belarge compared to the typical momenta inside the proton. In fact, these quarkmass parameters should strictly speaking be running masses mq(µ), where µ is thescale at which they are being probed.

Such a running mass effect is well-known andis readily incorporated in the program, as the accuracy one is interested in maydictate. Thus, with this understanding, further explicit reference to the runningmass effect is suppressed.The interactions realized by YFS2 (Eq.

(2.1)) involve only an exchange of γ andZ0 in the s-channel. For SSC-YFS2, realizing the more general interaction (3.1), γ,Z0 and W ± exchange in the t- and u-channels, accordingly, had to be introduced.This was done by generalizing the Born cross section to include the additionalchannels.

Moreover, in the case of quark interactions, a gluon exchange was addedin all three channels. The running strong coupling constantαs(µ) =12π(33 −2nf) ln{µ2/(ΛMSnf )2},(3.2)was used, where nf is the number of quark flavors below the energy level µ. Inour case, nf = 6, and therefore the QCD parameter ΛMS6is used.

It can easily be♮At present, the program cannot handle third-generation fermions.6

related to the experimentally measured parameter ΛMS4= 238 MeV :ΛMS6= ΛMS5 ΛMS5mt!2/21, ΛMS5= ΛMS4 ΛMS4mb!2/23. (3.3)The masses of the top and bottom quarks were set to mb = 5 GeV and mt =250 GeV, respectively, but the results are little affected by their precise values.Certain numerical problems arise at very high energies, because of the verysmall value of all ratios m/√s, where m is the mass of any interacting particle,and √s = 40 TeV is the energy of the incoming fermions in their center-of-massframe.

Certain formulas had to be rewritten so that such small numbers would notbe ignored by the computer when they should not be; if one is not careful, ratiosof the form 0/0 appear at various places. Working at SSC energies, however, hasthe advantage that all terms of order m2/s or higher can be dropped.

The error isnegligible and leads to a considerable simplification of formulas, and consequentlyto a reduction in computer time.Next we discuss the event-generation procedure. To perform the integral forthe total cross section, we first simplify the form of the differential cross section.Thus the exact cross section dσ is replaced by dσ′, so that the integralRdσ′ canbe performed analytically.

The exact cross section is then computed by rejectingevents according to their weights,w = dσdσ′ . (3.4)Apart from simplicity, we require that dσ′ lead to an efficient generation of events.In YFS2, dσ′ was chosen to be a constant.

In the present case, this is no longerpossible, because of the presence of the t-channel. The cross section has a singu-larity at t ≡(p1 −q1)2 = 0 of the form 1/t2.

To account for the singularity, anangle cutoffθ0 = 100 mrad is introduced. This is in accord with current detector7

capabilities, and can be changed at will. A crude cross section dσ′ of the formdσ′ = A + Bt2 ,(3.5)was chosen, where the constants A and B depend on the interaction.‡ When a u-channel also contributes, a similar term of the form 1/u2 must be added to accountfor the singularity at u ≡(p1 −q2)2 = 0.Finally, we comment on the choice of the reduction procedure which is neededfor the definition of the arguments of the YFS residuals βi (i = 0, 1, 2), as explainedin Section 2.

The reduction procedure is more delicate in the presence of the t-channel, due to the singularity at t = 0. One has to make sure that the weights (3.4)do not become uncontrollably large.This is managed by making t as large aspossible after the reduction.

It is not always possible to increase the reduced tso that the weight (3.4) remains below the maximum weight. The object of thisexercise is to minimize the error originating from the tail of the distribution ofweights above the maximum weight (which is set to 3, but can be changed if sodesired).

This is accomplished by a somewhat involved reduction procedure, whichis an adaptation of the similar procedure in BHLUMI1.xx [7].This concludes our discussion of the modifications in the YFS2 program nec-essary in order to realize interactions at SSC energies. Next, we present some ofour results.‡ A fictitious photon mass cutoffwas also tested, but it turned out to lead to a large weightrejection rate.8

4. Multiple-photon effects at SSC energiesIn this Section we present some results on the effects of multiple-photon initial-state radiation on the incoming qq and q¯q “beams” at SSC energies using our YFSMonte Carlo event generator SSC-YFS2 Fortran.

Our objective is to determinethe size of these effects with an eye toward their incorporation into SSC physicsevent generators. This latter step will be taken up elsewhere [2].We consider, as illustrated in Fig.

2 (the kinematics is summarized in thefigure),q(p1) +(¯q)(p2) −→q′(q1) +(¯q)′(q2) + γ(k1) + . .

. + γ(kn) ,(4.1)at √s = 40 TeV for q, q′ = u, d, s. For definiteness, we will illustrate our resultswith q = u, d, where we use mu = 5.1 ×10−6 TeV, md = 8.9 ×10−6 TeV, and view√s = 40 TeV as our worst-case scenario.

The more typical [8] value √s ≈1640 TeV≈6.7 TeV is also presented here for completeness.For these respective inputscenarios, we shall discuss the following distributions: the number of photons perevent, the value of v = (s−s′)/s, where s′ = (q1 +q2)2, and the squared transversemomentum of the outgoing nγ state. These distributions give us a view of theeffect of this multiple-photon radiation on the incoming quarks and (anti)quarksin the SSC environment, where one is really interested in p¯p →H + X, where His the Standard Model Higgs particle.Considering first the number of photons per event, we have the results in Fig.

3.There, we show that for the uu incoming beams, the mean number ⟨nγ⟩of radiatedphotons is 0.85±0.92 (it is similar for √s = 6.7 TeV). This should be compared tothe dd incoming state, where ⟨nγ⟩is 0.21 ± 0.45.

For reference, we recall [1] thatat LEP/SLC energies, the corresponding value of ⟨nγ⟩is, for the incoming e+e−state, ∼1.5 ± 1.0. Hence, we see here one immediate effect of the high energy ofthe SSC incoming beams: the initial uu-type state will radiate a significant numberof real photons, with a consequent change in the observed final-state character.

Inparticular, the issue of how much energy is lost to photon radiation is of immediate9

interest, for this energy is unavailable for Higgs production by uu (or dd) and,further, it may fake a signal of H →γγ if we are unlucky. Accordingly, we nowlook at the predicted distribution ofv ≡(s −s′)/s ,(4.2)where s′ = (q1 + q2)2 is the squared invariant final fermion pair mass.

If only onephoton is radiated, v is just the energy of this photon in the center-of-mass systemof the incoming beams (in units of the incoming beam energy).What we find for v is shown in Fig. 4 for the uu →uu + nγ case (the dd →dd + nγ case is similar).

We see the expected shape of v from Ref. [1], and itsaverage value is ⟨v⟩= 0.05 ± 0.09.Hence, ∼10% of the incoming energy isradiated into photons; this energy is not available for Higgs production and henceit is crucial to fold our radiation into the currently available SSC Higgs productionMonte Carlo event generators [9] and to complete the development of our own YFSmultiple-photon (-gluon) Higgs production Monte Carlo event generator, which isunder development and will appear elsewhere [2].Given that we know we have, in the SSC environment, significant multiple-photon radiation effects, the question of immediate interest is how often the trans-verse momenta of two photons are large enough that they could fake a H →γγsignal.

We will answer this very important question in detail in the not-too-distantfuture when our complete Higgs production YFS Monte Carlo event generators areavailable [2]. However, here we can begin to study this question by looking into thetransverse momentum distribution of our YFS multiple-photon radiation in, e.g.,uu →uu+nγ.

This is shown in Fig. 5, where we plot the total transverse momen-tum distribution of the respective YFS multiple-photon radiation.

What we findis that , for √s = 40 TeV, the average value of this total transverse momentum is(in the incoming uu center-of-mass system)p⊥,tot≡*|nXi=1⃗ki⊥|+= (0.0184 ± 0.0129)√s ,(4.3)10

where ki (i = 1, . .

. , n) are the four-momenta of the n photons.

(For √s = 6.7 TeV,this average is (0.0186 ± 0.0136)√s. )Hence, for the SDC acceptance cut of12| ln tan(θ/2)| ≡|η| < 2.8, or θi > 122 mrad, this means that there may besome possible background to H →γγ for, e.g., mH ≈150 GeV.

Such effects willbe discussed in detail elsewhere [2].Finally, concerning the overall normalization, we find that the Born cross sec-tion is corrected according to the results in Table 1. This shows clearly that thehigher-order effects change the normalization by approximately 5%.

This sets thelevel at which precise simulations of SSC physics must take the higher-order effectsfrom multiple photons into account.5. ConclusionsWe conclude that our initial study of YFS multiple-photon radiation in the SSCphysics environment shows that any Monte Carlo event generator which hopes toachieve an accuracy of order 10% in the SSC physics simulations must treat therespective effects in a complete way.

In this paper, we have computed these effectsfor incoming quark-(anti)quark states at SSC energies using the Monte Carlo eventgenerator SSC-YFS2 Fortran based on our original YFS2 Monte Carlo in Ref. [1].Specifically, using our SSC-YFS2 Monte Carlo event generator for q(¯q) →q′(¯q)′ + nγ, at √s = 40 TeV, we find that for an initial uu state, the mean num-ber of radiated photons is 0.85 ± 0.92, so that the multiple-photon character ofthe events must be taken into account in detailed detector simulation and physicsanalysis studies.

Further, the mean value of v = (s −s′)/s is 0.05 ± 0.09 and theaverage total squared transverse momentumDk2⊥,totEis 0.025±0.002 s. Hence, theimpact of these event characteristics on Higgs production in general and on theH →γγ scenario in particular must be assessed in detail. Such assessment willappear elsewhere [2].In conclusion, we can say that the initial platform for precision SSC elec-troweak physics simulations on an event-by-event basis using our YFS Monte Carlo11

approach [1] has now been established. We look forward with excitement to itscomplete development for all such electroweak phenomena and to its extension tothe SSC QCD processes as well [2].AcknowledgementsThe authors have benefited from the kind hospitality of Prof. F. Gilman ofthe SSC Laboratory, where part of this work was done.

The authors also thankProfs. G. Feldman, J. Dorfan and M. Breidenbach of Harvard and SLAC, andProf.

J. Ellis of CERN for the kind hospitality of the Mark II and SLD Collabo-rations and the CERN Theory Division, where the basic ideas in this paper wereconceived in the context of LEP/SLC Z0 physics.12

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FIGURE CAPTIONS1) The process e+(p1) + e−(p2) →f(q1) + ¯f(q2) + γ(k1) + . .

. + γ(kn).2) The SSC process q(p1) + (¯q)(p2) →q′(q1) + (¯q)′(q2) + γ(k1) + .

. .

+ γ(kn),where q, q′ = u, d, s.3) Histogram of the photon multiplicity in uu →uu + nγ for |η| < 2.8:(a) √s = 40 TeV; (b) √s = 6.7 TeV. Here, vmin = 10−6-we have shown inRef.

[1] that the cross section does not depend on vmin.4) v-distribution for uu →uu + nγ, where v = (s −s′)/s and s′ = (q1 + q2)2 isthe squared final uu invariant mass. Here, |η| < 2.8: (a) √s = 40 TeV; (b)√s = 6.7 TeV.5) Total transverse momentum distribution of the photons in uu →uu + nγ for|η| < 2.8 in units of s: (a) √s = 40 TeV; (b) √s = 6.7 TeV.TABLE CAPTIONS1: Sample output for uu →uu + nγ at √s = 40 TeV and |η| < 2.8.

The entriesin the table are largely explained therein: XSEC = cross section, WT =event weight, and BORN = Born cross section.14

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