Radiative Four-Fermion Processes at LEP2

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LEP2 at processes four–fermion Radiative n ftemi ol feetoekpyisa E2is LEP2 at the physics of electroweak properties the of of goals study the main the of One Introduction 1 2 3 wl eisre yteeditor) the by inserted be (will No. manuscript EPJ iatmnod iiaNcer erc nvri` iP Università di - Teorica e Nucleare Fisica di Dipartimento iatmnod iia-Uiestad err n NN-Se - INFN and Ferrara Università di - Fisica di Dipartimento NN-Szoed ai,vaA as ,Pva tl n Dipar and Italy Pavia, 6, Bassi A. via Pavia, di Sezione - INFN eypcla etr ftepoessudrcon- under processes the of feature peculiar very A O ( α e lcrwa orcin o4 to corrections electroweak ) + PACS. described. are analysis, etrso h ot al program the Carlo to differen Monte and paid the sections is of cross attention features integrated particular for state, results ligh final Explicit taken the are the couplings In in gauge masses. photon anomalous fermion quartic and of study triple The effect LEP, range. the energy including LEP2 the elements, on emphasis particular with Abstract. date version: Revised / date Received: e f − e rcse r loa neetn hsc sub- physics interesting an also are processes + 1 → .Moretti M. , e − W 21.i–1.0K 13.85.Hd – 13.40.Ks – 12.15.Ji e f → + e + nlsae.A elkon because known, well As states. final ) h rdcino orfrin lsavsbepoo nelec in photon visible a plus fermions four of production The + e W − e 4 − f → − + → → ν 2 γ 4 νγ ¯ .Nicrosini O. , r uligboko the of block building a are , 4 f f , W rcse r eddi or- in needed are processes , e e + W + oo rmteanalysis the from boson e W e − oos[] h center The [1]. bosons − ν + f → νγγ ¯ → W rcse,provid- processes, 4 eν W − 3 f .Osmo M. , nlsae,the states, final γ + WRAP rs section cross W e γ (single- pi pro- -pair reactions sdt efr h aclto n vial o experime for available and calculation the perform to used , 1 .Piccinini F. , W vaadIF ein iPva i .Bsi6 ai,Italy Pavia, 6, Bassi A. via Pavia, di Sezione - INFN and avia in iFraa err,Italy Ferrara, Ferrara, di zione ttsa E2eege 1] naltetertclstud- theoretical the all In [16]. energies LEP2 at states lmn 89.Snete,sm vn eeaosfrthe for generators event matrix 4 some tree-level of then, exact simulation Since the to [8,9]. of approaches element all calculation different automatic papers exploiting effects by the these mass for fermion In accounted as [6,7]. were well refs. as contributions in electroweak performed were cesses ol,as aclto ftemsiemti lmn of element matrix massive computational the above of e the calculation to a also addition scatter- [2] tools, ref. In the to details. referred of more is calculation for reader interested the The amplitudes. for ing algorithm recursive a developed: claayi ffrinms ffcsi 4 in effects mass fermion phenomenolog- of detailed analysis a ical by accompanied [15], literature n eta urn N)oe,mdae ytwo by mediated ones, (NC) current neutral and aino l 4 all of lation in[11]; tion a teto spi oC rcse,bcueo h larger the of of because processes, section CC particu- cross to work possibly paid of this is attention In effect lar (AGC). the couplings examine gauge to anomalous suitable principle in are fte-ee mltds(nldn emo ass and masses) fermion (including phase-space; calculation amplitudes automatic tree-level the of upon relying packages purpose C)rdaie4 radiative (CC) iiyo E,bigacsil nya h nriso a of energies the at sensi- only the outside accessible are being future and LEP, states of final tivity give six-fermion bosons to gauge massive rise only involving vertices Quartic states. + iet iFsc ulaeeToia-Uiestad Pavia Università di - Teorica e Nucleare Fisica di timento ildsrbtosaesonadcmetd The commented. and shown are distributions tial e h rtte-ee acltosof calculations tree-level first The − ftepeetmaueet efre at performed measurements present the of t → noacut u otepeec favisible a of presence the to Due account. into ramn fhge-re E corrections. QED higher-order of treatment e 3 + sbsdo h aclto featmatrix exact of calculation the on based is 4 e CompHEP f − RacoonWW + Helac/Phegas ierClie L) ohcagdcurrent charged Both (LC). Collider Linear f γ + f rnpsto olsosi analyzed, is collisions tron-positron γ W W f rcse aercnl perdi the in appeared recently have processes + γ rcse,mdae ytwo by mediated processes, 1]and [12] nlsae ntemsls approxima- massless the in states final γ 1] eeao ae ntecalcu- the on based generator a [10], vnsin events vnswt epc to respect with events grc4f 1] rga implementing program a [14], e + e 1] hc r general- are which [13], − e olsoshv been have collisions + f e − n 4 and → ntal 4 W f ZZγ f Z + + -bosons, -bosons, γ γ final final pro- 2 G. Montagna, M. Moretti, O. Nicrosini, M. Osmo, F. Piccinini: Radiative four–fermion processes at LEP2 ies devoted insofar to 4f + γ production, the effects of quartic anomalous gauge couplings (QAGC), which are 6 a window on the mechanism of spontaneous symmetry dΦ5 = (2π) dΦ3(P ; p5,QV1 ,QV2 )dΦ2(QV1 ; p1,p2) 2 2 (1) breaking [17] and are presently of special experimental dΦ2(QV2 ; p3,p4)dQV1 dQV2 , interest, have not been taken into account. Actually, recent phenomenological studies on the subject of QAGC at where V1 and V2 indicate the W gauge bosons, the mo- − high-energy e+e colliders have been performed by con- menta pi with i = 1,..., 4 stand for the momenta of the sidering three-vector boson WWγ,ZZγ,Zγγ production final state fermions and p5 is the photon momentum. The and treating W, Z particles in the on-shell approxima- eleven independent variables have been chosen to be: tion [18,19]. Anomalous quartic couplings in ννγγ¯ pro- – photon variables Eγ , θγ and φγ in the c.m. frame; duction via W W fusion have been analyzed in ref. [20]. 2 2 – invariant mass squared QV1 and QV2 ; Experimental searches for QAGC at LEP rely upon the – three θ and φ angle pairs in the rest frame of each theoretical results of refs. [18,19,20,21,22] and make use of decaying “particle”, namely in the frames given by the the computational tools of refs. [18,20]. conditions P p5 = 0, QV1 = 0 and QV2 = 0. In the light of the present situation and in view of fu- − ture measurements at LC, a full calculation of e+e− When the photon is emitted from the final state, which 4f + γ processes, including the effects of AGC and of the→ means for a CC process from a virtual W or from a vir- most important radiative corrections, is desirable. This tual charged fermion, the following decomposition can be task is accomplished in the present paper, by presenting conveniently considered: the new event generator WRAP (W Radiative process with ALPHA [9] and Pavia) for the simulation of 4f +γ processes + − 6 at e e colliders. The program is based on the calculation dΦ5 = (2π) dΦ2(P ; QV1 ,QV2 )dΦ3(QV1 ; p5,p1,p2) 2 2 (2) of exact matrix elements, including the effect of fermion dΦ2(QV2 ; p3,p4)dQV1 dQV2 . masses, both for CC and NC processes. Charged trilinear anomalous gauge couplings (TAGC) and the genuinely In the case of the photon emitted from an internal QAGC, i.e. those giving no contributions to trilinear ver- gauge boson the independent variables can be chosen as tices, are included in the calculation, as well as the large follows: effect of initial state radiation (ISR). A tuned compari- – photon variables Eγ , θγ and φγ in the c.m. frame; son between the predictions of a preliminary version of – invariant mass squared Q2 and Q2 ; WRAP RacoonWW V1 V2 and of the other two event generators and – one θ and φ W -angle pair in the c.m. frame; Helac/Phegas can be found in ref. [2]. – two θ and φ angle pair for p1 and p3 in the rest frame The paper is organized as follows. In Sect. 2 the main of the bosons V1 and V2, respectively. features of the the calculation are described. After the description of the treatment of the multi-particle phase- In the case of photon emission from a final state fermion space, the theoretical details concerning the calculation of the following independent variables can been adopted: the exact matrix elements, the implementation of anoma- – invariant mass squared Q2 and Q2 ; lous gauge couplings and ISR are given. A sample of nu- V1 V2 merical results as obtained by means of the Monte Carlo – one θ and φ W -angle pair in the c.m. frame; WRAP is presented in Sect. 3, paying particular attention to – one θ and φ angle pair for p3 in the rest frame of the the contribution of fermion masses, to the impact of ISR boson V2; and to the effects of AGC at LEP2 and LC energies. Con- – energies of p1 and p5 momenta in the c.m. frame; clusions and possible perspectives are drawn in Sect. 4. – azimuthal angle φ of p1 in the rest frame of V1; – φγ in the rest frame of the radiating fermion; – cos θγ−f in the c.m. frame , 2 Features of the calculation where θγ−f is the relative angle between the radiating fermion and the photon. An analogous phase-space parameterization has been 2.1 Phase-space integration implemented in WRAP for the case of NC processes, neglect- ing of course the channels related to the photonic emission The kinematics of the 2 5 particles processes has been from internal lines. The above phase-pace decompositions, treated generating the 5-body→ phase-space recursively, since iterated for each possible radiation pattern, give rise to the process can be seen as production and subsequent de- several channels, depending on the final state considered. cay of a pair of massive gauge bosons. In the previous formulas dΦn represents the element of Concerning CC like processes, the configurations of in- n-body phase-space given by: terest are related to a photon emitted from the initial state (see Fig. 1), from an intermediate W -boson (see Fig. 2) n n d3 p and from the final state charged fermions (see Fig. 3). 4 4 i dΦn(P ; p1,...,pn) = (2π) δ (P pi) 3 . − (2π) 2Ei As far as emission from the initial state is concerned, the Xi=1 iY=1 adopted phase-space decomposition reads as follows: (3)

G. Montagna, M. Moretti, O. Nicrosini, M. Osmo, F. Piccinini: Radiative four–fermion processes at LEP2 3

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Fig. 1. Example of Feynman diagrams for photon radiation from the initial state.

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Fig. 2. Example of Feynman diagrams for photon radiation from W bosons.

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Fig. 3. Example of Feynman diagrams for photon radiation from ﬁnal-state fermions.

The code works taking into account all the configu- 2.2 Tree-level matrix element rations discussed above according to a standard multi- channel Monte Carlo approach [23]. As already mentioned, the present work is based on the In order to perform an efficient event generation, the calculation of the fully massive Born matrix element of − peaking behaviour of the matrix element has been treated e+e 4f + γ processes. The exact matrix elements for → + − in the following way: CC and NC e e 4f + γ processes are available in → – the squared invariant masses of the massive gauge bosons WRAP. The calculation is performed by using ALPHA [9], an V are sampled according to a Breit-Wigner distribu- iterative algorithm for the automatic evaluation of tree- 2 level scattering amplitudes without using Feynman graphs tion centered around MV , while the photon propagator is sampled according to the 1/Q2 distribution; (see ref. [25] for a review of the method and of recent phenomenological applications). For the processes under – the infrared divergence is sampled according to a 1/Eγ distribution; consideration, a completely numerical approach turns out – the collinear peak arising from the photon emission to be particularly convenient not only for the very large due to an external charged fermion is taken under con- amount of contributing Feynman diagrams, but also be- trol by sampling it with a distribution proportional cause the calculation can be performed in the presence to 1/(1 β cos θ), where θ is the separation angle be- of fermion masses without any additional complication. tween the− radiating fermion and the photon, and β = This is of special importance for 4f +γ final states involving muons, where the separation angle between muon and 1 m2/E2, m and E being the mass and the energy photon can be realistically set to zero, and a calculation pof the− fermion respectively. taking into account of the finite muon mass is mandatory, For a realistic account of gauge boson properties, and to avoid collinear singularities. to avoid integration singularities, it is mandatory to include the gauge boson width in the propagators. The so- called fixed-width scheme [10,24] is adopted in WRAP. Actu- 2.3 Anomalous gauge couplings ally, as shown in ref. [10], the fixed-width scheme, even if it violates SU(2) gauge invariance, is a reliable U(1) gauge- Information about the structure of TGC and QGC can be restoring method and is able to guarantee predictions for obtained by the analysis of 4f + γ production processes. e+e− 4f + γ processes in good numerical agreement In particular, CC radiative 4f processes, although char- with a→ scheme preserving all the relevant Ward identities, acterized by a lower statistics, are potentially a comple- such as the complex-mass scheme [10]. mentary channel to the 4f final states in order to test the 4 G. Montagna, M. Moretti, O. Nicrosini, M. Osmo, F. Piccinini: Radiative four–fermion processes at LEP2 effect of TGC, because of the larger amount of diagrams In the above equation the Lorentz structure of the opera- involving trilinear gauge interactions. More importantly, tors is given by: 4f + γ processes are interesting in order to put bounds µν + −ρ on deviations from standard quartic gauge couplings. In W1 = aw1Fµν F Wρ W µρ +ν − the following, the theoretical details of the parameteriza- W2 = aw2Fµν F W Wρ + h.c. µν ρ tion adopted in order to keep under under control this Z1 = az1Fµν F ZρZ µρ ν important phenomenological issue are described. Z2 = az2Fµν F Z Zρ Z µν + −ρ W0 = awz0Fµν Z Wρ W (7) Z µρ +ν − Wc = awzcFµν Z W Wρ + h.c. Z +µν ρ − 2.3.1 Trilinear anomalous gauge couplings W1 = awz1Fµν W Z Wρ + h.c. Z +µρ ν − W2 = awz2Fµν W Z Wρ + h.c. Z +µρ −ν It is possible to take into account the effect of charged W3 = awz3Fµν W ZρW + h.c., TGC (anomalous and not) by means of the following la- −2 grangian [26,27]: where the ai are coefficients of dimension M . It is worth noticing that, by imposing appropriate relations between V µ − +ν + −ν iLTGC = gW W V [ g1 V Wµν W Wµν W the ai’s, symmetry properties, such as for instance SU(2)c + − µν λV µν −+ρ − custodial symmetry or SU(2) U(1) gauge invariance, can +κV W W V + 2 V W W µ ν mW µ ρν × V ρ −µ +ν −µ ρ +ν σ be guaranteed, as shown in ref. [19]. In the parameteriza- + ig εµνρσ ((∂ W ) W W (∂ W )) V ] 5 − tion adopted in ref. [19] the ai are real coefficients whose V = γ, Z, explicit expression can be directly read off from the corre- (4) sponding operator structure of ref. [19] itself. In particular, which represents the most general lagrangian describing the coefficients a0 and ac, originally introduced in ref. [21] trilinear W W V gauge interactions, with the exception of and related to the WWγγ and ZZγγ structure, can be the operators violating C, P and CP simmetries. LTGC obtained from the above ai coefficients by means of the has been implemented in ALPHA, and the presence of anoma- following relations lous couplings can be studied, as done at LEP [28], by 2 using the relations [27,29] e aw1 = 2 a0 8Λ2 − e 2 az1 = 2 2 a0 cW Z − 16 cos θwΛ ∆κγ = 2 (∆κZ ∆g1 ) λZ = λγ λ, (5) 2 (8) −sW − ≡ e aw2 = 2 ac 16Λ2 − e 2 2 Z Z az2 = 16 cos θwΛ ac , where ∆κV = κV 1 and ∆g1 = g1 1. The Standard − − − V Model (SM) Lagrangian is recovered for g1 = kV = 1, V where Λ represents a scale of new physics. As far as WWZγ λ = 0, g5 = 0. Triple anomalous neutral gauge couplings, considered in ref. [30] and looked for at LEP in vertex is concerned, an additional structure has been pro- e+e− Zγ,ZZ processes [31], are not presently taken posed in the literature [18,22], whose expression can be into account.→ derived from the above ai coefficients by means of the following relations

2 e 2 awzc = i 16 cos θwΛ an 2.3.2 Quartic anomalous gauge couplings 2 e 2 (9) awz2 = i θwΛ an 16 cos 2 e awz = i 2 an . Quartic gauge couplings involving at least one photon 3 − 16 cos θwΛ are analyzed at LEP [5]. In particular, W +W −γγ and + − W W Zγ vertices are probed in WWγ 4f + γ and On the experimental side, bounds on a0, ac and an cou- ννγγ¯ final states [3,4], while e+e− Zγγ→processes [32] plings are quoted by LEP collaborations via the analysis are investigated to put bounds on the→ZZγγ vertex, which of WWγ and ννγγ¯ final states [3,4,5]. It is worth notic- is a gauge interaction not predicted by SM at tree level. In ing, in passing, that, thanks to the implementation of the the present work the operators considered in ref. [19] for lagrangian of eq. (6) in the ALPHA code, an improved ver- genuine anomalous quartic couplings containing at least sion of the Monte Carlo generator NUNUGPV [34,35] is also one photon, namely W +W −Zγ, W +W −γγ and ZZγγ available for the study of QAGC in ννγγ¯ events. vertices, have been implemented in ALPHA, upgrading the The result of the implementation in ALPHA has been version used in ref. [33] for the analysis of QAGC in six- carefully cross-checked by an independent analytical cal- fermion final states at the energies of future linear collid- culation of all V1V2 V3V4 amplitudes, with Vi = γ,W,Z. ers. The implemented lagrangian include all the relevant The check has been→ performed for all the processes ob- − six-dimensional operators and reads as follows tained from W +W Zγ scattering, by permutating particles. → LQGC = W1 + W2 + Z1 + Z2 Z Z Z Z Z W0 + Wc + W1 + W2 + W3 . (6) G. Montagna, M. Moretti, O. Nicrosini, M. Osmo, F. Piccinini: Radiative four–fermion processes at LEP2 5

2.4 Initial-state radiation Table 1. Comparison between WRAP and RacoonWW predictions for the massless Born cross section of NC processes at √s = In order to match the precision of LEP measurements, 190 GeV. Input parameters and cuts as in ref. [10]. the most important radiative corrections have to be considered. Among them, it is well known that undetected Cross section (fb) WRAP RacoonWW initial-state radiation (ISR) plays a major rôle. It can be µ+µ−τ +τ −γ 6.76 0.03 6.78 0.03 + − ± ± taken into account in the leading log approximation by us- µ µ ντ ν¯τ γ 4.248 0.009 4.259 0.009 − ing the QED Structure Function (SF) approach, in terms µ+µ uuγ¯ 12.65 ± 0.03 12.70 ± 0.04 of collinear [36] or p⊥-dependent SF [34,35]. Following re- ± ± cent work done in ref. [35], both prescriptions are available in WRAP, for the reason explained below. When ISR is in- element for the considered 4f + γ final state. Therefore, cluded via collinear SF, the QED corrected cross section eq. (11) applies to the signature of four fermions plus an can be written as isolated hard photon, corrected by the effect of undetected soft and/or collinear radiation. Aiming to obtain a correct σ4f+1γ (s)= dx dx D(x ,s)D(x ,s) dσ4f+1γ (x x s), QED Z 1 2 1 2 0 1 2 evaluation of the size of the double counting effects, a (10) limit of the present treatment of undetected radiation is by convoluting the tree-level cross section with electron that only ISR is actually considered. This issue could be SF. However, due to the presence of an observed photon addressed in a more complete way by using, for example, a in the hard-scattering matrix element, the inclusion of ISR QED parton shower approach as developed in ref. [37], in needs some care. Actually, since the collinear SF can be order to describe the radiation from all external charged viewed as the result of an integration over the angular legs, thus including the contribution of undetected final- variables of the photon radiation, an overlapping between state radiation. the detected kernel photon and pre-emission photons at large angle may occur1. The consequence is that a double counting takes place if higher-order QED corrections are 3 Numerical results naively included by using collinear SF [35]. On the other hand, it is expected that the bulk of The aim of the present section is to give some details on the correction is well estimated by collinear SF, since the the technical precision of WRAP and discuss the impact of emission of a photon from an on-shell initial state fermion the effects due to fermion masses, ISR and AGC on ob- is almost collinear. However, in order to provide a more servables of experimental interest. appropriate treatment of photon corrections and give an In order to test the reliability and the theoretical ac- estimate of the double-counting effect, the SF method can curacy of the event generators, a detailed tuned compar- WRAP be improved by means of the use of p⊥-dependent SF, ison between the predictions of and other available i.e. by generating angular variables for the ISR photons programs have been carried out in the context of the four- according to 1/(p k), which is the leading behaviour for fermion working group of the LEP2 Monte Carlo work- radiation of momentum· k emitted by an on-shell fermion shop at CERN [2]. The comparisons, referred to integrated of momentum p. In such a scheme, the QED corrected cross sections and differential distributions of several CC cross section can be calculated as processes, showed perfect technical agreement. The comparison is here extended to NC processes, as shown in 4f+1γ (1) (2) Tab. 1, between the predictions of WRAP and RacoonWW σQED = dx1dx2 dcγ dcγ Z ZΩc with input parameters and cuts as in ref. [10]. As can be seen, also for NC final states the agreement is excellent. A D˜(x ,c(1); s)D˜(x ,c(2); s)dσ4f+1γ , (11) 1 γ 2 γ further comparison between the predictions of WRAP and those of ref. [16] is reported in Tab. 2, for several cross where D(x, cγ ; s) is the p⊥-dependent SF [34]. According to eq. (11), an “equivalent ” photon is generated and ac- sections of CC and NC processes, in the presence of finite cepted as an ISR contribution only if it satisfies a rejection fermion masses and in terms of the same input parame- algorithm based on the following requirements: ters and cuts as adopted in ref. [16]. Perfect agreement is registered for all the considered 4f + γ final states. – the energy of the “equivalent” photon is below the en- The phenomelogical analysis makes use of the following ergy threshold for the observed photon, for arbitrary input parameters: angles; or – the “equivalent” photon is collinear to a charged parti- −5 −2 cle (i.e. under the minimum separation angle required GF =1.16637 10 GeV MZ = 91.1867 GeV · 2 2 2 MW = 80.35 GeV sin θw =1 M /M in order to be detected), for arbitrary energies. − W Z ΓZ =2.49471 GeV ΓW =2.04277 GeV Within the angular acceptance of the observed photon, mµ =0.10565839 GeV ms =0.15 GeV the cross section is computed by means of the exact matrix mc =1.55 GeV 1 The same problem is discussed in detail in ref. [35] for the (12) − process e+e νν¯ + nγ. We refer the reader to ref. [35] for The form used for the propagator of the massive gauge more details on→ the strategy here adopted. bosons is, according to the fixed-width scheme, 1/(p2 ∼ − 6 G. Montagna, M. Moretti, O. Nicrosini, M. Osmo, F. Piccinini: Radiative four–fermion processes at LEP2

Table 2. Comparison between WRAP and the predictions of Table 4. Comparison between massive and massless Born + ref. [16] for the massive Born cross section of CC and NC pro- cross sections for the final state µ νµcs¯ + γ at √s = 200 GeV. cesses at √s = 190 GeV. Input parameters and cuts as in θγ−f , with f = q, µ is the minimum separation angle between ref. [16]. the photon and final state charged fermions; other cuts as in eq. (13). In the third column, the first result refers to the mas- Cross section (fb) WRAP ref. [16] sive case, and the second one to the massless case. The relative − difference is shown in the last column. See also ref. [2]. ude¯ ν¯eγ 220.1 0.5 220.3 0.7 − ± ± cse¯ ν¯eγ 217.5 0.4 218.2 0.7 − − + − ± ± ϑγ q (deg) ϑγ µ (deg) Cross Section (fb) δ (%) µ ν¯µe ν¯eγ 78.6 0.1 79.0 0.3 + − ± ± ◦ ◦ τ ν¯τ e ν¯eγ 77.6 0.2 77.5 0.2 5 1.0 90.157 0.036 1.92 0.08 − ± ± ± ± udµ¯ ν¯µγ 213.0 0.1 213.8 0.3 91.903 0.035 ¯ − ± ± ◦ ◦ ± udτ ν¯τ γ 208.7 0.4 209.3 0.5 5 0.1 104.777 0.046 9.31 0.09 + − ± ± τ ν¯τ µ ν¯µγ 75.2 0.1 75.1 0.2 ± ± ± ± 115.004 0.044 uds¯ cγ¯ 590.0 0.6 593 2 ◦ ◦ ± + − ± ± 5 0.0 105.438 0.045 µ µ ντ ν¯τ γ 5.32 0.02 5.32 0.03 ± − − τ +τ µ+µ γ 4.15 ± 0.02 4.18 ± 0.02 + − ± ± τ τ νµν¯µγ 3.175 0.006 3.167 0.007 ± ± ble 4 shows the difference between massive and massless cross section, with the minimal separation between quarks Table 3. ◦ Comparison between massive and massless Born and photon fixed at 5 and progressively relaxing the sep- cross section for the final state µ+ν cs¯ + γ at √s = 200 GeV. µ aration cut between muon and photon. It can be seen that θγ−f is the minimum separation angle between the photon and ◦ final state charged fermions; other cuts as in eq. (13). The first the massless calculation is still reliable for 1 of minimum line refers to the massive case, the second one to the massless separation, being the relative difference around 2%, but approximation. it becomes inadequate when the separation falls at some fraction of degree, the relative difference being of the order θγ−f (deg) Cross Section (fb) of 10%. Therefore, in particularly stringent experimental ◦ conditions, only a massive 4f + γ calculation can provide 5 74.294 0.029 reliable predictions in the presence of muons in the final 75.732 ± 0.022 ◦ ± state. 1 93.764 0.037 ± Figure 4 shows the line-shape of the cross sections 100.446 0.037 + ± of the radiative semi-leptonic processes µ νµud¯ + γ and + e νeud¯ + γ, as a function of the c.m. energy in the LEP2 range. The QED corrected cross section via collinear SF M 2 + iΓM). The processes considered are the radiative + ′ for the µ νµud¯ + γ final state is also plotted. The com- + − + semi-leptonic final states of the kind e e l νlqq¯ γ. parison shows that the contribution due to the additional → + The cuts adopted are: t-channel diagrams present in the e νeud¯ + γ final state is not particularly relevant for the adopted selection cri- cos θγ 0.985; Eγ 1 GeV; | |≤ ≥ teria, small being the differences between the cross sec- cos θl 0.985; El 5 GeV; tions of the two processes. Concerning ISR in the strictly | |≤◦ ≥ (13) θγ−f 5 ; collinear approximation, its impact on the cross section is M ′ ≥ 10 GeV, qq¯ ≥ at the level of 10 15%, which is a phenomenologically relevant effect in the− light of the LEP experimental ac- where θγ (l) is the photon(lepton) scattering angle, Eγ (l) is curacy. It is worth noticing that this result, obtained by the photon(lepton) energy, θγ−f is the angular separation means of a standard treatment of ISR as tipically adopted ′ between photon and final charged fermions, and Mqq¯ is in the experimental analysis of QAGC in radiative events ′ the qq¯ invariant mass. at LEP [3,4], just provides the bulk of the effect due to Table 3 shows the effect of fermion masses on inte- ISR but it is affected, as previously discussed and quanti- grated cross sections at √s = 200 GeV for two different fied below, by a double counting because of the presence photon-fermion separation angles. In the first row, the an- of a radiative process as hard-scattering reaction. gular separation θγ−f between photon and all charged fi- The contribution of initial-state photon radiation is nal state fermions is fixed at 5◦, while in the second row also shown in Figs. 5, as a function of the threshold en- ◦ min θγ−f = 1 . As expected, the relative difference between ergy Eγ of the observed photon. It can be noticed that the massless and massive cross section increases, going the reduction factor due to collinear ISR is around 12- from 2% of the first row to the 7% of the second row, 13%, almost independent of the photon detection thresh- because of the importance of fermion mass contributions old. However, as previously discussed, ISR in the collinear when the photon approaches the collinear region around SF scheme introduces a double counting effect when the an on–shell charged particle. In the case of a final state pre-emission “equivalent” photon enters the phase-space containing a muon, the separation angle ϑγ−f can be re- region of the kernel photon. In order to get an estimate alistically set to zero, because of different behaviour of of this overlapping contribution, the comparison of the photons and muons in the experimental apparatus. Ta- corrections due to the collinear SF and p⊥-dependent SF G. Montagna, M. Moretti, O. Nicrosini, M. Osmo, F. Piccinini: Radiative four–fermion processes at LEP2 7

+ − + + + + + Fig. 4. Cross section for the semi-leptonic processes e e l νludγ¯ with l = µ (dashed line) and l = e (dotted line). → + The solid line shows the QED corrected cross section via collinear SF for the µ νµudγ¯ ﬁnal state. See also ref. [2].

Fig. 5. Comparison between collinear (dashed line) and p⊥-dependent (solid line) SF on the cross section of the process + − + min e e µ νµudγ¯ , as a function of the energy threshold of the visible photon Eγ . The dotted line is the Born prediction. See also ref.→ [2]. 8 G. Montagna, M. Moretti, O. Nicrosini, M. Osmo, F. Piccinini: Radiative four–fermion processes at LEP2 is shown. Is is observed that the two prescriptions can culation of the process e+e− W +W −γ (performed in- min → differ at 5% level for Eγ close to 1-2 GeV, while the be- dependently and in agreement with the results of ref. [19]), comes smaller and smaller as Emin increases. Numerical by taking into account the suitable branching ratios of the γ + investigation points out that, as expected, the discrepancy W bosons decaying into µ νµ andud ¯ pairs. The dotted between collinear and p⊥-dependent SF is larger near the line is the prediction as obtained by WRAP, with additional soft and collinear region and at the level of some per cent, cuts on the invariant masses of the two fermionic pairs con- thus yielding an estimate of the size of the double-counting strained within 75 GeV and 85 GeV, in order to enhance, effect at the level of ISR. Therefore, in the presence of as much as possible, the contribution of diagrams with two particularly stringent experimental constraints sensitive resonant W bosons. It can be clearly noticed that, even to the soft and collinear emission, precise predictions de- in the presence of cuts on the invariant masses of the de- mand a treatment of ISR able to keep under control the cay products, the complete 4f + γ calculation differ from transverse degrees of freedom of photon radiation. the prediction of the WWγ approximation, thus proving Let us come to the discussion of the effects due to (a the importance of a full calculation for the extraction of sample of) AGC. Both integrated cross sections (Figs. 6- meaningful limits on QAGC. 9) and differential distributions (Figs. 10-13) are consid- In Figs. 10-13 the most important photonic distribu- ered. In Figs. 6-7 the (relative) effect of the TAGC λ on tions are displayed using the code as event generator with + − + the e e µ νµudγ¯ cross section is examined, by plot- the cuts of eq. (13) at a typical LEP2 energy √s = 192 GeV. ting the relative→ difference between the cross section in In each plot, the SM Born and the QED corrected predic- the presence of a non-vanishing λ coupling and the SM tions are compared with those obtained in the presence of cross section (λ = 0 ), as a function of the λ value at AGC. The values used for the anomalous couplings are: W 2 √s = 192 GeV. Fig. 6 shows a comparison of the effect of λ = 0.25 and k0 /Λ = 0.01. For the sake of compari- + − the λ coupling on the radiative µ νµud¯ +γ process and the son, all the data sample are normalized to the same lumi- corresponding 4f final state, which, as already remarked, nosity. The cos θγ distribution and the distribution of the differ for their content of trilinear gauge interactions. The cosine of the angle between the photon and the nearest numerical results for the 4f process have been obtained charged particle are shown in Fig. 10 and Fig. 11, respec- by means of the program WWGENPV [38]. For the considered tively. Typical peaking behaviour in the close-to-collinear λ values, the relative contribution is almost the same on regions is clearly registered. In such regions, a particularly W the two processes (obviously the cross sections are quite significative impact of the QACG k0 is also observed. different), giving a difference at 2 3% level only for ex- Fig. 12 and Fig. 13 refer to the Eγ and transverse pho- treme λ values. Therefore, in the presence− of standard cuts ton momentum p⊥ distribution, respectively, showing the on the observed photon, trilinear gauge interactions due characteristic infrared peak. As already noticed in ref. [19], to W radiation in radiative 4f processes doesn’t enhance these observables turn out to be particularly sensitive to the sensitivity to TAGC with respect to a pure 4f final the presence of a QAGC in the region of high energy and state. This conclusion is further corroborated by the re- p⊥, being the operator involved of derivative type with sults shown in Fig. 7, where the effect of the λ coupling respect to the photon field. In all the considered distribu- is studied for different photon cuts, with the aim of sup- tions, ISR introduce sizeable effects if compared with the pressing the mostly collinear fermion radiation by impos- deviations due to anomalous couplings. ing more and more severe cuts on the detected photon. As far as the parameterization of QAGC in terms of By comparing the relative deviations shown in Fig. 6 and a0,ac,an parameters is concerned, numerical results are Fig. 7, one can conclude that in radiative 4f processes W shown in Tab. 5 and Tab. 6, at √s = 200 GeV and √s = radiation can be hardly disentangled from the radiation 500 GeV, respectively. The cross sections in the presence off fermions, being the observed deviations almost at the of non-vanishing anomalous couplings are compared with same level for all the set of cuts considered. the pure SM predictions. By looking at Tab. 5 and Tab. 6, W it can be noticed that the sensitivity of the 4f+γ processes The effect of the QAGC k0 , as defined in ref. [19], is + to QAGC is much higher at the energies of a future LC shown in Figs. 8-9 for the process µ νµudγ¯ , as a function of the parameter kW at √s = 200 GeV. For the scale of than at LEP2, as a priori expected and already noticed in 0 the literature for the WWγ process [18,19]. new physics Λ, the value Λ = MW is used, as convention- ally done in the literature. Absolute and relative effects are shown in Fig. 8 and Fig. 9, respectively. In terms of the W 4 Conclusions coefficients ai of eq. (7) the k0 coupling can be expressed as: − 2 2 The production of four fermions plus an additional de- e g W + − aw1 = 2Λ2 k0 tected photon in e e collisions is studied at LEP to test 2 2 (14) −e g cos θw W 2 electroweak gauge boson couplings and in particular to de- awz0 = Λ sin θw k0 . rive bounds on QAGC. In order to provide predictions of The solid line refers to the complete 4f + γ calculation of phenomenological interest, an exact calculation of 4f + γ WRAP with input parameters and photon cuts as used in processes, including the effect of fermion masses, AGC and ref. [19] and the additional cuts on fermions as given by ISR has been performed. On the basis of the experimen- eq. (13). In order to compare with the results of ref. [19], tal accuracy, the contribution of fermion masses and ISR the dash-dotted line has been obtained by means of a cal- has been analyzed in comparison with typical deviations G. Montagna, M. Moretti, O. Nicrosini, M. Osmo, F. Piccinini: Radiative four–fermion processes at LEP2 9

+ − + Fig. 6. The relative eﬀect of TAGC λ on the cross section of the radiative process e e µ νµudγ¯ (solid line) and the corresponding 4f ﬁnal state (dashed line). →

+ − + Fig. 7. The relative eﬀect of TAGC λ on the cross section of the radiative process e e µ νµudγ¯ for diﬀerent photon selection criteria. → 10 G. Montagna, M. Moretti, O. Nicrosini, M. Osmo, F. Piccinini: Radiative four–fermion processes at LEP2

W + − + Fig. 8. The eﬀect of the QAGC k0 at √s = 200 GeV on the absolute cross section for the process e e µ νµud¯ + γ, → with Λ = MW . The solid line is obtained by means of the full calculation of WRAP, the dash-dotted one with the real WWγ + approximation, and the dotted line refers to the calculation of WRAP with the additional cuts 75 GeV M(µ νµ), M(¯ud) 85 GeV. ≤ ≤

W W Fig. 9. The ratio between the cross section in the presence of a QAGC k0 and the SM cross section, as a function of k0 , for the three diﬀerent cases as in Fig. 8. G. Montagna, M. Moretti, O. Nicrosini, M. Osmo, F. Piccinini: Radiative four–fermion processes at LEP2 11

+ − + Fig. 10. The cos θγ distribution for the process e e µ νµudγ¯ at √s = 192 GeV. The Born approximation (solid line),the → W 2 QED corrected calculation (dashed line), the predictions for λ = 0.25 (dotted line) and the ones for k0 /Λ = 0.01 (dash–dotted line) are shown. Cuts as in (13). −

+ − + Fig. 11. The cos θγf distribution for the process e e µ νµudγ¯ at √s = 192 GeV, where θγf is the angle between the photon and the nearest charged particle. The samples are→ the same of Fig. 10. 12 G. Montagna, M. Moretti, O. Nicrosini, M. Osmo, F. Piccinini: Radiative four–fermion processes at LEP2

+ − + Fig. 12. The Eγ distribution for the process e e µ νµudγ¯ at √s = 192 GeV for the same sample of events as in Fig. 10. →

+ − + Fig. 13. The distribution of the transverse momentum of the visible photon for the process e e µ νµudγ¯ at √s = 192 GeV. The events samples are the same of Fig. 10. → G. Montagna, M. Moretti, O. Nicrosini, M. Osmo, F. Piccinini: Radiative four–fermion processes at LEP2 13

2 2 2 Table 5. Effect of the QAGC a0/Λ ,ac/Λ ,an/Λ , with Λ = where to look for TAGC, if compared with 4f final states, + − − MW , on the cross section of the process e e udµ¯ ν¯µγ at which benefit of a higher statistics. On the contrary, these → √s = 200 GeV. radiative processes are significantly affected by QAGC. In particular it has been shown that difference in the effect QAGC Cross section (fb) of QAGC are present between the predictions of the com- Standard Model 76.0 0.1 plete calculations by means of WRAP and the ones obtained 2 ± a0/Λ = 0.01 77.0 0.1 in the limit of on-shell W bosons, which is the approxi- 2 − ± a0/Λ = +0.01 77.2 0.1 mation presently used in the literature. A more complete 2 ± + − ac/Λ = 0.01 75.5 0.1 investigation of QAGC in radiative events at e e collid- 2 − ± ac/Λ = +0.01 76.9 0.1 ers is currently in progress. 2 ± an/Λ = 0.01 76.0 0.1 Acknowledgements 2 − ± an/Λ = +0.01 76.0 0.1 The authors wish to thank F. Cavallari, D.G. Charl- ± ton, A. Denner, S. Dittmaier, M. Musy, C.G. Papadopou- los, M. Roth, D. Wackeroth, and the participants in the Table 6. The same as in Tab. 5 at √s = 500 GeV. CERN LEP2 Monte Carlo Workshop for useful discus- QAGC Cross section (fb) sions. A particular thank goes to the authors of RacoonWW for the very helpful collaboration in numerical compar- Standard Model 25.3 0.1 isons and to P. Bell, D.G. Charlton and M. Thomson for 2 ± a0/Λ = 0.001 83.8 0.3 2 − ± useful information and help concerning QAGC. a0/Λ = +0.001 88.0 0.2 2 ± ac/Λ = 0.001 41.3 0.2 2 − ± ac/Λ = +0.001 45.4 0.2 2 ± References an/Λ = 0.001 26.4 0.1 2 − ± an/Λ = +0.001 26.4 0.1 ± 1. Physics at LEP2, G. Altarelli, T. Sjöstrand and F. Zwirner eds., CERN 96-01, CERN, Geneva, 1996. 2. Four-Fermion Production in Electron-Positron Collisions, introduced by AGC. A new Monte-Carlo event generator M. Grünewald, G. 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