Infrared weak corrections to strongly interacting gauge bosons scattering
aa r X i v : . [ h e p - ph ] F e b Infrared weak corrections to strongly interacting gauge bosonsscattering
Paolo Ciafaloni ∗ and Alfredo Urbano † INFN - Sezione di Lecce and Universit`a del SalentoVia per Arnesano, I-73100 Lecce, Italy
Abstract
We evaluate the impact of electroweak corrections of infrared origin on longitudinal strongly interacting gaugebosons scattering, calculating all order resummed expressions at the double log level. As a working example, weconsider the Standard model with a heavy Higgs. At energies typical of forthcoming experiments (LHC,ILC,CLIC),the corrections are in the 10-40% range, the relative sign depending on the initial state considered and on whetheror not additional gauge bosons emission is included.
Hopefully, experiments at the Large Hadron Collider (LHC) will soon unveil the mystery surrounding the way in whichthe SU(2) ⊗ U(1) electroweak (EW) symmetry is implemented in fundamental physics. The mechanism of elementaryparticles masses generation, the profound reason why the EW symmetry is apparently respected by the interactionsbut broken by the spectrum of the theory, are all puzzles that have been awaiting for a clear answer for a long time,and will be most probably be clarified soon. In the framework of the Standard Model (SM), if the Higgs particleis light, i.e. close to the present experimental upper bound of 114.4 GeV [1], the Higgs sector, which also includesthe Goldstone bosons, is characterized by a perturbative coupling of the same order of the gauge couplings. Indeed,indications coming from present experimental data at the 100 GeV scale and below seem to favor a light SM Higgssituation of this kind, or a scenario similar to this one, like supersymmetric extensions of the SM itself. However, ascenario in which the sector responsible for Symmetry breaking is a strongly interacting one is not excluded, and manymodels of this kind have been considered also in recent times [2]. In this case the scattering of longitudinal gaugebosons, related to the Goldstone bosons interactions via the equivalence theorem, is strongly enhanced and providesa direct experimental probe of the physics that is responsible for symmetry breaking. The purpose of this paper is toevaluate the impact of EW radiative corrections of infrared (IR) origin on the scattering of longitudinal gauge bosons.As a prototype of a strongly interacting symmetry breaking sector, we consider the Standard Model with a heavyHiggs; a more general study will be considered elsewhere [5].There are several reasons why studying the impact of EW corrections of IR origin to strongly interacting longitu-dinal gauge bosons scattering appears to be a sensible idea. In first place, these corrections depend on the exchangeof quantas of energy M ≪ ω ≪ √ s , M being the IR cutoff scale ( M ∼ M W ∼ M Z ) and √ s the physical process scale,of the order of 1 TeV. Then, it is reasonable to expect that corrections of IR origin are somehow universal, dependingonly on the known ”low energy” SM physics and not on the unknown ”high energy” strong dynamics at the TeV scale.In second place, these corrections have been shown in the literature to be significant at the TeV scale: at one loop thepresence of double logs of IR origin produces relative corrections that can be as big as 30-40%[6]. Then, it might turnout to be important to include them into the analysis of longitudinal gauge bosons scattering.In the present work we mainly consider “fully inclusive” observables, i.e. observables that include gauge bosonsemissions in the final state, and we resum EW corrections at the Double Log level. It is by now an established factthat, despite na¨ıve expectations, even fully inclusive EW observables are affected by DLs of IR origin [7]. We alsoconsider exclusive observables, affected by Sudakov logs, for a final comparison. Since we consider the Higgs massto be heavy, of the order of 1 TeV, we cannot directly rely on the results obtained in the ”recovered SU(2) ⊗ U(1)symmetry limit” [6, 7], where all energies in the physical processes are considered to be much larger than the particlesmasses. We are then led to consider two different situations: M ≪ √ s < M H , and √ s ≫ M H > M W , that we describein some detail below. The case √ s ≫ M H > M W was considered in [8], where fermion-antifermion production was ∗ [email protected] † [email protected] √ s ≪ M H If the c.m. energy of the process is much smaller than the Higgs mass, √ s ≪ M H , the SU(2) ⊗ U(1) gauge is of nodirect use. In fact, this symmetry is badly broken in the mass spectrum by the heavy Higgs mass, and since the Higgs h transforms into the Goldstone bosons ϕ k , say, under an infinitesimal SU L (2) isospin transformation ‡ : h → h − ϕ k δ kl α Ll ϕ k → ϕ k + 12 hα Lk + 12 ϕ l ε lmk α Lm , (1)where α Lk =1 , , are the parameters of the transformation; then even the tree level hard cross sections are not relatedby isospin symmetry. This does not mean that a calculation of IR effects cannot be performed of course, but astraightforward calculation of resummed effects is out of question. Here we prefer to consider the g ′ → custodial symmetry under which the Higgs transforms as a singlet and the Goldstones as a triplet is validalso after symmetry breaking: h → h ϕ k → ϕ k + ϕ k + iα m ( T mV ) kl ϕ l , (2)where ( T mV ) kl ≡ iε kml are the custodial symmetry generators in the adjoint SU (2) V representation.Let us first investigate what are the relations on cross sections dictated by custodial symmetry. Since all ourquantities are inclusive over final states, the cross sections only depend on the initial legs through the so called overlapmatrix [7] (see fig. 2): O β ,α ; β ,α = h β , β |O| α , α i O = S † S dσ α ,α = O α ,α ; α ,α . (3)Because of t-channel custodial symmetry invariance, we have: h −→ T V , O H i = 0 ⇒ h T ′ V , T ′ V |O H | T V , T V i = C HT V δ T V T ′ V δ T V T ′ V , (4)where, following the signs convention reported in fig. 2, we have defined the total custodial generators on the t-channelthat coupling legs 1 and 1 ′ as T aV ≡ T aV, − T aV, ′ and where the action of the isospin generators on the overlap matrixis described by the relations: (cid:0) T aV, O H (cid:1) β ,α ; β ,α = X δ (cid:0) T aV, (cid:1) α δ O Hβ ,δ ; β ,α (5)and: (cid:0) T aV, ′ O H (cid:1) β ,α ; β ,α = X δ (cid:0) T aV, ′ (cid:1) β δ O Hδ ,α ; β ,α . (6)Using the states: | , i = 1 √ (cid:0) | ϕ − ϕ + i − | ϕ ϕ i + | ϕ + ϕ − i (cid:1) , (7) | , i = 1 √ (cid:0) | ϕ + ϕ − i − | ϕ − ϕ + i (cid:1) , (8) | , i = 1 √ (cid:0) | ϕ + ϕ − i + | ϕ − ϕ + i + | ϕ ϕ i (cid:1) (9)and specializing eq. (4) to h , |O| , i = 0 h , |O| , i = 0 h , |O| , i = 0 , (10)we obtain a system whose solutions are the SU (2) V constraints on the cross sections: σ ++ = σ −− (11) σ = σ − (12) σ = σ ++ + σ − + − σ . (13) ‡ In the Appendix the symbols appearing here and in the following are defined, and a discussion of the symmetry proprieties of theLagrangian is given. g ′ → σ H ++ = 132 πs (cid:26) g (cid:20) s + tt + t − ss + t (cid:21) − sv (cid:27) , (14) σ H = 132 πs (cid:26) g (cid:20) t − st + s − t + ss (cid:21) + tv (cid:27) + 132 πs (cid:26) g (cid:20) s + tt + 2 t + ss (cid:21) − t + sv (cid:27) , (15) σ H = 116 πs (cid:26) g (cid:20) s − ts + t − s + tt (cid:21) + sv (cid:27) , (16) σ H − + = 132 πs (cid:26) g (cid:20) s + tt + 2 t + ss (cid:21) − t + sv (cid:27) + 132 πs (cid:26) g (cid:20) s − ts + t + 2 t + ss (cid:21) − tv (cid:27) + 132 πs (cid:26) g (cid:20) s − ts + t − s + tt (cid:21) + sv (cid:27) , (17)where s, t, u are the Mandelstam variables as usually defined. The expression of the eikonal current that describes theemission of a soft gauge boson W a from the external legs of the overlap matrix is given by: J a,µV ( k ) = g (cid:20) p µ p · k (cid:0) T aV, − T aV, ′ (cid:1) + p µ p · k (cid:0) T aV, − T aV, ′ (cid:1)(cid:21) ; (18)squaring this current, summing over all the possible gauge bosons emitted a , we obtain the following insertion operatorwritten in the Feynman gauge: I ( k ) = g p · p (2 p · k )(2 p · k ) (cid:16) −→ T V , − −→ T V , ′ (cid:17) · (cid:16) −→ T V , − −→ T V , ′ (cid:17) . (19)Because of the conservation of the total custodial generator on the t-channel, we have: I ( k ) = − g p · p (2 p · k )(2 p · k ) |−→ T V | (20)and the resummed expression for the overlap matrix is: O = e S ( s,M W ) O H = exp (cid:20) − L W |−→ T V | (cid:21) O H , (21)where: L W = g Z d −→ k (2 π ) ω k p · p ( p · k )( p · k ) = α w π ln sM W . ( α w = g / π ) (22)At this point it’s straightforward to convert eq. (21) into a system of equations that are able to connect the electroweakcorrected cross sections and the tree level ones. In fact we just have to use the states classified in eqns. (7 ÷ h , |O| , i = h , | exp (cid:20) − T V ( T V + 1) L W (cid:21) O H | , i = exp (cid:20) − L W (cid:21) h , |O H | , i ; (23)Using eq. (7) and the relations between the overlap matrix elements and the usual cross sections, this equationbecomes: σ + − − σ ++ = e − L W / (cid:0) σ H + − − σ H ++ (cid:1) ; (24)reasoning in a similar way for the states | , i and | , i we can obtain two other fundamental relations: σ + − + σ ++ − σ + 2 σ = e − L W / (cid:0) σ H + − + σ H ++ − σ H + 2 σ H (cid:1) (25)3nd 2 σ + − + 2 σ ++ + 4 σ + σ = 2 σ H + − + 2 σ H ++ + 4 σ H + σ H . (26)Solving the system obtained from eqns. (24 ÷
26) we can write the following expressions for dressed independent crosssections: σ ± + = σ H ++ (cid:18) ± e − L W + 16 e − L W (cid:19) + σ H (cid:18) − e − L W (cid:19) (27)+ σ H − + (cid:18) ∓ e − L W + 16 e − L W (cid:19) ,σ = σ H ++ (cid:18) − e − L W (cid:19) + σ H − + (cid:18) − e − L W (cid:19) + σ H (cid:18)
13 + 23 e − L W (cid:19) (28)and, for completeness, the expression for σ that can be obtained from (13): σ = ( σ H − + + σ H ++ ) (cid:18)
13 + 23 e − L W (cid:19) + σ H (cid:18) − e − L W (cid:19) . (29) √ s ≫ M H > M W If the c.m. energy of the process is much higher than the Higgs mass, √ s ≫ M H > M W , in the limit g ′ → SU (2) L ⊗ SU (2) R symmetry (see Appendix). In order to identify themost useful way in which we can use this symmetry considering the resummation of the electroweak corrections, it’snecessary to take a look at the form of the eikonal current in this energy region; when √ s ≫ M H > M W the situationis complicated by the fact that, after the gauge boson emission, one can have as final state a gauge boson as well asan Higgs particle. Considering the usual eikonal approximation and the equivalence theorem it’s possible to constructthe current for the emission of a soft gauge boson of momentum k , Lorentz index µ and isospin index b out of alongitudinal gauge boson of isospin index a and momentum p : J µac ( k, b ; p ) = gp µ p · k [ iε abc Θ W − iδ ab Θ H ] , (30)where Θ W ≡ ϑ (cid:0) p · k − M W (cid:1) and Θ H ≡ ϑ (cid:0) p · k − M H (cid:1) are the usual Heaviside functions and where the secondterm on the right hand side take into account the presence of the Higgs into the final state; we can choose to use amore compact and useful matrix notation, as follows § : J µ ( k, b ; p ) = gp µ p · k (cid:2) T bV Θ W + T bH Θ H (cid:3) = gp µ p · k (cid:20) T bL Θ H + 12 T bV (Θ W − Θ H ) (cid:21) , (31)in the basis ( ϕ , ϕ , ϕ , h ) T where: (cid:0) T bV (cid:1) ac = (cid:18) iε abc
00 0 (cid:19) (cid:0) T bH (cid:1) ac = (cid:18) − iδ ab iδ bc (cid:19) T L = 12 ( T V + T H ) T R = 12 ( T V − T H ) . (32)In eq. (30) when the energy of the emitted boson ω is such that M H < ω < M W , the Higgs boson contribution isturned off and the current is the same as the one achieved in the previous paragraph; when ω > M H , contrarily, wehave to consider also an Higgs additional contribution that, as we can see in (31), affects the T aV term and forces tointroduce a further one proportional to T aL .The obtained expression for the eikonal emission current as explicit function of the operators T a =1 , , L and T a =1 , , V leads automatically to the correct way in which we must look to the states in the t-channel; in fact we have to considerthe diagonal subgroup of SU (2) L ⊗ SU (2) R generated through T aV,i ≡ T aL,i + T aR,i , on a single leg denoted by i , and then § In the following discussion we use the notation T a =1 , , V,L,R referring to the contribution of a single leg, without any other additional index i .
4o classify the states according to the quantum numbers of the total t-channel Casimir |−→ T V | operator; considering thenotation | T L , T R ; T V i , we are left with 6 physical overlap states: | ,
0; 0 i = 12 (cid:0) −| ϕ + ϕ − i − | hh i − | ϕ ϕ i − | ϕ − ϕ + i (cid:1) , (33) | ,
1; 1 i = 12 (cid:0) −| ϕ + ϕ − i + i | ϕ h i − i | hϕ i + | ϕ − ϕ + i (cid:1) , (34) | ,
0; 1 i = 12 (cid:0) −| ϕ + ϕ − i − i | ϕ h i + i | hϕ i + | ϕ − ϕ + i (cid:1) , (35) | ,
1; 2 i = 1 √ (cid:0) − | ϕ ϕ i + | ϕ + ϕ − i + | ϕ − ϕ + i (cid:1) , (36) | ,
1; 1 i = 1 √ i | hϕ i + i | ϕ h i ) , (37) | ,
1; 0 i = 12 √ (cid:0) | hh i − | ϕ ϕ i − | ϕ + ϕ − i − | ϕ − ϕ + i (cid:1) , (38)matching all to the eigenvalues T V = 0. Notice that the SU (2) V constraints in eqns. (11 ÷
13) between the cross sectionsare still valid; in addiction we shall have other relations characteristics of the SU (2) L ⊗ SU (2) R symmetry. Reasoningas in the previous paragraph, through the explicit evaluation of the overlap matrix element h ,
1; 0 |O| ,
0; 0 i = 0, weare able to write: σ hh = σ ++ + σ + − + σ − σ h + . (39)The tree level cross sections are: σ H ++ = πα W s (cid:20) s + tt − s − tt + s − M H M W (cid:21) , (40) σ H = πα W s " (cid:18) s − ts + t − s + tt − M H M W (cid:19) + 9 M H M W , (41) σ H − + = πα W s ((cid:18) t + ss + 2 s + tt + 2 M H M W (cid:19) + (cid:18) s − ts + t − t + ss − M H M W (cid:19) (42)+2 (cid:18) s − ts + t − s + tt − M H M W (cid:19) + 2 (cid:18) t + ss − s − tt + s − s + tt (cid:19) ) ,σ H = πα W s ((cid:18) t + ss + s − ts + t + M H M W (cid:19) + (cid:18) s + tt + 2 t + ss − M H M W (cid:19) + 2 (cid:18) t + ss − s + tt − s − ts + t (cid:19) ) , (43)in which as usual α W = g π . Once the expression of the eikonal current is known, it’s straightforward in the overlapformalism to obtain the expressions of the dressed cross sections. The procedure follows closely the one described inthe previous section, and we obtain (see also [8]): O = e S ( s,M H ,M W ) O H , (44)where S ( s, M H , M W ) is obtained through an energy ordered integration of the eikonal factor given by the square ofthe emission current described in eq. (31): S ( s, M H , M W ) = −−→ T L L H − −→ T V L W − L H ) ! , (45)5here: L W = α W π ln (cid:18) sM W (cid:19) ϑ ( √ s − M W ) , (46)and L H = α W π (cid:20) ln (cid:18) √ sM W (cid:19) − (cid:18) M H M W (cid:19)(cid:21) ϑ (cid:18) √ s − M H M W (cid:19) + (47) (cid:20) α W π ln (cid:18) √ sM H (cid:19)(cid:21) ϑ ( √ s − M H ) ϑ (cid:18) M H M W − √ s (cid:19) . Starting from (44) the pathway to obtain the explicit expressions for the dressed cross sections follows how pointed inthe previous paragraph, considering obviously the states classified in (33) as external physical states for the overlapmatrix. A straightforward calculation leads to: σ ± + = 14 (cid:0) σ H ++ + σ H − + + 2 σ H (cid:1) + 112 (cid:0) σ H ++ + σ H − + − σ H (cid:1) e − L H (48) ± (cid:0) σ H ++ − σ H − + + 2 I H (cid:1) e − ( L W − L H ) ± (cid:0) σ H ++ − σ H − + − I H (cid:1) e − L W + ( L W − L H ) + 16 (cid:0) σ H ++ + σ H − + − σ H (cid:1) e − L W + ( L W − L H ) ,σ = 14 (cid:0) σ H ++ + σ H − + + 2 σ H (cid:1) + 112 (cid:0) σ H ++ + σ H − + − σ H (cid:1) e − L H (49) − (cid:0) σ H ++ + σ H − + − σ H (cid:1) e − L W + ( L W − L H ) ,σ = 14 (cid:0) σ H ++ + σ H − + + 2 σ H (cid:1) + 112 (cid:0) σ H ++ + σ H − + − σ H (cid:1) e − L H (50)+ 23 (cid:0) σ H ++ + σ H − + − σ H (cid:1) e − L W + ( L W − L H ) ,σ h + = 14 (cid:0) σ H ++ + σ H − + + 2 σ H (cid:1) − (cid:0) σ H ++ + σ H − + − σ H (cid:1) e − L H , (51)where I H = ( σ − σ ∗ + ). We report below also the explicit expressions for these particular contributions, that are: σ H = πα W s ((cid:18) s + tt + 2 s − ts + t + M H M W (cid:19) + (cid:18) s + tt + s − ts + t − M H M W (cid:19) ) , (52) σ H ∗ + = πα W s ((cid:18) s + tt − t + ss − M H M W (cid:19) + (cid:18) t + ss − s − ts + t − M H M W (cid:19) ) . (53)From eqs. (48,49,50) it is easy to obtain the asymptotic ( √ s → ∞ ) behavior of the cross sections. Namely, someparticular combinations between the cross sections are radiative invariants, that is combinations that are free fromthe DL corrections; these invariants are ¶ :Θ = σ ++ + σ − + + σ h + + σ , (54)Θ = σ ++ − σ − + + 2 I , (55)In the limit √ s → ∞ it’s possible to establish a precise asymptotic behaviour of the electroweak corrected crosssections; in fact in this situation we have: lim √ s →∞ e − ( L W − L H ) = e − αWπ ln MHMW (56) ¶ notice that in the case at hand a direct calculation gives σ Hh + = σ H . √ s →∞ e − L W = 0 lim √ s →∞ e − L H = 0 , (57)so the cross sections in this limit become: σ ++ → (cid:0) σ H ++ + σ H − + + 2 σ H (cid:1) + 14 (cid:0) σ H ++ − σ H − + (cid:1) e − αWπ ln MHMW , (58) σ − + → (cid:0) σ H ++ + σ H − + + 2 σ H (cid:1) − (cid:0) σ H ++ − σ H − + (cid:1) e − αWπ ln MHMW , (59) σ , → (cid:0) σ H ++ + σ H − + + 2 σ H (cid:1) . (60)Finally, let us now consider resummed DL EW corrections of infrared origin for exclusive observables, i.e. observablesin which additional gauge bosons emission is forbidden. In this case, the treatment of Sudakov DLs is analogous to theknown results present in the literature [6], but we have to take into account the mass splitting between the weak scaleand the Higgs mass. The resummed cross section is obtained by multiplying each external leg i by an exponentialfactor as follows: σ Sud = σ H exp " − X i (cid:18) t iL ( t iL + 1) L H + t iV ( t iV + 1)4 ( L W − L H ) (cid:19) , (61)where, compared with (45), this last expression shows a sum over the charged external legs labeled by i ; t iL ( V ) isreferred to a single leg and not to a double leg composition. Our results are given in eqs. (27,28,29) for the case √ s < M H and eqs. (48,49,50) for the case √ s ≫ M H ≫ M W :they represent the all-order resummed expression for EW radiative corrections at the double log level. Here wehave considered fully inclusive observables (i.e., gauge bosons radiation in the final state is always included); thecorresponding tree level cross sections for longitudinal gauge bosons scattering are given in eqs. (14,15,16) for √ s < M H and eqs. (48,49,50) for √ s ≫ M H ≫ M W . The asymptotic behavior of the cross sections can be seen in fig. 1: forvery high energies every single cross section tends to a value which is a linear combination of the ”radiative invariants”defined in eq. (54). In this regime radiative corrections are of the same order of tree level values; notice however thatthis situation is valid for energies that are far too high for current or near future experiments. At the TeV scale relevantfor LHC and ILC (1 TeV) or CLIC (3 TeV) the situation is depicted in fig. 4. Inclusive radiative corrections are below10 % under the TeV scale, i.e. at the level of, or bigger than, NLO QCD corrections [10]. Relative corrections growtowards the 30% value as the invariant mass of the scattering gauge bosons reaches 3 TeV.It is particularly interesting to compare the EW corrections of IR origin for inclusive observables with the ones forexclusive observables (fig. 5 and 6). The latter are the ones usually considered in the literature: it is usually assumedthat additional gauge boson emission is either irrelevant and/or produces a final state that is distinguishable from theone produced by the hard scattering. However, it has been noticed for instance in [9] that at the LHC, even withthe actual experimental cuts a certain degree of weak boson emission may escape detection and needs to be included.We plot in fig. 5 and in fig. 6 the corrections to the fully inclusive cross sections σ and σ , labeled ”BN”, and tothe exclusive cross sections, which includes only virtual EW corrections, labeled ”Sudakov” [6]. The case of σ isparticularly interesting since radiative corrections range between -40 % and +25 % depending on the definition of theobservable. The relevant value for EW corrections of infrared origin will depend on the experimental setup and on thevarious cuts defining the observables; we think it is important to notice that in any case, for the “standard” exclusivedefinition there is a relevant suppression of the signal. Our result is compatible with the ones obtained in [4] in view ofthe different treatments (complete one loop vs. DL resummed, light vs. heavy Higgs and so on). Finally, in the caseof σ , which is interesting for a linear collider where the two final electrons are doubly tagged, radiative correctionsturn out to be negative both in the inclusive and exclusive case; however the exclusive case is more suppressed andreaches the -40% value at 3 TeV.Overall, electroweak radiative corrections to strongly interacting longitudinal bosons turn out to be potentiallyrelevant for next generation of colliders. Here we have chosen to use the SM heavy Higgs case as a prototype and wefind that these corrections are in the -40 % + 25 % range for energies below 3 TeV. The impact of these correctionson the analysis of the EW symmetry breaking sector through boson scattering depends on the model considered andon the details of the experimental cuts; we postpone a more refined study to a subsequent paper [5].7 Appendix
In this appendix we illustrate the main symmetry proprieties of the Standard Model Lagrangian in the limit g ′ → SU (2) L ⊗ SU (2) R extension.Since we work considering just the scalar sector, our starting point is the Lagrangian of the SU (2) gauged σ -model: L = − T r (cid:16)c W µν c W µν (cid:17) + 12 T r (cid:2) ( D µ Φ) † D µ Φ (cid:3) − µ T r (cid:0) Φ † Φ (cid:1) − λ (cid:2) T r (cid:0) Φ † Φ (cid:1)(cid:3) , (62)where the scalar content of the theory is organized into the following matrix:Φ ≡ √ h + iϕ i τ i ) = 1 √ (cid:18) h + iϕ ϕ − iϕ ϕ + iϕ h − iϕ (cid:19) (63)and where, as usual: c W µ ≡ W kµ τ k , c W µν = ∂ µ c W ν − ∂ ν c W µ + ig hc W µ , c W ν i ,D µ Φ = ∂ µ Φ + ig c W µ Φ . (64)Taking into account this notation, in which the matrix Φ acquires the standard “ doublet-antidoublet ” form, the gaugetransformations, restricted to the isospin SU (2) L case in the limit g ′ →
0, are: ig c W ′ µ = g L ( x ) ig c W µ g † L ( x ) + g L ( x ) ∂ µ g † L ( x ) , c W ′ µν = g L ( x ) c W µν g † L , Φ ′ = g L ( x )Φ , (65)where: g L ( x ) = exp (cid:16) iα k ( x ) τ k (cid:17) ∈ SU (2) L . (66)In order to clarify the transformation proprieties under the gauge symmetry, it’s straightforward to consider explicitly(65) for the scalar fields, obtaining: h → h − ϕ k δ kl α Ll ( x ) ,ϕ k → ϕ k + hα Lk ( x ) + ε klm ϕ l α Lm ( x ) . (67)At this point it’s possible to see that the Lagrangian in (62), because of the limit g ′ →
0, has a larger global SU (2) L ⊗ SU (2) R symmetry, under which the scalar fields transform as:Φ ′ = L Φ R † , (68)with L ∈ SU (2) L and R ∈ SU (2) R or, expanding the fields as in (63): h → h + ϕ k δ kl (cid:0) α Rl − α Ll (cid:1) ϕ k → ϕ k + h (cid:0) α Lk − α Rk (cid:1) + ε lmk ϕ l (cid:0) α Rm + α Lm (cid:1) (69)After electroweak symmetry breaking the global SU (2) L ⊗ SU (2) R symmetry is spontaneously broken into its diagonalcustodial subgroup SU (2) V ; the transformation laws under this custodial symmetry for the scalar fields are: h → hϕ k → ϕ k + ε lmk ϕ l α m = ϕ k + i ( T mV ) kl α m ϕ l . (70)8s a consequence the Higgs boson h is a singlet, while the three Goldstone bosons are a triplet, transforming accordingto the adjoint representation of SU (2) V .Once assumed these transformation proprieties it’s possible to consider the explicit interactions in the Lagrangian (62)that we have used during our work.Introducing the Higgs mass as function of the parameters in the scalar potential as M H = 2 λv , the interactions thatinvolve the scalar fields are described by: L = − M H v h − M H v h − M H v h ϕ k ϕ l δ kl − M H v hϕ k ϕ l δ kl − M H v ( ϕ k ϕ l δ kl ) ( ϕ m ϕ n δ mn ) , (71)while the gauge interactions of the scalar fields, written in the g ′ → L = g δ kl W lµ [ h ( ∂ µ ϕ k ) − ϕ k ( ∂ µ h )] + g ε klm ϕ k W lµ ( ∂ µ ϕ m ) + g v g µν hW kµ W lν δ kl . (72)Considering in particular the 3-vertex interactions L ( hϕW, ϕϕW ) of the previous Lagrangian, as in fig.2, it’s possibleto construct the eikonal current that describes the emission of a soft gauge boson from a longitudinal one, with thepossibility to have, as final state, a gauge boson as well as an Higgs boson, obtaining the current: J µ ( k, b ; p ) = gp µ p · k (cid:2) T bV Θ W + T bH Θ H (cid:3) = gp µ p · k (cid:20) T bL Θ H + 12 T bV (Θ W − Θ H ) (cid:21) , (73)with: (cid:0) T bL (cid:1) ac = 12 (cid:18) iε abc − iδ ab iδ bc (cid:19) = 12 ( T aV + T aH ) (cid:0) T bR (cid:1) ac ≡ (cid:18) iε abc + iδ ab − iδ bc (cid:19) = 12 ( T aV − T aH ) ; (74)at this point the proprieties of symmetry follow directly from the commutation relations: (cid:2) T aV , T bV (cid:3) = iε abc T cV (cid:2) T aH , T bH (cid:3) = iε abc T cH (cid:2) T aH , T bH (cid:3) = iε abc T cV , (75)and: (cid:2) T aL , T bL (cid:3) = iε abc T cL (cid:2) T aR , T bR (cid:3) = iε abc T cR (cid:2) T aL , T bR (cid:3) = 0 , (76)from which n T a =1 , , V , T a =1 , , H o are the six generators of the O (4) group and n T a =1 , , L , T a =1 , , R o are the generatorsof the SU (2) L ⊗ SU (2) R ∼ O (4) group; as a consequence, n T a =1 , , V o are the generators of the custodial SU (2) V diagonal subgroup. 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800 GeV ++-++ Figure 1: Case √ s ≫ M H > M W : ratio between the dressed cross sections and the invariant Θ , as defined in (54),as function of the c.m. energy. α α β β ig(p a − p c ) µ (T Lb ) ac ig/2(p a − p c ) µ (T Vb ) ac p a p c W µ b p c p a W µ b ω > M H ω < M H ϕ c ϕ a ϕ c ϕ a Figure 2: Left: Diagrammatic representation of the overlap matrix. The t-channel couples legs 1 and 1 ′ , while thechoice of the momentum sign on the external legs fixes the conservation law for the generators of the consideredsymmetry as: T a − T a ′ = T a ′ − T a . Right: Feynman rules for the scalar-scalar-gauge interactions from (72) in the twosituations ω > M H and ω < M H . In the first case we have chosen the basis in which ϕ ≡ h . T L and T V are thosedefined in (74). 11 .4 0.6 0.8 1.0 s , TeV - - ∆Σ ij I Σ H M ij % , Θ=Π (cid:144) ++-++
Figure 3: Case √ s ≪ M H : ratio between the dressed cross sections and the tree level ones as function of the c.m.energy, at fixed scattering angle θ = π/ M H = 800 GeV. s ,TeV - - ∆Σ ij I Σ H M ij % , M H =
800 GeV,
Θ= Π ++-++
Figure 4: Case √ s ≫ M H > M W : ratio between the dressed cross sections and the tree level ones as function of thec.m. energy, at fixed scattering angle θ = π/ M H = 800 GeV.12 .5 2.0 2.5 3.0 s ,TeV - - - - ∆Σ ij I Σ H M ij % , M H =
800 GeV,
Θ= Π + , Sudakov3 + , BN Figure 5: Case √ s ≫ M H > M W : Block-Nordsieck corrections vs. Sudakov corrections in the σ case. In theinclusive treatment the electroweak corrections change their sign compared to the exclusive case. s ,TeV - - - - ∆Σ ij I Σ H M ij % , M H =
800 GeV,
Θ= Π
Figure 6: Case √ s ≫ M H > M W : Block-Nordsieck corrections vs. Sudakov corrections in the σ33