The top quark right coupling in the tbW-vertex
aa r X i v : . [ h e p - ph ] D ec The top quark right coupling in the tbW-vertex
Gabriel A. González-Sprinberg ∗ and Jordi Vidal † Instituto de Física, Facultad de Ciencias, Universidad de la República,Iguá 4225, Montevideo 11600, Uruguay. Departament de Física Teòrica, Universitat de València, and Institutode Física Corpuscular (IFIC), Centro Mixto Universitat deValència-CSIC, E-46100 Burjassot, València, Spain.May 16, 2018
FTUV - 15 - 1006.2801
Abstract
The most general parametrization of the tbW vertex includes a right coupling V R that is zero at tree level in the standard model. This quantity may be measuredat the Large Hadron Collider where the physics of the top decay is currentlyinvestigated. This coupling is present in new physics models at tree level and/orthrough radiative corrections, so its measurement can be sensitive to non standardphysics. In this paper we compute the leading electroweak and QCD contributionsto the top V R coupling in the standard model. This value is the starting point inorder to separate the standard model effects and, then, search for new physics. Wealso propose observables that can be addressed at the LHC in order to measurethis coupling. These observables are defined in such a way that they do not receivetree level contributions from the standard model and are directly proportional tothe right coupling. Bounds on new physics models can be obtained through themeasurements of these observables. Top quark physics is now a high statistics physics, mainly due to the huge amounts ofdata coming from the Large Hadron Collider (LHC) run I and, now, run II [1, 2]. It isstrongly believed that, due to its very high mass, the top quark will be a window to newphysics [3]. This can be easily understood in the effective lagrangian approach, wherethe new physics contributions can be parametrized in a series expansion in terms ofthe parameter m t / Λ , where m t is the top quarks mass and Λ is the new physics scale.Besides, most of the top quark properties and couplings are known with a precision ∗ gabrielg@fisica.edu.uy † [email protected] t → bW + , is known. For instance, many extensions of the SM predict newdecay modes that can be accessible at LHC. The top quark was detected for the firsttime at TEVATRON [4, 5], where many of its physical properties where measured andsome bound on the anomalous W tb couplings were set [6–8]. Nowadays, top physics isintensively investigated in theoretical research [3] and at the LHC [1,2,9], as the readercan verified in the ATLAS and CMS web pages. In particular, the measurements ofthe different helicity components of the W in the top decay allows to study the tbW Lorentz vertex structure [10]. These studies were extended in recent years [11–15].There, the longitudinal and transverse helicities of the W coming from the top decaywere investigated and they show that a precise determination of the Lorentz formfactors of the vertex can be done with a suitable choice of observables. The mostgeneral parametrization of the on-shell vertex needs four couplings, but in the SMthree of them are zero at tree level while only the usual left, coupling V L is not zeroand with a value close to one [16]. This is not the case in extended models where, inaddition, some of these couplings can also be sensitive to new CP-violation mechanisms.The right top coupling is largely unknown and deserves a careful study. The othertwo couplings, usually called tensorial couplings, were investigated at the LHC [17]and will not be considered here. The predictions for the SM, for the two-Higgs-doubletmodel (2HDM) and other extended models where recently considered in refs. [18–20].In this paper we first compute the right coupling V R in the SM at leading order, anddefine appropriate observables in order to have direct access to it. The former oneloop calculation is needed in order to disentangle SM and new physics effects in theobservables. Next, we will introduce a set of observables that allows to perform aprecise search of the V R coupling. We obtain a combination of observables directlyproportional to it in such a way that they are not dominated by the leading tree levelSM contribution V L . These observables can be an important tool in order to measurenew physics contributions to V R .This paper is organized as follows. In the next section we introduce a precisedefinition of the right coupling and compute the first order QCD and electroweak(EW) contributions. In section 3 we present and discuss a set of observables thatcan be measured with LHC data, both in the polarization matrix and in the spincorrelations. Finally, we discuss the results and present our conclusions in section 4. tbW coupling in the SM Considering the most general Lorentz structure for on-shell particles, the M tbW am-plitude for the t ( p ) → b ( p ′ ) W + ( q ) decay can be written in the following way: M tbW = − e sin θ w √ ǫ µ ∗ × u b ( p ′ ) (cid:20) γ µ ( V L P L + V R P R ) + iσ µν q ν M W ( g L P L + g R P R ) (cid:21) u t ( p ) , (1)2here the outgoing W + momentum, mass and polarization vector are q = p − p ′ , M W and ǫ µ , respectively. The form factors V L and V R are the left and right couplings,respectively, while g L and g R are the so called tensorial or anomalous couplings.The expression (1) is the most general model independent parametrization for the tbW + vertex. Within the SM, the W couples only to left particles then, at tree level,all the couplings are zero except for V L that is given by the Kobayashi-Maskawa matixelement V L = V tb ≃ [16]. At one loop all of these couplings receive contributions fromthe SM. For example, the real and imaginary parts of the g L and g R couplings werecalculated in the SM [18] and in a general alligned-2HDM [19].The one loop electroweak and QCD contributions to the V R coupling can be calcu-lated just by considering the vertex corrections shown in figure 1 and extracting fromthem the Lorentz structure corresponding to the V R coupling. Note that this Lorentzstructure is not present in the SM lagrangian so, at one loop, there is no need for anycounter-term for this contribution. For this reason, the SM one-loop contribution to the V R coupling is finite. This is similar to what happens for the top g L,R couplings [18,19].We will denote each diagram by the label
ABC according with the particles running t bAB CW + µ Figure 1: One-loop contributions to the V R coupling in the t → bW + decay. in the loop. In addition to the usual notation for the SM particles we use the symbols w and w for the neutral and charge would-be Goldstone bosons, respectively, and H for the SM Higgs. There are 18 diagrams that have to be considered and the one loop V R value one gets from them is finite, without the need of renormalization as alreadystated. In particular, the one-loop contributions from diagrams tw w , tHw , bww , bwH , w tb and Htb are ultraviolet (UV) divergent. However, when summing them upby pairs ( i.e. tw w + tHw , bww + bwH and w tb + Htb ), the result is finite. Theelectroweak contributions of all the diagrams are given in the appendix A in terms ofparametric integrals. For the UV divergent diagrams we present the sum of the twodiagrams that cancels the divergence, as can be seen in eqs. (46), (52) and (57). There,the first (second) term, in each of these expressions, corresponds to the UV safe sectorof the first (second) diagram, while the third one corresponds to the sum of the UVdivergent part of the two diagrams that results in a finite contribution, as it should be.All the contributions can be written as: V ABCR = α V tb r b I ABC , (2)where r b = m b /m t and I ABC is an integral shown in appendix A for all the diagramsthat contribute to V R . As expected, all the contribution are proportional to the bottommass through r b . 3hen one of the particles circulating in the loop is a photon the integral can beperformed analytically. Then, the contribution can be written as V ABCR = α π V tb Q A I ABC , (3)with Q A being the charge of the A -quark circulating in the loop in units of | e | (for the γtb diagram, Q A = Q t · Q b ). The I ABC analytical expressions, as well as their limitsfor r b → , are shown in appendix B. All these expressions were used as a check of ourcomputations.The leading QCD contribution can be easily obtained from eq. (56) just by substi-tuting the couplings of the photon by the gluon ones, so we get: V gtbR = − α s π C F V tb I γtb , (4)with I γtb given in eq. (62) and (67), and C F = 4 / is the color factor.With the set of values of ref. [16], the numerical values for the contribution to V R of each diagram and the complete one-loop SM value are given in table 1, for V tb = 1 .In this table the contribution from the UV divergent diagrams are summed up toget a finite result; the first (second) quantity between brackets corresponds to the finitecontribution of the first (second) diagram, and the third one (which is logarithmic) isthe finite sum of the two UV divergent parts.As can be seen in table 1, even though the contribution of most of the diagrams isof the order of − for the real part of the V R coupling, the total EW contribution is,at the end, two orders of magnitude smaller due to the accidental cancellations amongthe diagrams. The QCD contribution is real and four orders of magnitude bigger thanthe real EW one so that the real part of the V R coupling is dominated by the former.The imaginary part, instead, remains of order − , and it is purely EW. In general, the LHC observables considered in the literature are not very sensitive tothe right coupling V R . This is due to the fact that this coupling comes from a lagrangianterm that has the same parity and chirality properties than the leading coupling V L so that the observables receive contributions from both terms. These observables arethe angular asymmetries in the W rest frame [10, 11, 21, 22], angular asymmetries inthe top rest frame [11, 22–24] and spin correlations [11, 22, 25, 26]. Our strategy willbe to define observables directly proportional to V R considering the dependencies onthe coupling terms. Similar ideas were widely applied when investigating tau physicsdipole moments [27, 28]. For top decays, one way to suppress the V L contribution is todefine observables where only right polarized quarks contribute, but this polarizationis not accessible to the present facilities and experiments. Given the results shown inthe previous section, from now on we always assume that the imaginary part of the V R coupling is negligible. 4 able 1: Contribution to V R from the different diagrams Diagram Contribution to V R tZW . × − tγW − . × − tHW tw w (cid:2) ( − . × − ) + +(0 . × − ) = − . × − tHw +(0 . × − ) (cid:3) tZw . × − tγw . × − bW Z (1 .
12 + 8 . i ) × − bW γ (8 . − . i ) × − bW H bww (cid:2) ( − .
72 + 3 . i × − ) + +( − . − . i × − ) = (1 . − . i ) × − bwH +(1 . − . i × − ) (cid:3) bwZ (0 .
00 + 0 . i ) × − bwγ ( − .
47 + 2 . i ) × − Ztb − . × − γtb − . × − w tb (cid:2) ( − . × − ) + +(0 . × − ) = − . × − Htb +( − . × − ) (cid:3) Σ( EW ) (0 .
06 + 6 . i ) × − gtb ( QCD ) 2 . × − ( QCD + EW ) (2 .
68 + 0 . i ) × − W rest frame Top properties were studied in previous works by means of the observables that we justmentioned. One of the first possibilities are the angular asymmetries for the t → W + b decay, with the W + decaying leptonicaly. The normalized charged lepton angulardistribution in the W rest frame can be written as: d Γ d cos θ l = 38 (1 + cos θ l ) F + + 38 (1 − cos θ l ) F − + 34 sin θ l F , (5)where F , F ± are the normalized partial widths of the top decay into the W helicitystates, and θ l is the angle between the charged lepton momentum in the W rest frameand the W momentum in the t rest frame. Then, the asymmetries are defined, in termsof a new parameter z , as follows [10, 21]: A z = N (cos θ l > z ) − N (cos θ l < z ) N (cos θ l > z ) + N (cos θ l < z ) . (6)5he z parameter allows to separate the helicity fractions F , F ± . The value z = 0 gives the usual forward-backwards asymmetry while z = ± (2 / − defines the A ∓ asymmetries, respectively. The first one, A depends only on the F ± helicity fractionswhile A ± depends on both F and F ± .The helicity fractions, F and F ± , can be computed in terms of the tbW couplingsgiven in eq. (1), and they can be found, for example, in ref. [22]. Performing the θ l integration of eq. (6) for a given value of z , the numerator of the asymmetry A z , interms of V L and V R is: V L (cid:0) . z + 2 . z − . z − . (cid:1) + 0 . zV L V R + V R (cid:0) . z − . z − . z + 2 . (cid:1) . (7)One can now choose the value of the z parameter, within the range ( − , , to makezero the leading V L contribution. In this way we get the maximum sensitivity to V R .The coefficient of V L and V R from eq. (7), are plotted in figure 2. There, it can be seen - - - z Coefficient of V L Coefficient of V R Figure 2: Plot of the coefficients of V L and V R from eq. (7) in terms of z . - - V R A z A + A - A R Figure 3: Dependence on the V R couplingfor A R (blue-solid), eq. (8), and for theusual A + (purple-dashed) and A − (black-dot-dashed) asymmetries, eq. (6). that for z = z R = − . the coefficient of V L cancels, leaving the V R term, in eq. (7),as the leading one. For this z R the new asymmetry A R ≡ A Z ( z = z R ) is proportionalto V R and has the form: A R ( V L , V R ) = − . V L V R + 0 . V R .
12 ( V L + V R ) − . V L V R ≃ − . V R , (8)where the last expression is the value of the asymmetry taking V L = 1 and assuming | V R | << . As can be seen this asymmetry is directly proportional to V R so that anon-zero measurement of it is a direct test of V R = 0 .Note that the leading contribution to the usual A ± asymmetries comes from V L so that in order to be sensitive to V R one needs to subtract this SM central value. Infigure 3 we show the V R dependence of A R and also, for comparison, the dependence of A ± , with the V L leading contribution subtracted. As can be seen there A R may allowmore precise bounds on V R . Besides, it is directly proportional to V R and eliminatesother uncertainties that the V L dependence in A ± may introduce.6or polarized top it is possible to define asymmetries with respect to the normal andtransverse spin directions. These were studied in ref. [11], but they are not sensitive tothe V R coupling so we are not going to consider them in our analysis. Angular distribution of the decay products for the weak process t → W + b carriesinformation about the spin of the decaying top. Then, angular asymmetries can bebuilt to test the Lorentz structure of the vertex. We will follow the same proceduresdescribed previously, in order to optimize the sensitivity of the observables to V R but,in this case, for the asymmetries in the top rest frame.For the top decay t → W + b → l + νb, q ¯ q ′ b , the angular distribution of the product X = l + , ν, q, ¯ q ′ , W + , b , in the top rest frame, is given by: d Γ d cos θ X = 12 (1 + α X cos θ X ) , (9)where θ X is the angle between the momentum of X and the top spin direction, and α X are the spin-analyzer powers, given in references [11,22–24,29] in terms of the couplingsshown in eq. (1). Then, an asymmetry can be defined as: A zX ≡ N (cos θ X > z ) − N (cos θ X < z ) N (cos θ X > z ) + N (cos θ X < z ) = 12 (cid:2) α X (1 − z ) − z (cid:3) . (10)For z = 0 , one gets the usual forward-backward asymmetry: A X ≡ N (cos θ X > − N (cos θ X < N (cos θ X >
0) + N (cos θ X <
0) = α X . (11)Sensitivity to V R and V L for this asymmetry have already been given for X = l + , b, ν in the t -channel single top production in refs. [11, 30]. In order to define a new asym-metry directly proportional to the V R coupling one can again extract the SM leadingcontribution, given by the V L term in the A zX asymmetry, and make it zero. For the X = l , ν , b cases, the eq. (10) is: A zl = −
12 ( V L + V R − . V L V R ) × " V L (cid:18) − z − z + 12 (cid:19) + V R (cid:0) − . z − z + 0 . (cid:1) +0 . V L V R (cid:18) z + z − (cid:19) , (12) A zb = 0 . V L + V R − . V L V R × (cid:2) V L (cid:0) z − . z − (cid:1) + V R (cid:0) − z − . z + 1 (cid:1) + 0 . zV L V R (cid:3) , (13)7 zν = 0 . V L + V R − . V L V R × (cid:2) V L (cid:0) z − . z − (cid:1) + V R (cid:0) . z − . z − . (cid:1) + V L V R (cid:0) − . z + 0 . z + 0 . (cid:1) (cid:3) . (14) - - - - z Coefficient of V L from A lz Coefficient of V L from A bz Coefficient of V L from A Ν z Figure 4: Plot of the coefficients of V L , interms of z , from eqs. (12), (13) and (14). - - - - - - - V R A l z l Figure 5: Dependence on the V R couplingfor the A z l l asymmetry, eq. (16). Figure 4 shows the behavior of the V L coefficient from the A l , A b and A ν asymme-tries. We can choose z in order to make the leading coefficients of eqs. (12), (13) and(14) to be zero. These z values are: z l = √ − , z b = − . , z ν = − . , (15)and then, the new asymmetries are: A z l l = − . V R V L + V R − . V L V R ≃ − . V R , (16) A z b b = 0 . V R − . V L V R V L + V R − . V L V R ≃ − . V R , (17) A z ν ν = 0 . V L V R − . V R V L + V R − . V L V R ≃ . V R , (18)where in the last expression we show the value of the asymmetries for V L = 1 andassuming | V R | << . From eq. (12) one can see that unfortunately, for the A l asym-metry, the same value of z that cancels the coefficient of V L also cancels the V L V R term, in such a way that a poorer sensitivity to the coupling will be expected from thisobservable because the surviving term is V R instead of V L V R .In figures 5, 6 and 7 we show the dependence on V R for the new A z l l , A z b b and A z ν ν observables. We have explicitly checked that the usual forward-backward asymmetriesfor the same decay product have a very similar behavior as the ones shown in thefigures, once the leading V L contribution is removed. However, the new observablesagain have the advantage that are proportional to V R so that their measurement mayprovide a direct bound or measurement of the V R coupling with no assumption on thevalue of the V L leading term. 8 - V R A b z b Figure 6: Dependence on the V R couplingfor the A z b b asymmetry, eq. (17) - - - - - - - V R A Ν z Ν Figure 7: Dependence on the V R couplingfor the A z ν ν asymmetry, eq. (18). t ¯ t production The top–antitop spin correlations depend on the Lorentz structure of the tbW vertex.This structure can be studied through the measurement of the angular distributions ofthe decay products for the t → W + b and ¯ t → W − ¯ b processes, that carry informationon the top and antitop spin correlation terms. In particular, using the notation ofprevious section, the double angular distribution – of the decay products X , from top,and ¯ X ′ , from antitop – can be written as [25, 26]: σ dσd cos θ X d cos θ ¯ X ′ = 14 (1 + C α X α ¯ X ′ cos θ X cos θ ¯ X ′ ) , (19)where θ X ( θ ¯ X ′ ) is the angle between the momentum of the decay product X ( ¯ X ′ ) andthe momentum of the top (antitop) quark in the t ¯ t center-of-mass frame, α X and α ¯ X ′ are the spin analyzer powers of particles X and ¯ X ′ , respectively. C is a coefficientthat weights the spin correlation between the quark and the antiquark. Then, one candefine the asymmetry A z z X ¯ X ′ ≡ σ " Z z d (cos θ X ) Z z d (cos θ ¯ X ′ ) dσd cos θ X d cos θ ¯ X ′ + Z z − d (cos θ X ) Z z − d (cos θ ¯ X ′ ) dσd cos θ X d cos θ ¯ X ′ − Z z − d (cos θ X ) Z z d (cos θ ¯ X ′ ) dσd cos θ X d cos θ ¯ X ′ − Z z d (cos θ X ) Z z − d (cos θ ¯ X ′ ) dσd cos θ X d cos θ ¯ X ′ . (20)The different regions of integration for this asymmetry are plotted in figure 8. There,the number of events on each of the regions is collected and the sign of their contributionto the asymmetry is also indicated. The particular case z = z = 0 corresponds to theusual spin correlation asymmetry: A X ¯ X ′ = N (cos θ X cos θ ¯ X ′ > − N (cos θ X cos θ ¯ X ′ < N (cos θ X cos θ ¯ X ′ >
0) + N (cos θ X cos θ ¯ X ′ < . (21)9 z , z L H + LH + LH - L H - L - - - - Θ X C o s Θ X ¢ Figure 8: Regions of integration and signof the contribution to the spin correlationasymmetry A z z X ¯ X ′ defined in eq. (20), de-pending on the point ( z , z ) z + C = = = z - C = = = - - - - z z Figure 9: Values of z and z that makeszero the V L leading terms of the A z z ll ′ asym-metry, eq. (22), for different values of thespin correlation coefficient C . For the A z z X ¯ X ′ asymmetry, eq. (20), there is a range of values of z and z that cancelthe leading V L contribution, in such a way that the observable becomes proportionalto the V R coupling. For the X = l , ¯ X ′ = l ′ case, these values satisfy the relation: z ± = − z ± p C (1 − z ) + 4 z C (1 − z ) . (22)Similarly to what happened for A zl , in eq. (12), for values of z and z satisfying theprevious equation, the coefficient of V L V R in the asymmetry also accidentally cancels.Then, it remains the V R coupling as the leading contribution. Solutions to eq. (22) areplotted in figure 9 where it can be seen that the values of z and z are restricted tobe in the first quadrant (for the z +2 solution) or in the third one (for the z − solution).In addition to the trivial solutions of eq. (22), i.e. z = ± , z = 0 , that reproducesthe usual A ll ′ asymmetry, one can improve the computation by finding the values of z and z that, satisfying eq. (22), also maximize the coefficient of the leading V R termin eq.(20), namely: . V R (cid:2) z z + 0 . C (cid:0) z (cid:0) − z (cid:1) + z − (cid:1)(cid:3) . (23)This can be trivially done and the result is z = z = z ll ′ ≡ ± r C − C √ C, (24)that correspond to the intersection of the curves of figure 9 with the diagonal line ofthe first and third quadrants. These values of z , z give a maximum sensitivity of the A z z ll ′ asymmetry to the V R coupling.In figure 10 we show the dependence on the V R coupling for the A z z ll ′ asymmetryin eq. (20), for C = 0 . [11] , and z = z = z ll ′ = 0 . , given by eq. (24).The same procedure followed here can be used for other decay products of the W ± .For t ¯ t → l ν b ¯ ν l ′ ¯ b final state, the values of z and z that cancels the coefficient of10 - - V R A ll ¢ z ll ¢ Figure 10: Dependence on V R coupling for A z z ll ′ in eq. (20), for C = 0 . and z = z = z ll ′ = 0 . . - - V R A Ν l ¢ z Ν l ¢ Figure 11: Dependence on the V R couplingfor A z z νl ′ , eq. (20), for C = 0 . and z = − z = z νl ′ = 0 . . the leading V L term in the A z z νl ′ asymmetry ( X = ν , ¯ X ′ = l ′ in eq. (20)), satisfy aquadratic equation z z = − β (1 − z )(1 − z ) , β = 0 . C, (25)and the solution is z ± = 12 β (1 − z ) h z ± p z + 4 β (1 − z ) i . (26)Here, − ≤ z ≤ for the z +2 solution and ≤ z ≤ for the z − solution of eq. (26),in order to satisfy | z ± | < . In that case, these set of z , values do not cancel the V R term of the asymmetry, that remains as the leading one: − . (cid:20) z z − C − z )(1 − z ) (cid:21) V R . (27)One can easily find the values that maximize this coefficient and simultaneously verifyeq. (26): z = − z = z νl ′ ≡ ± r β − β p β (28)In figure 11 we show the dependence of the spin correlation asymmetry A z νl ′ z νl ′ νl ′ on V R ,for z , z given by eq. (28) with C = 0 . . Note that the sensitivity of both asymmetriesshown in figures 10 and 11 is rather similar.Another spin correlation considered in the literature is the angular distribution ofthe top (antitop) decay products defined as [31]: σ dσd cos ϕ X ¯ X ′ = 14 (1 + D α X α ¯ X ′ cos ϕ X ¯ X ′ ) , (29)where ϕ X ¯ X ′ is the angle between the momentum of the X particle in the t rest frame,and that of the X ′ one, in the ¯ t rest frame. D is the spin correlation coefficient. Then,one can construct the following asymmetry: ˜ A zX ¯ X ′ ≡ N (cos ϕ X ¯ X ′ > z ) − N (cos ϕ X ¯ X ′ < z ) N (cos ϕ X ¯ X ′ > z ) + N (cos ϕ X ¯ X ′ < z ) = 12 (cid:0) α X α ¯ X ′ (1 − z ) − z (cid:1) . (30)11or z = 0 one gets the usual forward-backward spin correlation asymmetry ˜ A X ¯ X ′ = 12 D α X α ¯ X ′ . (31)For both W ± leptonic decays, X = l and ¯ X ′ = l ′ , and following similar procedures asin the previous sections, we can find the value of z that cancels the V L leading term ineq. (30): z = ˜ z ll ′ ≡ D (cid:16) − √ D (cid:17) . (32)This value also cancels the V R term so that the V R contribution is the dominant one.Then, for D = − . [11], the ˜ A zll ′ asymmetry is: ˜ A ˜ z ll ′ ll ′ = − . V L V R + 0 . V L V R − . V R ( V L − . V L V R + V R ) ≃ − . V R (33)where again, in the last term, we show the asymmetry for V L = 1 and | V R | << . Thisasymmetry can be seen in figure 12. - - - - - - - V R A Ž ll ’ z Figure 12: Dependence on the V R couplingfor the ˜ A zll ′ asymmetry, eq. (33), for D = − . and z = ˜ z ll ′ given by eq. (32). - - - - - - - V R A Ž Ν l ’ z Ν l ’ Figure 13: Dependence on the V R couplingfor the ˜ A zνl ′ asymmetry, eq. (35), for D = − . and z = z = ˜ z νl ′ given by eq. (34). Analogously, for W leptonic decays with neutrino-lepton final state, X = ν , ¯ X ′ = l ′ ,the z -value that cancels the V L term in eq. (30) is z = ˜ z νl ′ = − . D − s (cid:18) D . (cid:19) , (34)and then, the ˜ A zνl ′ asymmetry, for D = − . , is: ˜ A ˜ z νl ′ νl ′ = 0 . V L V R − . V L V R + 0 . V L V R ( V L − . V L V R + V R ) ≃ . V R . (35)12he last term in this equation is given for V L = 1 and | V R | << . This asymmetryis depicted in figure 13. Note that in this case the sensitivity to V R is lower than theone obtained in some of the previous observables due to the small coefficients in thenumerator of eq. (35).We checked that the expressions of the observables defined for the V R coupling inthis section show a behavior comparable to the one obtained from the expressions ofthe observables defined in the literature, once the V L leading contribution is removed.Moreover, the observables defined here have the advantage of being directly propor-tional to the V R term that we want to test. We computed the SM one-loop QCD and electroweak contribution to V R . Due to acci-dental cancellations between the diagrams, the leading contribution is mainly comingfrom QCD, it is real and of the order of − . Any measurement of observables thatmay lead to V R above to − should be interpreted as new physics effects.We also have proposed new observables that may provide a direct measurement ofthe right coupling. We found that for several angular asymmetries considered in theliterature it is possible to define new observables, with an optimal choice of parameters,in such a way that they become a direct probe of V R . These observables include angularasymmetries in the W rest frame, angular asymmetries in the top rest frame and alsospin correlations. All the new observables defined in this paper are proportional to V R and then suitable for a direct determination of this coupling. In some cases the behaviorshown by the expressions for our observables, presents a better sensitivity to V R (like itis shown in Figure 3). While the asymmetries usually considered in the literature haveleading contributions from V L , the new asymmetries that we have studied here have asa leading term the V R right coupling we are interested in. These asymmetries can bemeasured with LHC data, where a huge number of top events are being collected, inorder to obtain a direct measurement on the standard model contribution to the righttop quark. This work has been supported, in part, by the Ministerio de Ciencia e Innovación, Spain,under grants FPA2011-23897 and FPA2011-23596; by Ministerio de Economía y Com-petitividad, Spain, under grants FPA2014-54459-P and SEV-2014-0398; by GeneralitatValenciana, Spain, under grant PROMETEOII2014-087; and by CSIC and Pedeciba,Uruguay. 13 ppendix A Diagram contributions
Using the following definitions for the denominators: A Z = x (cid:2)(cid:0) ( y − r b + 1 (cid:1) y − r w ( y − (cid:3) − r z ( x − , (36) B Z = x (cid:8)(cid:2) ( x ( y −
1) + 1) r b + x − (cid:3) y − r z ( y − (cid:9) − r w ( x −
1) [ x ( y −
1) + 1] , (37) C Z = ( x − xy − r b − r w ( x − x ( y −
1) + r z xy + x ( y − xy − , (38) { A γ , B γ , C γ } = { A Z , B Z , C Z } ( r z → , (39) { A H , B H , C H } = { A Z , B Z , C Z } ( r z → r h ) . (40)with r x ≡ m x m t (41)and taking the usual definition for the SM couplings a t = − a b = 1 , v b = − s w and v t = 1 − s w , (42)the expression for the contribution of each diagram is given by: I tZW = − πs w × Z dx Z dy x y [ v t (1 + 2 xy ) − a t (5 − xy )] A Z , (43) I tγW = − π Q t × Z dx Z dy x y (1 + 2 xy ) A γ , (44) I tHW = 0 , (45) I tw w + I tHw = 116 πs w r w × Z dx Z dy ( − x y [1 + y − r b (1 − y )] A Z +(1 − rb ) x y (1 − y ) A H + x log A H A Z ) , (46) I tZw = 132 πc w × Z dx Z dy x y ( a t − v t ) A Z , (47) I tγw = − π Q t × Z dx Z dy x yA γ , (48) I bW Z = 132 πs w × Z dx Z dy x y [ v b (1 + 2 xy ) − a b (5 − xy )] B Z , (49) I bW γ = 18 π Q b × Z dx Z dy x y (1 + 2 xy ) B γ , (50)14 bW H = 0 , (51) I bww + I bwH = 116 πs w r w × Z dx Z dy ( x y [1 − x − r b (1 − x (1 − y ))] B Z − (1 − r b ) x y (1 − x ) B H + x log B H B Z ) , (52) I bwZ = 132 πc w × Z dx Z dy x y ( v b − a b ) B Z , (53) I bwγ = − π Q b × Z dx Z dy x yB γ , (54) I Ztb = 132 πc w s w × Z dx Z dy x [ a b (1 + xy ) − v b (1 − xy )] [ a t (1 + xy ) − v t (1 − xy )] C Z , (55) I γtb = 12 π Q b Q t × Z dx Z dy x (1 − xy ) C γ , (56) I w tb + I Htb = 116 πs w × Z dx Z dy x (1 − x )(1 − y ) ( − C Z + 1 C H + xr w log C Z C H ) . (57) Appendix B Contribution of diagrams with a photon I tγW = − r b ( " (1 − r w − r b − ∆)(1 − r w − r b − ∆)4 r b ∆ log (cid:18) r w − r w (cid:19) + " ∆ → − ∆ , (58) I tγw = " − r b ∆ (1 − r b − r w − ∆) log (cid:18) r w − r b + ∆2 r w (cid:19) + " ∆ → − ∆ , (59)15 bW γ = 2 r b + ( − − r b − r w + ∆) r b ∆(1 + r b − r w + ∆) " (1 − r w ) − r b (1 + r w ) + ∆(1 − r w ) − r b (1 − r w + ∆) log (cid:18) r b − r w − r b + ∆ (cid:19) + ( ∆ → − ∆ ) + " π i r b (1 − r w − r b + ∆)(1 − r w + ∆)(1 − r w + r b + ∆) ∆ − " ∆ → − ∆ , (60) I bwγ = − r b ( π i − r b − r w + ∆∆(1 + r b − r w + ∆) ! − ∆ → − ∆ ! + " − r b − r w + ∆∆(1 + r b − r w + ∆) log (cid:18) r b − r b − r w + ∆ (cid:19) + " ∆ → − ∆ , (61) I γtb = 2 r b ∆ log (cid:18) − r w + r b + ∆1 − r w + r b − ∆ (cid:19) , (62)with ∆ = q (1 − r w ) + r b − r b (1 − r w ) . In the limit r b → , the formulas get a simplest expression: I tγW r b → −−−−−→ − r b (1 − r w ) (cid:2) (3 − r w + 5 r w ) + 4 r w (1 − r w ) log( r w ) (cid:3) , (63) I tγw r b → −−−−−→ r b (1 − r w ) (cid:2) − r w + r w log( r w ) (cid:3) , (64) I bW γ r b → −−−−−→ r b " πi (2 + r w + r w )1 − r w + 2 r w (1 − r w + 2 r w − r w )(1 − r w ) log( r w )+ 1 − − r w (5 − r w + 2 r w − r w + r w )(1 − r w ) log(1 − r w ) , (65) I bwγ r b → −−−−−→ − r b − r w " π i (1 + r w ) + log (cid:18) r b − r w (cid:19) + r w log (cid:18) r w − r w (cid:19) , (66) I γtb r b → −−−−−→ − r b − r w log (cid:18) r b − r w (cid:19) , (67)16 eferences [1] F.-P. Schilling, Top Quark Physics at the LHC: A Review of the First TwoYears , Int. J. Mod. Phys.
A27 (2012) 1230016, [ arXiv:1206.4484 ].[2] R. Hawkings,
Top quark physics at the LHC , Comptes Rendus Physique (2015) 424–434.[3] W. Bernreuther, Top quark physics at the LHC , J. Phys. G (2008) 083001,[ arXiv:0805.1333 ].[4] CDF Collaboration
Collaboration, F. Abe et al.,
Observation of top quarkproduction in ¯ pp collisions , Phys. Rev. Lett. (1995) 2626–2631,[ hep-ex/9503002 ].[5] D0 Collaboration
Collaboration, S. Abachi et al.,
Observation of the topquark , Phys. Rev. Lett. (1995) 2632–2637, [ hep-ex/9503003 ].[6] C. Deterre, W helicity and constraints on the W tb vertex at the Tevatron , NuovoCim.
C035N3 (2012) 125–129, [ arXiv:1203.6802 ].[7]
D0 Collaboration
Collaboration, V. M. Abazov et al.,
Search for anomalous
W tb couplings in single top quark production in p ¯ p collisions at √ s = 1 . TeV , Phys. Lett. B (2012) 21–26, [ arXiv:1110.4592 ].[8] D0 Collaboration, V. M. Abazov et al.,
Search for anomalous Wtb couplings insingle top quark production , Phys. Rev. Lett. (2008) 221801,[ arXiv:0807.1692 ].[9] C. Bernardo, N. F. Castro, M. C. N. Fiolhais, H. Gonçalves, A. G. C. Guerra,M. Oliveira, and A. Onofre,
Studying the
W tb vertex structure using recent LHCresults , Phys. Rev.
D90 (2014), no. 11 113007, [ arXiv:1408.7063 ].[10] F. del Aguila and J. Aguilar-Saavedra,
Precise determination of the Wtbcouplings at CERN LHC , Phys.Rev.
D67 (2003) 014009, [ hep-ph/0208171 ].[11] J. Aguilar-Saavedra and J. Bernabéu,
W polarisation beyond helicity fractions intop quark decays , Nucl. Phys. B (2010) 349–378, [ arXiv:1005.5382 ].[12] J. Drobnak, S. Fajfer, and J. F. Kamenik,
New physics in t − > bW decay atnext-to-leading order in QCD , Phys. Rev.
D82 (2010) 114008,[ arXiv:1010.2402 ].[13] S. D. Rindani and P. Sharma,
Probing anomalous tbW couplings in single-topproduction using top polarization at the Large Hadron Collider , JHEP (2011) 082, [ arXiv:1107.2597 ].[14] Q.-H. Cao, B. Yan, J.-H. Yu, and C. Zhang,
A General Analysis of
W tb anomalous Couplings , arXiv:1504.0378 .1715] A. V. Prasath, R. M. Godbole, and S. D. Rindani, Longitudinal top polarisationmeasurement and anomalous
W tb coupling , Eur. Phys. J.
C75 (2015), no. 9 402,[ arXiv:1405.1264 ].[16]
Particle Data Group
Collaboration, K. Olive et al.,
Review of ParticlePhysics , Chin.Phys.
C38 (2014) 090001.[17] M. Moreno Llácer,
Search for CP violation in single top quark events with theATLAS detector at LHC . PhD thesis, Valencia U., IFIC, 2014.[18] G. A. González-Sprinberg, R. Martinez, and J. Vidal,
Top quark tensor couplings , JHEP (2011) 094, [ arXiv:1105.5601 ]. [Erratum: JHEP05,117(2013)].[19] L. Duarte, G. A. González-Sprinberg, and J. Vidal, Top quark anomalous tensorcouplings in the two-Higgs-doublet models , JHEP (2013) 114,[ arXiv:1308.3652 ].[20] W. Bernreuther, P. Gonzalez, and M. Wiebusch,
The Top Quark Decay Vertex inStandard Model Extensions , Eur. Phys. J. C (2009) 197–211,[ arXiv:0812.1643 ].[21] B. Lampe, Forward - backward asymmetry in top quark semileptonic decay , Nucl.Phys.
B454 (1995) 506–526.[22] J. Aguilar-Saavedra, J. Carvalho, N. F. Castro, F. Veloso, and A. Onofre,
Probing anomalous Wtb couplings in top pair decays , Eur.Phys.J.
C50 (2007)519–533, [ hep-ph/0605190 ].[23] B. Grzadkowski and Z. Hioki,
New hints for testing anomalous top quarkinteractions at future linear colliders , Phys.Lett.
B476 (2000) 87–94,[ hep-ph/9911505 ].[24] R. M. Godbole, S. D. Rindani, and R. K. Singh,
Lepton distribution as a probe ofnew physics in production and decay of the t quark and its polarization , JHEP (2006) 021, [ hep-ph/0605100 ].[25] T. Stelzer and S. Willenbrock,
Spin correlation in top quark production at hadroncolliders , Phys.Lett.
B374 (1996) 169–172, [ hep-ph/9512292 ].[26] G. Mahlon and S. J. Parke,
Angular correlations in top quark pair production anddecay at hadron colliders , Phys.Rev.
D53 (1996) 4886–4896, [ hep-ph/9512264 ].[27] J. Bernabéu, G. González-Sprinberg, and J. Vidal,
Normal and transverse singletau polarization at the Z peak , Phys. Lett. B (1994) 168–174.[28] J. Bernabeu, G. González-Sprinberg, M. Tung, and J. Vidal,
The Tau weakmagnetic dipole moment , Nucl.Phys.
B436 (1995) 474–486, [ hep-ph/9411289 ].1829] B. Grzadkowski and Z. Hioki,
Decoupling of anomalous top decay vertices inangular distribution of secondary particles , Phys.Lett.
B557 (2003) 55–59,[ hep-ph/0208079 ].[30] J. Aguilar-Saavedra, J. Carvalho, N. Castro, M. Fiolhais, A. Onofre, et al.,
Studyof ATLAS sensitivity to asymmetries in single top events , Nuovo Cim.
B123 (2008) 1323–1324.[31] W. Bernreuther, A. Brandenburg, Z. Si, and P. Uwer,
Top quark pair productionand decay at hadron colliders , Nucl.Phys.
B690 (2004) 81–137,[ hep-ph/0403035hep-ph/0403035