Probing the anomalous triple gauge boson couplings in $e^+e^-\to W^+W^-$ using $W$ polarizations with polarized beams

We study the anomalous $W^+W^-V$ ($V=\gamma,Z$) couplings in $e^+e^-\to W^+W^-$ using the complete set of polarization observables of $W$ boson with longitudinally polarized electron ($e^-$) and positron ($e^+$) beams. For the effective $W^+W^-V$ couplings, we use the most general Lorentz invariant form factor parametrization as well as $SU(2)\times U(1)$ invariant dimension-$6$ effective operators. We estimate simultaneous limits on the anomalous couplings in both the parametrizations using Markov-Chain--Monte-Carlo (MCMC) method for an $e^+e^-$ collider running at centre of mass energy of $\sqrt{s}=500$ GeV and integrated luminosity of ${\cal L}=100$ fb$^{-1}$. The best limits on the anomalous couplings are obtained for $e^-$ and $e^+$ polarization being $(\pm 0.8,\mp 0.6)$ and they are better than the best available limit from various experiments.


Introduction
The non-abelian gauge symmetry SU (2) × U (1) of the Standard Model (SM) allows the W W V (V = γ, Z) couplings after the Electro-Weak Symmetry Breaking (EWSB) by Higgs field, discovered at the LHC [1]. To test the EWSB, the W W V couplings have to be measured precisely, which is still lacking. We intend to study the measurement of these couplings using polarization observables of spin-1 boson [2][3][4][5][6][7][8]. To test the SM W W V couplings one has to hypothesize beyond the SM (BSM) couplings and make sure they do not appear at all or are severely constrained. One approach is to consider SU (2) × U (1) invariant higher dimension effective operators which provide the W W V form factors after EWSB [9]. The effective Lagrangian considering the higher dimension operators can be written as where Φ is the Higgs doublet field and Here g and g are the SU (2) and U (1) couplings, respectively. Among these operators O W W W , O W and O B are CP -even, while O W W W and O W are CP -odd. These effective operators, after EWSB, also provides ZZV , HZV couplings which can be examined in various processes, e.g. ZV production, W Z production, HV production processes. The couplings in these processes may contain some other effective operator as well.
The other alternative to step beyond the SM W W V structure is to consider the most general Lorentz invariant effective form factors in a model independent way. A Lagrangian for the above parametrization is given by [12] Here W ± µν = ∂ µ W ± ν − ∂ ν W ± µ , V µν = ∂ µ V ν − ∂ ν V µ , V µν = 1/2 µνρσ V ρσ , and the overall coupling constants are defined as g W W γ = −g sin θ W and g W W Z = −g cos θ W , θ W being the weak mixing angle. In the SM g V 1 = 1, κ V = 1 and other couplings are zero. The anomalous part in g V 1 , κ V would be ∆g V 1 = g V 1 − 1, ∆κ V = κ V − 1, respectively. The couplings g V 1 , κ V and λ V are CP -even (both C and P -even), while g V 4 (odd in C, even in P ), κ V and λ V (even in C, odd in P ) are CP -odd. On the other hand g V 5 is both C and P -odd making it CP -even. We label these set of 14 anomalous couplings to be c L i as given in Eq. (A.2) in appendix A for later uses.
The W + W − production is one of the important processes to be studied at the future International Linear Collider (ILC) [60][61][62] for precision test [63] as well as for BSM physics. This process has been studied earlier for SM phenomenology as well as for various BSM physics with and without beam polarization [12,[64][65][66][67][68]. Here we intend to study W W V anomalous couplings in e + e − → W + W − at √ s = 500 GeV and integrated luminosity of L = 100 fb −1 using the cross section, forward-backward asymmetry and 8 polarizations asymmetries of W − for a set of choices of longitudinally polarized e + and e − beams in the channel W − → l −ν l (l = e, µ) 1 and W + → hadrons. The polarization of Z and W are being used widely recently for various BSM studies [69][70][71][72][73][74][75] along with studies with anomalous gauge boson couplings [3,7,76,77]. Recently the polarizations of W/Z has been measured in W Z production at the LHC [78]. Besides the final state polarizations, the initial state beam polarizations at the ILC can be used to enhance the relevant signal  [38] to background ratio [63,66,68,79,80]. It also has the ability to distinguish between CP -even and CP -odd couplings [63,[81][82][83][84][85][86][87][88][89][90]. We note that an e + e − machine will run with longitudinal beam polarizations switching between (η 3 , is the longitudinal polarization of e − ( e + ). For integrated luminosity of 100 fb −1 , one will have half the luminosity available for each polarization configurations. The most common observables, the cross section for example, studied in literature with beam polarizations are the total cross section We find that combining the two opposite beam polarizations at the level of χ 2 rather than combining them as in Eq. (1.7) & (1.8), we can limit the anomalous couplings better in this analysis, see appendix C for explanation. We note that there exist 64 polarization correlations [12] apart from 8 + 8 polarizations for W + and W − . The measurement of these correlations requires identification of light quark flavours in the above channel, which is not possible, hence we are not including polarization correlations in our analysis. In the case of both the W s decaying leptonicaly, there are two missing neutrinos and reconstruction of polarization observables suffers combinatorial ambiguity. Here we aim to work with a set of observables that can be reconstructed uniquely and test their ability to probe the anomalous couplings including partial contribution up to O(Λ −4 ) 2 .
The rest of the paper is arranged in the following way. In Sect. 2 we introduce the complete set polarization observables of a spin-1 particle along with the forward-backward asymmetry and study the effect of beam polarizations on the observables. In Sect. 3 we use the vertex form factors for the Lagrangian in Eq. (1.4) and obtained expressions for all the observables. In this section, we cross-validate analytical results against the numerical result from MadGraph5 [91] for sanity checking. We also study the cos θ (of W ) dependences of the observables and study their sensitivity on the anomalous couplings. In this section, we also estimate simultaneous limits on c L i , c O i and the translated limits on c  Figure 1: Feynman diagram of e + e − → W + W − , (a) t-channel and (b) s-channel with anomalous W + W − V (V = γ, Z) vertex contribution shown as blob.

Observables and effect of beam polarizations
We study W + W − production at ILC running at √ s = 500 GeV and integrated luminosity L = 100 fb −1 using longitudinal polarization of e − and e + beams giving 50 fb −1 to each choice of beam polarization. The Feynman diagram for the process is shown in Fig. 1 where Fig. 1(a) corresponds to the ν e mediated t-channel diagram and the Fig. 1(b) corresponds to the V (Z, γ) mediated s-channel diagram containing the anomalous triple gauge boson couplings (aTGC) contributions represented by the shaded blob. The decay mode is chosen to be where q u and q d are up-type and down-type quarks, respectively. We use complete set of eight spin-1 observables of W − boson [6,7]. The W boson being a spin-1 particle, its normalised production density matrix in the spin basis can be written as [2,5] ρ(λ, λ ) = 1 3 where p = {p x , p y , p z } is the vector polarization of a spin-1 particle, S = {S x , S y , S z } is the spin basis and T ij (i, j = x, y, z) is the 2 nd -rank symmetric traceless tensor, λ and λ are helicities of the particle. The tensor T ij has 5 independent elements, which are T xy , T xz , T yz , T xx − T yy and T zz . Combining the ρ(λ, λ ) with normalised decay density matrix of the particle to a pair of fermion f , the differential cross section would be [5] 1 σ Here θ f , φ f are the polar and the azimuthal orientation of the fermion f , in the rest frame of the particle (W ) with its would be momentum along the z-direction. In this case α = −1 and δ = 0. The vector polarizations p and independent tensor polarizations T ij are calculable from the asymmetries constructed from the decay angular distribution of lepton (here l − ). For example p x can be calculated from the asymmetry A x as The asymmetries corresponding to all other polarizations, vector polarizations p y , p z and independent tensor polarizations for details.
Owing to the t-channel process (Fig. 1a) and absence of a u-channel process, like in ZV production [7,76], the W ± produced are not forward-backward symmetric. We include forward-backward asymmetry, defined as of the W − to the set of observables making total of ten observables including the cross section as well. Here θ W − is the production angle of the W − w.r.t. the e − beam direction and σ W + W − is the production cross section. These asymmetries can be measured in a real collider from the final state lepton l − . One has to calculate the asymmetries in the rest frame of W − which require the missinḡ ν l momenta to be reconstructed. At an e + e − collider, as studied here, reconstructing the missingν l is possible because only one missing particle is involved and no Parton distribution function (PDF) is involved, i.e., initial momentas are known. But for a collider where PDF is involved, reconstructing the actual missing momenta may not be possible. We explore the dependence of the cross section and asymmetries on longitudinal polarization η 3 of e − and ξ 3 of e + . In Fig. 2 we show the production cross section σ W + W − and A x as a function of beam polarization as an example. The cross section decreases along η 3 = −ξ 3 path from 20 pb on the left-top corner to 7.2 pb at the unpolarized point and further to 1 pb in the right-bottom corner. This is because of the W ± couples to left chiral e − i.e., it requires e − to be negatively polarized and e + to be positively polarized for the higher cross section. The variation of A f b (not shown) with beam polarization is the same as cross section but very slow above the line η 3 = ξ 3 . From this, we can expect that a positive η 3 and a negative ξ 3 will reduce the SM contributions to observables increasing the S/ √ B ratio (S = signal, B = background). Some other asymmetries like A x have opposite dependence on the beam polarizations compared to the cross section, its modulus reduces for negative η 3 and positive ξ 3 . Figure 3: The W W V vertex showing anomalous contribution represented as blob on top of SM. The momentum P is incoming to the vertex while, q andq are outgoing from the vertex.
3 Probe to the anomalous couplings [12,14] and it is given by where P, q,q are the four-momenta of V, W − , W + , respectively. The momentum conventions are shown in Fig. 3. The form factor f i s has been obtained from the Lagrangian in Eq. (1.4) using FeynRules [92] to be We use the vertex factors in Eq.  Table 5 in appendix A. The CP -even couplings in CP -even observables σ, A x , A z , A xz , A x 2 −y 2 , A zz appear in linear as well as in quadratic form but do not appear in the CP -odd observables A y , A xy , A yz . On the other hand CP -odd couplings appears linearly in CP -odd observables and quadratically in CP -even observables. Thus the CP -even couplings may have double patch in their confidence interval leading to asymmetric limits which will be discussed in Sect. 3.1. On the other hand the CP -odd couplings will have a single patch in their confidence interval and will poses symmetric limits.

Sensitivity of observables on anomalous couplings and their binning
The sensitivity of an observables O depending on anomalous couplings f with beam polarization η 3 , ξ 3 is given by where δO = (δO stat. ) 2 + (δO sys. ) 2 is the estimated error in O. The error for the cross section would be, whereas the estimated error in the asymmetries would be, Here L is the luminosity of the data set, σ and A are the systematic fractional error in the cross section and asymmetries, respectively. We take L = 50 fb while the ∆κ γ being CP -even (linear as well as quadratic terms present), it has two patches in the sensitivity curve, as noted earlier. The CP -odd observable A y provides the tightest one parameter limit on g Z 4 . The tightest 1σ limit on ∆κ γ is obtained using A f b , while at 2σ level, a combination of A f b and A x provide the tightest limit.
Here, we have a total of 14 different anomalous couplings to be measured, while we only have 10 observables. A certain combination of large couplings may mimic the SM within the statistical errors. To avoid these we need more number of observables to be included in the analysis. We achieve this by dividing cos θ W − into eight bins and calculate the cross section and polarization asymmetries in all of them. In Fig. 6 the cross section and the polarization asymmetries A z , A x , and A y are shown as a function of cos θ W − for the SM and some aTGC couplings for both polarized and unpolarized beams. The SM values for unpolarized case is shown in dotted (blue) lines, SM with polarization of (η 3 , ξ 3 ) = (+0.6, −0.6) is shown in dashed (black) lines. The solid (red) lines correspond to unpolarized aTGC values while dashed-dotted (green) lines represent polarized aTGC values of observables. For the cross section (left-top-panel) we take ∆g γ 1 to be 0.1 and all other couplings to zero in the case of both polarized and unpolarized beam. We see that the fractional deviation from the SM value is larger in the most backward bin (cos θ W − ∈ (−1.0, −0.75)) and gradually reduces in the forward direction. The deviation is even larger in case of beam polarization. The sensitivity of the cross section on ∆g γ 1 is thus expected to be high in the most backward bin. In the case of asymmetries A z (right-top-panel), A xz (left-bottom-panel) and A y (rightbottom-panel) the aTGC is assumed to be ∆κ Z = 0.05, λ Z = 0.05 and g Z 4 = 0.05, respectively, while others are kept at zero. The change in the asymmetries due to aTGC is larger in the backward bin for both polarized and unpolarized case. We note that the asymmetries may not have the highest sensitivity in the most backward bin but some other bin. We consider the cross section and eight polarization asymmetries in all 8 bins, i.e., we have 72 observables in our analysis.
One parameter sensitivity of the set of 9 observables in all 8 bin has been studied. We show sensitivity of A y on g Z 4 and of A z on ∆κ γ in the 8 bin in Fig. 7 as representative. The tightest limits based on sensitivity (coming from one bin) is roughly twice as tight as compared to the unbin case in Fig. 5. Thus we expect simultaneous limits on all the couplings to be tighter when using binned observables. We perform a set of MCMC analyses with a different set of observables for different kinematical cuts with unpolarized beams to understand their roles in providing limits on the anomalous couplings. These analyses are listed in Table 2. The corresponding 14dimensional rectangular volume 3 made out of 95% Bayesian confidence interval (BCI) on the anomalous couplings are also listed in Table 2 in the last column. The simplest analysis would be to consider only the cross section in the full cos θ W − domain and perform MCMC analysis which is named as σ-ubinned. The typical 95% limits on the parameters range   from ∼ ±0.04 to ±0.25 giving the volume of limits to be 4.4 × 10 −11 . As we have polarizations asymmetries, the straight forward analysis would be to consider all observables for the full domain of cos θ W − . This analysis is named Unbinned where limits on anomalous couplings get constrained better reducing the volume of limits by a factor of 10 compared to the σ-ubinned. To see how binning improve the limits we perform an analysis named σ-binned using only the cross section in 8 bin. We see the analysis σ-binned is better than σ-unbinned and comparable to the analysis Unbinned. To see the strength of the polarization asymmetries, we perform an analysis named Pol.-binned using just the polarization asymmetries in 8 bin. We see that this analysis is much better than the σ-binned. The most natural and complete analysis would be to consider all the observables after binning. The analysis is named as Binned which has limits much better than any analysis. The comparison between the analysis σ-binned, Pol.-binned and Binned is shown in Fig. 8 in the panel λ γ -λ Z in two-parameter (left-panel) as well as in multi-parameter (right-panel) σ+Pol. analysis using MCMC as representative. The right-panel reflects the Table 2 and even in the two parameter analysis (left-panel) by keeping all other parameter to zero the behaviour is same, i.e, the bounded region for χ 2 = 4 is smaller in Pol.-binned (Pol.) than σ-binned (σ) and smallest for Binned (σ+Pol.). We also calculate one parameter limit on all the couplings at 95 % C.L. considering all the binned observables with unpolarized beams in the effective vertex formalism as well as in the effective operator approach and list them in the last column of Tables 3 & 4, respectively for comparison. In the next subsection, we study the effect of beam polarization on the limit of the anomalous couplings.

Effect of beam polarizations to the limits on the anomalous couplings
We perform MCMC analysis to estimate simultaneous limits on the anomalous couplings using the binned observables in both effective vertex formalism with 14 independent couplings and effective operator approach with 5 independent couplings for a set of chosen beam polarizations (η 3 , ξ 3 ) to be (0, 0), (+0.  6) where N runs over all the observables. The 95 % simultaneous limits for the chosen set of beam polarizations combined according to Eq. (3.6) are shown in Table 3 for effective vertex formalism (c L i ) and in Table 4 for effective operator approach (c O i ). The corresponding   Tables 3 & 4. translated limit to the vertex factor couplings c Lg i are also shown in the Table 4 using relation from Eq. (1.5). While presenting limits the following notation is used high low ≡ [low, high] with low being lower limit and high being upper limit. A pictorial visualization of the limits shown in Table 3 & and 4 is given in Fig. 9 for the easy comparisons. The limits on the couplings get tighter as the beam polarization is increased along η 3 = −ξ 3 path and become tightest at the extreme beam polarization (±0.8, ∓0.8). However, the choice (±0.8, ∓0.6) is best to put constraints on the couplings within the technological reach [93,94].
To show the effect of beam polarization the marginalised 1D projection for the couplings λ γ , ∆g Z 1 and ∆κ Z as well as 2D projection at 95 % C.L. on λ γ -λ Z , ∆g Z 1 -κ Z and ∆κ γ -∆κ Z planes are shown in Fig. 10 for the effective vertex formalism (c L i ) as representative. We observe that as the beam polarization is increased from (0, 0) to (±0.8, ∓0.8) the contours   Fig. 11. In this case the couplings c W and c B has two patches up-to beam polarization (±0.2, ∓0.2) and become one single patch starting at beam polarization (±0.3, ∓0.3) centred around SM values. As the beam polarization is increased along the η 3 = −ξ 3 line the measurement of the anomalous couplings gets improved. The set of beam polarization chosen here are mostly along η 3 = −ξ 3 line, but some choice off to the line might provide the same results. A discussion on the choice of beam polarization is given in the next section.

On the choice of beam polarizations
In the previous section, we found that (±η 3 , ±ξ 3 ) = (±0.8, ±0.6) is the best choice of beam polarization to provide simultaneous limits on the anomalous couplings obtained by MCMC analysis. Here, we discuss the average likelihood or the weighted volume of the parameter to cross-examine the beam polarization choices made in the previous section. Here f is the coupling vector and V f is the volume of parameters space over which the average is done and L(V f ; η 3 , ξ 3 ) corresponds to the volume of the parameter space that is statistically consistent with the SM . One naively expects the limits to be tightest when L(V f ; η 3 , ξ 3 ) is minimum. We calculate the above quantity as a function of (±η 3 , ±ξ 3 ) for Binned

Conclusion
In conclusion, we studied anomalous triple gauge boson couplings in e + e − → W + W − with longitudinally polarized beams using W boson polarization observables together with the total cross section and the forward-backward asymmetry for √ s = 500 GeV and luminosity of L = 100 fb −1 . We have 14 anomalous couplings, whereas we have only 10 observables to measure them. So we binned all the observables (A f b excluded) in 8 regions of the cos θ W − to increase the number of observables to measure the couplings. We estimated simultaneous limit on all the couplings for several chosen set of beam polarization in both effective vertex formalism and effective operator approach. The limits on couplings are tighter when SU (2) × U (1) symmetry is assumed. We show the consistency between the best choice of beam polarizations and minimum likelihood averaged over the anomalous couplings. We find that the polarization (±0.8, ∓0.6) to be the best to provide the tightest constraint on the anomalous couplings in both approaches at the ILC within the technological reach. Our one parameter limits with unpolarized beams and simultaneous limits for best polarization choice are much better than the one parameter limits from experiment, see Table 4. Our analysis consider certain simplifying assumptions, such as the absence of initial-state/finalstate radiation and detector effects. While the former might dilute the limits by little amount the later is expected to have no effects on the results as only the leptonic channel is assumed and no flavour tagging or reconstruction is required.

B Note on linear approximation
If the cross section σ is express as a function of couplings c i as, linear approximation for the BSM operator will be possible if the quadratic contributions are much smaller than the linear contribution, i.e., As an example, consider the λ Z dependent unpolarized cross section given by The linear approximation is valid for |λ Z | 0.004. However, the limit on λ Z is ±0.36 at 1σ level at 100 fb −1 (2% systematic is used) assuming linear approximation of Eq. (B.3), which is much beyond the validity of the linear approximation. To derive a sensible limit one needs to include the quadratic term which appear at O(Λ −4 ). However, at O(Λ −4 ) one also has contribution from dimension-8 operators at linear order. Our present analysis include quadratic contributions in dimension-6 operators and does not include dimension-8 contributions to compare our result with current LHC constrain, Table 1.   To reduce the systematic errors in analysis due to luminosity the beam polarization is flipped between two opposite choices frequently giving half the total luminosity to both the polarization choices in an e + -e − collider. One can, in principle, use the observables, e.g., the total cross section (σ T ) or their difference (σ A ) as in Eqs. (1.7) & (1.8), respectively or for the two opposite polarization choices (σ &σ) separately for a suitable analysis. In this work, we have combined the opposite beam polarization at the level of χ 2 as given in Eq. (3.6) not at the level of observables as the former constraints the couplings better than any combinations and of-course the individuals. To depict this, we present the χ 2 = 4 contours of the unbinned cross sections in Fig. 13 (left-panel) for beam polarization (+0.6, −0.6) (σ) and (−0.6, +0.6) (σ) and the combinations σ T and σ A along with the combined χ 2 in the λ γ -λ Z plane for L = 50 fb −1 luminosity to each polarization choice as representative. A systematic error of 2% is used as a benchmark in the cross section. The nature of the contours can be explain as follows: In the W W production, the aTGC contributions appear only in the s-channel (see Fig. 1) where initial state e + e − couples through γ/Z boson and both left and right chiral electrons contribute almost equally. The t-channel diagram, however, is pure background and receives contribution only from left chiral electrons. As a result theσ (big-dashed/black) contains more background than σ (solid/green) leading to a weaker limit on the couplings. Further, inclusion ofσ into σ T (dotted/blue) and σ A (dashed-dotted/red) reduces the signal to background ratio and hence they are less sensitive to the couplings. The total χ 2 for the combined beam polarization shown in dashed (magenta) is, of course, the best to constrain the couplings. This behaviour is reverified with the simultaneous analysis using the binned cross section and polarization asymmetries (72 observables in the Binned case) and depicted in Fig. 13 (right-panel) in the same λ γ -λ Z plane showing the 95 % C.L. contours for beam polarizations (+0.6, −0.6), (−0.6, +0.6), and their combinations (±0.6, ∓0.6). Thus we choose to combine the opposite beam polarization choices at the level of χ 2 rather than combining them at the level of observables.   Table 5: Dependences of observables (numerators) on anomalous couplings in the form of c L i (linear), (c L i ) 2 (quadratic) and c L i c L j , i = j (interference) in the process e + e − → W + W − . Here V ∈ {γ, Z}. The " " (checkmark) represents the presence and "-" (bigdash) corresponds to absence.