Helicity Selection Rules and Non-Interference for BSM Amplitudes

Precision studies of scattering processes at colliders provide powerful indirect constraints on new physics. We study the helicity structure of scattering amplitudes in the SM and in the context of an effective Lagrangian description of BSM dynamics. Our analysis reveals a novel set of helicity selection rules according to which, in the majority of 2 to 2 scattering processes at high energy, the SM and the leading BSM effects do not interfere. In such situations, the naive expectation that dimension-6 operators represent the leading BSM contribution is compromised, as corrections from dimension-8 operators can become equally (if not more) important well within the validity of the effective field theory approach.


I. INTRODUCTION
Standard Model (SM) precision tests represent an important strategy in the search for new physics. Effective field theories provide a suitable theoretical framework which allows these tests to be performed model-independently while maintaining a simple connection to explicit UV theories. The effective field theory (EFT) approach is especially convenient to organize hierarchically possible departures from the SM. Models in which a large separation exists between the new physics scale Λ and the electroweak (EW) scale can be expanded in powers of fields and derivatives 1 L = L SM +L 6 +L 8 +· · · , L D = . In most theories, D = 6 terms are expected to capture the leading beyond-the-SM (BSM) effects. (In the presence of approximate symmetries or other selection rules, effects from D = 6 operators can be suppressed compared to those from D = 8 or higher-dimensional operators, see [1,2].) This motivates searches for generic new physics, as parametrized by L 6 only [3][4][5]. In particular, when departures from the SM are small, as typically occurs in weakly-coupled theories, the leading corrections to the cross section are expected to arise at order 1/Λ 2 from the interference between the SM and D = 6 operators. Aim of this note is to assess the validity of this naive expectation by analyzing the relative importance of the contributions to scattering amplitudes from the different terms in Eq. (1).
Precision searches can be divided into two categories: i) those exploiting the resonantly enhanced production of a SM state (such as measurements at the Z-pole or single-Higgs production); ii) those exploiting the high-energy E m W behavior of nonresonant processes (including e + e − → ff at LEP2 and W + W − production). This second mode of exploration is ubiquitous in the LHC experimental program [4][5][6][7][8], as an obvious consequence of its highenergy reach, and it will be the focus of this work.
We anticipate our main result in Table I: in the high-energy (massless) limit and working at tree level, SM and D = 6 BSM contributions to 2 → 2 scattering processes involving at least one transversely-polarized vector boson appear in different helicity amplitudes and thus do not interfere. This non-interference rule contradicts the naive expectation and implies that in these processes D = 6 and D = 8 operators contribute at the same order in the 1/Λ expansion if masses and loop corrections are neglected. It follows that in many cases of interest analyses based on an EFT truncated at the D = 6 level are incomplete in the high-energy region away from threshold.

II. HELICITY SELECTION RULES AND NON-INTERFERENCE
When departures from the SM are small, the leading BSM contribution comes from the SM-BSM in- terference term in the amplitude squared. Obviously, interference is possible only if SM and BSM give nonvanishing contribution to the same helicity amplitude. In this section we study the helicity structure of scattering amplitudes at tree-level, in the SM and at leading order in the effective field theory expansion, i.e. at the level of D = 6 operators. We will denote the corresponding new-physics contribution as BSM 6 in the following. We focus first on the phenomenologically relevant case of 2 → 2 scatterings and work in the massless limit; the massive case and higher-points amplitudes are discussed below. We use the spinor-helicity formalism (see Refs. [9,10] for a review), where the fundamental objects which define the scattering amplitudes are Weyl spinors ψ α andψα, transforming as (1/2, 0) (undotted indices) and (0, 1/2) (dotted indices) representations of SU (2) × SU (2) SO(3, 1), and Lorentz vectors A µ σ µ αα , transforming as (1/2, 1/2). 2 In this language, the field strength is written as in terms of its self-dual and anti-self dual parts F andF (transforming respectively as (1, 0) and (0, 1) representations). 2 We will not distinguish between fermions and anti-fermions except where explicitly mentioned, as this distinction is not crucial to our analysis. We will denote a Weyl fermion or anti-fermion of helicity + (−) with ψ + (ψ − ). When indicating a scattering amplitude, the symbol ψ will stand for either ψ + or ψ − . Our analysis will be in terms of complex momenta p ∈ C: this allows one to make sense of 3-point amplitudes on-shell, even though these vanish for massless states with real kinematics. We will need three wellknown results, that we summarize here and discuss in the Appendices, see Refs. [9][10][11]. These are: 1. Consider an amplitude A n with n external legs (n-point amplitude), and let A m and A m be any two sub-amplitudes, with m + m − 2 = n, see Fig. 1. We define the net helicity of an on-shell amplitude, h(A), as the sum of the helicities of all its external states, where all momenta are taken to be outgoing. Then one has: for all possible sub-amplitudes A m and A m . This relation is a consequence of the fact that the amplitude has a pole when the intermediate line goes on-shell, and that in this limit it factorizes into the product of the two sub-amplitudes. While in the SM there are no exceptions to Eq. (3), in the D = 6 effective theory this relation fails if an effective operator gives a contribution to the vertex attached to the intermediate line that vanishes on shell. In this case the pole from the propagator is canceled by the vertex, and factorization does not hold. When this occurs the operator can be rewritten through the equations of motion in terms of other operators with more fields. We will discuss below how this complication is avoided.
2. Dimensional analysis, Little group scaling and the 3-particle special kinematics fix completely the form of the 3-point amplitudes, and in particular relate their total helicity h(A 3 ) to the dimensionality of the coupling g characterizing the 3-point vertex: For instance, the triple gauge interaction of the SM is characterized by a dimensionless coupling, and the corresponding 3-point on-shell amplitude has |h| = 1.
and thus generates a 3-point amplitude with |h| = 3.

3.
Helicity selection rules in the SM force the following 4-point amplitudes with |h| = 2 to vanish: These relations can be proved by means of the Supersymmetric Ward Identities (SWI) [12,13], as sketched in Appendix B. In the limit in which all uptype or all down-type Yukawa couplings vanish, the SM Lagrangian can be uplifted to a supersymmetric one, for which SWI hold. Such theory has in addition an R-parity implying that the supersymmetric partners do not contribute at tree level to scattering amplitudes with external SM legs only. As a consequence, Eq. (5) holds for the SM.

A. The Standard Model
Within the SM, it follows from property 2 that 3-point vertices associated with marginal couplings have The three-scalar vertex (which would have vanishing total helicity) is absent in the SM in the massless limit (i.e. in the limit of unbroken EW symmetry). With the exception of the quartic scalar vertex, which has trivially total helicity h(A 4 ) = 0, all 4-point on-shell vertices in the SM can be made vanish with a suitable definition of polarization vectors (this is a direct consequence of gauge invariance, see for example [10]). Properties 1 and 3 then imply that all 4-point amplitudes with fermions or gauge fields have vanishing total helicity, unless they involve both up-and down-type Yukawa interactions [9,14,15]. The only exceptions are in fact the amplitudes ψ + ψ + ψ + ψ + and ψ − ψ − ψ − ψ − , which receive a contribution (proportional to the product of up-and down-type Yukawas) from the Higgs exchange and have |h(A 4 )| = 2. These known results are summarized in Table I.

B. Beyond the Standard Model
Local operators entering at a given order in the 1/Λ expansion of an EFT can be redefined by making use of the equations of motion (EoM) derived at lower order. For example, it is always possible to rewrite D = 6 operators by using the EoM of the renormalizable D=4 Lagrangian; the new effective Lagrangian will differ from the original one by D = 8 terms. This freedom allows one to systematically replace operators with more derivatives in terms of operators involving more fields. At the D = 6 level, this procedure leads to the so called Warsaw basis of operators introduced in Ref. [16]. This basis is particularly convenient to study 2 → 2 scattering processes for two reasons. First, the number of operators with two and three fields (bivalent and trivalent operators) is reduced to a minimum. In particular, there are no bivalent operator and only two trivalent ones, i.e. O 3W and O 3 W = tr(W µν W ν ρ W ρµ ); all the other D = 6 operators have at least four fields, see Table II. A second, more important, reason why the Warsaw basis is convenient is because condition 2 requires that sub-amplitudes do not receive contributions that are vanishing on shell (but different from zero off shell). Such contributions would be proportional to inverse SM propagators and thus arise from local operators that vanish on the D = 4 EoM. Eliminating higherderivative redundant operators proportional to the EoM automatically guarantees that the amplitudes factorize and Eq. (3) is fulfilled. As an example, consider the operator O B = (i/2)H † ↔ D µ H ∂ ν B µν , which appears in the SILH basis of Ref. [17]. It gives a vanishing contribution to the on-shell HHB vertex (even for complex momenta), but contributes off shell to processes like HH → HH or HH → ψψ. Indeed, by using the EoM it can be eliminated in favor of operators of the form D 2 H 4 or H 2 Dψ 2 , that contribute to the previous processes via contact interactions.
In order to determine the helicity of an amplitude generated through the insertion of some operator O, it is useful to introduce the holomorphic and antiholomorphic weights of O, as defined by Ref. [18]. The point is that, as a consequence of Eq. (3) and the fact that h(A SM 3 ) = ±1, building amplitudes with more SM interactions cannot decrease w(A) and w(A), so that the weight is always determined by the amplitude with the smallest number of SM vertices. Since they are defined in terms of on-shell amplitudes, weights offer various advantages. First of all, they are gauge-invariant quantities characterizing also operators, involving covariant derivatives, that contribute to different processes with different helicities. Moreover, they are well defined even for operators whose contribution to a given amplitude vanishes on shell (as for O B discussed above). Finally, one can easily deduce from Eqs. (7) and (8) that the helicity of n-point amplitudes with one O insertion is constrained to be in the rangē Using these ingredients we can constrain the total helicity of BSM 6 contributions to 2 → 2 scattering amplitudes. Let us start with the unique trivalent F 3 andF 3 structures of O 3W and O 3 W . Given the dimensionality of their coefficients, [c i ] = −2, Eq. (4) fixes the helicity of their contribution to the 3-point amplitude up to a sign: |h(A 3 )| = 3. It is in fact not difficult to show that F αβ andFαβ generate states with helicity +1 and −1 respectively (see [19,20]), which implies that h(A 3 ) = +3 for F 3 and h(A 3 ) = −3 forF 3 . From Eq. (8) it follows that the weights of F 3 andF 3 are respectively (w,w) = (0, 6) and (w,w) = (6, 0). Equation (9) thus constrains the helicity of a 4-point function with one insertion of either of these operators to be in the range 2 ≤ |h(A 4 )| ≤ 4 (more precisely, 2 ≤ h(A 4 ) ≤ 4 for F 3 and −4 ≤ h(A 4 ) ≤ −2 forF 3 ). Considering that h(A 4 ) = 0 in the SM (for the amplitudes under consideration), this shows that no SM-BSM interference is possible in this case. It is useful to directly verify the constraint of Eq. (9) for some specific amplitudes. Starting from a 3-point amplitude with one F 3 insertion, for example, a 4-point one is obtained by adding a SM cubic vertex (which has h(A SM 3 ) = ±1 as shown previously). Then Eq. (3) implies that the 4-point amplitude with only vectors, V V V V , has |h| = 4, 2 [19,21] (notice that F 3 is not supersymmetrizable and condition 3 does not hold [11,22]). Similarly, the helicity of an amplitude V V ψψ, is |h| = 2. Both results agree with the bound of Eq. (9).
Apart from F 3 andF 3 , the remaining D = 6 operators of the Warsaw basis do not contribute to 3point amplitudes in the massless (high-energy) limit. Those contributing to 4-point amplitudes are listed in the second and third row of Table II. The helicity of the 4-point amplitudes in this case can be directly determined from the corresponding operators by noticing that F αβ (Fαβ) creates states with helicity +1 (−1), ψ α (ψα) creates fermions or antifermions with helicity +1/2 (−1/2), and the helicity of scalars trivially vanishes. For example, an operator F ψ 2 φ can excite states with net helicity h min = +2, which equals the helicity of the corresponding 4-point amplitude. The results are reported in Table II: Table I.
In addition to the helicity selection rules derived above, 2 → 2 tree-level scattering amplitudes are constrained by additional selection rules in the massless limit. In particular, a simple one follows from weak isospin conservation. In the Warsaw basis, the only BSM contribution to the amplitudes V V ψψ, V V V V comes from F 3 ,F 3 , and it can always be written as the product of two 3-point amplitudes with a vector propagator (V V ψψ receives no quartic contribution from D = 6 operators, while the quartic V V V V vertex can always be made vanish through a suitable choice of polarization vectors). The propagation of a vector boson implies a well defined SU (2) L isospin structure of the external states produced at each vertex: they transform in the 3 ∈ 3⊗3, which is totally antisymmetric and thus does not include pairs of identical bosons. For this reason amplitudes like ZZZZ, γγγγ, ψψZZ and ψψγγ can only be generated by D = 8 operators. It is worth mentioning another selection rule which characterizes the D = 6 effective theory in the massless limit. Its Lagrangian as a direct consequence of SU (2) L invariance and of the SM quantum numbers (it is not possible to form operators which are singlets of SU (2) L with an odd number of ψ L and H fields). It follows that the amplitudes V V V φ and V φφφ identically vanish (in the massless limit), while the helicities of the fermion and anti-fermion in V ψψφ are forced to be the same. Notice that these same conclusions are also a consequence of the helicity selection rules, since by the arguments presented above no 4-point amplitude has total helicity |h| = 1, 3 in the SM or at the D = 6 level. Using Eq. (10) might still be useful, however, as a quicker way to determine if a given amplitude vanishes, independently of helicity arguments. To summarize, we have shown that working at tree level and in the massless (i.e. high-energy) limit, the BSM contribution never interferes with the SM one in 2 → 2 scattering amplitudes involving at least one transversely-polarized vector boson. Interference is possible, instead, for amplitudes involving only scalars (including longitudinally-polarized vector bosons) and fermions, such as ψψ → ψψ, ψψ → φφ and φφ → φφ. We will comment on the practical implications of these results in Section III, but first we discuss how our analysis generalizes to the massive case and to higher-point scattering amplitudes.

C. Higher-point Amplitudes
The helicity of amplitudes with 5 or more external legs can be easily determined in the SM by starting from that of 4-point amplitudes, given in Table I, using the addition rule (3) and knowing that 3-point vertices change helicity by ±1 unit (Eq. (6)). We find For example, one has h(A SM 5 ) = ±1, h(A SM 6 ) = 0, ±2 respectively for 5-point and 6-point SM amplitudes. By making use of the helicity selection rules for 4-point amplitudes Eq. (5), combined with Eq. (3), one reproduces the well known result that the first non-vanishing amplitudes with largest total helicity are the Maximal Helicity Violating ones [9,10].
The helicity of BSM 6 amplitudes, including those with 5 or more external legs, can be constrained by using Eq. (9) and the weights reported in Table II. We find: with h even (odd) for n even (odd) where n min is given in Table II for the various operators.
Notice that, both for the SM and BSM 6 , the total helicity is even (odd) if n is even (odd). This follows from Little group scaling and the even dimensionality of the coupling constants ([g] = 0 in the SM and [c i ] = −2 in BSM 6 ), see Appendix A. In this respect (as well as to derive Eq. (11)), it is crucial that no scalar cubic vertex is present in the SM in the limit of unbroken EW symmetry. This selection rule automatically implies that amplitudes such as V V V φ or V V φφφ must vanish since they would have necessarily a total helicity with the wrong parity (a similar conclusion follows from chiral invariance, as seen above).
From Eq. (12) it follows that all D = 6 operators contribute to amplitudes with h(A 5 ) = ±1 or h(A 6 ) = 0, ±2 (the operator φ 6 contributes only to 6point amplitudes), and can thus potentially interfere with the SM. Having the same total helicity is in fact a necessary but not sufficient condition for the SM and BSM amplitudes to have interference. The same net helicity can indeed be distributed differently on the external legs, in which case no interference occurs. As an example consider the 5-point amplitudes ψ − ψ − φV + g + and ψ + ψ + φV + g − , both with h = +1, where ψ is a quark, g a gluon and V = W, Z, γ. The SM contributes only to the first as a consequence of the addition rule (3) and the fact that 4-point subamplitudes have necessarily h = 0. The second amplitude instead can be generated by the insertion of an EW dipole operator F ψ 2 φ, which forces the helicity of the vector boson to have the same sign as that of the quarks. Moreover, in the case in which both the SM and BSM 6 amplitudes have the same external helicities, additional selection rules can forbid the interference. As an example, consider the amplitude Higgs boson and the corresponding physical scattering gg → ttH), which is non-vanishing in the SM in the massless limit. The operator O 3 G contributes to the same helicity amplitude, but no interference occurs because O 3 G is CP odd. Except for these particular cases, however, D = 6 operators will in general interfere with the SM at the level of n ≥ 5 amplitudes. Non-interference seems therefore a peculiarity of 4-point amplitudes.

D. Finite-Mass Effects and Radiative Corrections
The non-interference between SM and BSM 6 amplitudes holds for 2 → 2 scatterings in the massless limit and at tree level. There are two main subleading effects which correct this result in real scattering processes: finite-mass corrections and radiative effects (1-loop corrections and real emissions).
Finite-mass effects can be easily included in our analysis. They can be parametrized in terms of where m V (ψ) is the vector (fermion) mass and E the energy. Finite-mass effects have been extensively studied in the literature, see [23][24][25][26][27]. In this note we are interested, in particular, to determine at which order in ε V,ψ the leading correction to a given amplitude appears. To this aim, the most effective procedure is to consider higher-point amplitudes with additional Higgs bosons in the external legs. Restricting to the phase space region where no momentum is transferred through a Higgs line corresponds in fact to setting the Higgs field to its vev. One can thus identify the contribution to a given amplitude at order k in the ε expansion by considering a higher-point amplitude with k insertions of the Higgs. Since the gauge and Yukawa interactions of the Higgs violate helicity by ±1 unit, this procedure allows one to easily determine the leading contribution to the n = 4 amplitudes that are vanishing in the massless limit. For example, a transversely polarized vector can be turned into a longitudinal one (or vice versa) at order ε V through the insertion of a vertex φ * ∂ µ φA µ , by setting φ * to its vev. This follows from the Equivalence Theorem [28,29], which states that, at leading order in the ε V expansion, a longitudinal polarization can be replaced by the corresponding would-be Nambu-Goldstone boson. Notice that it is not possible to flip the helicity of a given vector line by inserting two Higgs vevs (e.g. making use of two consecutive φ * ∂ µ φA µ interactions or inserting one φ * φA µ A µ vertex), since the V V φφ sub-amplitude has vanishing total helicity in the SM. For fermions, the Yukawa interaction ψ α ψ α φ (ψαψαφ) has total helicity h = +1 (−1) and its insertion leads to a flip of the fermion chirality at order ε ψ , according to Eq. (3).
In general, the SM-BSM 6 interference in an n = 4 amplitude will arise at order k SM +k BSM in ε if nonvanishing SM and BSM 6 amplitudes exist with respectively k SM and k BSM additional external Higgs fields. The power k SM + k BSM is always even, as a consequence of the fact that the total helicity of an n-point amplitude is even (odd) if n is even (odd). Hence, the interference in n = 4 scattering amplitudes with one or more transversely polarized vector bosons is suppressed at least by two powers of ε. As an example of the case k SM = 0, k BSM = 2 consider the amplitude V + V + V − V − : it is non-zero in the SM but its BSM 6 contribution vanishes in the massless limit. The 6-point amplitude V + V + V − V − φφ, on the other hand, is generated by F 3 and F 2 φ 2 . At order ε 2 V this leads to a contribution to the n = 4 amplitude and thus to an interference with the SM. As a second example, the amplitude V + V + V + V + has interference at order ε 4 V for k SM = 4, k BSM = 0. The SM contribution arises from V + V + V + V + φφφφ after taking four Higgs vevs, while the BSM one is generated already in the massless limit from insertions of F 3 . Finally, an example with k SM = 1, k BSM = 1 is given by the amplitude V + ψ + ψ − φ, which vanishes in the massless limit both in the SM and beyond. The 5-point amplitude V + ψ + ψ − φφ, on the other hand, is generated through one insertion of any of the operators F 3 , F 2 φ 2 , F ψ 2 φ and ψψφ 2 D, and is non-vanishing also in the SM. The interference arises at order ε 2 V (in the case of F 3 , F 2 φ 2 and F ψ 2 φ) and ε V ε ψ (for ψψφ 2 D).
Contributions from fermion mass insertions are in general subdominant compared to those from vector mass insertions, with the exception of processes involving the top-quark, for which ε ψ ≈ ε V , like bW → th [30] and tW → tW [31]. In processes involving gluons instead of EW vector bosons, top quark mass insertions are in fact the only way to get interference between SM and BSM 6 . An example is given by the scattering gg → tt [32][33][34], where the operators F 3 and F ψ 2 φ (where F is a gluonic field strength) interfere at order ε 2 ψ with the SM. We summarize our results for processes involving EW vector bosons in Table III, where we report the order in ε V at which a given helicity amplitude appears, in the SM and BSM 6 . For simplicity we work in the limit of vanishing Yukawa couplings and do not specify which D = 6 operators give rise to an interference. In all cases the interference term in the amplitude squared goes like a constant in the high-energy limit, E m W , except for the processes in the first line of the two panels, where it scales as 1/E 2 .
Let us now consider radiative corrections. In general, 1-loop corrections to 4-point amplitudes violate the helicity selection rules discussed in Sections II A and II B, and thus generate a non-vanishing interference. The most relevant contribution arises from QCD corrections to amplitudes with external quarks or gluons. Pure EW loop corrections have a similar effect but are numerically smaller. The emission of an extra gluon transforms a 4-point to a 5-point amplitude and can also lead to interference. It is well known (see [9]) that in the limit in which one parton becomes soft, an n-point color-ordered amplitude factorizes into the product of the (n − 1)point amplitude made of the remaining hard partons times a singular eikonal factor. Because of the helicity selection rules controlling 4-point amplitudes, it is thus clear that soft emissions of extra gluons or quarks cannot lead to interference; that is: the interference vanishes in the soft limit. Similarly, when two partons i and j become collinear, an n-point colorordered amplitude factorizes into the (n − 1)-point amplitude obtained by replacing the ij pair with an "effective" parton carrying its momentum, times a singular splitting function (see [9]). Starting with a 5-point amplitude and taking the collinear limit, the selection rules acting on 4-point amplitudes force the helicity of the effective parton to be opposite in the SM and BSM 6 cases. This implies that the SM-BSM 6 interference term, once integrated over the full phase space, is non-singular, that is: the collinear singularity of the amplitude squared vanishes in the total cross section at order 1/Λ 2 [21]. The absence of soft and collinear singularities in real emission processes in turn implies that SM and BSM 6 amplitudes which vanish at tree level are IR-finite at 1-loop [21]. Hence, although 1-loop QCD corrections and real emissions of extra gluons do lead to interference between SM and BSM 6 amplitudes, no logarithmic enhancement is present in the collinear and soft limits. This means that the interference is suppressed by a factor α s /4π, where α s is evaluated at the high-energy scale characterizing the scattering process.
Summarizing, interference between SM and BSM 6 can arise in 2 → 2 exclusive processes as a result of 1-loop corrections and finite-mass effects, with a relative suppression of order, respectively, α s /4π (or α em /4π for processes without colored particles) and m 2 W /E 2 . Mass effects dominate at lower energies, while radiative corrections take over at energies E m W 4π/α s . Similar conclusions hold in the case of processes where the final state is defined inclusively with respect to the emission of additional QCD radiation. In this case the leading SM-BSM 6 interference arises also from amplitudes with an additional gluon, while the pure SM contribution stems at lowest perturbative order.
A way to access the 1/Λ 2 corrections from D = 6 operators without any relative suppression of the signal compared to the SM irreducible background, is instead considering exclusive 2 → 3 scattering processes, where the additional particle could be a hard gluon. In this case, as discussed in Section II C, the interference term arises at tree-level also in the massless limit, so that both the SM and SM-BSM 6 interference are equally suppressed. This strategy was for example proposed by the authors of Ref. [21], who suggested to constraint the operator O 3G using three-jet events to avoid the non-interference of 4-point amplitudes already noticed in [35]. For final states with one extra hard gluon, the gain in signal/ √ background, compared to 2 → 2 loop processes, is only of order 4π/α s . Moreover, additional radiation does not necessarily guarantee interference in the 2 → 3 amplitude. For instance, in the specific case of the EW process qq → V V, V φ (with V = W, Z, γ), simple inspection of the treelevel Feynman diagrams further reveals that no in-terference arises by adding one extra gluon. 3 The emission of two gluons does seem to give interference, but in that case the significance is further suppressed and there is no parametric gain compared to 2 → 2 processes. Despite the above considerations, studying 2 → 3 processes seems in general a promising strategy to constrain D = 6 operators, and deserves further investigation.

III. PHENOMENOLOGICAL IMPLICATIONS
The helicity selection rules of Section II imply that the contributions from operators F 3 , F 2 φ 2 , F ψ 2 φ and their conjugates never interfere with the SM ones in 2 → 2 scattering process at high energy and tree level, and that the interference is suppressed by powers of (m W /E) 2 or α s /4π. What does this imply on the phenomenology at the colliders ?
The impact of BSM precision searches performed using the EFT approach can be easily quantified in the context of theories characterized by a single microscopic scale Λ and a single (new) coupling g * [17]. This provides a power counting prescription to estimate the size of the effective coefficients c i in Eq. (1) in terms of the parameters of the UV theory: where n i counts the number of fields in O . The corresponding estimate of the coefficient of each operator is reported in the last column of Table II. Such power counting smoothly interpolates between the strong coupling limit g * ∼ 4π, where it is equivalent to Naive Dimensional Analysis [36], and the weak coupling limit g * g SM . Additional symmetries and selection rules can lower the estimates of Eq. (14) [1,2,17].
To appreciate the relevance of non-interference, let us first discuss the BSM amplitudes which do interfere with the SM, such as A(φφφφ) in the scattering of 4 longitudinally-polarized vector bosons. This process receives a contribution from the operator O H = (1/2)(∂ µ |H| 2 ) 2 , with estimated coefficient c H ≈ g 2 * /Λ 2 , which may capture for example the effect of Higgs compositeness or the virtual exchange of heavy vectors coupled to the Higgs current with strength g * . At the D = 8 level, higher-derivative operators will also contribute with estimated coefficients c (8) ≈ g 2 * /Λ 4 (for example they may capture higher-order terms in the p 2 /Λ 2 expansion of the propagator of the heavy vectors). The corresponding contributions to the V L V L → V L V L scattering cross section are, schematically, Since E Λ for the EFT expansion to make sense, D = 8 effects are always subdominant, while the BSM 6 -SM interference term always dominates for weakly-coupled theories. In the case of stronglycoupled theories, g * > g SM , the BSM contribution is larger than the SM one at energies E Λ (g SM /g * ), where the (BSM 6 ) 2 term dominates. We illustrate this situation in the left panel of Fig. 2. Similar arguments hold for ψψ → ψψ [37] and ψψ → φφ [38].
As an example where the non-interference is at work, consider the scatterings V T V T → V L V L (with its crossings) and V T V T → V T V T . We will be inclusive on the transverse polarizations, implying a sum (or average) over them in the following discussion. We will later highlight the advantages of an angular distribution analysis able to select the finalstate polarizations. Let us discuss first the scattering V T V T → V L V L . In this case the largest BSM correction potentially comes from operators of the form F 2 φ 2 and F 3 , whereas φ 4 D 2 and φ 6 contribute only at subleading level in ε V . The helicity selection rules of Section II imply that the interference with the SM is suppressed and arises at order ε 2 V in the mass insertion or via 1-loop EW corrections. It turns out that the latter effect is always subdominant in the following discussion and will be thus neglected for simplicity. The naive estimate of the various terms entering the cross section is different, according to Eq. (14), for the operators F 2 φ 2 and F 3 . Assuming that only F 2 φ 2 contributes, one finds, schematically, The importance of the various terms is illustrated in the central panel of Fig. 2. For small enough energy, where the BSM gives a small perturbation to the SM prediction, the BSM 6 -SM interference dominates. The suppression of the latter has however an important impact on the behavior at higher energies. If g * > g SM , it implies a precocious onset of the regime where the (BSM 6 ) 2 term must be included: for (m W Λ g/g * ) 1/2 < E < Λ g/g * , corresponding to the light blue region of the Figure, the SM still dominates but the (BSM 6 ) 2 term gives the largest correction; for higher energies (BSM 6 ) 2 eventually dominates the cross section. For weak or super-weak UV completions, g * < g SM , the largest correction to the SM prediction comes from D = 8 operators, in particular from the interference BSM 8 -SM, as soon as the energy is larger than ∼ √ m W Λ (light orange region in the Figure). In this case, an EFT analysis in terms of D = 6 operators alone is insufficient.
Yet a different energy behavior is found for the scattering V T V T → V T V T , where F 3 gives the leading correction, while the operators F 2 φ 2 , φ 4 D 2 and φ 6 contribute at sub-leading order in ε V . (Similar conclusions are in fact obtained also for V T V T → V L V L in the case in which only F 3 contributes.) Because the coefficient of F 3 scales with only one power of g * according to Eq. (14), the size of the D = 6 terms (both (BSM 6 ) 2 and the BSM 6 -SM interference) is suppressed compared to Eq. (16). The cor-rection from D = 8 operators might not carry a similar suppression, as it happens for example for the F 2F 2 operator, whose coefficient has a naive estimate c (8) ≈ g 2 * /Λ 4 . The different contributions to the cross section can thus be schematically summarized as follows: Independently of the size of the interference term, this expression shows that as soon as the D = 6 effects become bigger than the SM (for E > Λ(g/g * ) 1/2 ), the D = 8 contribution takes over and dominates the cross section [1]. Non-interference implies a precocious onset of the regime where D = 8 operators must be included: for energies E > √ m W Λ (g/g * ) 1/4 the dominant correction to the SM comes both from (BSM 6 ) 2 and from the BSM 8 -SM interference. The situation is illustrated in the right panel of Fig. 2. We conclude that, for the scattering V T V T → V T V T , inclusion of D = 8 operators is crucial in a vast energy region above threshold.
So far we have considered processes where the transverse polarizations of the vector bosons are treated inclusively, i.e. they are summed over in the final state and averaged in the initial one. This in practice corresponded to sum over two different kinds of helicity amplitudes in each process, one in which the SM arises at higher order in ε V , the other in which it is the BSM 6 amplitude to be suppressed. As an example, consider the amplitudes A(φφV ± V ± ) and A(φφV ± V ∓ ) which have been both included to obtain the estimate of the cross section of V L V L → V T V T Eq. (16). This suggests that an experimental analysis able to distinguish the polarizations in the final state could be used to select those processes where the SM amplitude arises at subleading order in the mass insertion (while the BSM 6 contribution is unsuppressed). This would increase the significance of the signal compared to the irreducible SM background. Another example is the process ψψ → V ± V ± , relevant for the study of anomalous triple gauge couplings, where the polarizations of the final-state vector bosons are equal (while those of the fermions are averaged over). In this process the leading SM amplitude arises at order ε V , while the BSM 6 one is unsuppressed. Selecting the final-state polarizations through an angular distribution analysis can thus improve the sensitivity on new physics. More in general, an exclusive approach to the final state can lead to a parametric enhancement of the signal significance compared to the naive estimates which follow from our analysis of Section II. An interesting example in this sense is given by the proposal of Ref. [21] to study three-jet events by exploiting the distribution of collinear jet pairs under azimuthal rotations as a way to (parametrically) enhance the sensitivity on the operator O 3G .
We conclude our discussion on the impact of noninterference by noticing an interesting fact: with the exception of ψψψψ, the SM amplitudes that do interfere with the BSM are accidentally suppressed in their contribution to inclusive cross sections. Indeed, the contribution of the V L V L → V L V L amplitude to the V V → V V inclusive cross section is accidentally suppressed in the SM by a factor ∼ 1/500 with respect to V T V T → V T V T [39]. Similarly, in the SM the contribution of ψψ → V L V L is only ∼ 1/10 of the ψψ → V V total cross section [40]. Therefore, despite arising at leading order in the high-energy limit, the SM-BSM 6 interference is anyway suppressed by the fact that the SM amplitude is small. Since at the LHC current experimental studies mostly focus on unpolarized cross sections, this implies an additional obstacle in extracting useful information on D = 6 operators through their interference with the SM.

IV. CONCLUSIONS
In this paper we have shown that in a theory where the SM is extended by D = 6 effective operators, tree-level 4-point amplitudes are subject to helicity selection rules in the massless limit. These forbid the interference between SM and D = 6 BSM contributions for all amplitudes involving at least one transversely-polarized vector boson. Such non-interference was noticed before in the literature for few specific operators and processes (see [21,32,35]). Our analysis extends the result in a systematic way to all the D = 6 operators, identifying the exceptions in which interference can instead arise. At the phenomenological level, our analysis implies that the BSM effects that are naively expected to be dominant in an EFT approach, i.e. those captured by the interference of D = 6 effective operators with the SM, are suppressed in the high-energy limit. The interference only arises at next-to-leading order in an expansion in mass over energy and in the 1-loop perturbative parameter α s /4π (or α em /4π for processes not involving colored particles). Interestingly, some of the remaining amplitudes which do feature interference are accidentally small in the SM, implying anyway a small interference. This leads to a reduced sensitivity on new physics, especially in the case of analyses that are inclusive on the polarizations of the final-state particles. Furthermore, in many cases of interest and in particular when the underlying theory is weakly coupled, a generic EFT analysis in terms of D = 6 operators alone is insufficient, as D = 8 ones give an equally large (if not larger) contribution.
defining the scattering amplitudes are the spinors |p ȧ and |p] a transforming as (1/2, 0) and (0, 1/2) under SU (2) × SU (2) SO(3, 1). They are independent solutions of the massless Dirac equation: where the subscript ± corresponds to an helicity h = ±1/2. Dotted and undotted indices are raised/lowered with the 2-index Levi-Civita tensor. A (1/2, 1/2) Lorentz vector is written in terms of the spinors as − p = |p [p| + |p] p|, while the polarization vectors for spin-1 massless bosons are where q is a reference vector whose arbitrariness reflects gauge invariance. The products of angle and square spinors pq ≡ p|ȧ|q ȧ and [pq] ≡ [p| a |q] a satisfy the properties for any p and q. In many theories the basic building blocks for all scattering amplitudes are 3-point amplitudes. Momentum conservation in the 3-point vertex (p µ 1 +p µ 2 + p µ 3 ) = 0 and the on-shell condition p 2 i = 0 imply p i · p j = 0, which in bra-ket notation reads 12 [12] which leaves the momentum (p i ) aḃ = −|p i ] a p i |˙b invariant. Under such transformation the polarization vector of a spin-1 particle scales as t −2hi i if it has helicity h i = ±1. An on-shell tree-level amplitude thus scales as t −2hi under the rescaling of a particle i with helicity h i , and as t −2h , with h = i h i , when all particles are rescaled. We have seen that the special 3-particle kinematics described below Eq. (A4) implies that a 3-point amplitude must depend either on square or angle brackets. Little group scaling and the request of locality then fix completely the form of the amplitude to be (at tree level) where [g] is the sum of the dimensions of the couplings contributing to the amplitude. For [g] even, in particular, it follows that h(A n ) has the same parity as n.

Appendix B: Supersymmetric Ward Identities
As long as all up-type or all down-type Yukawa couplings vanish, the SM fields and interactions can be embedded in a supersymmetric Lagrangian with R-parity. When both kinds are non-vanishing, however, holomorphy of the superpotential requires the introduction of an additional Higgs doublet or explicit supersymmetry breaking. Most SM tree-level amplitudes (all those not involving simultaneously up-and down-type Yukawas) can thus be written in supersymmetric form. R-parity implies that no supersymmetric state propagates in the internal lines, so that these amplitudes are effectively supersymmetric. This feature is generically lost in BSM 6 , although some operators can still be uplifted to a supersymmetric form [41].
Supersymmetry implies important relations between scattering amplitudes [12] (see [9,19] for a review). Since the supercharge Q(ξ) =ξ α Q α annihilates the vacuum for a generic spinor parameter ξ, the following Supersymmetric Ward Identities (SWI) hold for amplitudes made of n arbitrary fields Φ i : where O n ≡ Φ 1 · · · Φ n . For a scalar φ and a Weyl fermion ψ in the same chiral supermultiplet, and a Eq. (B2) holds also for fields with opposite helicity ± → ∓ provided one replaces φ ↔ φ † and ξk → −[ξk]. For n = 4, taking O 4 = λ + 1 V + 2 V + 3 V + 4 in Eq. (B1) gives Since (supersymmetric-)gauge interactions conserve helicity, amplitudes involving two gauginos with the same helicity and two gauge fields are vanishing at tree level. Then Eq. (B3) implies in Eq. (B3) and choosing ξ = k 1 and ξ = k 2 , one obtains A 4 (V − 1 V + 2 V + 3 V + 4 ) = 0 = The second term in this equation vanishes as a consequence of the Z 2 chiral symmetry (10). If no cubic scalar term is present in the theory, as we will assume, the same symmetry argument also ensures that the last amplitude vanishes. Choosing ξ = k 1 then implies A 4 (φ † 1 φ 2 V + 3 V + 4 ) = 0, while ξ = k 3 gives A 4 (ψ + 1 φ 2 λ + 3 V + 4 ) = 0. Since supersymmetry commutes with the SM gauge group G SM and the color and Lorentz structures factorize in helicity amplitudes, it follows that the above results hold for fermions in generic representations of G SM , and not only for gauginos in the adjoint; this proves Eq. (5). Relations for higherpoint amplitudes can be obtained by similar arguments or simply through Eq. (3). Finally, notice that the operators ψ 2ψ2 , ψψφ 2 D and φ 4 D 2 can be uplift into a supersymmetric form [41], so that their introduction in the theory will not change the SWI.