Searches for deviations from Standard Model (SM) predictions in processes involving interactions between known particles are a well established technique to study possible contributions from Beyond the Standard Model (BSM) physics. In this paper we address the question how much we can learn about the scale of new physics and its strength using the Effective Field Theory (EFT) approach to \(W^+W^+\) scattering if a statistically significant deviation from the SM predictions is observed in the expected LHC data for the process \(pp \rightarrow 2 jets + W^+W^+ \). Our specific focus is on the proper use of the EFT in its range of validity. With this in mind, we discuss the practical usefulness of the EFT language to describe vector boson scattering (VBS) data and whether or not this can indeed be the right framework to observe the first hints of new physics at the LHC.
The EFT is in principle a model independent tool to describe BSM physics below the thresholds for new states. One supplements the SM Lagrangian by higher dimension operators
$$\begin{aligned} {\mathcal {L}}={\mathcal {L}}_{SM} +\Sigma _i\frac{C_i^{(6)}}{\Lambda ^2_i}{\mathcal {O}}_i^{(6)} +\Sigma _i\frac{C_i^{(8)}}{\Lambda ^4_i}{\mathcal {O}}_i^{(8)} +\cdots , \end{aligned}$$
(1.1)
where the C’s are some “coupling constants” and \(\Lambda \)’s are the decoupled new mass scales. The mass scale is a feature of the UV completion of the full theory and thus is assumed common to all the coefficients
$$\begin{aligned} f_i^{(6)}=\frac{C_i^{(6)}}{\Lambda ^2} , \quad f_i^{(8)}=\frac{C_i^{(8)}}{\Lambda ^4},\cdots , \end{aligned}$$
(1.2)
which are free parameters because the full theory is unknown. One should stress that the usefulness of any EFT analysis of a given process relies on the assumption that only few terms in the expansion of Eq. (1.1) give for that process an adequate approximation to the underlying UV theory. The necessary condition obviously is that the energy scale of the considered process, \(E<\Lambda \). However, the effective parameters in the expansion Eq. (1.1) are the f’s and not the scale \(\Lambda \) itself. Neither \(\Lambda \) nor the C’s are known without referring to specific UV complete models. Even for \(E<<\Lambda \) a simple counting of powers of \(E/\Lambda \) can be misleading as far as the contribution of various operators to a given process is concerned. The latter depends also on the relative magnitude of the couplings C, e.g., \(C_i^{(6)}\) versus \(C_i^{(8)}\) and/or within each of those sets of operators, separately, [1,2,3,4], as well as on the interference patterns in various amplitudes calculated from the Lagrangian Eq. (1.1) [5].
For instance, the contribution of dimension-6 (\(\hbox {D}=6\)) operators to a given process can be suppressed compared to dimension-8 (\(\hbox {D}=8\)) operators contrary to a naive \((E/\Lambda )\) power counting [3,4,5,6] or, vice versa, the [\(\hbox {SM}\ \times \hbox {D}=8\)] interference contribution can be subleading with respect to the \([D=6]^2\) one [2, 4, 7]. Clearly, the assumption about the choice of operators in the truncation in Eq. (1.1) used to analyze a process of our interest introduces a strong model dependent aspect of that analysis: one is implicitly assuming that there exist a class of UV complete models such that the chosen truncation is a good approximation. It is convenient to introduce the concept of EFT “models” defined by the choice of operators \({\mathcal {O}}_i\) and the values of \(f_i\). The question of this paper is then about the discovery potential at the LHC for BSM physics described by various EFT “models”.
The crucial question is what the range of validity can be of a given EFT “model”. There is no precise answer to this question unless one starts with a specific theory and derives Eq. (1.1) by decoupling the new degrees of freedom. However, in addition to the obvious constraint that the EFT approach can be valid only for the energy scale \(E<\Lambda \) (unfortunately with unknown value of \(\Lambda \)), for theoretical consistency the partial wave amplitudes should satisfy the perturbative unitarity condition. The latter requirement translates into the condition \(E^2<\Lambda ^2\le s^U\), where \(s^U\equiv s^U(f_i)\) is the perturbative partial wave unitarity bound as a function of the chosen operators and the values of the coefficients \(f_i\)’s. Thus, the value of \(\Lambda ^2_{max}=s^U\) gives the upper bound on the validity of the EFT based “model”. Since the magnitude of the expected (or observed) experimental effects also depends on the same \(f_i\), one has a frame for a consistent use of the EFT “model” to describe the data once they are available. For a BSM discovery in the EFT framework, proper usage of the “model” is a vital issue. It makes no physical sense to extend the EFT “model” beyond its range of applicability, set by the condition \(E<\Lambda \). We shall illustrate this logic in more detail in the following.
A common practice in the LHC data analyses in the EFT framework is to derive uncorrelated limits on one operator at a time while setting all the remaining Wilson coefficients to zero. This in fact means choosing different EFT “models”: such limits are valid only under the assumption that just one chosen operator dominates BSM effects in the studied process in the available energy range. In this paper we will consider only variations of single dimension-8 operators.Footnote 1 However, the strategy we present can be extended to the case of many operators at a time, including dimension-6 (keeping in mind that varying more than one operator substantially complicates the analysis). For a given EFT “model”
$$\begin{aligned} \sigma \propto |A_{full}|^2=|A_{SM}|^2+(A_{SM}\times A_{BSM}^*+hc)+|A_{BSM}|^2. \end{aligned}$$
(1.3)
We focus on the process
$$\begin{aligned} pp\rightarrow 2 jets + W^+ W^+ \rightarrow 2 jets + l^+\nu +l'^+\nu ^\prime , \end{aligned}$$
(1.4)
where l and \(l'\) stand for any combination of electrons and muons. The process depends on the \(W^+W^+\) scattering amplitude (the gauge bosons can of course be virtual). The EFT “models” can be maximally valid up to certain invariant mass \(M=\sqrt{s}\) of the \(W^+W^+\) system
$$\begin{aligned} M<\Lambda \le M^U(f_i), \end{aligned}$$
(1.5)
where \(M^U(f_i)\) is fixed by the partial wave perturbative unitarity constraint, \((M^U(f_i))^2=s^U(f_i)\).
The differential cross section \(\frac{d\sigma }{dM}\) reads (actual calculations must include also all non-VBS diagrams leading to the same final states):
$$\begin{aligned}&\frac{d\sigma }{dM}\sim \Sigma _{ijkl} \int dx_1 dx_2 q_i(x_1) q_j(x_2) | {\mathcal {M}}(ij\nonumber \\&\quad \rightarrow klW^+W^+)|^2 d\Omega \; \delta (M-\sqrt{(p_{W^+}+p_{W^+})^2}), \end{aligned}$$
(1.6)
where \(q_i(x)\) is the PDF for parton i, the sum runs over partons in the initial (ij) and final (kl) states and over helicities, the amplitude \({\mathcal {M}}\) is for the parton level process \(ij\rightarrow klW^+W^+\) and \(d\Omega \) denotes the final state phase space integration. The special role of the distribution \(\frac{d\sigma }{dM}\) follows from the fact that it is straightforward to impose the cutoff \(M\le \Lambda \), Eq. (1.5), for the WW scattering amplitude. The differential cross section, \(\frac{d\sigma }{dM}\), is therefore a very sensitive and straightforward test of new physics defined by a given EFT “model”. Unfortunately, the \(W^+W^+\) invariant mass in the purely leptonic W decay channel is not directly accessible experimentally and one has to investigate various experimental distributions of the charged particles. The problem here is that the kinematic range of those distributions is not related to the EFT “model” validity cutoff \(M<\Lambda \) and if \(\Lambda < M_{max}\), where \(M_{max}\) is the kinematic limit accessible at the LHC for the WW system, there is necessarily also a contribution to those distributions from the region \(\Lambda< M < M_{max}\). The question is then: in case a deviation from SM predictions is indeed observed, how to verify a “model” defined by a single higher-dimension operator \({\mathcal {O}}_i^{(k)}\) and a given value of \(f_i\) by fitting it to a set of experimental distributions \(D_i\) and in what range of \(f_i\) such a fit is really meaningful [7]. Before we address this question, it is in order to comment on the perturbative partial wave unitarity constraint.
It is worthwhile to stress several interesting points.
-
1.
For a given EFT “model”, the unitarity bound is very different for the \(J=0\) partial wave of different helicity amplitudes and depends on their individual energy dependence (some of them remain even constant and never violate unitarity, see Appendix). Our \(M^U\) has to be taken as the lowest unitarity bound, universally for all helicity amplitudes, because it is the lowest bound that determines the scale \(\Lambda _{max}\). More precisely, one should take the value obtained from diagonalization of the matrix of the \(J=0\) partial waves in the helicity space.
-
2.
Correct assessment of the EFT “model” validity range in the \(W^+W^+\) scattering process requires also consideration of the \(W^+W^-\) scattering amplitudes which by construction probe the same couplings and are sensitive to exactly the same operators. For most higher dimension operators, this actually significantly reduces their range of validity in \(W^+W^+\) analyses. Conversely, the WZ and ZZ processes can be assumed to contain uknown contributions from additional operators which adjust the value of \(\Lambda \) consistently.
-
3.
It is interesting to note that for the \(f_i\) values of practical interest the deviations from SM predictions in the total cross sections become sizable only in a narrow range of energies just below the value of \(M^U\), where the \(|A_{BSM}|^2\) term in Eq. (1.3) takes over. However, for most dimension-8 operators the contribution of the interference term is not completely negligible (see Appendix for details). Even if deviations from the SM are dominated by the helicity combinations that reach the unitarity bound first, the total unpolarized cross sections up to \(M=M^{U}\) get important contributions also from amplitudes which are still far from their own unitarity limits.
In the Appendix we illustrate various aspects of those bounds by presenting the results of analytical calculations for two dimension-8 operators, one contributing mainly to the scattering of longitudinally polarized gauge bosons and one to transversely polarized.
We now come back to the problem of testing the EFT “models” when the \(W^+W^+\) invariant mass is not accesible experimentally. Let us define the BSM signal as the deviation from the SM prediction in the distribution of some observable \(D_i\).
$$\begin{aligned} S=D_i^{model}-D_i^{SM}. \end{aligned}$$
(1.7)
The first quantitative estimate of the signal can be written as
$$\begin{aligned} D_i^{model}=\int ^{\Lambda }_{2M_W}\frac{d\sigma }{dM}|_{model} dM +\int _{\Lambda }^{M_{max}}\frac{d\sigma }{dM}|_{SM} dM. \end{aligned}$$
(1.8)
It defines signal coming uniquely from the operator that defines the “model” in its range of validity and assumes only the SM contribution in the region \(M>\Lambda \). Realistically one expects some BSM contribution also from the region above \(\Lambda \). While this additional contribution may enhance the signal and thus our sensitivity to new physics, it may also preclude proper description of the data in the EFT language. Such description in terms of a particular EFT “model” makes sense if and only if this contribution is small enough when compared to the contribution from the region controlled by the EFT “model”. The latter depends on the value of \(\Lambda \) and \(f_i\), and the former on the unknown physics for \(M>\Lambda \), which regularizes the scattering amplitudes and makes them consistent with partial wave unitarity. Ideally, one would conclude that the EFT “model” is tested for values of \((\Lambda \le M^U, f_i)\) such that the signals computed from Eq. (1.8) are statistically consistent (say, within 2 standard deviations) with the signals computed when the tail \(\Lambda >M^U\) is modeled in any way that preserves unitarity of the amplitudes, i.e., the contribution from this region is sufficiently suppressed kinematically by parton distributions. This requirement is of course impossible to impose in practice, but for a rough quantitative estimate of the magnitude of this contribution, one can assume that all the helicity amplitudes above \(\Lambda \) remain constant at their respective values they reach at \(\Lambda \), and that \(\Lambda \) is common to all the helicity amplitudes. For \(\Lambda = \Lambda _{max}\), this prescription regularizes the helicity amplitudes that violate unitarity at \(M^U\) and also properly accounts for the contributions of the helicity amplitudes that remain constant with energy. It gives a reasonable approximation to the total unpolarized cross sections for \(M>M^U\), at least after some averaging over M. More elaborated regularization techniques can also be checked here. The full contribution to a given distribution \(D_i\) is then taken as
$$\begin{aligned} D_i^{model}=\int ^{\Lambda }_{2M_W}\frac{d\sigma }{dM}|_{model} dM +\int _{\Lambda }^{M_{max}}\frac{d\sigma }{dM}|_{A=const} dM. \end{aligned}$$
(1.9)
BSM observability imposes some minimum value of f to obtain the required signal statistical significance. It can be derived based on Eq. (1.9) [or Eq. (1.8)]. On the other hand, description in the EFT language imposes some maximum value of f such that signal estimates computed from Eqs. (1.8) and (1.9) remain statistically consistent. Large difference between the two computations implies significant sensitivity to the region above \(\Lambda \). It impedes a meaningful data description in the EFT language and also suggests we are more likely to observe the new physics directly.
Assuming \(\Lambda =M^U\), we get a finite interval of possible f values, bounded from two sides, for which BSM discovery and correct EFT description are both plausible. In the more general case when \(\Lambda <M^U\), i.e., new physics states may appear before our EFT “model” reaches its unitarity limit, respective limits on f depend on the actual value of \(\Lambda \). We thus obtain a 2-dimensional region in the plane \((\Lambda , f_i)\), which is shown in the cartoon plot in . 1. This region is bounded from above by the unitarity bound \(M^U(f_i)\) (solid blue curve), from the left by the signal significance criterion (dashed black curve) and from the right by the EFT consistency criterion (dotted black curve). The EFT could be the right framework to search for BSM physics as long as these three criteria do not mutually exclude each other, i.e., graphically, the “triangle” shown in our cartoon plot is not empty. In Sect. 3 we will verify whether such “triangles” indeed exist for the individual dimension-8 operators.
Thus, our preferred strategy for data analysis is as follows:
-
1.
From collected data measure a distribution \(D_i\) (possibly in more than one dimension) that offers the highest sensitivity to the studied operator(s).
-
2.
If deviations from the SM are indeed observed,Footnote 2 fit particular values of \((\Lambda \le M^U, f_i)\) based on EFT simulated templates in which the contribution from the region \(M > \Lambda \) is taken into account according to Eq. (1.9) or using some more elaborated regularization methods.
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3.
Fixing \(f_i\) and \(\Lambda \) to the fit values, recalculate the \(D_i\) template so that the region \(M > \Lambda \) is populated only by the SM contribution Eq. (1.8).
-
4.
Check statistical consistency between the original simulated \(D_i\) template and the one based on Eq. (1.8).
-
5.
Physics conclusions from the obtained \((\Lambda , f_i)\) values can only be drawn if such consistency is found. In addition, stability of the result against different regularization methods provides a measure of uncertainty of the procedure – too much sensitivity to the region above \(\Lambda \) means the procedure is destined to fail and so the physical conclusion is that data cannot be described with the studied operator.