Perturbative unitarity bounds on di-boson scalar resonances

We consider the constraints implied by partial wave unitarity on new physics in the form of di-boson resonances at LHC. We derive the scale where the effective description in terms of the SM supplemented by a single scalar resonance is expected to break down depending on the resonance width and signal cross-section.


Introduction
Perturbative unitarity is a powerful theoretical tool for inferring the range of validity of a given effective field theory (EFT), with notable examples of applications both in the physics of strong and electroweak interactions.Perhaps most famously, constraints imposed by perturbative unitarity in WW scattering have been used in the past to infer an upper bound on the Higgs boson mass or, alternatively, on the scale where the standard model (SM) description of weak interactions would need to be completed in the ultraviolet (UV) in terms of some new strongly coupled dynamics [1].
The recently rekindled interest in new physics (NP) in the form of (possibly broad) di-photon resonances [2,3] at the LHC motivated us to reconsider the implications of perturbative unitarity for EFT interpretations of resonances decaying to di-boson final states.In particular, focusing on promptly produced scalar SM singlets decaying to two SM gauge bosons we aim to address the following question: at which maximal energies do we expect the effective description in terms of the SM supplemented by a single scalar to break down?Assuming that a scalar resonance is within the reach of the present LHC run, can a next generation hadron collider potentially probe the on-shell effects of new degrees of freedom responsible for the restoration of unitarity?
After a brief recap of partial wave unitarity arguments in Sect.2, we apply them in Sect. 3 to the EFT case where a di-boson resonance is the only new degree of freedom beyond the SM.The scale of unitarity violation is hence interpreted as an upper bound on the mass scale of new degrees of freedom UV completing the effective low-energy description and regularizing (unitarizing) the amplitudes' growth.The present contribution is largely based on Ref. [4], to which the reader is referred for further details.a e-mail: luca.di-luzio@durham.ac.uk b e-mail: jernej.kamenik@cern.chc e-mail: marco.nardecchia@cern.chLet us denote by T f i ( √ s, cos θ) the matrix element of a 2 → 2 scattering amplitude in momentum space, defined via (2π) 4 δ (4) where T is the interacting part of the S -matrix, S = 1+iT .The dependence of the scattering amplitude on cos θ is eliminated by projecting it onto partial waves of total angular momentum J a J f i = where d J µ i µ f is the J-th Wigner d-function appearing in the Jacob-Wick expansion [5], while µ i = λ i1 − λ i2 and µ f = λ f 1 − λ f 2 are defined in terms of the helicities of the initial (λ i1 , λ i2 ) and final (λ f 1 , λ f 2 ) states.The function β(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2yz − 2zx is a kinematical factor related to the momentum (to the fourth power) of a given particle in the center of mass frame.The right hand side of Eq. ( 2) must be further multiplied by a 1 √ 2 factor for any identical pair of particles either in the initial or final state.
When restricted to a same-helicity state (zero total spin), the Wigner d-functions reduce to the Legendre polynomials, i.e. d J 00 = P J .In practice, we will only focus on J = 0 (d 0 00 = P 0 = 1), since higher partial waves typically give smaller amplitudes, unless J = 0 amplitudes are suppressed or vanish for symmetry reasons.Hence, the quantity we are interested in is In the high-energy limit, √ s → ∞, one has β 1/4 f β 1/4 i /s → 1.The unitarity condition on the S -matrix, where the sum over h is restricted to 2-particle states, which slightly underestimates the left hand side.For i = f Eq. ( 4) reduces to Im Hence, a J ii must lie inside the circle in the Argand plane defined by which implies Under the assumption that the tree-level amplitude is real, Eq. ( 7) suggests the following perturbativity criterium |Re (a J ii ) Born | ≤ In fact, a Born value of Re a J ii = 1 2 and Im a J ii = 0 needs at least a correction of 40% in order to restore unitarity, thus signalling the breakdown of the (E/Λ) power expansion of the EFT, where E is the typical energy involved in a process and Λ is the EFT cut-off scale.

Effective field theory of a scalar resonance
We consider the EFT of a gauge singlet spin-0 resonance, S with mass M S , coupled to the SM fields.Assuming CP invariance, we choose S to transform as a scalar. 1 Apart from renormalizable terms coupling S to the Higgs in the scalar potential, the d = 5 Lagrangian reads L (5)  int.= − where we have suppressed flavor indices.This parametrization makes it clear that the leading interactions of a scalar singlet with the SM fields, directly relevant for di-boson resonances at the LHC, are all due to non-renormalizable d = 5 operators.Their effects are thus expected to be enhanced at high energies eventually leading to the breakdown of perturbative unitarity.In order to quantify this simple observation in the following subsections we evaluate the relevant scattering amplitudes involving SM gauge bosons, Higgs and quarks at the respective leading orders in perturbation theory.Moreover, since we are interested in studying 2 → 2 scattering processes at energies √ s M S v 246 GeV, we can safely set all the massive parameters (including M S ) to zero and work within the unbroken SM theory.This also implies that we can neglect any h-S mixing effects and set the masses of the final state SM particles to zero.We distinguish between two classes of tree-level processes characterized by a different energy scaling of the amplitude: scalar mediated scatterings and d = 5 contact interactions.

Scalar mediated boson scattering
Let us start, as an example, by considering the γγ → γγ scattering amplitude due to the effective operator In the (++, −−) helicity basis we find [4] where in the last step we took the high-energy limit, and only the s-channel survives at high energies.
The projection on the J = 0 partial waves is obtained by applying Eq. ( 3) and by multiplying by a 1/2 factor which takes into account the presence of identical particles both in the initial and final states.In the high-energy limit we get which, confronted with Eq. ( 8), leads to the tree-level unitarity bound As a of fact, the bound above can be made stronger if one considers the full VV → V V scattering matrix, where V and V are any of the 8 + 3 + 1 (transversely polarized) SM gauge bosons of the effective Lagrangian in Eq. ( 9).In such a case, the previous calculation is readily generalized in the high-energy limit where only the s-channel survives.To this end, we note that a scattering amplitude in the s-channel can be written as where a i and a j are obtained by cutting any i → j diagram in two parts along the s-channel propagator.
The matrix in Eq. ( 14) has rank 1 and its non-zero eigenvalue is given by the trace.Hence, denoting by ã0 the eigenvalue of the VV → V V scattering matrix, in the high-energy limit we get Correspondingly, the tree-level unitarity bound is given by We remark that in deriving these bounds we consider only the transverse polarizations of the W and Z gauge bosons.Generally, scattering amplitudes involving longitudinally polarized massive vector bosons can grow as positive powers of E/m W,Z implying apparently stronger dependence on s.However, as it can be easily verified (through an explicit calculation of the processes at hand or more generally via a clever gauge choice [6]), the scattering amplitudes involving longitudinally polarized states sourced by the gauge field strengths in Eq. ( 9) are suppressed by powers of m W,Z /E and thus do not lead to relevant unitarity constraints at high s.

Fermion-scalar contact interactions
Next we consider the contact interaction where we have explicitly factored out the color and S U(2) L group structure.In this case the leading scattering process is Qd → S H.By explicitly writing the polarization and gauge indices in the amplitude, one finds Only the ++ and −− polarizations survive.By explicit evaluation we get [4] T At high energies the J = 0 partial wave is obtained by considering the color singlet channel for a state in the linear combination 1 √ 2 |Qd + |S H , which gives Correspondingly, the tree-level unitarity bound reads √ s 8πΛ d .
Similarly, from the other two contact interactions in the last row of Eq. ( 9) we get √ s 8πΛ u,e .

Unitarity bounds
As an exemplification we consider a scalar resonance S with mass M S and total width Γ S appearing in a di-photon final state at the LHC. 2 Expanding the effective Lagrangian in Eq. ( 9) around the broken electroweak (EW) vacuum, the part relevant for S production at the LHC is whose operators give rise to the decay widths The matching between the operators in Eq. ( 23) and Eq. ( 9) then yields In the narrow width approximation the prompt S production at the LHC can also be fully parametrized in terms of the relevant decay widths where √ s is the LHC pp collision energy and C PP parametrize the relevant parton luminosities.For illustration purposes in the following we consider in turn either gg and γγ induced processes or alternatively bb and γγ rates at a benchmark mass of M S = 750 GeV.The remaining possibilities

EPJFigure 1 :
Figure 1: of unitarity violation Λ U in TeV in the (B γγ , σ γγ ) plane (cf.Eqs.(37)-(38)).Upper/lower plots corresponding to gg/bb production, while left/right plots to the large/small width scenario.As reference values we assume M S = 750 GeV and f (r) = 30.The red curve denotes the new physics scale accessible at a futuristic 100 TeV collider, Λ = 20 TeV, while the three horizontal lines from top to bottom are three reference cross-sections, namely 6, 0.6 and 0.2 fb.The yellow triangle on the top-left of each figure is the region in parameter space where Γ S /M S > 10 %.