Very rare, exclusive, hadronic decays in QCD factorization

We study exclusive hadronic decays of the electroweak bosons Z, W and h in the framework of QCD factorization. We show that the theory uncertainties in these channels are remarkably small compared to past applications of the QCD factorization framework. While the branching ratios are small, many of the modes are accessible at future colliders. The Higgs decays exhibit interesting dependences on the couplings due to the interferences of different diagram topologies, making the h → Vγ decays possible probes of the quark Yukawa couplings and the h → VZ decays probes of the coupling between the Higgs boson, a photon and a Z-boson. Talk based on work in collaboration with Stefan Alte, Yuval Grossman and Matthias Neubert [1–4].


Introduction
The Standard Model (SM) of particle physics is in remarkable agreement with observation and the discovery of the Higgs boson in 2012 [5,6] is arguably the biggest triumph of particle physics in recent times. Still many questions remain unanswered and thus the SM has to be extended. When doing so, the Higgs sector is usually modified, making it a crucial experimental task to measure the parameters in this sector as accurately as possible. The exclusive hadronic decays of the form h → Mγ, h → MZ and h → MW can be used to probe several different couplings of the Higgs boson to the SM fermions and gauge bosons [2,7,8].
Even within the SM, particle physics still faces the challenge of obtaining rigorous control over non-perturbative effects in QCD at low energy scales. The framework of QCD factorization is a wellestablished approach to hard exclusive processes with indiviual final state hadrons [9][10][11][12][13]. Within this framework, amplitudes are given as convolutions of hard-scattering functions with non-local hadronic matrix elements, which encode the non-perturbative physics at the low scale. The amplitudes will formally be given as expansions in the ratio of the two scales. Past applications struggled with the fact that this factorization scale was not high enough for power corrections to be neglected and that it was difficult to disentangle these uncertainties from those in the poorly-known hadronic input parameters. In our case, the scale is set by the decaying heavy boson and power corrections are well under control. The decays Z → Mγ and W → Mγ can be used to test that factorization approach in a theoretically clean environment [1,3].

QCD factorization
In the framework of QCD factorization, the amplitude for a hard exclusive process can be written as a convolution of a hard scattering function with a hadronic function, where the result will formally be an expansion of the scale separation parameter λ ∼ Λ QCD /μ hard . While the original derivation of the factorization formula dates back to the early 1980s, it can be rephrased in the language of Soft-Collinear Effective Theory (SCET) [14][15][16][17]. The factorization formula for the decays of h/Z → MV the final state meson M and the gauge boson V can be written as: where T q H (x, μ) is the hard-scattering kernel, φ q M (x, μ) is the light-cone distribution amplitude (LCDA) and the sum runs over the active quark flavors. Power corrections of higher orders in λ are tiny in our case due to the high scale μ hard .
The LCDAs are expanded in the set of Gegenbauer polynomials where a M,q n (μ) are the scale-dependent Gegenbauer moments [9,13,[18][19][20][21]. By evolving these parameters from the hadronic scale to the high scale using the renormalization group, large logarithms of the form α s log λ are resummed to all orders. It is noteworthy that the evolution to an arbitrarily high scale lets all moments a M,q n vanish, decreasing the sensitivity of our prediction to these poorly-known hadronic parameters.

Radiative hadronic decays of Z-bosons
We can now write down the amplitude for the decays Z → Mγ. We will neglect the modes Z → η ( ) γ here since the flavor-singlet components lead to non-trivial matching conditions that are too involved to explain here [3]. The amplitude of a Z-boson decaying into a meson M with momentum k and a photon with momentum q can be written as: We can express the form factors as sums over the Gegenbauer moments and the hard scattering coefficients: Decay mode Branching ratio where Q ( ) are combinations of the meson decay constants and the quark couplings to the Z-boson and the photon. Evaluating diagrams like the ones shown in figure 1, we find the hard scattering coefficients: This is the leading result in our expansion in the scale separation parameter λ, with subleading corrections starting at O(m 2 V /m 2 Z ), that we can safely neglect. We present our phenomenological results in table 1.
Similar results are obtained for the W → Mγ decays. We omit a discussion of these for briefness and refer the reader to ref. [1] for details.

Exclusive hadronic decays of the Higgs boson
In order to investigate the hadronic decays of the Higgs boson and understand to what extend new physics can manifest itself in them, we work in an effective Lagrangian framework, where we allow a rescaling of the relevant parameters in the SM as well as including higher-dimensional operators:  Table 1. Predicted branching fractions for various Z → Mγ decays, including error estimates due to scale dependence (subscript "μ") and the uncertainties in the meson decay constants (" f "), and the shape parameters of the LCDA ("φ"). In the SM, the parameters κ W , κ Z and κ f are equal to 1 whereas all other κ's vanish.
In the following, we only briefly state the main results of our analysis and direct the interested reader towards refs. [2,4].

Radiative hadronic decays of the Higgs
The case of h → Vγ is similar to the Z-boson decays, with the difference of an important additional contribution, where the Higgs decays into a pair of photons or into a photon and a Z-boson. Here, either the off-shell photon or Z-boson converts to the final state meson. The hγγ-and hγZ-couplings are generated at 1-loop in the SM or can occur at tree-level in our effective Lagrangian [22]. Generally, these "indirect" contributions are dominant over the ones that directly involve the Yukawa couplings of the valence quarks of the vector meson V. When one wants to use these decays to probe said Yukawa couplings, it is useful to normalize the branching ratios to the branching ratio of h → γγ, since all NP effects in the indirect contribution will affect h → γγ in the same way (up to small corrections due to the off-shellness of the photon and the Z-boson diagram). Expanding in small parameters, we find: The parameters Δ V andΔ V contain the direct contributions and the corrections to the indirect contributions due to the off-shellness of the photon and the additional diagram with the Z-boson: The direct form factors F V direct can written in a similar form to the ones in eq. (4). The parameter r CP vanishes in the SM and contains the various CP-odd couplings to the indirect amplitude and the h → γγ rate. The detailed expressions for both F V and r CP can be found in ref. [2]. Table 2 quotes our SM branching ratios. Note that in the case of h → Υ(1S ) γ the direct and indirect contributions cancel almost exactly, suppressing this decay by three orders of magnitude. This cancellation leads to a spectacular sensitivity of this branching ratio on κ b andκ b , as demonstrated in

Weak radiative Higgs decays
Weak Higgs decays of the form h → VZ can be used to probe the coupling of the Higgs to a photon and a Z-boson, both loop-induced and from the couplings κ γZ andκ γZ . There are three important contributions, shown in figure 3. While naive dimensional analysis suggests that the tree-level diagram involving the hZZ gauge coupling is the dominant amplitude, the diagram with the loop-induced hγZ vertex is equally important. This is because the loop-suppression is overcome by the photon propagator being almost on its pole. For pseudoscalar mesons, the photon diagram does not exist and thus the decays h → PZ can only serve as a SM reference values. The current limits on the parameters κ γZ andκ γZ from CMS [23] and ATLAS [24] imply upper bounds on the decay rates of 9 and 11 times the SM value, respectively, both at 95% confidence level. The stronger bound from CMS implies the constraint (under the assumption that all other parameters are at their SM value)

Conclusions
Exclusive hadronic decays of heavy electroweak bosons are a theoretically clean playground for the framework of QCD factorization. The hard scale μ hard is set by the decaying bosons and thus very high, rendering the impact of power corrections of O(Λ QCD /μ hard ) tiny. Additionally, the hadronic uncertainties are comparably small thanks to the renormalization group evolution of the LCDA. The price to pay is the smallness of the branching ratios. For the Z-decays, the most promising experimental environment would be a future lepton collider running at √ s = m 2 Z , whereas the Higgs decays are an interesting physics target for the high-luminosity run of the LHC or a future 100 TeV proton collider.
This report skips over a large amount of technical and phenomenological details, like a detailed derivation of the factorization theorem, the renormalization group evolution of the hadronic parameters and the special case of flavor-singlet mesons as well as the W ± → M ± γ and h → M ± W ∓ decays [1][2][3]. We nevertheless hope to convince the reader that the physics opportunities in a dedicated program for the decays considered here are compelling, both within and beyond the standard model.