Scattering of Goldstone Bosons and resonance production in a Composite Higgs model on the lattice

We calculate the coupling between a vector resonance and two Goldstone bosons in $SU(2)$ gauge theory with $N_f=2$ Dirac fermions in the fundamental representation. The considered theory can be used to construct a minimal Composite Higgs models. The coupling is related to the width of the vector resonance and we determine it by simulating the scattering of two Goldstone bosons where the resonance is produced. The resulting coupling is $g_{\rm{VPP}}=7.8\pm 0.6$, not far from $g_{\rho\pi \pi}\simeq 6$ in QCD. This is the first lattice calculation of the resonance properties for a minimal UV completion. This coupling controls the production cross section of the lightest expected resonance at the LHC and enters into other tests of the Standard Model, from Vector Boson Fusion to electroweak precision tests. Our prediction is crucial to constrain the model using lattice input and for understanding the behavior of the vector meson production cross section as a function of the underlying gauge theory. We also extract the coupling $g_{\rm{VPP}}^{\rm{KSRF}} =9.4 \pm 0.6$ assuming the vector-dominance and find that this phenomenological estimate slightly overestimates the value of the coupling.


Introduction
Among the numerous approaches to address the shortcomings of the Standard Model, new strongly coupled sectors provide a number of interesting mechanisms that address fundamental issues like the naturalness problem or the origin of the Higgs' field.
Pseudo Nambu-Goldstone Boson (PNGB) Composite Higgs models aim at identifying the Higgs degrees of freedom with the Goldstone bosons of a new strongly coupled sector [1][2][3]. In this framework, the fundamental Higgs field of the Standard Model is an effective low energy degree of freedom at the electroweak scale of a new strongly interacting sector featuring spontaneous chiral symmetry breaking. The mechanism alleviates the naturalness problem. The coupling of the new strongly interaction sector with the Standard Model breaks explicitly the flavour symmetry of the underlying gauge theory. This breaking is responsible for a non-trivial potential for the pseudo-Goldstone bosons which provides the Higgs with a mass and triggers electroweak symmetry breaking.
These models are constrained by comparing predictions at the electroweak scale with the LHC data. Typically the predictions depends on low energy constants (LECs) of the effective theory and on free parameters that are related to the model building itself. The vast majority of models are tested without specifying an underlying strongly interacting sector, and by assuming that the LECs are free parameters. While the approach allows to investigate entire classes of UV completion, it neglects correlations among LECs and additional information provided by a quantitative understanding of the strong dynamics. The lattice approach allows to make first-principle predictions of the low energy parameters and can therefore provide stringent constraints on some scenarios.
Strongly interacting theories are expected to feature a rich spectrum of resonances that modify the phenomenology at colliders. In minimal scenarios, vector resonance are expected to mix with the electroweak bosons and can therefore be produced via vector boson fusion or via the Drell-Yan production mechanism, see for instance [4,5]. These processes are controlled by the coupling g VPP of the vector resonance (V) to two pseudoscalar Goldstone bosons (P). The phenomenology of new vector resonances is attracting considerable attention by the community [6][7][8][9][10][11][12].
In this work, we perform the first ab-initio calculation of the coupling g VPP between a resonance and a pair of Goldstone bosons using lattice techniques in isolation of the Standard Model for a minimal UV completion. We consider SU (2) gauge theory with N f = 2 fundamental Dirac fermions. The theory features an extended SU(4) flavour symmetry that spontaneously breaks to Sp (4). It is used to build a PNGB Composite Higgs models in [13] and was recently reviewed in [14]. In this model the physical Higgs boson is a mixture of PNGBs and of the scalar state of the strong sector. The phenomenology of the model has been shown to be viable in view of the LHC data in [15], and the mixing between the scalar resonance and the Higgs can relax the bounds on the model [9]. The phenomenology of vector resonances for theories sharing the same chiral symmetry breaking pattern is investigated in detail as a function vector meson coupling constant in [16]. The authors derived the bound from di-lepton and di-boson searches at the LHC, as well as the dependence of the electroweak precision parameters as a function of the coupling constant. Our work can therefore be used to constrain a minimal model more efficiently.
We determine the V → PP coupling by extracting the scattering amplitude for PP → V → PP scattering in the vector channel via the rigorous Lüscher formalism [17,18]. We apply techniques that have been widely used for ππ → ρ → ππ scattering in QCD, for example [19][20][21][22][23][24][25]. The coupling g VPP is inferred from a Breit Wigner parametrization of the scattering amplitude. This work represents the first fully-fledged scattering calculation of the coupling g VPP in a 4D gauge theory with fermions beyond QCD. It complements previous studies addressing the phenomenology of models based on the same underlying gauge theory [26][27][28][29][30][31].
The coupling g VPP was previously estimated via the phenomenological relation g KSRF VPP = m V /F PS by Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin (KSRF) [32,33] that assumes the vector-meson dominance. These calculations were based on simulations where the vector state was stable. It was employed for QCD-like theories based on SU(3) [34,35] as well as various groups SU and Sp [35][36][37][38][39]. We refer to [34] for a discussion of the estimation of g KSRF VPP for a number of gauge theories. This relation turns out to be satisfied in QCD but has however never been rigorously tested beyond before this work. It is therefore crucial to guide model builders and searches for Beyond the Standard Model Physics.

Lattice setup
In the numerical simulations of SU (2) gauge theory with N f = 2, we choose a cloverimproved Wilson action for the two flavours of Dirac fermions [40,41] and the tree-level Symanzik improved action for the gauge sector [42]: Here U µ is the gauge field, ψ is the doublet of u and d fermions, P µν is the plaquette, R µν is a 2×1 rectangular loop,F µν is the Wilson clover term and β is the inverse lattice gauge coupling. The coefficient of the gauge action are set to c 0 = 5/3 and c 0 + 8c 1 = 1 [42]. The improvement coefficient c SW = 1 is set to its tree-level value in our simulations. The presence of bare mass term am 0 and the Wilson term explicitly breaks the SU(4) flavour symmetry to a Sp(4) subgroup. We use periodic boundary conditions in all four directions for the fermions.
The results presented in this work are obtained at β = 1.45. For each ensemble, we compute the pseudoscalar meson mass m PS , and vector meson mass m V by fitting at large euclidean time the effective mass of appropriate two-points correlation function. We also determine the bare quark mass m PCAC defined through the Partially Conserved Axial Current relation and the bare pseudoscalar decay constant F bare PS , which renormalizes multiplicatively with the renormalization factor Z A . The normalization of F PS corresponds to the convention where the pion decay constant in QCD is 130 MeV. More details concerning our calculation of these quantities can be found in [26] where we use the same strategy and normalization.
To compare our results with our previous results we convert the lattice quantities in physical units by setting the scale using the Wilson Flow observable w 0 [43]. The scale w 0 is defined by W( 1 is the flowed action density at flow time t. The reference value W ref is dimensionless and set to 1.0, consistently with our previous definition. The value of w 0 /a is determined for a range of fermion masses and chirally extrapolated using NNLO expansion in terms of m 2 PS [44]. We employ ensembles with m PS L > 4, where the finite volume effects are small. The data and are shown in Fig. 1, together with the best fitting curve, and the 1σ error band in gray. The final results reads w χ 0 /a (β = 1.45) = 3.08(2) using the NNLO fit which has a χ 2 /ndo f = 6.5/4. In order to compare our result with the KSRF relation g VPP = m V /F PS , it is necessary to renormalize the pseudoscalar decay constant with Z A . We performed the non perturbative determination of Z A in the RI'-MOM scheme [45], using the same strategy as in our previous setup [26]. The correlators are estimated using momentum sources introduced in [46]. To interpolate between the lattice momenta we furthermore use twisted boundary conditions [47,48]. The amputated vertex function is then defined as follows: where S −1 (p) is the inverse propagator in spin and color space. Defining, it can be shown that in the chiral limit Λ A (p 2 = µ 2 ) −→ Z A (β, µ 2 ). The chiral extrapolation at fixed p 2 is performed using linear extrapolation in m PCAC and our estimate of Λ A (p 2 ) in the chiral limit is shown in Fig. 2. We used (w χ 0 p) 2 = 7 > (w χ 0 m V ) 2 ∼ 1 as a reference scale and our final estimate reads Z A (β = 1.45, (w χ 0 p) 2 = 7) = 0.8022 (3). The statistical errors are estimated using bootstrap resampling.
A range of simulations performed at β = 1.45 allowed us to determine simulation parameters so that m V > 2m PS , therefore allowing the decay of the vector meson into two Goldstone bosons. We used a single bare mass m 0 = −0.6050 to perform the calculation of the phase-shift of the two Goldstone boson scattering process, and two lattice volumes of 16 3 × 32 and 24 3 × 48. The value of the m PS , m PCAC and m naive V are summarized in Table 1. The topological charge was monitored during the runs, our simulations explore a range of topological sectors. In average the topological charge is compatible with zero. The scattering study employs the quark propagators that are combined from the propagators with periodic and anti-periodic boundary conditions in time in order to effectively double the time extent of the lattice, as detailed for example in [22].

Vector resonance from scattering of two pseudoscalars
The vector resonance V can decay into two pseudoscalar mesons P as shown based on the symmetries in the next subsection. The vector meson mass extracted from the two-point correlator on the chosen gauge ensemble is above the decay threshold 2m PS and the decay is therefore kinematically allowed. A resonance is not an eigenstate of the Hamiltonian and its properties have to be inferred from the study of the scattering process PP → V → PP. We will show that the investigation of this resonance has similarities with the study of the ρ-vector resonance in QCD which has been studied through ππ → ρ → ππ in numerous lattice QCD investigations, for example [19][20][21][22][23][24][25]. The main strategy is briefly reviewed here. The discrete eigen-energies E n of two pseudoscalars in the finite box of size L and with periodic boundary conditions in space are determined. These are different from the noninteracting energies E n.i. = (m 2 PS + p 2 1 ) 1/2 + (m 2 PS + p 2 2 ) 1/2 of two pseudoscalars with momenta p 1,2 = 2π L n 1,2 due to mutual interactions of pseudoscalars. The energy-shifts ∆E = E n − E n.i give information on their interactions and therefore phase-shift in partial wave l. Lüscher's formalism provides a rigorous relation between the eigen-energy E CM of two-particles in a finite box and their infinite-volume scattering amplitude S l (E CM ) = e 2iδ l (E CM ) at this energy E CM in center-of-momentum frame [17,50]. We will extract the phase shift for seven values of energy by considering the lowest two eigenstates E n=1,2 in two different inertial frames and two different volumes 2 . The energy-dependence of the phase shift will have a resonance shape. The resonance mass, width and the V → PP coupling g VPP will be extracted from this dependence via the Breit-Wigner type fit.

Goldstone Bosons scattering: flavour structure
We consider the scattering channels of the two-Goldstone boson scattering in SU (2) gauge theory with N f = 2 fundamental fermions. The goal of this section is to show that one can employ the interpolators with the flavor structure analogous to QCD. For this purpose we demonstrate that the operators π − π + − π + π − and ρ µ belong to the ten-dimensional irreducible representation of Sp (4), where π and ρ are bilinears We follow the convention used for instance in [51] and summarized in appendix A, where the Pauli matrix acts in colour space and C = iγ 0 γ 2 is the conjugation charge matrix. It can be shown that the massless Lagrangian is symmetric under SU(4) transformation, while the mass term can be shown to be Sp(4) invariant.
Let's focus on the multiplet Π i=1,...,5 and the decoupled V a=1,..,10 µ defined as √ 2α a T a is a skew-symmetric matrix which can be written as a linear combination of the generators T a=1,...,10 of SO(5) defined in the appendix. The multiplet Π therefore transforms as a 5−dimensional irreducible representation of Sp (4).
The transformation of the tensor product of two Goldstone Bosons operators Π ⊗ Π can then be worked out. In this work, we focus on the antisymmetric part of the tensor: 2), using properties of the conjugation charge matrix and of the skew-symmetric matrix (−iσ 2 ) acting in color space, we find that F 3 = 1 4 (π − π + − π + π − ). In a similar manner the bilinears V a µ transform as V a −→ V a + i f abc α b S c and V 3 µ = ρ µ . In other words, we have shown that the usual 3-dimensional irrep (I = 1) of the isospin group is contained in the 10-dimensional representation of Sp (4). Regarding discrete symmetries, it straightforward to check that F 3 and ρ µ have negative parity and negative charge conjugation.

Operators and correlators
The aim is to extract eigen-energies of the system with the quantum numbers of the vector resonance, namely J P = 1 − . Given the flavour structure discussed in the previous section, we employ the two following operators: To compute the phase shift for different values of E CM we consider two inertial frames with total momenta P tot = p = (0, 0, 1) and (1, 1, 0) in units of 2π/L. The operators with momenta (0, 0, 1) transform according to the irreducible representation A − 2 of the group D 4h within the notation of [18]. The operators with momenta (1, 1, 0) transform according to B − 1 of the group D 2h in the notation of [21]. The eigen-energies are obtained by computing in each frame the 2 × 2 correlation matrix which renders the Wick contractions analogous to those in QCD. We evaluate those with the stochastic U(1) noise technique used in simulations [19,20] of ρ-meson decay in QCD.

Eigen-energies
The time-dependence of the correlation matrix C ij (t) = 0|O i (t)O † j (0)|0 = n 0|O i |n e −E n t n|O † j |0 contains information on the eigen-energies E n . We extract those using the widely used GEVP variational method [52][53][54] with reference time t 0 = 4. The effective energies E e f f n (t) = log[λ n (t)/λ n (t + 1)] for seven eigenstates are given in Fig. 3 and they are related to E n in the plateau region. They correspond to lowest two eigenstates for two inertial frames and two volumes; the first excited state for N L = 16 and P tot = (1, 1, 0) has a large statistical uncertainty and will not be used in analysis. We do not find an indication for non-exponential time-dependence at large t before the signal is lost in the noise since the size of the time-direction has been effectively doubled by combining periodic and antiperiodic propagators in time. The eigen-energies E n are extracted from the correlated one-exponential fits (3.7) of eigenvalues λ n (t) in the plateau region, that are indicated in the Fig. 3 and tabulated in 2.  Table 2. Seven eigen-states of PP system and the resulting information related to the scattering phase shift δ l for partial-wave l = 1.

Scattering amplitude and phase shift
The eigen-energies E n in Fig. 3 (solid lines) are shifted with respect to non-interacting energy E n.i. = (m 2 PS + p 2 ) 1/2 + m PS of two-pseudoscalar system P(p)P(0) (dot-dashed lines). These non-zero shifts are essential for extracting the scattering information.
The relation between eigen-energy E CM of two particles and infinite-volume scattering phase shift δ(E CM ) at that energy was derived by Lüscher [17] for P tot = 0. This relation was generalized in [18,21] for inertial frames with P tot = (0, 0, 1) and (1, 1, 0) that are employed here. In this case, the eigenstate with energy E n corresponds to the energy E CM = (E 2 n − P 2 tot ) 1/2 in the center-of-momentum frame, where pseudoscalars P(p * )P(−p * ) have back-to-back momenta p * = (E 2 CM /4 − m 2 PS ) 1/2 and dimension-less momenta q = p * L/(2π). The relation between eigen-energy (or corresponding q) and the scattering phase-shift at that energy (or q) is given by for P tot = (0, 0, 1) (3.8) for P tot = (1, 1, 0) , for the relevant irreducible representations of the symmetry groups listed after Eq. (3.5).
Every eigen-energy E n renders certain E CM , q and phase shift, which are tabulated for seven eigenstates in Table 2. The dependence of the phase-shift on E CM is plotted in Fig.  4a. It increases from the small values, past the resonance value 90 • to the large values.

Mass m R , width Γ and the coupling g VPP of the vector resonance
The energy dependence of the phase shift for PP scattering in Fig. 4a shows a behavior expected for a narrow resonance that lies slightly above threshold: it rises from small values, through the resonance value 90 • at about m R a = E CM a ≃ 0.45, to the large values close to 180 • . It is expected to have a Breit-Wigner type resonance form in the vicinity of a narrow vector resonance The resonance width Γ(E CM ) is parametrized in terms of the V → PP coupling g VPP and the phase space for partial wave l = 1, where p * is the momentum of the pseudoscalar in CM frame. The resonance phase shift (3.9) implies that p 3 This quantity is shown in Fig. 4b for PP scattering from our lattice simulation. It is falling with energy, it crosses through the resonance value cot δ 1 = 0 at E CM = m R and roughly supports the linear dependence on E 2 CM . Finally, we proceed to determine the vector resonance parameters from the phase shift that shows a resonant shape. We aim to determine the mass m R and the coupling g VPP that parametrizes the width rather than the width itself. The width is strongly dependent on m PS through the phase space, while the dependence of g VPP on m PS is expected to be much milder (the dependence of g ρππ on m π in QCD is very mild as evidenced in the review of various lattice results [24,25]). The m R and g VPP can be read-off from zero and the slope for the quantity (3.10) in Fig. 4b. In order to take the correlations properly into account, we extract m R and g VPP by minimizing the correlated χ 2 ( 3.11) with N k = 7 for seven eigenstates. Here E k=1,..,N k are energies of seven eigenstates determined on the lattice. Cov(k, k ′ ) is their 7 × 7 covariance matrix, where Cov(k, k ′ ) = 0 if eigenstates k and k ′ correspond to ensembles with different N L . The E par k (m R , g VPP ) is the value of E k determined analytically for given m R and g VPP based on the parametrization (3.10) and δ 1 from Lüscher-type relation (3.8).
The final parameters of the vector resonance based on minimization of χ 2 (3.11) are m R a = 0.445 ± 0.008 ± 0.002 g VPP = 7.8 ± 0.5 ± 0.1 cor(m R , g VPP ) = 0.35 . (3.12) The first error is statistical and arises from the errors on the energies encoded in the covariance matrix, while the second error accounts for the uncertainty in the pseudoscalar mass.
The correlation between resonance parameters is provided by cor = The phase-shift dependence based on these resonance parameters (3.12) is plotted by red solid lines in Fig. 4. The results (3.12) are based on seven eigenstates from both volumes N L = 24, 16. Restricting the fit to four eigenstates on the larger volume N L = 24, one gets compatible results m R a = 0.442 ± 0.009 ± 0.002, g VPP = 7.7 ± 0.5 ± 0.1 and cor = 0.37. Note that the calculation is performed at the finite pseudoscalar meson mass and at finite lattice spacing. However it should be noted that lattice results on g ρππ in QCD do not show large discretization errors and that its dependence on m π is very mild: the compilation [24,25] of lattice simulations by several groups shows that the coupling is g ρππ ≃ 6 for a wide range of m π and lattice spacings. We therefore believe our calculation provides a first estimate of the coupling g VPP in the chiral limit of SU(2) gauge theory with two fundamental flavors of Dirac fermions.

Comparison of the coupling g VPP with the KSRF relation
The V → PP coupling is often determined phenomenologically via the KSRF relation g KSRF VPP = m V /F PS based on the vector-meson dominance [32,33]. This relation gives us g KSRF VPP = m naive V /(Z A F bare PS ) = 9.4(6) at finite m PS , V = 24 3 × 48 and Z A provided in Section 2. The central value g KSRF VPP is 20% larger than the value g VPP (3.12) rigorously extracted from the scattering. Note furthermore that using our previous calculation with an unimproved action [26], and using the value of m V and F PS extrapolated to the chiral and continuum limit, we found g KSRF VPP = 13.1(2.2). Our calculation therefore suggests that the KSRF relation overestimates the value of the coupling g VPP for SU(2) gauge theory with two fundamental flavors of Dirac fermions.

Conclusions
We determined the coupling g VPP between a vector resonance and a pair of Goldstone bosons in a minimal realization of a composite Higgs model. This is the first lattice result for the coupling of a decaying resonance in a such 4-dimensional gauge theories with fermions. The calculation is performed by considering the scattering of two Goldstone bosons in the channel of the vector resonance. Assuming that the chiral dependence of g VPP is mild (like in QCD) and that the discretization error are negligible, we obtain a rigorous estimate of the coupling in the continuum and in the chiral limit based on the non-perturbative calculation. The result suggests a rather large value of the coupling but smaller than the one obtained using the KSRF relation. Additional numerical simulations would be required to obtain a more definite answer. The coupling is relevant to constrain the phenomenology of a vector resonance in composite Higgs model at the LHC, and bring additional constraints on Vector Boson Scattering and on electroweak precision tests.

A Conventions
We follow the convention used in [51], where The generators are normalized so that : tr S a S b = 1 2 δ ab , tr X i X j = 1 2 δ ij , tr S a X i = 0 .