M-Theory Exotic Scalar Glueball Decays to Mesons at Finite Coupling

Using the pull-back of the perturbed type IIA metric corresponding to the perturbation of arXiv:hep-th/1306.4339's M-theory uplift of arXiv:hep-th/0902.1540's UV-complete top-down type IIB holographic dual of large-$N$ thermal QCD, at finite coupling, we obtain the interaction Lagrangian corresponding to exotic scalar glueball($G_E$)-$\rho/\pi$-meson interaction, linear in the exotic scalar glueball and up to quartic order in the $\pi$ mesons. In the Lagrangian, the coupling constants are determined as (radial integrals of) arXiv:hep-th/1306.4339's M-theory uplift's metric components and six radial functions appearing in the M-theory metric perturbations. Assuming $M_G>2M_\rho$, we then compute $\rho\rightarrow2\pi, G_E\rightarrow2\pi, 2\rho, \rho+2\pi$ decay widths as well as the direct and indirect (mediated via $\rho$ mesons) $G_E\rightarrow4\pi$ decays. For numerics, we choose $f0[1710]$ and compare with previous calculations. We emphasize that our results can be made to match PDG data (and improvements thereof) exactly by appropriate tuning of some constants of integration appearing in the solution of the M-theory metric perturbations and the $\rho$ and $\pi$ meson radial profile functions - a flexibility that our calculations permits.


Gravity dual
K The type IIB supergravity solution involving resolved warped deformed conifold is given as: where the black hole functions g i in the above limit are of the form: K The warp factor that includes the back-reaction is given in IR as: logr 1 + 3gsN f 2π K Motivated however by, e.g., (a) asymptotically the type IIB background of K. Dasgupta et al [2009] and its delocalized type IIA mirror of M. Dhuria et al [2013] consist of AdS 5 and (b)Gauge theory operators corresponds to the solution to the Laplace equation on T 1,1 S. Gubser [1998] (the operator TrF 2 which shares the quantum numbers of the 0 ++ glueball couples to the dilaton and TrF 4 which also shares the quantum numbers of the 0 ++ glueball couples to trace of metric fluctuations and the four-form potential, both in the internal angular directions), • type IIB dilaton fluctuations, which we refer to as 0 ++ glueball • type IIB complexified two-form fluctuations that couple to d abc Tr (F a µρ F b ρ λ F c λ ν ), which we refer to as 0 −− glueball • type IIA one-form fluctuations that couple to Tr (F ∧ F ), which we refer to as 0 −+ glueball • M-theory metric's scalar fluctuations which we refer to as another (lighter) 0 ++ glueball • M-theory metric's vector fluctuations which we refer to as 1 ++ glueball, and • M-theory metric's tensor fluctuations which we refer to as 2 ++ glueball.
K The M-theory metric for D=11 K M-theory metric fluctuations corresponding to scalar glueball G E with J PC = 0 ++ ,Hashimoto et al [2009] : Solutions for the functions q i=1,2,...,6 VY et al [2018] K Solution for functions q i 's can be obtained by solving the EOM's obtained from the 11-D action which can be written using K EOM upto linear order in perturbation is given as: Here,M,N,P 2 ,P 3 takes value from 0 to 10 while, R (1) MN is perturbed part of the Ricci tensor and G 2 = GÂBĈD GÂBĈD where,Â,B,Ĉ,D takes value from 0 to 10.  in the (θ 2 , T 3 (x, y , z))-subspace near can be written as: where the T 3 is formed of x, y , z; θ 1 = K The Buscher triple-T-type-IIA dual of the NS-NS B: K Considering the branch of Ouyang embedding where (θ 1 , x) = (0, 0) and, taking z = z(r ) and defining Σ 0 (r ; gs, N f , N, M) and Σ 1 (r ; gs, N f , N, M) as the embedding functions, the pulled back metric + NS-NS B appearing the DBI action for N f D6(x 0,1,2,3 , r , θ 2 , y )-branes becomes: K Analogus to Dasgupta et al [2015], z =constant is a solution for EOM. Choosing z = ±C π 2 , one can choose the D6/D6 branes to be at 'antipodal points' (in the toroidal analog of the spherical ψ coordinate).
K Vector meson spectrum is obtained by considering gauge fluctuations of a background gauge field along the world volume of the embedded flavor D6-brane.
K Turning on a gauge field fluctuationF σ 3 2 about a small background gauge field F 0 σ 3 2 and the backround i * (g + B): . 14 / 31 K Keeping terms quadratic inF and using KK expansion for gauge fields . We obtain following EOMs: K while the form for the φn(Z ) can be obtained by imposing normalization condition. Which gives φn = m −1 nψn for all n ≥ 1 while, for n=0 we get Further for well-behaved ψ 1 (Z ) near Z = 0 one requires to set c 2 ψ 1 = 0. Therefore: and To satisfy Neumann boundary condition at Z = 0, one will hence set:

Glueball Mesons interaction lagrangian
K The DBI action for D6 branes is written in terms of the 10 dimensional type-IIA metric and dilaton field. The glueball modes and dilaton field for type-IIA background were obtained in terms of 11-D M theory metric perturbations using witten's relation. The perturbed type-IIA field components and dilaton are given as: where a, b run fro 1 to 3 corresponding tho the spatial part of the metric.
K Substituting all the expressions for the type IIA metric components g IIA MN and the M-theory perturbations h MN into the D6-brane DBI action and, keeping terms only upto linear order we get three different type of terms as: Here O(h 0 ) represent term wthout any perturbation while O(h) represents term with linear order in perturbation. In both the terms subscripts d,F,φ corresponds to part of the integrand of the DBI action from which they are obtained, O d corresponds to term obtained from −det(ι * (g + B)), O φ corresponds to the term e −φ and, O F corresponds to the term of type g −1 Fg −1 F . K Hencce, one can write the glueball-meson interaction Lagrangian up to quadratic order in magnetic fields: At quadratic order in field strength tensor these are the only interaction terms. Terms with higher order in ρµ and π can be obtained in the same manner by keeping higher order terms of F in the DBI action. Assuming that, dZ , the coefficients c i s setting q 2 q 6 (r ) = 0, are giver as under: g (q 5 (Z ) + (q 1 (Z ) − q 2 (Z ) − q 4 (Z ) + 2q 6 (Z ))) , which for b ∼ 0.6 yields :

Decay Widths
We calculated the decay widths for following processes assuming f0[1710] as glueball candidate with M G > 2M ρ .
K The decay width summed over a = 1, 2, 3 is: K In our paper, we have assumed | log r h | = fr h 3 log N, 0 < fr h < 1, or equivalently r h = N − fr h 3 . From PDG-2018, the 2π-decay width per unit mass associated with f 0[1710] is ∼ 10 −2 . Therefore by a convenient choice of K For onshell decay for G E → ρρ. The differential width is given by K The relevant interaction term in the action and decay width is given by: where:

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Direct glueball decay to 4π 0 sVY et al [2108] K For coupling to four π 0 we need to expand the DBI action upto quartic order in Fµν . The action restricted to quartic order, reads K Putting everything together and setting q 2 (Z ) = q 6 (Z ) = 0, one gets the following interaction Lagrangian corresponding to the direct G E → 4π decay: where: g UV s N 17/20 M UV 2 r h 8 α θ 1 α 2 K We note that the combination of constants of integration appearing in the solutions to the EOMs of φ 0 (Z), ψ 1 (Z) and q 1,2,3,4,5,6 (Z) in the IR and UV: -involving C 4 φ 0 c 1q 4 and(C φ 0 UV C φ 0 ) 4 (c 2q 1 UV − 3.015c 2q 4 UV ) appearing inn Γ G E →4π 0 -involving C 4 ψ 1 c 1q 4 and(c 2 ψ 1 UV c ψ 1 ) (c 2q 1 UV − 3.015c 2q 4 UV ) appearing in can be tuned and equality of these two combinations can be effected such that one can reproduce the PDG value of Thanks