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 [1]’s M-theory uplift of [2]’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(GE)-ρ/π-meson interaction, linear in the exotic scalar glueball and up to quartic order in the π mesons. In the Lagrangian the coupling constants are determined as (radial integrals of) [1]’s M-theory uplift’s metric components and six radial functions appearing in the M-theory metric perturbations. Assuming MG> 2Mρ, we then compute ρ → 2π, GE → 2π, 2ρ, ρ + 2π decay widths as well as the direct and indirect (mediated via ρ mesons) GE → 4π decays. For numerics, we choose f 0[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 ρ and π meson radial profile functions — a flexibility that our calculations permits.


Introduction
The non-abelian nature of QCD makes it possible to form color-neutral bound states of gauge bosons known as glueballs (gg, ggg, etc). In pure Yang-Mills theory these are the only possible particle states. Glueballs are represented by quantum numbers J P C , where J denotes total angular momentum, P denotes parity, and C denotes charge conjugation. Their spectrum has been studied in detail in lattice gauge theory. Despite the theoretical proof of existence of glueballs their experimental identification remains difficult. This difficulty in the identification arises mainly becasue of lack of information about coupling of glueballs with quark-antiquark states in strongly coupled QCD. Lattice simulation of QCD provides a reliable means of studying the glueballs, but lattice simulation of QCD with dynamical quarks are notoriously difficult. Lattice QCD predicts the mass of the lightest scalar glueball to be around 1600-1800 MeV. appendices to supplement the main text: appendix A gives the metric components of the M-theory uplift of [1] near θ 1 ∼ N − 1 5 , θ 2 ∼ N − 3 10 and appendix B is the potential appearing when the ρ-meson's radial profile function's equation of motion is rewritten as a Schrödinger-like equation in the MQGP limit.
2 Background: large-N thermal QCD at finite gauge coupling from M-theory In this section, we will provide a lightning review of the type IIB background of [2] -a UV complete holographic dual of large-N thermal QCD -discuss the 'MQGP' limit of [1] along with the motivation for considering this limit, issues as discussed in [1] pertaining to construction of delocalized S(trominger) Y(au) Z(aslow) mirror and approximate supersymmetry along with (an appendix-supplemented) discussion on the SYZ mirror in fact being independent of angular delocalization, construction explicit SU(3) and G 2 structures respectively of type IIB/IIA and M-theory uplift as constructed for the first time in [12,13]. Let us start with the UV-complete holographic dual of large-N thermal QCD as constructed in Dasgupta-Mia et al. [2]. To include fundamental quarks at non-zero temperature in the context of type IIB string theory, the authors of [2] considered N D3-branes placed at the tip of six-dimensional conifold, M D5-branes wrapping the vanishing S 2 and M D5-branes distributed along the resolved S 2 placed at anti-podal points relative to the M D5-branes. Let us denote the average D5/D5 separation by R D5/D5 . On the gravity side, the domain of the radial coordinate, in [2], is divided into the IR, the IR-UV interpolating region and the UV with the D5-branes placed at the outer boundary of the IR-UV interpolating region/inner boundary of the UV region. Roughly, r > R D5/D5 , would be the UV. The N f D7-branes are holomorphically embedded via Ouyang embedding in the resolved conifold geometry in the brane construction. They are present in the UV, the IR-UV interpolating region and they dip into the (confining) IR (but do not touch the D3-branes with the shortest D3 − D7 string corresponding to the lightest quark). In addtion, N f D7-branes are present in the UV and the UV-IR interpolating region. This brane construct ensures UV conformality and chiral symmetry breaking in the IR. Let us understand this in some more detail. In the UV, one has SU(N + M ) × SU(N + M ) color gauge group and SU(N f ) × SU(N f ) flavor gauge group. There occurs a partial Higgsing of SU(N + M ) × SU(N + M ) to SU(N + M ) × SU(N ) as one goes from r > R D5/D5 to r < R D5/D5 [14]. The reason is that in the IR, the D5-branes are integrated out resulting in the reduction of the rank of one of the product gauge groups (which is SU(N + number of D5−branes)×SU(N +number of D5−branes); the number of D5-branes drops off in the IR to zero). By the same token, the D5-branes are integrated in the UV resulting in the conformal Klebanov-Witten-like SU(M +N )×SU(M +N ) color gauge group [5]. The two gauge couplings, g SU(N +M ) and g SU(N ) , were shown in [4] to flow logarithmically and oppositely via: 4π 2 1 g 2 SU(N +M ) 2πα ′ S 2 B 2 . One thus sees that S 2 B 2 , in the UV, is the obstruction to obtaining conformality which is why M D5-branes were included in [2] to cancel the net D5-brane charge in the UV.

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Further, the N f flavor D7-branes which appear in the dilaton profile, enter the RG flow of the gauge couplings. This therefore needs to be annulled by N f D7-branes which is the reason for their inclusion in the UV in [2]. The RG flow equations for the gauge coupling g SU(N +M ) -corresponding to the gauge group of a relatively higher rank -can be used to show that the same flows towards strong coupling, and the SU(N ) gauge coupling flows towards weak coupling. One can show that the strongly coupled SU(N +M ) is Seiberg dual to weakly coupled SU(N − (M − N f )); the addition of the flavor branes hence decelerates the reduction in the rank of the gauge group under Seiberg duality. One then performs a Seiberg duality cascade such that N decreases to 0 but there is a finite M left at the end. One will thus be left with an SU(M ) gauge theory with N f flavors which confines in the IR. It was the finite temperature version of this SU(M ) gauge theory that was looked at by the authors of [2]. So, at the end of the duality cascade in the IR, number of colors N c is identified with M , which in the 'MQGP limit' can be tuned to equal 3. The number of colors N c = N eff (r) + M eff (r), where N eff (r) = Base of Resolved Warped Deformed Conifold F 5 and M eff = S 3F3 (the S 3 being dual to e ψ ∧ (sin θ 1 dθ 1 ∧ dφ 1 − B 1 sin θ 2 ∧ dφ 2 ), wherein B 1 is an asymmetry factor defined in [2], and e ψ ≡ dψ + cos θ 1 dφ 1 + cos θ 2 dφ 2 ) wherẽ ) , α ≫ 1 [15]. Further, the flavor group SU(N f ) × SU(N f ), is broken in the IR to SU(N f ) because the IR has only N f D7-branes. The gravity dual is given by a resolved warped deformed conifold wherein the D3-branes and the D5-branes are replaced by fluxes in the IR, and the back-reactions are included in the warp factor and fluxes.
It was argued in [12] that the length scale on the gravity side in the IR will be given by: which implies that in the IR, relative to KS, there is a color-flavor enhancement of the length scale. Hence, in the IR, even for N IR c = M = 3 and N f = 2 (light flavors) upon inclusion of higher order terms in M and N f , L ≫ L KS (∼ L Planck ) in the MQGP limit involving g s ∼ < 1, implying that the stringy corrections are suppressed and one can trust supergravity calculations.
Hence, the type IIB model of [2] make it an ideal holographic dual of thermal QCD because: (i) it is UV conformal (Landau poles are absent), (ii) it is IR confining with required chiral symmetry breaking in the IR, (iii) the quarks transform in the fundamental representation of flavor and color groups, and (iv) it is defined for the full range of temperature -both low and high.
In [1], the authors considered the following limit:

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The motivation for considering the MQGP limit which was discussed in detail in [12], is summarized now. The usual AdS/CFT limit involves g YM → 0, N → ∞ such that the 't Hooft coupling g 2 YM N is very large. However, for strongly coupled thermal systems like sQGP, this limit is not relevant as it is expected that g YM is finite, and N c = 3 [32]. From the discussion in the paragraph preceding (2.1), one recollects that at the end of the Seiberg duality cascade in the IR, N c = M . Note that in the MQGP limit (2.2), M can be set to equal 3. Further, in the MQGP limit, g s < ∼ 1. The finiteness of g s requires one to construct the M theory uplift of [2]. These were precisely the reasons for coining 'MQGP limit' in [1]. In fact this was the reason why the type IIA mirror was first constructed in [1] a la delocalized Strominger-Yau-Zaslow mirror symmetry, and then its M-theory uplift obtained in the same paper.
In order to be able to implement quantum mirror symmetry a la SYZ [16], one needs a special Lagrangian (sLag) T 3 fibered over a large base. Defining delocalized T-duality/local T 3 (x, y, z) coordinates [1]: it was shown in [13,18] that the aforementioned T 3 is the T 2 -invariant sLag of [17] for a deformed/resolved conifold. Hence, the local T 3 of (2.3) is the sLag needed to effect the construction of the SYZ mirror. In the 'delocalized limit' [19] ψ = ψ , under the transformation: and an appropriate shift in ψ, it was shown in [1] that one introduces a local isometry along ψ in the resolved warped deformed conifold in the gravity dual in [2]; of course this is not true globally. Now, to be able to construct the SYZ mirror, one also needs to ensure a large base of the T 3 (x, y, z) fibration. This is effected via: [20]: for appropriately chosen large values of f 1,2 (θ 1,2 ). The guiding priciple behind choosing such large values of f 1,2 (θ 1,2 ), as given in [1], is that one requires the metric obtained after SYZ-mirror transformation applied to the non-Kähler resolved warped deformed conifold to be like a non-Kähler warped resolved conifold at least locally. This was explicitly demonstrated in [12] and appropriate values of f 1,2 (θ 1,2 ) obtained therein. The aforementioned delocalization procedure used to construct the type IIA mirror of the UV-complete [2]'s type IIB holographic dual of large-N thermal QCD a la SYZ triple-T-duality prescription and its M-theory uplift as worked out in [1], is in fact, not restricted to fixed-ψ mirrors. To understand this, let us look at the example of the mirror of a D5brane wrapping the resolved S 2 with fluxes as studied in [21] -in paricular sections 5 JHEP09(2018)133 and 6 therein. In the large-complex structure limit and after a fixed-ψ coordinate rotation, the SYZ mirror was found in [21] to be D6-brane wrapping a non-Kähler deformed conifold. As shown in (section 6 of) [21], an explicit G 2 structure can be constructed in terms of which the M-theory uplift of the type IIA mirror could be rewritten, which is valid ∀ψ. Hence, the type IIA mirror in section 6 of [21] obtained from arbitrary-ψ M theory metric, will be the same as the fixed-ψ type IIA mirror of section 5 of [21] that was obtained using delocalization. Thus, the fixed ψ value chosen to effect the abovementioned delocalized SYZ mirror, could simply be replaced by an arbitrary ψ, implying the type IIA mirror is effectively free of delocalization.
Let us understand what SYZ mirror transformation via triple T-duality does to the brane construct. A single T-duality along a direction orthogonal to the D3-branes world volume, e.g., z of T 3 (x, y, z), yields D4 branes that are straddling a pair of N S5-branes with world-volume coordinates, let us say, denoted by (θ 1 , x) and (θ 2 , y). A second T-duality along x and a third T-duality along y would yield a Taub-NUT space from each of the two N S5-branes [22]. The D7-branes yield D6-branes which get uplifted to Kaluza-Klein monopoles in M-theory [23] which also involve Taub-NUT spaces. Globally, one expects the eleven-dimensional uplift would involve a seven-fold of G 2 -structure, analogous to the uplift of D5-branes wrapping a two-cycle in a resolved warped conifold [20].
We will now briefly review G = SU(3), G 2 -structures of the holographic type IIB dual of [2], its delocalized type IIA SYZ mirror and its M-theory uplift constructed in [1]. In [18], it was shown that the five SU(3) structure torsion classes, in the MQGP limit, were given by (schematically): 3 ), such that: in the UV-IR interpolating region/UV, implying a Klebanov-Strassler-like supersymmetry [24]. Locally, around θ 1 ∼ 1 , the type IIA torsion classes of the delocalized SYZ type IIA mirror metric, were worked out in [12] to be: indicative of supersymmetry after constructing the delocalized SYZ mirror. Apart from quantifying the departure from SU(3) holonomy due to intrinsic contorsion arising from the NS-NS three-form H, via the evaluation of the SU(3) structure torsion classes, to our knowledge for the first time in the context of holographic thermal QCD at finite gauge coupling in [12]:

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(i) the existence of approximate supersymmetry of the type IIB holographic dual of [2] in the MQGP limit near the coordinate branch θ 1 = θ 2 = 0 was explicitly shown, which apart from the existence of a special Lagrangian three-cycle, is essential for construction of the local SYZ type IIA mirror; (ii) it was shown that the large-N suppression of the deviation of the type IIB resolved warped deformed conifold from being a complex manifold, is lost on being dualitychased to type IIA, and that a fine tuning in W IIA 2 can ensure that the local type IIA mirror is complex; (iii) for the local type IIA SU(3) mirror, the possibility of surviving approximate supersymmetry was explicitly shown which is essential as SYZ mirror is supersymmetric.
We can get a one-form type IIA potential from the triple T-dual (along x, y, z) of the type IIB F 1,3,5 in [1] and using which the following D = 11 metric was obtained in [1] (u ≡ r h r ): (2.10) The torsion tensor associated with the G 2 structure of a seven fold, possesses 49 components and can be split into torsion components as: where T 1 is a function and gives the 1 component of T . We also have T 7 , which is a 1-form and hence gives the 7 component, and, T 14 ∈ Λ 2 14 gives the 14 component. Further, T 27 is traceless symmetric, and gives the 27 component. Writing T i as W i , we can split W as (2.12) From [25], we see that a G 2 structure can be defined as: where A, B, C = 1, . . . , 6, 10; a, b, c, = 1, . . . , 6, and f ABC are the structure constants of the imaginary octonions. Using the same, the G 2 -structure torsion classes were worked out in [12] around θ 1 ∼ 1 (schematically): (2.14) Hence, the approach of the seven-fold, locally, to having a G 2 holonomy (W G 2 is accelerated in the MQGP limit. As stated earlier, the global uplift to M-theory of the type IIB background of [2] is expected to involve a seven-fold of G 2 structure (not G 2 -holonomy due to non-zero M theory four-form fluxes). It is therefore extremely important to be able to see this, at least locally. It is in this sense that the results of [1] are of great significance as one explicitly sees in the context of holographic thermal QCD at finite gauge coupling, though locally, the aforementioned G 2 structure having worked out the non-trivial G 2 -structure torsion classes.

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Let us now argue that in the MQGP limit, apart from the gluon-bound states, i.e. glueballs, and the light (ρ/π) mesons, all other scalar mesons are integrated out. As per [31], supersymmetry can be broken by imposing anti-periodic boundary conditions for fermions along the x 0 -circle (which at finite temperature has periodicity given by the reciprocal of the Hawking temperature). This is expected to generate fermionic masses of the order of the reciprocal of the S 1 t radius R r h and scalar masses of the order of g 2 s N Rr h . We will now argue that R r h is very small implying scalar mesons (apart from the lightest ρ-vector and pionic pseudo-scalar mesons) are very heavy and are hence integrated out, and effectively the 3+1-dimensional QCD-like theory thus reduces to 2+1 dimensions. From (A.3), one sees that working with a near-horizon coordinate χ : We therefore read off the radius of the temporal direction: One hence sees that R r h is very small and hence the assertion.

Glueballs from M-theory metric perturbations
To start off our study of glueball decays into meson, we first need to understand how glueballs are obtained in the M-theory background. Glueballs are gauge invariant composite states in the Yang-Mills theory and their duals corresponds to the supergravity fluctuations in the near horizon geometry of brane solutions. The M-theory metric for D=11 was written out in (2.10). Here g IIA M N and φ IIA corresponds to the metric components and dilaton in type IIA string background respectively; A's are the one form potential in type IIA background. The M-theory metric components up to NLO in N near 10 , φ 1,2 = 0/2π, whereat an explicit G 2 structure was worked out in [12], are given in (A.3). The general M-theory metric fluctuations corresponding to 'exotic' scalar glueball with J P C = 0 ++ in terms of the three dimensional spacetime x 1 , x 2 , x 3 can be written following [30,33] as:

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Here G E (x 1 , x 2 , x 3 ) is the glueball field in the 2+1 dimensional spacetime and, M g is the mass of the glueball. The explicit expression for functions q i=1,2,...,6 can be obtained by solving their EOM's obtained from 11-D action. The 11-d action, using C 3 ∧ G 4 ∧ G 4 = 0 [1], is given as: the first order perturbation of whose EOM yields: (3.2) Here, hatted letters like M, N etc go from 0 to 10 while, R (1) M N is perturbed part of the Ricci tensor. Putting in the expressions for each of the components following coupled eom's were obtained, 1 Defining: the q 1 (r) EOM, near r = r h , can be written as:

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whose solution is given as: (3.14) We conclude that for the solution to vanish in the UV region one requires c 1 = 0, then the solution can be approximated as: (3.16) Defining:

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(3.16) near r = r h can be written as: whose solution is given by: So, to be able impose Neumann boundary condition q ′ 5 (r = r h ) = 0, one needs to set c 2 = 0 and c 1 q5 = N −α 5 , α 5 > 0, In the UV region(r > r h ), (3.16) can be approximated as: whose solution after a large r and large N expnasion can be written as: (3.20)

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Defining: (3.20) near r = r h can be written as: whose solution is given by: To be able to impose Neumann boundary condition at r = r h , one needs to set c q 3 = 0. For c 1 q 1 only α 4 , β 3 , and γ 3 gives a non-zero value, In the UV region (r > r h ), the (3.20) can be approximated as: whose solution after taking an expansion around large r and large n can be written as: This yields an EOM for q 4 (r) which is identical to that for q 1 (r) for both UV and IR region.
• δR[r, r] q 1 ′ (r) = 0. (3.27) This along with the δR[t, t] EOM implies that c 1 q 1 is vanishingly small. In section 6, we set it to zero while calculating decay widths associated with decays of the exotic scalar glueball.

Meson sector
To start off our study of glueball-meson interaction in the type IIA background we first have to understand how the mesons are obtained in the theory. The meson sector in the type IIA dual background of top-down holographic type IIB setup [1] is given by the flavor D6-branes action. We first need to understand how the D6 branes are embedded in the mirror(constructed in [1]) of the resolved warped deformed conifold of [2]. To obtain the pullback metric and the pullback NS-NS flux on the D6 branes, we choose the first branch of the Ouyang embedding where (θ 1 , x) = (0, 0) and we consider the 'z' coordinate as a function of r, i.e z(r) [9]. In [7] a diagonal metric {t, x 1 , x 2 , x 3 , r, θ 1 , θ 2 ,x,ỹ,z} was JHEP09(2018)133 used to obtain the mirror of the Ouyang embedding, but it turns out that the embedding conditions remains same even with the nondiagonal basis {t, x 1 , x 2 , x 3 , r, θ 1 , θ 2 , x, y, z}. For θ 1 = α θ 1 N − 1 5 and θ 2 = α θ 2 N − 3 10 one will assume that the embedding of the D6-brane will be given by ι : Σ 1,6 t, R 1,2 , r, θ 2 ∼ α θ 2 N 3 10 , y ֒→ M 1,9 , effected by: z = z(r). As obtained in [7] one sees that z=constant is still a solution and by choosing z = ±C π 2 , one can choose the D6/D6-branes to be at "antipodal" points along the z coordinate.
As done in [9] after redefining (r,z) in terms of new variables (Y,Z): the constant embedding of the D6(D6)-branes corresponds to z = π 2 for C = 1 for D6branes and z = − π 2 for C = −1 forD6-branes, both corresponding to Y = 0. Vector mesons are obtained by considering gauge fluctuations of a background gauge field along the world volume of the embedded flavor D6 branes. Turning on a gauge field fluctuation F σ 3 2 about a small background gauge field F 0 σ 3 2 and the backround i * (g +B). This implies: .

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Here ι * g and ι * B are the pulled back metric and NS-NS B on the D6-brane respectively. Writing the Klauza-Klein modes for the gauge fields in a 2+1 dimensional minkowski spacetime consisting of x 1,2,3 as, one obtains: The terms quadratic in ψ/ψ in (4.6) are given as: where: After integrating by parts once, and utilizing the EOM for ρ (n) µ , one writes: which yields the following equations of motion:

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The normalization condition of ψ n are given as Thus the action for vector meson part for all n ≥ 1can be wriiten as To normalize the kinetic term for π n , we impose the normalization condition for all n corresponding to π n which ranges from 0 to ∞ From (4.11), it is seen that we can choose φ n = m −1 nψ n for all n ≥ 1. For n = 0 corresponding to φ 0 we choose its form such as it is orthogonal toψ n for all n ≥ 1. By writing Thus the cross component in (4.6) vanishes for n = 0, and the remaining cross components can be absorbed in the ρ n µ by following a specific gauge transformation given as, Then the action becomes: (4.14)

Radial profile function ψ 1 (Z) for ρ-meson
Up to NLO in N :

Glueball-meson interaction lagrangian
The couplings appearing in the DBI action after ignoring the derivatives and possible indices can be written as: 1) The interaction terms written above are generic results for single glueball case. The flavor structure remains same for the case involving multi-glueball vertices. In subsequent sections we will be considering the n = 1, 0 modes respectively in the KK expansion of A µ , A Z . Substituting all the fluctuations for the metric in the D-6 brane action gives us the glueballmeson couplings. We only consider the interaction terms that are linear in glueball field G E , since we are interested in glueball decays.
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

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perturbed type-IIA field components and dilaton are given as: where a, b run from 1 to 3 corresponding to the spatial part of the metric. Substituting all the expressions for the type IIA metric components g IIA M N and the M-theory perturbations h M N into the D6-brane DBI action and, working 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 F g −1 F . Contributions to the interaction lagrangian from these three different terms were obtained as:

(5.4)
Putting everything together: implying: JHEP09(2018)133 yielding: Hence, one can write the following glueball-meson interaction Lagrangian up to quartic order in the meson fields: where: At quadratic order in field strength tensor these are the only interaction terms. Terms with higher order in ρ µ and π can be obtained in the same manner by keeping higher order terms of F in the DBI action. Assuming that in (5.10), 3b dZ, the coefficients c i s setting q 6 (r) = 0, are giver as under: ×(log(e Z r h )) 72a 2 r h e Z log(e Z r h )+3a 2 +2r h 2 e 2Z M 2 g (q5(Z)+(q1(Z)−q2(Z)−q4(Z)+2q6(Z))) , which for b ∼ 0.6 yields: which for b ∼ 0.6 yields: which for b ∼ 0.6 yields: × 3a 2 +72a 2 r h e Z (log(e Z r h ))+2r h 2 e 2Z log(N )

Decay widths
In this section, using standard techniques in scattering theory (specially in dealing with multi-particle phase-space integrals: see [28,29] 2 ), in the following sub-sections, we calculate decay widths for G E → 2π, G E → 2ρ, ρ → 2π, G E → 4π 0 , G E → ρ + 2π as well as indirect four-π decay with associated with G E → ρ+2π → 4π as well as G E → 2ρ → 4π assuming M G > 2M ρ for definiteness and specifically concentrating on the potential glueball candidate f 0[1710]. JHEP09(2018)133 The decay width for two body decay is given as, where M is the amplitude for the decay, and p is the final momentum of one of the identical particles in the decay product. The relevant coupling for the 2π decay in the rest frame of the glueball is given by following terms in the interaction lagrangian Considering a specific adjoint index for the pion π a (a=1,2,3). M for two pions π 1 and π 2 as final state particles in the rest frame of glueball is given as, where the factor of 2 is for the symmetry of exchanging the two final state particles. Pions are massless which gives k 0 = |k| = m/2 for both the particles, so we obtain The decay width summed over a = 1, 2, 3 is: (6.5) In our paper, we have assumed | log [26], the 2π-decay width per unit mass associated with f 0[1710] is ∼ 10 −2 . Therefore by a convenient choise of : Λ G E →2π ∼ 10 -implying a constraint on a linear combination of C 2 φ 0 c 1 q 4 and C UV ) -one obtains: Γ G E →2π m 0 = 10 −2 -clearly an exact match with the PDG-2018 results is also similarly possible.

ρ → 2π
The relevant interaction term in the action is given by: where:

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We will demand Γ ρ→2π = 149 MeV ( [26]); replacing MeV by r h π √ 4πgsN , this implies a constraint on C 2 φ 0 (c ψ 1 ) and C UV φ 0 2 c UV 2 ψ 1 : (6.14) 6.4 Direct glueball decay to 4π 0 s For coupling to four π 0 we need to expand the DBI action upto quartic order in F µν . The action restricted to quartic order, reads Inserting the metric fluctuations corresponding to the glueball and keeping the terms which are quartic in φ 0 (Z) gives the interaction term  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:  Writing: Assuming the experimental value for -not yet known in [26] -is 10 −5+required 2 , (6.27) for N = 10 2 , implies the following constaint:  5. The combination of constants of integration appearing in the solutions to the EOMS of φ 0 (Z), ψ 1 (Z) in the IR and UV, using (7.1): C 2 φ 0 c ψ 1 and C φ 0 UV (C φ 0 ) 2 c UV 2 ψ 1 (c ψ 1 ), can be adjusted to reproduce the PDG value of Γ ρ→2π exactly.

Acknowledgments
VY is supported by a Junior Research Fellowship (JRF) from the University Grants Commission, Govt. of India. AM was partly supported by IIT Roorkee. He would also like to thank the theory group at McGill University, and Keshav Dasgupta in particular, for the wonderful hospitality and support during the final stages of the work and for discussions. We would like to thank Karunava Sil for valuable participation in the earlier stages of this work, and Sharad Mishra in verifying some of the results in 6.1 as part of his Masters project. We would like to thank M.Dhuria for bringing [29] to our attention. AM would like to dedicate this paper to the memory of his Ph.D. advisor, the late, Professor D.S. Koltun, a nuclear theorist, who initiated him into Hadronic Physics via Chiral Perturbation Theory.

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.