Color dependence of tensor and scalar glueball masses in Yang-Mills theories
Ed Bennett, Jack Holligan, Deog Ki Hong, Jong-Wan Lee, C.-J. David Lin, Biagio Lucini, Maurizio Piai, Davide Vadacchino
PPNUTP-20/A02
On the colour dependence of tensor and scalar glueball masses in Yang-Mills theories
Ed Bennett, Jack Holligan,
2, 3
Deog Ki Hong, Jong-Wan Lee,
4, 5
C.-J. David Lin,
6, 7
Biagio Lucini,
8, 1
Maurizio Piai, and Davide Vadacchino Swansea Academy of Advanced Computing, Swansea University, Bay Campus, SA1 8EN, Swansea, Wales, UK Department of Physics, College of Science, Swansea University, Singleton Park, SA2 8PP, Swansea, Wales, UK The Institute for Computational Cosmology (ICC),Department of Physics, South Road, Durham, DH1 3LE, UK Department of Physics, Pusan National University, Busan 46241, Korea Extreme Physics Institute, Pusan National University, Busan 46241, Korea Institute of Physics, National Chiao-Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan Centre for High Energy Physics, Chung-Yuan Christian University, Chung-Li 32023, Taiwan Department of Mathematics, College of Science,Swansea University, Bay Campus, SA1 8EN, Swansea, Wales, UK INFN, Sezione di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy (Dated: April 24, 2020)We report the masses of the lightest spin-0 and spin-2 glueballs obtained in an extensive latticestudy of the continuum and infinite volume limits of Sp ( N c ) gauge theories for N c = 2 , , , N c limit. We compute the ratio ofscalar and tensor masses, and observe evidence that this ratio is independent of N c . Other latticestudies of Yang-Mills theories at the same space-time dimension provide a compatible ratio. Wefurther compare these results to various analytical ones and discuss them in view of symmetry-basedarguments related to the breaking of scale invariance in the underlying dynamics, showing that aconstant ratio might emerge in a scenario in which the 0 ++ glueball is interpreted as a dilaton state. I. INTRODUCTION In D = 3 + 1 space-time dimensions, Yang-Mills (YM)theories are classically scale-invariant. At high energiesthe theory is perturbative, and governed by a trivial fixedpoint—this is the essence of asymptotic freedom. Scalesymmetry is anomalous though, broken by quantum ef-fects that make the theory flow away from its trivial fixedpoint, and introduce an intrinsic scale Λ, via dimensionaltransmutation.At high energy, the massless gluons, carrying colourcharges, are the natural choice of degrees of freedomto describe small perturbations around the trivial fixedpoint. Yang-Mills theories are believed to confine at lowenergies O (Λ). Low-energy excitations are colour sin-glets, called glueballs, and their spectrum is gapped. Thephenomena associated with the transition to the confinedphase are intrinsically non-perturbative and difficult tostudy.In Ref. [1], some of us started an extensive study of Sp ( N c ) gauge theories, which includes calculating themasses of the glueballs in the YM theory. The spectrumof Sp (4) glueballs was one of the most robust results ofthat exploratory and agenda setting paper. We updatethe measurements for the Sp (4) group, by doubling thesize of the combined statistical ensemble, and then pro-ceed to the next step of this programme, by performingdetailed studies of the YM theory (with no matter con-tent) with gauge groups Sp (2), Sp (6), and Sp (8) (seealso preliminary results in Ref. [2]). We report here our results for the lightest scalar and tensor glueballs.Understanding the glueball spectrum is tantamount tosolving the YM theory, and uncovering the mechanism ofconfinement. Reference [3] suggested that the quantity R ≡ m ++ m ++ , (1)defined as the ratio of masses of the glueballs with quan-tum number J P C = 2 ++ and J P C = 0 ++ , captures someuniversal, intrinsic properties of YM theories, in the sensethat it depends only on the dimensionality of the space-time and of the operators of the field theory. We devotethis paper to these specific observables. A comprehen-sive report on the physics of Sp ( N c ) YM theories, whichdetails the results for excited states and for extended ob-jects, is in preparation [4]. II. GLUEBALL MASSES: NEW LATTICERESULTS
We report at the top of Table I our new lattice mea-surements of glueball masses in D = 3 + 1 dimensionsfor Sp ( N c ) YM theories. The algorithm employed in ourlattice calculations adopts the Wilson action, and the lo-cal updates are based upon a combination of Heat Bathand Over Relaxation, by supplementing the Cabibbo-Marinari update with a simple re-symplectisation pro-cedure, as described in Ref. [1]. a r X i v : . [ h e p - l a t ] A p r D Group Reference m √ σ = m A ++1 √ σ m E ++ √ σ m T ++1 √ σ m √ σ R Sp (2) [4] . ( ) . ( ) . ( ) . ( ) . ( )3 + 1 Sp (4) [1, 4] . ( ) . ( ) . ( ) . ( ) . ( )3 + 1 Sp (6) [4] . ( ) . ( ) . ( ) . ( ) . ( )3 + 1 Sp (8) [4] . ( ) . ( ) . ( ) . ( ) . ( )3 + 1 Sp ( ∞ ) [4] . ( ) . ( ) . ( ) . ( ) . ( )3 + 1 SU (2) Table 14 [6] 3 . . . ( )3 + 1 SU (3) Table 14 [6] 3 . . . ( )3 + 1 SU (4) Table 14 [6] 3 . . . ( )3 + 1 SU (6) Table 14 [6] 3 . . . ( )3 + 1 SU (8) Table 14 [6] 3 . . . ( )3 + 1 SU ( ∞ ) Table 14 [6] 3 . . . ( )2 + 1 SO (3) Table 28 [7] 3 . . . ( )2 + 1 SO (4) Table 28 [7] 3 . . . ( )2 + 1 SO (5) Table 28 [7] 3 . . . ( )2 + 1 SO (6) Table 28 [7] 3 . . . ( )2 + 1 SO (7) Table 29 [7] 3 . . . ( )2 + 1 SO (8) Table 29 [7] 3 . . . ( )2 + 1 SO (12) Table 29 [7] 3 . . . ( )2 + 1 SO (16) Table 29 [7] 3 . . . ( )2 + 1 SO ( ∞ ) Table 31 [7] 4 . . . ( )2 + 1 SO ( ∞ ) Table 31 [7] 4 . . . ( )2 + 1 SU (2) Table B3 [8] 4 . . . ( )2 + 1 SU (3) Table B4 [8] 4 . . . ( )2 + 1 SU (4) Table B5 [8] 4 . . . ( )2 + 1 SU (6) Table B6 [8] 4 . . . ( )2 + 1 SU (8) Table B7 [8] 4 . . . ( )2 + 1 SU (12) Table B8 [8] 4 . . . ( )2 + 1 SU (16) Table B9 [8] 4 . . . ( )2 + 1 SU ( ∞ ) Tables B10,B11 [8] 4 . . . ( )TABLE I: Lattice measurements of the masses of the glueballs, as described in the main text. In bold face are the calculationsperformed for this letter, while the other numerical values are lifted from the literature, as indicated. In the case of Sp (4), newmeasurements have been combined with those from Ref. [1], doubling the combined statistics. We restrict attention to the ratio m G / √ σ betweenglueball masses m G and the square root of the stringtension σ . The notation G = E ++ , A ++1 , T ++1 , refers ex-plicitly to the representations of the octahedral group,which describes the symmetry of the discretised space-time, and to P and C quantum numbers, as in Ref. [5]—although we interchange the roles of T and T . In themeasurements, we combine the smearing and blockingof Ref. [6] with the extended basis of operators in thevariational approach of Ref. [5].The errors are due to statistical uncertainties. We per-form continuum-limit extrapolations with a conventionallinear fit to the dependence on a , where a is the latticespacing. We also report a simple large- N c extrapolation,in which we include corrections O (1 /N c ) to m G / √ σ . Wefind that the uncertainty in the string tension σ is muchsmaller than in the masses m G . Other technical details,including comments on the systematics and on finite sizeeffects, will appear in Ref. [4].We identify m A ++1 = m ++ . As m E ++ and m T ++1 are compatible with each other, and they both relate to thesymmetric tensors in the continuum theory [5], we com-pute m ++ as the weighted average of the two. Finally,the error on the ratio R is obtained by simple propaga-tion. The error is overestimated, as we ignore correla-tions, in particular because of the common dependenceon σ , but we expect such effects to be small, and not toaffect our discussion.Figure 1 shows that the ratio R for the sequence of Sp ( N c ) YM theories is compatible with a constant. Thisconfirms that O (1 /N c ) effects, if present, are smaller thanthe current uncertainties, the magnitude of which variesbetween ∼
2% for Sp (4) and 5% for Sp (8). III. GLUEBALL MASSES: EARLIER LATTICERESULTS
We include in Table I and Figure 1 our measurements(denoted Sp ( N c ) ), together with lattice results by othercollaborations, for various classes of YM theories. R N c GPPZ/YM AdS × S B conf8 RomansYM Witten SU ( N c ) Sp ( N c ) SU ( N c ) SO ( N c ) FIG. 1: Numerical and analytical results for the ratio R de-fined in Eq. (1). Different shaped markers denote the latticemeasurements with continuum extrapolations in D = 3 + 1dimensions for Sp ( N c ) and for SU ( N c ) [6], as well as in D = 2 + 1 dimensions for SO ( N c ) [7] and SU ( N c ) [8]. Ex-trapolations to the N c → ∞ limit are also included. Dif-ferently rendered lines at R = √ , . , . , . , .
74, arethe holographic calculations in the GPPZ model [13], the cir-cle reduction of AdS × S [18, 19], the holographic model B conf8 in Ref. [34], the Witten model [18, 29], and the cir-cle reduction of Romans supergravity [29, 31], respectively.With R = √ , .
64 we report the field theoretical results fromRefs. [17] and [40], for YM theories in D = 3+1 and D = 2+1dimensions, respectively. More details can be found in themain text. The spectrum of YM glueballs in D = 3 + 1 dimen-sions with SU ( N c ) group (denoted SU ( N c ) ) was stud-ied in Refs. [5, 6]. In the former, the authors use a singlevalue of the lattice parameters for each value of N c , with-out studying the approach to the continuum limit. Con-versely, Ref. [6] reports continuum limits for the glueballmasses expressed in units of the string tension σ , but thevariational method uses a smaller basis of operators ofthe octahedral group in respect to our work, and the T channel is not measured. As long as we restrict attentionto the lightest states in the spectrum (the 0 ++ and 2 ++ ground states), at the same lattice spacing the results ofthe two approaches are in good agreement, and hence wecompare the Sp ( N c ) sequence of measurements, as wellas their extrapolation to large N c , to those of Ref. [6].As visible in Fig. 1, the agreement in the ratio R acrossthe gauge groups is excellent.We also summarise the lattice measurements for SO ( N c ) in D = 2 + 1 dimensions ( SO ( N c ) ), taken fromTables 28, 29 and 31 of Ref. [7] (see also Fig. 26 therein).We include only continuum limit results, and two differ-ent types of large- N c extrapolations. Finally, we collect results for SU ( N c ) theories in D = 2 + 1 dimensions( SU ( N c ) ) from Tables B3-B11 of Ref. [8]. The extrapo-lation to SU ( ∞ ) has been performed by including 1 /N c as well as 1 /N c corrections.Lattice results on R show the emergence of a regularpattern, that depends only on the dimensionality D ofthe system. The group sequence ( SU ( N c ), Sp ( N c ) or SO ( N c )) and the number of colours N c do not appear toaffect R , within current uncertainties—with some devia-tion from this pattern in D = 2+1 dimensions for SU (3), SO (3) and SU (2). We have at our disposal preliminaryresults for excited states and states with different quan-tum numbers in Sp ( N c ) theories (to appear in Ref. [4]),and we did not find significant evidence of similar regularpatterns, reinforcing the notion that the lightest 0 ++ and2 ++ glueballs play a special role in YM theories. IV. GLUEBALL MASSES: A BRIEF SURVEYOF ANALYTICAL RESULTS
In Fig. 1, we compare the result of lattice measure-ments of the ratio R to two classes of semi-analyticalcalculations, performed either via gauge-gravity dualitiesarising in the context of supergravity, or via alternativefield-theory methods. In all these models, the ratio R isknown only in the strict large- N c limit, as 1 /N c correc-tions are ignored.The GPPZ model was proposed in Ref. [9] (see alsoRefs. [10–12]) as a simple, classical supergravity dualof mass-deformed, large- N c , N = 4 Super-Yang-Mills.The geometry is singular and asymptotically approachesAdS . The spectrum of fluctuations yields R = √ N c field-theorystudy in Ref. [17] (see Table 1 therein), which in Fig. 1we denote as YM . A closely related model is stud-ied in Ref. [18], that reports a holographic calculationbased upon the circle reduction of the system yieldingthe AdS × S background (see also Ref. [19]). The re-sult in this case is R = 1 .
46. The close proximity betweenthe results of these two holographic calculations (bothof which use geometries that are asymptotically AdS ),Bochicchio’s field-theoretical approach [17], and latticecalculations in Sp ( N c ) and SU ( N c ) is remarkable.Witten’s holographic model of confinement [20] isbased upon S × S × S reduction of eleven-dimensionalsupergravity [21–24]. In the asymptotically AdS back-ground geometry, one S shrinks to zero size. Thestatic quark-antiquark potential is computed holographi-cally [25, 26], and yields linear confinement. Adaptationsto model quenched QCD were proposed in Refs. [27, 28].The spectrum of glueballs yields R = 1 .
74 [18] (see alsoRef. [29]). An alternative model, based on circle reduc-tion of Romans supergravity [30], has geometry that isasymptotically AdS , and again the circle shrinks. In thiscase, R = 1 .
61 [31] (see also Refs. [29, 32, 33]). For bothcelebrated models, Fig. 1 shows that R is not compatiblewith the lattice results, with current uncertainties.The literature on the holographic dual of three-dimensional confining theories is more limited. InRef. [34] the model dubbed B conf8 is the gravity dual ofa non-trivial, asymptotically free theory in 2 + 1 dimen-sions [35–38], and yields R (cid:39) .
57. A completely differentfield-theory approach to YM theories in 2 + 1 dimensionsis used to compute glueball masses in Refs. [39, 40] (wedenote it as YM in Fig. 1). From the latter of the two,we read that R (cid:39) .
64. This result is valid only in thestrict N c → + ∞ limit, although the analysis in Ref. [40]could potentially be extended to finite N c . Both theseapproaches ( B conf8 and YM in Fig. 1) slightly underesti-mate R in respect to the lattice results for SU ( N c ) and SO ( N c ). V. DISCUSSIONS AND UNIVERSAL RATIO
If the ratio between the masses of the lightest spin-2and spin-0 glueballs is universal for (pure) YM theories,there should be underlying principles that hold for all ofthem. We argue (see also Ref. [41]) that scale symmetryand perturbative unitarity are such principles.When the YM theory undergoes the phase transitionto the confining phase, the vacuum energy density E vac islowered, breaking scale invariance spontaneously, to yield E vac ≡ (cid:10) T µµ (cid:11) < , (2)with T µν the energy-momentum tensor.As the vacuum is not invariant under scale transforma-tions, the dilatation current D µ = x ν T µν creates a state,called a dilaton, out of the vacuum, which we write as (cid:104) | D µ ( x ) | σ ( p ) (cid:105) ≡ if D p µ e − ip · x , (3)where f D is the dilaton decay constant. If the two-pointfunction of dilatation currents is dominated by the dila-ton pole at low energy, for p → (cid:90) x e ip · x (cid:104) | T[ T µµ ( x ) T νν (0)] | (cid:105) ≈ f D m D = − E vac , (4)with m D being the dilaton mass. Under this assumption,we identify the ground-state glueball with the dilaton,because it is the lightest particle and both of them havethe same quantum numbers as the vacuum. How goodthis approximation is can only be assessed a posteriori.The Lagrangian density of the dilaton low-energy ef-fective field theory (EFT) is the subject of a vast lit-erature. The potential must break scale invariance ex-plicitly, and contain non-marginal operators. Depar-tures from marginality might be encoded in a logarithmicfield-dependent potential, as advocated in Refs. [42, 43].(More general, power-law potentials have also been con-sidered [44–51]). We dispense with such level of detailin the context of this discussion. It is natural to as-sume that the intrinsic, dynamically generated scale Λ sets E vac ∼ Λ and f D ∼ Λ. Therefore, from Eq. (4) andtaking 16 E vac = − βf D , we may write f D m D = βf D . (5)The numerical constant β is an intrinsic constant of theYM theory, and depends on the gauge group. It measuresthe size of explicit breaking of scale symmetry, sets thestrength of the self-interaction of the dilaton, and is theexpansion parameter of the EFT. The parameter β is notguaranteed to be small. Lattice calculations find that thespin-2 glueball is the lowest excited state, and has mass ofthe same order of magnitude as that of the ground-stateglueball.The dilaton EFT yields the amplitude M σ , for thescattering process σ ( p )+ σ ( p ) → σ ( p )+ σ ( p ) betweendilaton particles. For center-of-mass energies E (cid:29) m D ,we borrow Eq. (3.3) from Ref. [52] (see also Ref. [53]) andwrite M σ ∼ − α f D (cid:0) s + t + u (cid:1) + O (cid:18) m D f D (cid:19) , (6)in terms of the Mandelstam variables s = ( p + p ) , t = ( p − p ) , and u = ( p − p ) . Here α is a dimension-less constant characterising the theory. The scatteringamplitude violates perturbative unitarity at E ∼ αf D ,To achieve partial unitarity restoration, and raise thisbound, we introduce the spin-2 glueball in the EFT. Weassume that the spin-2 glueball couples to the energy-momentum tensor of the dilaton T µνD .The Lagrangian density of the massive spin-2 glueball h µν can be derived by identifying it with the expansionof the spacetime metric around the flat spacetime as in g µν = η µν + 2 κh µν , to obtain L G = L kin G − κ h µν T µνD + · · · , (7)where the first term is the so-called Fierz-Pauli kinetic-term for the massive spin-2 fields, κ is the (universal)coupling of the spin-2 glueballs and the ellipsis denotesthe higher order terms. Again, the assumptions under-neath this identification can be assessed a posteriori.The propagator of the massive spin-2 field of mass m T is then given by [54] (cid:90) x e ip · x (cid:104) | T { h µν ( x ) h αβ (0) } | (cid:105) = iP µναβ p − m T + i(cid:15) , (8)where 2 P µναβ = ˜ η µα ˜ η νβ + ˜ η µβ ˜ η να − ˜ η µν ˜ η αβ with ˜ η µν = η µν − p µ p ν /m T . The contribution of the diagrams withinternal exchange of the spin-2 particles changes thestructure of the amplitude, and partially restores pertur-bative unitarity to hold at the scale E ∼ ( κf D ) − · m T and slightly above, where κf D measures the strength ofthe spin-2 coupling to the dilaton, compared to the dila-ton self-coupling. For this to happen, one must requirethat αf D ∼ ( κf D ) − · m T , or m T ≡ gf D ∼ ακ f D .The dimensionless constant g ∼ ακ f D depends onthe microscopic details of the theory, as β . Combiningthis with Eq. (5), we write the mass ratio of the spin-2glueball and the ground-state glueball as R ≡ m T m D = gβ . (9)In the mass ratio between the lightest spin-2 and spin-0 glueball the dependence on microscopic details shoulddecouple as suggested by the lattice data. As the EFTcaptures the long-distance dynamics based on symme-try (and perturbative unitarity) considerations, that arecommon to all YM theories, it should describe all low-energy (pure) YM theories.The lattice data we summarised suggests the ratio R in D = 2 + 1 is also universal. It has been noted elsewherethat the similarities between the physics of confinementin D = 2 + 1 and in D = 3 + 1 dimensions turn out to bemuch deeper than naively expected (see e.g. Ref. [55]).On this basis, we argue that also in D = 2+1 dimensionsthe constant ratio is controlled by spontaneous as well asexplicit breaking of scale invariance through confinement,which, by generating a mass gap, changes the would-bepower law behaviour of gluon correlators, at distancesmuch larger than the intrinsic length scale set by thedimensional gauge coupling. VI. OUTLOOK
Our lattice measurements of the masses of the lightestscalar and tensor glueballs for Sp ( N c ) gauge theories in D = 3 + 1 dimensions show no discernible dependence on N c in the ratio R defined by Eq. (1). We compared thisfinding with lattice measurements taken from the litera-ture, and compiled a (non exhaustive) list of other cal-culations, that use holography or alternative field theorymethods. We found supporting empirical evidence thatthe ratio R might be a universal quantity in YM theo-ries, in the sense that it appears to depend only on thedimensionality of the system, not its microscopic details.This intriguing feature might be connected with thespecial role that the lightest scalar glueball and the light-est tensor glueball play in respect to scale invariance.As we argued in Section V, it might be explained underthe approximation that these two particles can be identi-fied with those sourced by the dilatation operator and bythe energy-momentum tensor. This approximation relieson two separate assumptions: that the explicit breakingof scale invariance is small compared to its spontaneousbreaking, and that single particle exchange saturates the2-point correlation functions build with the dilatation op-erator and the energy-momentum tensor.Our arguments highlight the distinguishing features ofthe two particles that are the main topic of this letter.More theoretical work would be useful, to better under-stand the role of these two particles, and whether theempirical evidence we uncovered points to an exact rela-tion, or, if otherwise, to estimate the size of deviations. It would also be very useful to have lattice data on Yang-Mills theories with other gauge groups, and we hope suchcalculations will be performed in the future. Acknowledgments [1] E. Bennett, D. K. Hong, J. W. Lee, C.-J. D. Lin, B. Lu-cini, M. Piai and D. Vadacchino, JHEP , 185 (2018),[arXiv:1712.04220 [hep-lat]].[2] J. Holligan, E. Bennett, D. K. Hong, J. W. Lee, C.-J. D. Lin, B. Lucini, M. Piai and D. Vadacchino,arXiv:1912.09788 [hep-lat].[3] A. Athenodorou, E. Bennett, G. Bergner, D. Elander,C.-J. D. Lin, B. Lucini and M. Piai, JHEP , 114(2016), [arXiv:1605.04258 [hep-th]].[4] J. Holligan, E. Bennett, D. K. Hong, J. W. Lee, C.-J. D. Lin, B. Lucini, M. Piai and D. Vadacchino, inpreparation.[5] B. Lucini, A. Rago and E. Rinaldi, JHEP , 119(2010) [arXiv:1007.3879 [hep-lat]].[6] B. Lucini, M. Teper and U. Wenger, JHEP , 012(2004) [hep-lat/0404008].[7] R. Lau and M. Teper, JHEP , 022 (2017)[arXiv:1701.06941 [hep-lat]].[8] A. Athenodorou and M. Teper, JHEP , 015 (2017)[arXiv:1609.03873 [hep-lat]].[9] L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni,Nucl. Phys. B , 451 (2000) [hep-th/9909047].[10] L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni,JHEP , 022 (1998) [hep-th/9810126].[11] J. Distler and F. Zamora, Adv. Theor. Math. Phys. ,1405 (1999) [hep-th/9810206].[12] K. Pilch and N. P. Warner, Adv. Theor. Math. Phys. ,627 (2002) [hep-th/0006066].[13] W. Mueck and M. Prisco, JHEP , 037 (2004) [hep-th/0402068].[14] R. Apreda, D. E. Crooks, N. J. Evans and M. Petrini,JHEP , 065 (2004) [hep-th/0308006].[15] D. Elander and M. Piai, Nucl. Phys. B , 241 (2012)[arXiv:1112.2915 [hep-ph]].[16] D. Elander and M. Piai, Nucl. Phys. B , 779 (2013)[arXiv:1208.0546 [hep-ph]].[17] M. Bochicchio, arXiv:1308.2925 [hep-th].[18] R. C. Brower, S. D. Mathur and C. I. Tan, Nucl. Phys.B , 249 (2000) [hep-th/0003115].[19] D. Elander, M. Piai and J. Roughley, arXiv:2004.05656[hep-th].[20] E. Witten, Adv. Theor. Math. Phys. , 505 (1998) [hep-th/9803131].[21] H. Nastase, D. Vaman and P. van Nieuwenhuizen, Phys.Lett. B , 96 (1999) [hep-th/9905075].[22] M. Pernici, K. Pilch and P. van Nieuwenhuizen, Phys.Lett. , 103 (1984).[23] M. Pernici, K. Pilch, P. van Nieuwenhuizen andN. P. Warner, Nucl. Phys. B , 381 (1985).[24] H. Lu and C. N. Pope, Phys. Lett. B , 67 (1999)[hep-th/9906168].[25] J. M. Maldacena, Phys. Rev. Lett. , 4859 (1998) [hep-th/9803002].[26] S. J. Rey and J. T. Yee, Eur. Phys. J. C , 379 (2001)[hep-th/9803001].[27] T. Sakai and S. Sugimoto, Prog. Theor. Phys. , 843(2005) [hep-th/0412141].[28] T. Sakai and S. Sugimoto, Prog. Theor. Phys. , 1083 (2005) [hep-th/0507073].[29] D. Elander, A. F. Faedo, C. Hoyos, D. Mateos andM. Piai, JHEP , 003 (2014) [arXiv:1312.7160 [hep-th]].[30] L. J. Romans, Nucl. Phys. B , 691 (1986).[31] C. K. Wen and H. X. Yang, Mod. Phys. Lett. A , 997(2005) [hep-th/0404152].[32] S. Kuperstein and J. Sonnenschein, JHEP , 026(2004) [hep-th/0411009].[33] D. Elander, M. Piai and J. Roughley, JHEP , 101(2019) [arXiv:1811.01010 [hep-th]].[34] D. Elander, A. F. Faedo, D. Mateos, D. Pravos andJ. G. Subils, JHEP , 175 (2019) [arXiv:1810.04656[hep-th]].[35] G. W. Gibbons, D. N. Page and C. N. Pope, Commun.Math. Phys. , 529 (1990).[36] A. Hashimoto, S. Hirano and P. Ouyang, JHEP ,101 (2011) [arXiv:1004.0903 [hep-th]].[37] M. Cvetic, G. W. Gibbons, H. Lu and C. N. Pope, J.Geom. Phys. , 350 (2004) [math/0105119 [math-dg]].[38] A. F. Faedo, D. Mateos, D. Pravos and J. G. Subils,JHEP , 153 (2017) [arXiv:1702.05988 [hep-th]].[39] R. G. Leigh, D. Minic and A. Yelnikov, Phys. Rev. Lett. , 222001 (2006) [hep-th/0512111].[40] R. G. Leigh, D. Minic and A. Yelnikov, Phys. Rev. D ,065018 (2007) [hep-th/0604060].[41] D. K. Hong, J. W. Lee, B. Lucini, M. Piaiand D. Vadacchino, Phys. Lett. B , 89 (2017)[arXiv:1705.00286 [hep-th]].[42] J. Schechter, Phys. Rev. D , 3393 (1980).[43] A. A. Migdal and M. A. Shifman, Phys. Lett. , 445(1982).[44] R. Rattazzi and A. Zaffaroni, JHEP , 021 (2001)[hep-th/0012248].[45] Z. Chacko and R. K. Mishra, Phys. Rev. D , no. 11,115006 (2013) [arXiv:1209.3022 [hep-ph]].[46] T. Appelquist, J. Ingoldby and M. Piai, JHEP , 035(2017) [arXiv:1702.04410 [hep-ph]].[47] T. Appelquist, J. Ingoldby and M. Piai, JHEP , 039(2018) [arXiv:1711.00067 [hep-ph]].[48] O. Cat`a, R. J. Crewther and L. C. Tunstall, Phys. Rev.D , no. 9, 095007 (2019) [arXiv:1803.08513 [hep-ph]].[49] O. Cat`a and C. Mueller, Nucl. Phys. B , 114938(2020) [arXiv:1906.01879 [hep-ph]].[50] T. Appelquist, J. Ingoldby and M. Piai, arXiv:1908.00895[hep-ph].[51] Z. Fodor, K. Holland, J. Kuti and C. H. Wong,arXiv:2002.05163 [hep-lat].[52] Z. Komargodski and A. Schwimmer, JHEP , 099(2011) doi:10.1007/JHEP12(2011)099 [arXiv:1107.3987[hep-th]].[53] D. K. Hong et al. in preparation.[54] M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A , 211(1939). doi:10.1098/rspa.1939.0140[55] M. J. Teper, Phys. Rev. D59