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PoS(QCD-TNT-III)023 http://pos.sissa.it/ ) Yang-Mills theory in N † limit and in candidate theories of strongly interacting dynamics beyond the standard ∗ N [email protected] Speaker. In memory of Pierre van Baal, who greatly contributed to Lattice Gauge Theories. the large- model (in which the lowest-lying playsimplications the for role our of theoretical the understanding Higgs ) ofcoupled are gauge reviewed theories in and with QCD-like their a theories (near-)conformal and dynamics are in discussed. strongly Recent numerical calculations of the spectrum in QCD, in SU( ∗ † Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

From and to hadronic2-6 : September, A 2013 bridge too far? European Centre for Theoretical Studies in(Italy) Nuclear and Related Areas (ECT*), Villazzano, Trento Biagio Lucini Physics Department, Swansea University, Singleton Park,E-mail: Swansea SA2 8PP, UK Glueballs from the Lattice PoS(QCD-TNT-III)023 ], 4 chan- . The ) gauge PC N J Biagio Lucini ] for a recent 1 glueballs framework. Among SU( ) Yang-Mills theories is the existence N ( ab-initio , are said to be walking, since the coupling runs 2 . Walking, and more in general infrared conformal 1 ] and references therein for more recent developments). 3 technicolour ] confirming the main results and providing some extra insights. The 5 ] for an early pedagogical review of the main results). 7 ] for a recent review and [ ] (see [ 2 6 limit [ N Although the coupling is not a physical quantity, the concept of running can be formalised in terms of spectral In addition to the theoretical and experimental interest for glueballs in QCD, more recently One of the implications of colour confinement in SU As for other non-perturbative features of QCD, numerical calculations with Monte Carlo meth- 1 observables. Those calculations should measure theglueballs and effective of couplings for (which will the be relateddescribing production to the and the interactions diagonal annihilation of elements of of those the states) effectiveAlthough Hamiltonian as progress well in as this the couplings directionThus, between has most glueballs been of and achieved, mesons. current a latticemesons calculation calculations and assume of that this glueballs the type separately, mixing is themuch is different still hope small from unviable. being this and simplified that compute case. thelarge In of statistics, real addition, which since interacting in glueball spectrum measurements turn willprovided often requires demand not in a a look the huge computational quenched effort, approximation,SU(3) the Yang-Mills in theory. glueball this The spectrum case reference is calculationwith amounting often some in to subsequent this work studying approximation [ the is still problem provided in by the [ there has been an increasing interestinfrared in and glueball-like are states asymptotically in free theories inregion that the in like ultraviolet, which but QCD unlike the confine QCD physics in have the isThese an governed intermediate theories, by energy normally an referred infrared as fixed point in an enlarged parameter space. , which atframework first that sight provides looks a quite controlledlarge- interpolation drastic, from can QCD in to fact SU(3) be Yang-Mills: formulated the in ’t Hooft a very slow in this intermediate energy range review of the theoretical andlack of experimental experimental status evidence of of glueballsis this in no the field). quantum spectrum can number The be that fundamental traced distinguishes back reason glueballs to for from the fact isosinglet the that mesons there in the same ods in the theory discretised on aballs (see lattice e.g. provide invaluable [ insights on the nature of glue- Glueballs from the Lattice 1. Introduction nel. Hence, physical statesconditions could naturally one be would thought call ofThis glueballs as mixing and admixtures can states of be that states assumed thatples to are in be nearby naturally the idealised states identified result in as of theone isosinglet some spectrum could off-diagonal mesons. classify with effective the Hamiltonian the states that same that cou- However, quantum are it coupled is numbers. by a this dynamical If Hamiltonian problem the under to mixing -rich establish and the is -rich. strength small, of then the mixing terms. of massive gauge-invariant colour singletcoupling states being of with order purely one, gluonicturbative a content, techniques quantitative the understanding and of can the onlygroups, glueball phenomenologically, be spectrum a achieved transcend special in role per- is an One played expects by that SU(3), glueball which states isexperimental also the appear attempt gauge in aimed of QCD. at QCD. However, their to identification date, in glueballs have the eluded QCD any spectrum (see e.g. [ PoS(QCD-TNT-III)023 ) g a i ( µ (2.1) (2.2) U and ˆ µ Biagio Lucini ) gauge theories N are two directions, ν and µ along the direction quantitative in the , where the † symbol indicates . ˆ , µ ) ˆ µ  is the real part of the trace of the +  ) i x i + ( d i µν ) ( x µν U ( ab-initio G 3 for QCD). Gauge transformations are U ) µ i Tr ( A e = Tr µ ˆ µ e R N U + i † is related to the continuum (bare) coupling R i ) i Z ( 1 β N 3 by G iga − µ to  1 ) Yang-Mills and their relevance for QCD, providing A )  N i ( ν µ > is an element of SU(3) (or of the gauge group, in the most Pexp ∑ µ U , ) to its nearest neighbour i i ( i β ) = µ ] and references therein). Our interest here is in the fact that in i 8 ( = U we briefly review the lattice formulation of SU( µ w defined on the lattice points and taking values in the gauge group. 2 U S ) i ( labels a point in the discretised spacetime, i G the number of colours (i.e. summarises the main lessons learnt from those calculations and points out 5 N . A link variable , with ˆ µ , with the latter being the path ordered product of the oriented link variables 2 g µν / N 2 discusses numerical results in SU( . Finally, Sect. 3 = 4 The lattice is the most reliable method for obtaining In this contribution, we give a brief overview of recent lattice results. This work does not aim β plaquette U represented by functions In the previous equation, and the sum extends over all points and directions. (with reverse orientation corresponding to Hermitianthe conjugation) lattice. around Positive an orientations elementary of square theof of links connecting the nearest versors neighbour sites followgeneral the case), direction related to the continuum field by They act on link variables by sending some of the current and future directions of lattice calculations of glueball2. properties. Extracting glueball masses from lattice calculations and of QCD, with a particular emphasisSect. on widely employed techniques to extract glueball masses. non-perturbative regime of QCD. The calculation startsdimensional with spacetime the grid QCD in regularised such onsector, the a a four- simplest way choice that is the gauge Wilson invariance action: is preserved. In the pure gauge being the lattice spacing. The lattice coupling also an explicit example ofpotentially a relevant QCD for calculation. explaining CalculationsSect. dynamically of the electroweak glueball spectrum breaking in are theories reported in at providing a comprehensive review, noris does to discuss it some pretend of to theand be most complete. their recent possible results The for main implications glueballthe purpose for masses field here obtained our is from understanding provided lattice in of calculations structured the glueballs. as quoted follows. references A In and Sect. more in works detailed cited account therein. of The rest of this paper is Glueballs from the Lattice gauge theories, can providediscussion a of dynamical this mechanism mechanism goes of beyond theto electroweak scope the of symmetry this specialised breaking. work, literature and (e.g. we A referthis [ the full framework interested the reader lowest-lying isosinglet scalarwith plays the a recently special discovered role,dynamical Higgs since problem it boson. that should can be Once be identified addressed again, with the lattice calculations. existence of such a light scalar is a with the integral taken from point PoS(QCD-TNT-III)023 . In with (2.4) (2.5) (2.7) (2.3) (2.6) ) τ O ( is gauge C w reproduces , we define S x Biagio Lucini ~ w S depending only , , which, by , where we have a i ) O t ) , τ x ~ + embodies the spinorial 0 = ( t ( i M O . Hence, if we want to have ) 0 t ( ) is amenable to Monte Carlo † . O w 2.4 h S is a discrete sum over Hamiltonian , − doublers ) w e . S ) = E O , − ( and the space coordinates E τ f e ( ) d t N f C t τ , C ) , 0, at the lowest order in N E j x ) − ( M . The path integral of the full theory is ψ e → M O µ ) i j depends on the point a U E x ~ M Det ∑ ( i O 4 , as ( Det t s ( C ψ µ 1 V µ U Z = U is the path integral measure of all links (which on the D f only. Using the spectral representation, ) = D S µ t Z ) = ( τ U Z τ 1 O Z ( D = C = Z i tales the form of a correlator of an appropriate observable ), the of an operator O O h 2.4 are Euclidean indices, with all other indices kept implicit. j is the number of flavours, f and the spatial volume. The correlator of interest is N i s V Using the path integral ( For , on the lattice one can not preserve at the same time chiral symmetry and ultralo- , the zero-momentum operator at timeslice ) comes after the Grasmann integral over the fermions. Eq. ( t ( lattice is well definedM and corresponds to a product of Haar measures) and the determinant of where For glueball calculations, At the lowest end ofeigenstates the spectrum, with the eigenvalues spectral corresponding function to physical masses of states contributing to where structure and depends linearly on the gauge fields numerical simulations. State of the art numerical results require modern . on the link variables is obtained as the quantum numbers of theconventionally state separated of the interest. Euclidean If time coordinate with O translational invariance, is a function of cality of the action without generating spurious copies called Glueballs from the Lattice Hermitian conjugation. It is straightforward toinvariant. see that with It these is definitions the alsothe action easy Yang-Mills continuum to action. see Using that,fluctuations asymptotic in do freedom, not the it spoil limit is the possible correctness of to the show continuum that limit. quantum an action coupling a finiteof number having of a neighbour generic points, numberactions we with of an have flavours arbitrary to or number give of exact upthe flavours chiral sites can either (or, also symmetry. equivalently, the of be adding possibility formulated, Chirally an at symmetric extrapect) the dimension). all fermionic expenses It lattice of is coupling fermionic worth all actions stressing that reproduceused (as the more one same proficiently would continuum than ex- physics, others but incialistic some particular of physical technical situations. them details, can Without the going be further actionsin into describing can spe- be the written -gauge as interactions quadratic we forms in are terms interested of the fermion fields: PoS(QCD-TNT-III)023 1 . If − ) (2.8) k τ would ( τ C Biagio Lucini 1). This procedure (normalised so that . For simplicity, we ) > by considering traced τ C contributes to i τ ( ) O i j t , C x , are still defined on lattice ( 1; although a higher O = τ ]. fat links 10 , 9 . ) τ is the dimension of (generally, τ ( transformation laws, we consider all possible τ C M ∞ PC J → 5 lim τ − = , where ) J 1 m ( can be performed only if the Hamiltonian eigenstates are , where each operator is the trace of the ordered product over M i ) O Φ τ + 0 t ( ,..., j ) 1 O . is chosen in such a way that the contribution of higher excitations is ( ) in the channel with corresponding to that irreducible representation: 1 , one obtains a variational estimate of the mass for the excitation 0 J t PC k O ( Φ J m † i Φ O h ) to ) = , only the lowest state with the quantum number of 2.8 τ ∞ ( i j C → τ is taken to transform under an irreducible representation of the rotational group, its energy is the identity) is then diagonalised for a fixed ) ) t In practice, things are more complicated. For a start, only a few low states are below the onset The same base paths can give an increased basis of operators by replacing the links with more Having built a set of loops with well defined 0 , ( x ( i j be desirable, rarely the statistics is sufficient to get anconsider accurate a signal case for in whichtechnical the complications that eigenvalues are could be non-degenerate,ordered dealt the from with the more quite one general corresponding easily. to case The theApplying only higher eigenvectors Eq. posing to are ( the assumed one to corresponding towith be the quantum lower numbers eigenvalue. of a continuum spectral density, whichenergy. starts This with requires the us multi- to identify bound scattering statethe states spatial of and lattice lowest to has possible project not them the out full fromof rotational our the symmetry, basis. but three-dimensional Second, has the cube, symmetryexcluding of the the ). octahedral invariance group group, Hence, one which needs has to a classify finite the number operators of according elements to the (24 transformations of extended objects. These new variables, often referred to as Since the lattice has aof finite glueballs extent, as the previous long formula as can be reliably usedknown, to but extract these the are mass exactlyusing what a we variational are method. after.path In However, ordered practice, we we products can use of approximate afixed link these contour large variables eigenstates is set transformed over of under closed the operators that contours rotational group irreducible of and representations various the results are shapes aresymmetries, constructed. and combined we in sizes. build such combinations Using a that paths Each way have also transformed well under defined parity the and appropriate conjugations. links, with a modification of thefrom original a link neighbourhood variables that of accounts each alsooriginal link. for ones. contributions The The coming new physical links ideathe behind transform links this under are procedure ultraviolet gauge quantities. is transformations that If asto we the use want are to variables extended access that objects, the have while properties supportvariables of on with physical the blocked typical we variables, need physical as length. inblocking a In transformations block-spin practice, can transformation one be in replaces combined statistical original this and mechanics. idea. iterated. The These results There we are review various in ways this of work implementing follow Refs. [ correlators some fixed base loop of eitherof the original iterations links of or the the link blocking variables obtained transformations. after a The given number correlation matrix defines the eigenstates minimised. An optimal choice of O will be the lowest mass Glueballs from the Lattice the limit C PoS(QCD-TNT-III)023 , N 2 g = λ Biagio Lucini torelon states is sent to infinity and extrapolated theory by a N N ] has appeared, in which the construction of the 6 12 . If the coefficients of this series are of order one, one 2 N / ]). Various approaches have been developed over the years, 11 . However, this still poses a computational challenge that limits π is sent to zero with the constraint that the ’t Hoof coupling ) and back to QCD g ∞ ) Yang-Mills theory. As the number of colours ]. Recently, the work [ N 5 ) are already quite accurate, at least in the pure Yang-Mills case. Specific , 4 ++ limit. In this context, pure SU(3) is seen as a particular case of the N and the 2 ++ For a long time, performing calculations at quark masses that are low enough for being phe- In fact, a natural framework to interpolate from SU(3) to QCD, and hence to ascertain the and glueball calculations forthe the 0 ground state andcalculations a have few targeted excitations other in quantumas some states, for spin providing instance channels a in (e.g. rathervariational comprehensive [ basis set is of fully results, automatised.the channels, This in which method we allows canfrom identify us scattering the to and physical perform torelon spectrum. contributions a and Thisfor full hence work the remove calculation them – main in results to all presented which in we the refer next for section. further details –3. is the From foundation SU(3) to SU( nomenologically relevant has beennumerical a algorithms big and practical computational problemmass power in close allow to lattice that calculations of QCD. to the Recent physical be advances done in at a pseudoscalar importance of glueball calculations in pure’t gauge Hooft for large- the full dynamical theory, is provided by the Glueballs from the Lattice the octahedral group and then decomposewith the more latter than one according continuum to spin the contributinggroup. continuum to each spin Then, irreducible contributions, on representation a of the finite octahedral operators lattice, winding there around are the spurious periodic contributions spatial to directions.need the These to glueball contributions, be spectrum called related removed to fromvariational. the While spectrum. this is Finally, nothigher one a excitations big has is problem to less for keep and thedecreases less in lowest-lying as accurate, states, mind we since go the that higher variational the the in estimatecalculations number calculation mass. of on of is These the operators issues lattice in have (see the been e.g. variational known [ since basis the early days of glueball the statistics. The number of configurationsthe that can spectrum be obtained at are an generallymasses acceptable sufficient require accuracy for getting higher level. statistics However, numerical (atfor calculations least the of by different glueball a statistical requirements factor to ofderstood, get and ten) an can Monte error be Carlo of explained simulations. in thepace, terms same The still of order reason the QCD of required dynamics. magnitude computational is Whilecomputational well the time demands, un- field it is is still highly progressing makes non-trivial atwhich sense to a the to fast obtain. quarks resort are to treated Because the as ofstudying so-called external these the quenched sources. pure high approximation, In Yang-Mills in glueball theory. calculations Athow of this these SU(3), stage, calculations this could it amounts be to is relevant an for open real-world problem QCD. to establish whether and generic family of SU( the coupling of the theory remains constant, at the perturbative level thecan theory be simplifies, expressed and in one can a prove power that series observables in 1 would expect the physics in SU(3) to only differ from that of the large- quantity of the order of 10% in each observable. PoS(QCD-TNT-III)023 ++ (3.1) to set σ √ Biagio Lucini ]. For various observables, this ) gauge theories can be achieved ] and references therein). , . N 13 , [ ) ) 14 3. In this sense, and following the channel, the lowest mass in the 2 ) 7 0 2 . . 1 2 2 . 1 2 N = 2 ( ( 1 N N / ++ ) group and assuming the validity of the ( 3 7 N . . N 1 N . 0 2 2 − limit of SU( ) + ) ) continuum limit of the lowest-lying glueball ) + of the lowest-lying glueball masses. N N 8 14 17 7 ( ( ( ∞ 28 78 93 . . . → ). Using the square root of the tension 3 4 5 N 1 = = = ∗ ++ ++ σ σ σ ++ 2 0 0 √ √ √ m m m ] (see Fig. 9 series being convergent at Extrapolations to excitation are respectively N ++ limit, the correction coefficient being of order one. This is consistent with the Figure 1: N The most precise determination of the SU( Numerically, the determination of the large- value can be related using only the leading correction (see [ procedure suggests that at the levelN of accuracy of a few percents the SU(3) value and the infinite- The emerging picture is that,by at their level large- of accuracyexpectation of of around 5-10%, the glueballs large- are well described masses has been provided in [ the scale, the quoted resultschannel for and the the first lowest 0 mass in the 0 Glueballs from the Lattice by studying the values of observables for various SU( diagrammatic argument that suggests a power series in1 PoS(QCD-TNT-III)023 limit at N -- 2 T Biagio Lucini -+ 2 T extrapolation of masses +- N 2 T ] and references therein) 3 (as recent lattice calcu- ++ 15 2 = T Ground States Excitations [Lucini,Teper,Wenger 2004] N -+ 1 T limit, the suppression of mixing at N +- 1 T ++ 1 ] shows – albeit at fixed lattice spacing – that T based approaches to glueballs and QCD-like 2 limit is that mesons and glueballs do not mix 12 N "# M -- 2 !! N 8 E –36– ]. -+ 16 E +- E .Theyellowboxesrepresentthelarge ∞ Chew-Frautschi plot of the glueball spectrum being the deconfinement temperature. ++ c = E T , ). Hence, it makes sense to look at the physics of glueballs in the ) N c ]), then glueball calculations in SU(3) gauge theory can still provide a +- ∞ T 2 6 A 17 ( / 1 -+ Figure 21: PC = + + PC = + - PC = - + PC = - - 1 = A ). This is good news for large- a 2 -suppressed with respect to the Yang-Mills leading term. Hence, if SU(3) is close ++ N 1 theory. A calculation performed in [ / A ∞ 1 0 3 2 0 2 4 6 8 10 12 14 16 18

1 = 0.5 1.5 2.5 0 3 2 The spectrum at am Glueball masses in various channels (ground states and some excitations) in the large- J ) and the contribution given by quarks stays negligible for can justify an approach to glueballs that neglects their mixing with mesons. Very recently, N ∞ N Another relevant consequence of the large- When dynamical fermions are considered, it can be shown that they provide a leading contri- large- high statistics numerical calculations of the glueball spectrum in a full QCD setup have been per- for infinite number of colours. If SU(3) is close to the large- useful guidance to experiments.calculations. This provides a motivation to perform high precision quenched fixed lattice spacing similar conclusions seem also tosimulations be (Fig. valid for allstrong glueball interactions, states like currently for accessible instance to theor, more numerical gauge-string recently, duality topological (see field e.g. theory [ [ fact that the correctionsthat are SU(3) of is the close order to of SU( magnitude one would naively expect, one can say Figure 2: simpler bution that is 1 to SU( lations seem to confirm [ Glueballs from the Lattice Figure 20: obtained in ref. [38]. PoS(QCD-TNT-III)023 Biagio Lucini + + 3 + − 3 − + 3 − PDG expt Anisovich et al. Glueball, this work − ]. The central results of the latter 3 3 − − 2 − + 2 + − 2 + 9 ], from which the plot has been taken. + 3 2 ]. A current focus of lattice calculations is to reduce + ), since these can not be explained by the simple quark + 19 + 1 − − + 1 − and 0 − 1 − , which compares the results of the simulations (open points) with + − 3 − 0 + − 0 − + 0 + + 0 ] and extended to include more spectral states in [

18

1 2 7 5 3 6 4 Glueball masses in full QCD from a lattice calculation with dynamical fermions (open points) M GeV M QCD provides an exemplar realisation of confinement and with the calculation are shown in Fig. experimental states. Most of the glueballbe states accessible seem to to the be PANDA at experiment quite [ high mass, in a region that will Figure 3: physics determined only by one massthe scale. particular However, realisation it has of beeninfrared QCD conjectured in is for asymptotically a not free long the non-Abelian timebe only gauge that approximately theories. possibility infrared for Another conformal. possibility havingfermions is confinement close that Infrared in the to conformal the theory the theories deep point are at realised which for the a system number ceases of to be asymptotically free. The range of compared to experimental results, as described in [ formed in [ both the statistical error and theand systematic improving error spin (e.g. identification). by Intum taking particular, numbers into lattice (as account for calculations mixing instance are with 2 model. focusing mesons on exotic quan- 4. Glueballs in near-conformal gauge theories Glueballs from the Lattice PoS(QCD-TNT-III)023

. For

PCAC

values at m at values = = ∞ ], which has Biagio Lucini 20 ]. 24 conformal window . A walking or a conformal novel interaction PCAC 10 a m near-conformal ] and in lattice gauge theories in a strongly coupled regime far or 22 ). This feature is particularly important in the light of the recent 4 ]. In particular, lattice gauge theories at strong coupling can provide = string tension) σ 23 ( walking glueball mass glueball mass 1/2 ++ ++ σ PS meson mass PS meson 0 2 ], see e.g. Fig. 0.0 0.2 0.4 0.6 0.8 1.0

21 1.0 2.0 3.0 0.5 1.5 2.5 a M a The lowest-lying states in the spectrum of SU(2) with two adjoint Dirac quarks [ , 20 As with other strong interactions, the lattice approach can provide essential information on discovery of the Higgsanomalously particle, light which near a in point this inyet framework which this scale will mechanism invariance at be gets work restored. identified inexamples Although with gauge in we theories gauge-string a have that duality not scalar can [ seen have that phenomenological is relevance, there are from the continuum limit [ useful insights on the structure of conformal or nearly-conformal gauge theories [ number of flavours for which the theory is infrared conformal is called Figure 4: an infrared fixed point. the physical properties of thosebeyond theories. the Because , of recently their those potentiallations. theories relevance have One for been of widely the very studied first with objectivesallows numerical has to simu- been pin the down determination of some the candidatebe conformal able models window, which to for disentangle phenomenology. features As offermions a (near-)conformality have fundamental from a a step, finite typical one (albeit confining must behaviour small)spectrum when mass. whose the mass One ratios of the domass signatures not hierarchy of depend (i.e., a on conformal unlike the gauge inmass fermion theory QCD, [ mass, is glueball possibly a masses presenting can an inverted be lower than the Glueballs from the Lattice theories just below the onset of theand conformal for window a the infrared wide fixed range pointthis of is regime only energies are approximate, observables said have to a be could mild provide a dependence dynamical on explanation the of energy. electroweak symmetry Theories breaking. in PoS(QCD-TNT-III)023 , glue- Phys. , , D73 (2009) 1–49, Biagio Lucini E18 Phys.Rev. , (1974) 461. (2009) 264–267, Discovering ]. ]) now starting to emerge B72 186 20 Int. J. Mod. Phys. , Towards the glueball spectrum from Nucl. Phys. , ]. Y. Chen, A. Alexandru, S. Dong, T. Draper, ]. arXiv:1208.1858 ]. Nucl.Phys.Proc.Suppl. , 11 The Physics of Glueballs ]. ]. Whether electroweak symmetry breaking is described (2012) 170, [ The glueball spectrum from an anisotropic lattice study 21 hep-ph/0409183 1210 arXiv:1104.1255 Glueball Regge trajectories and the : A Lattice study hep-lat/9901004 JHEP , ]. ]. isosinglet states with a non-negligible contribution coming from the pure (2011) 81, [ ,. ++ Glueball spectrum and matrix elements on anisotropic lattices hep-lat/0510074 (2005) 344–354, [ 126 1/N Lattice status of gluonia/glueballs A planar diagram theory for strong interactions B605 (1999) 034509, [ D60 arXiv:0810.4453 arXiv:0809.2561 Eur.Phys.J.Plus [ [ (2006) 014516, [ I. Horvath, et al., unquenched lattice QCD Rev. Phys.Lett. The results reviewed here have been obtained in various collaborations with L. Del Debbio, E. The experimental identification of glueballs is one of the most urgent open problems in QCD. of a new is current the subject of a fervid theoretical and phenomenological [8] J. Andersen, O. Antipin, G. Azuelos, L. Del Debbio, E. Del Nobile, et al., [7] S. R. Coleman, [6] G. ’t Hooft, [2] C. McNeile, [3] E. Gregory, A. Irving, B. Lucini, C. McNeile, A. Rago, et al., [4] C. J. Morningstar and M. J. Peardon, [1] V. Mathieu, N. Kochelev, and V. Vento, [5] H. B. Meyer and M. J. Teper, Gregory, A. Irving, C. McNeile, A. Patella,A C. special Pica. thanks A. goes Rago, to E.crucial Craig Rinaldi, involvement M. McNeile, in Teper Antonio and more Rago U. recent Wenger. andby glueball Enrico the calculations. Rinaldi Royal for This their Society work constantPerfomance has grant Computing and been UF09003 Wales for partially and computational supported resources. the STFC grant ST/G000506/1.References We also thank High as a key signature of those theories [ ball investigation, with the lattice proving to be once more an invaluable quantitative tool. Acknowledgments glue dynamics) are starting toconformal play or a near-conformal gauge key theories role and inglueball in understanding their spectrum the phenomenological (originally signatures, dynamics with identified of a for strongly-coupled light one of those models in [ by a gauge theory in the near conformal-phase and the Higgs particle should be seen as a Thanks to recent progress, latticeoretical calculations predictions are for getting glueball closer and massesquantification closer in of to the all providing mixing possible firm with channels.bers the- mesons, and a a Further more systematic progress accurate study will reconstruction ofglueballs of include states the (or a continuum better, with spin 0 exotic from quantum the lattice num- data. More recently, Glueballs from the Lattice 5. Conclusions PoS(QCD-TNT-III)023 , , , . JHEP , ]. Nucl.Phys. , Biagio Lucini Glueball mass (2010) 119, ]. L. Del Debbio, Confining strings ]. 1008 (2010) 034501, (2013) 93–163, ]. ]. arXiv:1308.2925 JHEP D82 , , 526 (2013) 034513, arXiv:1305.6006 Mesons in large-N QCD First results for SU(2) Yang-Mills . Light composite scalar in D87 ]. Absence of chiral symmetry breaking (2005) 161–166, Phys. Rev. arXiv:0907.3896 36 , Multiscale confining dynamics from Conformal versus confining scenario in Phys.Rept. , hep-lat/0107007 . The Glueball Spectrum in SU(3) ]. B. Lucini and M. Teper, . Phys.Rev. ]. Physics Performance Report for PANDA: , (2013) 162001, [ arXiv:1309.1614 Introductory lectures to large-N QCD hep-lat/0404008 111 ]. E. Tomboulis, Infrared conformality and bulk critical points: SU(2) 12 . Few Body Syst. arXiv:0903.3905 (2009) 074507, [ , The infrared dynamics of Minimal Walking Technicolor , arXiv:1212.2600 (2001) 105019, [ ]. G. S. Bali, L. Castagnini, B. Lucini, and M. Panero, . D80 (2013) 106, [ Conformality in many-flavour lattice QCD at strong coupling (2004) 012, [ Glueballs and k-strings in SU(N) gauge theories: Calculations D64 Glueball masses in the large N limit arXiv:1309.3638 , Phys.Rev.Lett. 1311 hep-lat/0103027 arXiv:1311.4155 , 0406 arXiv:1004.3206 , Phys.Rev. , JHEP SU(N) gauge theories at large N , JHEP Phys. Rev. arXiv:1208.2148 Collaboration, M. Lutz et al., SU(N) gauge theories in four-dimensions: Exploring the approach to N = Glueball Spectroscopy in Four-Dimensional SU(3) . 1. , , On the glueball spectrum of walking backgrounds from wrapped-D5 (2013) 164–180, [ ]. ]. ]. ]. B. Lucini and M. Panero, arXiv:1312.7160 , ]. (2001) 050, [ arXiv:1304.4437 B871 arXiv:1311.7559 (1983) 109. C. Michael and M. Teper, , Glueball and meson spectrum in large-N massless QCD (2010) 014510, [ 0106 (2013) 051, [ B221 The Large N limit from the lattice D82 Collaboration, C. M. Richards, A. C. Irving, E. B. Gregory, and C. McNeile, JHEP , Nucl.Phys. 1302 (1989) 347. (2013) 071, [ , arXiv:1211.4842 arXiv:1005.2473 arXiv:1210.4997 arXiv:1007.3879 hep-ph/0410016 in multi-flavor strongly coupled lattice gauge[ theories [ [ twelve-flavor QCD on the lattice JHEP Large-N mesons Strong Interaction Studies with UKQCD in SU(N) gauge theories measurements from improved staggered fermion simulations PANDA Collaboration SU(2) with adjoint fermions B. Lucini, A. Patella, C. Pica,Phys.Rev. and A. Rago, A. Athenodorou, E. Bennett, G. Bergner,with B. one Lucini, adjoint and Dirac A. Fermion Patella, duals with heavy adjoint quarks infinity holographic RG flows with improved operators [ Nucl.Phys. B314 phenomenology and lattice results 1306 [ [9] B. Lucini, M. Teper, and U. Wenger, [23] B. Lucini, A. Patella, A. Rago, and E. Rinaldi, [18] [21] Y. Aoki, T. Aoyama, M. Kurachi, T. Maskawa, K.-i. Nagai, et al., [22] D. Elander and M. Piai, [19] [10] B. Lucini, [14] B. Lucini and M. Panero, [15] D. Elander, A. F. Faedo, C. Hoyos, D. Mateos, and M. Piai, [17] G. S. Bali, F. Bursa, L. Castagnini, S. Collins, L. Del Debbio, et al., [20] L. Del Debbio, B. Lucini, A. Patella, C. Pica, and A. Rago, [24] P. de Forcrand, S. Kim, and W. Unger, [13] B. Lucini and M. Teper, [11] B. Berg and A. Billoire, [16] M. Bochicchio, Glueballs from the Lattice [12] B. Lucini, A. Rago, and E. Rinaldi,