JHEP04(2019)150 d = 1 su- Springer N April 16, 2019 April 24, 2019 March 15, 2019 : : : January 21, 2019 : , Istvan Montvay, a , Revised Accepted Published Received f , Published for SISSA by Simon Kuberski, https://doi.org/10.1007/JHEP04(2019)150 a [email protected] , and Philipp Scior sectors. Extrapolated to the continuum limit, the e ½ [email protected] Henning Gerber, , [email protected] c,a , [email protected] , . 3 Stefano Piemonte 1901.02416 a The Authors. c Lattice Quantum Field Theory, Supersymmetric Gauge Theory
We have calculated the masses of bound states numerically in Georg Bergner, , [email protected] a,b [email protected] [email protected] Universit¨atsstr.31, D-93040 Regensburg, Germany Faculty of Physics, UniversityUniversit¨atsstr.25, of D-33615 Bielefeld, Bielefeld, Germany E-mail: [email protected] Lahore 54000, Pakistan Institute for Theoretical Physics, UniversityMax-Wien-Platz of 1, Jena, D-07743 Jena, Germany Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22607 Hamburg, Germany Institute for Theoretical Physics, University of Regensburg, Institute for Theoretical Physics, UniversityWilhelm-Klemm-Str. of 9, M¨unster, D-48149 M¨unster,Germany Department of Physics, Government College University Lahore, b c e d a f Open Access Article funded by SCOAP allows us to investigatepredictions the in mixing the literature. content of the physicalKeywords: states, which we compare to ArXiv ePrint: operators, we are ablein to the obtain scalar, the pseudoscalarcorresponding masses and of particles spin- the appear ground topredicted states chiral be supermultiplets. and approximately first The massand extended excited degenerate mesonic operator states operators and basis leads including to to both improved fit results glueball-like into compared to the earlier studies, and moreover Abstract: persymmetric Yang-Mills theory withvariational method gauge with group an SU(2). operator basis consisting Using of the smeared suitably Wilson loops optimised and mesonic Sajid Ali, Gernot M¨unster, Variational analysis of low-lying statessupersymmetric in Yang-Mills theory JHEP04(2019)150 4 3 4 17 6 – 1 – 11 8 10 11 5 8 7 8 ] to include also glueball operators which can mix with the meson 13 3 , = 1 SYM to form supermultiplets of degenerate masses. Based on super- 2 1 18 N = 1 supersymmetric Yang-Mills (SYM) theory is the minimal supersymmetric ex- bound state of gluons and gluinos, called gluino-glue. The original analysis has been ] to be a chiral supermultiplet formed out of a scalar and a pseudoscalar meson and a ½ N 1 4.4 Number of stochastic estimators 3.4 Smearing techniques 4.1 Jacobi smearing 4.2 Presmearing of4.3 the gauge fields Variational operator basis 3.1 Computation of3.2 the masses using Interpolating the operators 3.3 variational method Stochastic estimator technique (SET) extended in refs. [ operators carrying the same quantumare numbers. therefore created The by scalar linear andmass combinations pseudoscalar of hierarchy bound meson-like and states and glueball-like theeffective operators. amount Lagrangian The and of are mixing in are general unconstrained difficult by to supersymmetry. predict theoretically from the tension of the puretheir gauge fermionic sector superpartners, of the QCD, gluinos.bound describing states Unbroken the of supersymmetry strong requires interactions the ofsymmetric physical effective gluons actions, and the lowest-lying supermultipletref. has [ originally been proposed in spin- 1 Introduction The 7 Mixing between glueballs and mesons 8 Conclusions 5 Extended variational basis 6 Supermultiplets 4 Optimising the methods 3 Techniques for the determination of light states Contents 1 Introduction 2 Supersymmetric Yang-Mills theory on the lattice JHEP04(2019)150 ]. The results reveal the expected 8 – 4 ]. . The physical results, the improved 9 5 , and 4 , – 2 – = 1 SYM in the continuum and on the lattice is 3 N = 1 SYM on a space-time lattice are required to verify = 1 SYM. An alternative determination of supersymmetry = 1 SYM with SU(2) gauge symmetry has been the main N N N . 7 and 6 This paper is organised as follows: It should be noted that our studies, and corresponding investigations in QCD as well, In this work we have optimised our methods for SYM with gauge group SU(2). The In addition to our results on excited states, we confirm our earlier results concerning the The formation of supermultiplets at the level of the excited states is strong evidence for In the present work we have optimised our techniques to extract the masses with respect Numerical simulations of introduced in the next section.aspects, which The current are paper explainedsignal is in for particularly sections focused the on groundsections the states, technical the mixing, and the excited states, are finally presented in same methods successfully topublished SYM soon. with gauge Our group techniquesgations SU(3), are are for also being which applicable done. the to results lattice will QCD, be where similarare investi- quite challenging due tomeson the contributions. noisy signals from glueball operators and the disconnected mixing of glueball and mesonthe operators glueball and in the the meson groundof contents. state, the Eventually and conjectures this we concerning might are lead the able to low to a energy determine better multiplets understanding oftechniques the are, theory. however, also suitable for other theories. Currently we are applying the on the lattice. Thespectrum of present excited work states represents of theviolations beyond first the step ground state towards levelin an would terms be investigation the of of weighted the sum the derivative of of all energy the differences Witten index, see [ results concerning the possible formation of a supermultiplet of excited statesthe on the fact lattice. that the unavoidablecontrol. breaking The of states supersymmetry with on higherthe the masses predicted lattice are degeneracy can of affected these be stronger states by kept is lattice under more artefacts, sensitive and to hence the breaking of supersymmetry ators, is crucial to extractsystematic the errors physical stemming states from in excited the stateWe reanalyse contributions scalar can the channel. be gauge Using controlled configurations much these ofmore better. techniques precise previous results investigations for yielding the more lightest reliableexcited masses. states and in In the addition, scalar, we pseudoscalar, are and able fermionic to channel. extract the In this masses way of we provide first compatible values, which hints at mixing of the statesto in our this previous channel. work.the This two allows lowest for supermultiplets the from first thearticle lattice time we gauge to explain ensembles these investigate we the methods haveconstruction mixing generated. and of properties their an In enlarged optimisations. of this correlation We matrix show, including in both, particular, the meson that and the glueball oper- and extend the analyticalthe predictions about lightest bound the low statessubject energy of effective of theory. several publications Thedegeneracy by study our of of collaboration the [ membersthe of scalar the glueball and chiral the multiplet scalar in meson the masses, extrapolated continuum to limit. the continuum In limit, particular, have JHEP04(2019)150 → a (2.1) defined 0 w used for the a ] and is corrected ], chiral symmetry 17 10 , a λ a ¯ λ g 2 m where the “a” indicates the adjoint + 0 ]. A mild sign problem of our lattice a η ) λ 16 0. The lattice spacing , µ 4 D ( → µ γ g a – 3 – ¯ λ ]. The extrapolation to zero lattice spacing 1 2 13 + a µν F ], and to extrapolate to the point where it vanishes. For ] for further details. a 6 ]. In order to suppress lattice discretisation effects, we use µν . The gluino mass term breaks supersymmetry softly and 12 c F λ 15 1 4 = 1 SYM in Euclidean space-time, , b µ to be a Majorana fermion field in the adjoint representation of A = N 14 , λ L abc 8 g f ]. Another way is to employ the adjoint pion mass, which is defined in of the + a 11 L λ µ ∂ and the pseudoscalar meson is called a- , which effectively removes the doublers from the physical spectrum, but at 0 W = f D a ) = 1 SYM is asymptotically free, and the running of the strong coupling has been λ bound state of gluons and gluinos, called gluino-glue, which has no analogy in QCD. µ ½ The production of the gauge field configurations is performed with the two-step poly- N On the lattice, the fermion part of the action is discretised using the Wilson-Dirac D the Wilson-Dirac operator, see [ nomial hybrid Monte Carlo (PHMC)formulation algorithm arises [ fromby the reweighting. integration of the Majorana fermions [ investigated non-perturbatively in ref. [ 0 corresponds toextrapolation the to weak the continuum coupling limitfrom is limit the measured gradient in flow terms [ a of tree the level infrared Symanzik scale improved action together with stout smearing on the gauge links in non-zero lattice spacings, lattice artefactsminations. introduce a Because small the mismatch adjoint between thesewe pion deter- use mass this can be quantity to measuredthe define quite point precisely where our and supersymmetry extrapolations effectively, should to be the restored chiral in point. the This continuum corresponds limit. to the same time explicitlyand breaks supersymmetry chiral can symmetry. bethe As recovered bare simultaneously shown in gluino in the mass. ref.termine continuum [ the limit This point by tuning tuning where canWard of the be identities renormalised [ achieved gluino in massa different vanishes partially ways. from quenched the approach One [ supersymmetric way is to de- same quantum numbers. In additionspin- to these particles, the low-energy spectrum contains a operator the renormalised gluino mass mustthis be work tuned we focus to on zerospectrum the to theory of recover with the SYM gauge full group areconstituent supersymmetry. SU(2). similar fermions In The to are light gluinos the mesonsis and in mesons called the not in a- particle quarks. QCD, Due apartrepresentation. to from The this mesons the are similarity, difference the expected that to scalar mix the meson with the corresponding glueballs with the is similar torequires the the one gluino of field the one-flavour gauge QCD. group. Correspondingly Theby the ( main gauge covariant difference derivative is is the that adjoint one, supersymmetry given The Lagrangian 2 Supersymmetric Yang-Mills theory on the lattice JHEP04(2019)150 ] 18 (3.5) (3.1) (3.2) (3.3) (3.4) = 0 or 0 t . In ref. [ ) n ( } i ~v . O { 3.3 , ) 0 t − t ( n , ) satisfy . m ) 2. This, however, leads to very 0 n ) ∆ ( t/ 0 − . , t, t ) ~v e ( , t | 0 ) ≥ ) E 0 n n t 0 ( t, t ( t m O + 1 ( (0) λ ) † C j t n − ) ( ( ) O 0 l 1 + ) n λ t ( m ( t, t | λ i ) ( 0 ) ] to use n t O n – 4 – 6= ( − D l t 19 λ ( n = min ) = ln = m ) = 0 t n ) − and their corresponding eigenvectors ( n must be chosen in order to extract information about e ) ( t, t m ij ( i n ~v ( ) ∝ ∆ C ) O n λ t ) ( eff ( 0 m C t, t ( ) we explain in detail the techniques and the tunings used within n ( 5 λ for the correlators considered here and we therefore use to ) n →∞ lim 3 ( t λ have the same quantum numbers under the lattice symmetry group as i O ), are the masses of the physical states. In order to suppress contributions of higher t ( n C m The mesons in SYM are all flavour-singlet ones, which require techniques for the mea- Our approach is to use a set of basic interpolators of the physical states to which = 1 when necessary. A first estimate can be obtained from the effective mass 0 excitations, it has beennoisy suggested in eigenvalues ref. [ t with where matrix provides generalised eigenvalues it has been shown that the generalised eigenvalues of these operators. Tothe extract correlation the matrix masses is of built physical from states the using time-slice the correlators variational of method, the set The solution of the generalised eigenvalue problem (GEVP) associated with the correlation A set of interpolatingphysical operators states within thethe variational operators method. Inthe principle physical state the of onlyinfluence interest. requirement on In the is precision practice that of the the choice results, of and therefore this we variational explain basis in has detail a the optimisation crucial surement of all-to-all propagators.mation is We optimal have for found our that lattice a volumes.3.1 preconditioned This stochastic is explained esti- in Computation section of the masses using the variational method and parameters. we then apply smearingmethod. techniques In sections to createthis approach. the operator basis used in the variational In order to gain insights intogluino-glue the particles structure as of well the as low mixtures energyas of spectrum well mesons of as and SYM, the glueballs, consisting glueball of wequantities determine and can their meson masses be content in reliablyvariational the extracted scalar method from and for the the gauge pseudoscalar which states. field we ensembles These have using systematically the optimised well-known the required techniques 3 Techniques for the determination of light states JHEP04(2019)150 (3.6) (3.7) (3.8) (3.9) (3.10) , ) to an 0 ) t, t ( ) y, x n ( ( 1 . ] λ − W ν -glueball. D , γ + µ − )Γ γ . [ is the parity conjugate ) i 2 x x, y , C , ( ( 1 = λ )] P 5 )]] − W x γ ( x µν ) D ( σ 31 x C Γ ( P , and P ¯ λ h O ) + Tr [ (spin and group indices suppressed) 2 Tr ) = x with ( y x − − k ( 23 0 )] η ) i to P x ) a- ( x x C ( y, y ) + ( β 1 x λ ( ) . – 5 – − W [Tr [ x 12 plane. For the pseudoscalar glueball we use h 1 D ,O ( j P ) Γ O ij j i x - ∈ X P x ( i ) used in the interpolating field of 0 R h λ x Tr ) ( x Tr ) ( ) = ) = Tr [ ijk ¯ λ x αβ ij C x ( ( σ x, x + ( ) = − 1 ++ =1 x 3 − W ( gb gb X 0 . . i ], i η 20 (3.15) (3.13) (3.12) (3.14) (3.11) , since 5 2. The γ √ and eigen- / i i) λ , is prohibitively . | 1 i i η W u ) and for the final ). t t 1 D ih ( ( i − p 1 4 γ w C D | i C a = S . i N s + =1 . S i ) and X E ) both provide valid correlators t t ( N N αβ lowest eigenvalues 4 ( 1 4 γ . p ) γ = C E with . t / C | ( N 1 i 4 i ⊥ , v γ η C ih O i S v + | + i ⊥ N i are computed. The truncated eigenvector s 1 1 λ αβ p p δ / S E i = ) D 1 =1 – 6 – t N X N i 5 X ( i γ 1 S η complex numbers of the form ( ≈ 1 O C N 4 1 + i Z − p + η ) = i D | t η i by using the stochastic estimator technique (SET) [ ( S i v are ) and to the even part N 1 X t αβ gg ih i ( − W i i s 1 S C v 1 η | D C N i S 1 i λ N X E S =1 N i 1 X N ≈ = 1 we use its Hermitean version, obtained by multiplication with is then approximated by 1 − p p − p 1 D D is calculated using a conjugate gradient solver. − p D of the Hermitean operator D i ] for further details. The preconditioned approximation converges faster to the i η i 21 v | 1 − p D The two approximations are easily combined: the noise vectors of SET are just pro- For the truncated eigenmode approximation the The idea of SET is to solve the Dirac equation on a set of source noise vectors jected to the subspaceend orthogonal both to contributions the are onebetter summed, spanned condition gaining number by of also the the a lowest projected eigenvectors. speedup operator. of In the the inversions due to the approximation of the inverse matrix is and vectors The entries of thepropagator vectors the singular value decomposition provides a better approximation. fulfilling the relation which is improved bying, a see truncated [ eigenmodeexact approximation result and and even-odd the numerical precondition- is computation of much the faster eigenspace of thanDirac the the matrix preconditioned matrix one of the full matrix. For the inversion of the preconditioned 3.3 Stochastic estimatorThe technique calculation (SET) of correlators includingerators more requires than the estimation one of gluino thea field fermion complete in propagator from the determination all interpolating of to op- the allexpensive, lattice inverse we points. of approximate Since the Dirac-Wilson operator Projections to the odd part to be used inresults the we use GEVP. a weighted For sum the of tuning the of results our from methods both we use The gluino-glue correlation matrix consists of an odd and an even part under time reversal JHEP04(2019)150 ] . F 23 is a 4 J (3.20) (3.16) (3.17) (3.18) (3.19) and C 1 with the − W 1 D − W , D † 4 for smearing . F . The eigenspace . = 0 i ) 4.4 , i ˆ ! . − APE ) . ~x ( 3.2 † ~x,~y ( † i,ba ˆ F ˜ | ba U i ) i ˆ i u . − † H i h ( ~y,~x i F δ J 1 δ. w =1 | † N i − W X F F is a Jacobi smearing coefficient, ) + i 1 + ~x a J − FD W ( κ S ~x,~y = N δ i,ba FD + 1 ˜ i,~x 5 for smearing the glueball interpolators. U X . – 7 – E i − W ˆ = + N ˜ βα D δ δ = 0 1 ~y,~x J δ − W → N J Tr h ˜ 1 D C APE − W ≈ 3 =1 i X i D ) are APE smeared adjoint gauge links. The tuning of the 1 ) = J x can be computed exactly. However, in order to simplify the ] on the gauge links in order to create smeared gluino-glue − ( W κ i ˜ ee 22 ˜ D U and the smearing of the gauge links is explained in section ~x,~y h ( M ) = J Tr C ~x,~y βb,αa ( F ba and H J κ , ) with the smearing operator J N 3.15 of the Wilson-Dirac matrix is just the identity. More generally, the inverse of F λ is the integer Jacobi smearing level, = ee J ˜ λ N M Smearing the fermions fields is equivalent to replacing the propagator For the construction of the fermion source we use gauge invariant Jacobi smearing [ We use APE-smearing [ We have optimised the parameters of this approximation in order to reduce the noise For the case of plain Wilson fermions, investigated in the current study, the even-even → The connected piece isare calculated subsequently using applied: standard delta sources on which For the disconnectedequation piece, ( this translates to smearing the source and sink vectors in Here normalisation constant and parameters smeared propagator with and smeared glueball operators.the gluino-glue We interpolators use and a smearing parameter λ 3.4 Smearing techniques The different interpolating fields used in thesmearing variational techniques method to are constructed the by basic applying interpolating fields defined in and to speed up thea computations. reliable The estimation tuning of ofof the the the disconnected number preconditioned piece of matrix is noise computedis vectors discussed in required also in the for section used measurement of formeson disconnected or a the contributions gluino-glue deflation correlators. ofdependent, Concerning e. speedup, the g. the inversions optimisation our is in runs also on other machine KNL measurements, based like machines require the quite connected different parameters. part the block diagonal matrix smearing procedure, we have usedinverse a of large this number of matrix. stochastic These sources can to be approximate the computed very efficiently. JHEP04(2019)150 5 . 75 . (4.2) (4.1) . We = 0 of the 2 = 1 β APE APE N , . The efficiency J = 20 C . For values smaller c J and APE κ meson, see figure is chosen such that the = 80. The results are , while for values larger J N 0 J f κ J C N → ∞ J . N , the smearing level J J N C , J κ between 10 and 80 all feature sufficient . , 2 87 . . APE lead again to a degradation of the signal, see = 0 N = 0 – 8 – J J κ C APE N ]. The normalisation factor converges in the limit 24 F , and values of . Using Stout smearing instead of APE smearing didn’t improve = 100 [ 3 and the Jacobi smearing levels J ] there is a critical value of the parameter , APE N . With this choice the smearing radius is less than 6 lattices spacings 2 N ˜ 24 U c J κ 14925. There are in total 6800 thermalised configurations of which we have mea- . it diverges. In order to smear efficiently and at the same time to avoid large , the smearing operator c J c J = 160 in figures = 0 κ κ κ For the tuning of the smearing levels we have used SU(2) gauge ensembles at In principle, the final results do not depend on the values for As explained in [ APE leads to anot noise very reduction sensitive for to noise Jacobi suppression. smearing levels LargerN values up for to the noise suppression in our tests and we therefore stay with APE smearing. The application of Jacobitional smearing noise to defined the signal with of unsmeared thehave disconnected observed gauge contribution that to this links the noise a- introduces cansmearing. addi- be We suppressed have by tested using APE smearedfor and gauge the Stout fields in smearing preparation the together of Jacobi with the different smearing gauge levels field. Our results indicate that values stated here. and sured every 64th for our tests. We use 404.2 stochastic estimators for the Presmearing disconnected of pieces. the gauge fields could, however, depend oncould the re-tune them choice for of everywith ensemble. the gauge In values group our of SU(2) experience and theit from SU(3), smearing different is the ensembles parameters dependency not of on and SYM necessary these one to parameters is always very find mild, and the optimal values. We have therefore kept fixed the which is just above up to smearing levels values of the correlation functionssmearing stay levels. at the We use same order of magnitude for large and small than than numerical errors we choose a value 4.1 Jacobi smearing Aiming to achieve aatically good optimise signal-to-noise the Jacobi ratiosmeared smearing for gauge parameters the fields meson measurements, we system- Significant sources of possible systematicing errors techniques, are the non-optimal operator parametersdisconnected basis of propagator in the pieces, the smear- and variationalwe method, the discuss the fitting their stochastic methods influence estimation and and optimisation. of parameters. In this section 4 Optimising the methods JHEP04(2019)150 . 0 80 t =80 =40 in the = 10 APE APE ˜ 70 t 12] plot- U N N , [3 = 20. Using . 60 0 8 ∈ J t t N = = 20 is chosen, correlator (right) 50 t =20 =0 0 f 6 APE APE J 40 N N fitted APE t N N 30 4 20 =160 =80 =40 =20 =10 =5 =0 ). Higher Jacobi smearing levels 2 APE APE APE APE APE APE APE 10 80. N N N N N N N 3.17 ≤ 0 correlator using different levels of APE 0 0 η APE 045 040 035 030 025 020 015 010 1 2 7 6 5 4 3 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 N eff > ) t ( C ∆ < am ≤ ). However, when smeared gauge fields are 3 – 9 – correlator averaged over the interval 80 80 correlator. It does not suppress the noise that is 0 =160 =80 η 0 APE APE 70 70 f correlators, the smearing level N N 0 in the smearing kernel ( η 60 12] plotted against the Jacobi smearing level using different 60 ˜ U , [3 50 50 correlator, using smeared gauge fields in the Jacobi smearing ∈ and a- =20 =40 0 t J 0 J APE APE η 40 f N 40 N N N 30 30 ). Right: effective mass of the a- 20 3.17 20 =10 =5 =0 =160 =80 =40 =20 =10 =5 =0 APE APE APE APE APE APE APE APE APE APE 10 N N N 10 N N N N N N N as long as it is in a suitable range 10 . Jackknife-error of the disconnected piece (left) and of the full a- . Left: jackknife-error of the a- 0 0 000 005 010 015 020 025 030 035 040 00 05 10 15 20 25 30 35 ...... APE Considering both a- In the case of the a- 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 > ) t ( C ∆ < > ) t ( C ∆ < N used instead of unsmeared ones,contributions to Jacobi the smearing correlator. more effectively Again,of suppresses the excited results state are not very sensitive to the exact value as it appears to be a good choice for suppressing noise and excited state contributions. has a different effectintroduced than by on Jacobi the smearing a- (see figure Figure 3 ted against the Jacobismearing smearing kernel level, ( usingsmeared different gauge smearing fields levels in for thesmeared the Jacobi gauge gauge-field smearing. fields The morewith Jacobi effectively unsmeared smearing suppresses gauge level excited fields is state fixed (blue). contributions to The than correlator Jacobi has smearing been normalised to 1 at averaged over thesmearing interval levels for theare gauge-field affected by larger errors.significantly. Using The smeared gauge disconnected fields piece in and the Jacobi the smearing full suppresses correlator this have error been normalised to 1 at Figure 2 JHEP04(2019)150 are significantly higher than the t are used as estimators. Note that we 2 additional smearing levels uniformly t smearing levels of the full set are taken − n n smearing levels of the full set are taken into n – 10 – ). The results from the described procedure are shown 6 here to demonstrate the suppression of excited states by t , where Jacobi smearing is used for the meson interpolators } 95 of operators we have picked the following sets: n ,..., together with the best estimate for the mass obtained by fitting the 15 , = 35) and the largest one, and 6 5 , 2 additional smearing levels uniformly distributed in between. 0 − and n ∈ { 4 Smearing N ( distributed between them. account. and into account. Mid-large, uniformly distributed: out of the full set we chose a medium level Large smearing levels: only the largest Uniformly distributed: out of the full set we chose the smallest and the largest one Small smearing levels: only the smallest The results for the dependency of the effective masses on the smearing levels is similar To judge the quality of a mass determination with a chosen correlation matrix, the For each number 4. 2. 3. 1. Smearing these excited state contributions.state The contributions, uniformly but distributed not subsets quiteof as also much the suppress as excited effective the large masseslevels. smearing is levels. We almost Note conclude that constant that themass for it error estimations. the is different optimal combinations to of use the a smearing set of rather large smearing levels for the eigenvalues using the full set of interpolating operators. for all three particles: smallesthigher smearing excitations levels so obviously lead that tobest the unwanted estimate. effective contributions of masses Using at only the small largest smearing levels leads to the strongest suppression of effective masses of theuse lowest a two states rather atchoosing small fixed relevant value operators. of The fittingchosen intervals more for carefully the (cf. final section in extraction figures of the masses are most efficient estimation of the lowest two states in each sector. to find those smearingFor this levels purpose which we areN consider most the relevant for setand the of APE smearing extraction operators is of constructedwe used the have from for systematically the masses. the chosen glueball different smearing and subsets levels gluino-glue to interpolators. analyse which From smearing these levels sets allow the A variational basis forterpolating the operators. GEVP canvariational There be basis from is built many a different from smearingsuch trade-off different levels operators. and smearing between Each the levels new the gaincorrelator of of smearing cost new and level the information required requires for from in- additional the to inversions connected for build piece the a of gluino-glue the large meson correlator. Therefore it is important 4.3 Variational operator basis JHEP04(2019)150 ) t ( 1 C 10 ground state excited state 8 mid-large uniformly distributed uniformly distributed 6 . Therefore the variational number of operators 4 λ Γ ¯ λ 2 small smearing levels large smearing levels fitted 5 4 3 4 2 0 8 6 ...... 0 0 0 1 1 1 0 0 eff eff am 2) = t ( am 2) = t ( – 11 – 10 ground state excited state 8 uniformly distributed mid-large uniformly distributed 6 4 number of operators (right) from the GEVP based on correlation matrices of different sizes. Results for 0 η 2 channels the relevant operators can be constructed from glueball-like combina- + ) can be enlarged to include the most general mixing between glueball and me- . Effective masses of the lowest and first excited state of the gluino-glue using only − , and the scalar correlator requires more stochastic estimators than the pseudoscalar fitted large smearing levels small smearing levels 0 3.2 5 or 0 9 8 7 6 5 . . . . . 28 26 24 22 . . . . 0 0 0 0 0 eff 0 0 0 0 3) = t ( am eff 2) = t ( am ++ tion properties, independently of0 their fermion ortions gluon of field-content. Wilson loops In andbasis the meson-like ( case operators of of the the son form operators. 5 Extended variational basis In general, an operator transforminglattice according symmetry to group a has given awith irreducible non-zero same representation overlap quantum of with numbers. the all the Mixing eigenstates occurs of if the two Hamiltonian operators share the same transforma- figure one. At aroundstochastic 20 error stochastic is smaller estimators than the 15%for of error the the unsmeared is gauge scalar noise) dominated correlator. for by all the tested smearing gauge levels noise except (the With the optimised Jacobi smearingof parameters we SET also estimators tested the onevery influence fourth the of configuration the error of number of ourof the test stochastic estimators ensemble. disconnected where The the pieces. error aimthe from is gauge For the to noise. stochastic this find estimators We estimation is a find much we value smaller that for than used higher the smearing number levels require less stochastic estimators, see the matrices from the least-smearedoperators operators are are shown shown in in blue. orange, The results final from result the of most-smeared the fit4.4 to the eigenvalues is Number shown for of comparison. stochastic estimators Figure 4 (left) and the a- JHEP04(2019)150 10] , 40 =40 =20 =3 =0 (5.1) ) are [3 J J J J N N N N ∈ 5.1 35 t 30 25 . stands for meson-like 20 E , which means that the E (0) 7 15 (0) meson number of stochastic estimators O † meson 10 † meson O ) O t ) ( t 5 ( gb 200 175 150 125 100 075 050 meson ...... O 0 0 0 0 0 0 0 O > ) t ( C ∆ < ED ED – 12 – (0) 40 (0) =0 =3 =20 =40 † gb J J J J † gb O N N N N ) O t 35 ) ( t ( In this channel the estimation of masses does not improve gb . Using more than one glueball or more than one meson 30 meson O the results are similar. The disconnected pieces are normalised to 6 D O t D 25 The results of our calculations show that the enlarged variational stands for glueball-like operators and 20 ) = t channel. ( gb + ) is a submatrix consisting of the correlators among different interpolat- C O 15 − t number of stochastic estimators ( C channel. 10 . ++ . Jackknife error estimates of the disconnected piece averaged over the interval 0 5 t = 05 10 15 20 25 30 35 40 ...... t Taking both kinds of operators into account, the full correlation matrix has the fol- 0 0 0 0 0 0 0 0 > ) t ( C ∆ < when the enlarged basisanalysed also sufficiently includes by glueball usingglueball only operators. operators meson do The operators, notobservation lowest see mix that two significantly figure the states with entries canzero these in (within be states. the rather off-diagonal large This blocks errors). is of in the accord correlation with matrix the ( operator does not improve thisWe estimation, therefore but provides conclude better that accessthe it to variational the is basis excited crucial to states. reliably to extract include the both ground meson statePseudoscalar and in 0 the glueball scalar operators channel. in Scalar 0 basis leads to a moreor effective suppression glueball of operators excited states alone.glueball contributions operator than using Even appears meson to thethe be minimal sufficient scalar choice to channel, of extract see the using figure masses only of one the lowest meson two and states in one Each entry of ing fields, where operators. Since the mesoncomponents, and it the is glueball expected operators that arematrix their constructed we mutual from expect overlap quite is to different small. obtain significantly Using improved this signals. larger correlation for the scalar (left)estimators. and For the other pseudoscalar values of 1 (right) at meson as a function of the number of stochastic lowing form Figure 5 JHEP04(2019)150 40). , of the 10 ) 20 n , ( 3 λ , M:40; G:48 M:20,40; G:48 channel (in 8 . 8 ++ = (0 . J 1 N 6 t -channel are accessible ++ 4 ), and the use of smeared gauge 2 t ( 4 γ ]. The parameters of the configura- C 5 95) and the glueball smearing levels 1 2 3 4 0 − − − − 10 10 10 10 10 λ ) n ( ,..., 15 , – 13 – = (5 8 APE 7 N ground state excited state 6 5 large smearing levels large smearing levels 0 ++ f 4 gb a- 64). Note, that in contrast to the analysis above, we used a more 3 number of operators -channel. In orange the results from using a mixed basis including one meson 2 ++ ,..., can be used as the lower boundary of the fitting intervals. Consequently, we 8 fitted Mixed , 1 min t . Left: the ground state and the first excited state of the 0 = (4 0 9 8 7 6 5 For this investigation the full set of smearing levels under consideration consists of The main improvements compared to the previous analysis are the use of the variational . . . . . 45 40 35 30 25 . . . . . 0 0 0 0 0 eff 0 0 0 0 0 2) = t ( am APE eff 2) = t ( am we were also able to extract the masses ofthe the gluino-glue first excited smearing states, levels N see figure conservative choice of smearing levels for the meson operators, namely the inclusion of the additionalfields gluino-glue in correlator the definitionfrom of higher Jacobi smearing. states tosuppressed Due so the to that these correlators the improvements, and effectivevalues the mass of noise contributions plateaus stemming can from benot estimated the only obtained more Jacobi more reliably reliable smearing and and are smaller more precise estimates of the ground state masses, but gauge configurations of ourtions, previous on investigations which [ the present investigation is based, aremethod compiled in in all table three channelsprevious including glueball investigations and the meson variational operators method in was the 0 only applied to the glueball analyses), signal for the ground state and the first excited state, but a signal6 for a second excited state Supermultiplets appears. Using the methods explained above we have reanalysed the low-lying spectrum using the glueball operators (blue) doesn’t improveoperators the projection are to used the state. (orangecompared to If and only the green) meson mixed or excitedGEVP basis. only contributions in glueball Right: are the shown 0 notoperator are as (smearing the effectively first level suppressed threea 40) as generalised further and eigenvalues operator one (mesonic, glueball smearing level operator 20) (smearing to level the 48) mixed basis are (also shown. blue) doesn’t Adding change the Figure 6 with the mixed basis of one meson and one glueball operator (blue). Adding further meson or JHEP04(2019)150 for min t 5 of the . 7 14415 e. g. . ≈ . The errors 0 η = 0 d.o.f channel since this / , κ 2 9 χ . 40) and two glueball ++ , separately. = 1 + − β = (20 J 10 N d.o.f of the fit. Using the mixed / 2 excited states ground state χ 8 = 3 for the excited state in the scalar M:20,40; G:48 6 min t t ∆ 1433. This resulted in a . and the glueball gb 0 – 14 – . For the ensemble η 0 4 = 0 η fitted M:20,40 -channel from a basis consisting of two mesonic operators , κ 9 + . 2 − = 1 = 7 for ground and excited states of the a- β 0 min 6 8 0 2 0 1 2 3 4 5 6 t ...... 0 0 1 1 0 0 0 0 0 0 0 eff eff am am 48) in this channel. As explained above, including the glueball , = (24 = 4 for the ground state and AP E N min t . Effective masses in the 0 Due to the increased precision, we observed that in the chiral extrapolation the gluino- The fit intervals were determined at each ensemble individually based on the formation glue mass deviated stronglylargest from adjoint the pion assumed mass linearlinear at behaviour fit to at the the chiral limit. ensemblealso with contributes We therefore to the neglected the this fact data that point the for the updated gluino-glue. result for This the gluino-glue mass is smaller than basis of glueball andthe meson operators scalar allowed channel to compared includewe smaller used to time the separations a- channel whereas we used were obtained statistically usingin the order standard to Jackknife properly procedure take together into account with autocorrelations. binning operators operator in the GEVPhave for analysed the the pseudoscalar pseudoscalar channel meson does a- not improve the results,of thus effective we mass plateaus and the stabilisation of the It is preferable toassures use that not the more eigenvaluesand than of additional four the operators smearing do GEVPtherefore levels not are use in reduce clearly combinations the the separated of 0 contamination in two by meson the excited operators, states region further. e.g. of the We fit level 48) (orange).glueball operator The is estimation included.basis of In are the the shown ground upper (orange).(it state panel aligns the They (lower well first are panel) with andglueball of the doesn’t ground second excited similar change state. excited state mass. states when of of the the Apparently the purely one mixed mesonic is basis the (blue)), excited and meson the state other one is the Figure 7 (smearing levels 20 and 40) (blue), and a basis including an additional glueball operator (smearing JHEP04(2019)150 ]. Such 25 (1) 0 η 5 . (1) − 0 ++(1) 0 a gg 4 . 0 (0) 0 -glueball ground state and the excited η +(0) + (0) − − ++(0) − ] the extracted ground state masses turn gg a 0 gb 3 . 5 0 0 a/w – 15 – 2 . 0 1 . 0 gluino-glue channels as explained in the text. + 2 0 . / 0 0 5 0 5 0 5 0 5 0 ...... 0 0 1 1 2 2 3 3 4 0 m w and 1 + . The ground state masses, extrapolated to the continuum limit, agree − 2 both have similar masses, but they appear as two independent states in , 0 0 1433 in the chiral extrapolation. η . ++ = 0 . Extrapolation to the continuum of the masses of the ground states and first excited , κ 9 . Compared to our previous investigations [ In the continuum limit, the two-particle decay threshold is well below the mass of The results for the masses, extrapolated to the chiral and to the continuum limit, are = 1 out to be slightlyerrors smaller. of The the differencescorrect to old the estimation results, old of which resultsβ are the can be now fitting viewed much intervals as better and systematic under the control. exclusion of This the includes gluino-glue the mass at particles interacting on athe periodic resonance box, properties, fulfilling such acalculations as quantisation the condition require, phase that however, shift, determines sectors a of the within very excited the good supermultiplet currentapproximation [ precision statistics. are that already a The is good excited out estimate energy of of levels the reach within mass in of the the the single-hadron excited singlet supermultiplet. the excited supermultiplet,allowed. meaning that Since decays theoverlap into with operators particles two-particle presently of states,channels used the the would in full ground require consideration the state tosuch of GEVP as are include the two very glueballs in effects with likely our of oppositespectrum have momenta the variational would or only strong basis a be glueball small decay also enlarged and a and multi-particle gluino-glue. would operators, The include resulting also energy levels corresponding to two- excited states also seem totimes form the a ground mass state degeneratestate mass. supermultiplet of Interestingly of the the approximately a- 0 three the variational method which do not mix. the one from ourinstead, previous which investigation. gave We a also consistent tested result, to but use a a slightly second largercollected order error. in polynomial table quite well with each other within errors. Similarly to the ground states, the respective Figure 8 states in the 0 JHEP04(2019)150 ++ (0) 0 am ) m ++ (1)( 0 c 0 η (0) a- ) am g ++ (1)( 0 c (0) gg ) m am ++ (0)( 0 c π ) a- g ++ – 16 – am (0)( 0 c ++ #Conf (1) 0 am 64 10648 0.1498(32) 0.2255(72) 0.1978(50) 0.204(19) 64 20704 0.17780(63) 0.2482(44) 0.2126(43) 0.288(20) 64 10144 0.21330(92) 0.2842(25) 0.2447(58) 0.204(21) 64 10176 0.28694(42) 0.3312(46) 0.3165(34) 0.336(24) 64 1692 0.1263(65) 0.3347(75) — 0.287(59) 64 4956 0.1615(14) 0.3575(57) 0.258(23) 0.376(22) 48 10228 0.1897(26) 0.336(21) 0.265(20) 0.346(45) 48 6428 0.1968(20) 0.3714(69) 0.270(17) 0.330(30) 48 9816 0.2026(18) 0.3709(90) 0.292(14) 0.395(24) 48 9300 0.2380(14) 0.4043(88) 0.319(12) 0.463(15) 48 1612 0.201(10) 0.633(31) 0.378(43) 0.458(68) 48 1324 0.3208(38) 0.730(16) 0.459(35) 0.540(75) 48 1404 0.5757(41) 0.929(28) 0.626(13) 0.98(12) T × × × × × × × × × × × × × × 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 η (1) a- am ]. For consistency we use only the ensembles with one level of stout smearing. In 6 – 4 (1) gg . Summary of the ensembles, masses of the lowest (superscript 0) and excited (superscript 1) am β κ L M 0.505(14) 0.589(30) 0.435(19) 0.789(41) 0.577(45) 0.581(48) 0.793(31) L 0.503(13) 0.497(27) 0.440(15) 0.797(37) 0.543(36) 0.499(40) 0.828(16) K 0.560(10) 0.550(48) 0.451(16) 0.927(19) 0.352(27) 0.320(37) 0.901(11) J 0.618(12) 0.708(15) 0.524(12) 0.920(13) 0.316(17) 0.266(34) 0.903(10) I 0.695(75) — 0.69(10) 0.49(11) 0.686(55) 0.754(58) 0.685(76) H 0.725(31) 0.688(35) 0.673(38) 0.650(84) 0.702(31) 0.70(10) 0.734(25) G 0.621(44) 0.669(29) 0.688(58) 0.715(93) 0.654(39) 0.700(54) 0.732(30) F 0.663(39) 0.764(21) 0.645(29) 0.718(67) 0.516(38) 0.523(44) 0.827(19) E 0.756(23) 0.689(28) 0.715(44) 0.685(98) 0.604(35) 0.591(79) 0.784(24) D 0.751(31) 0.724(24) 0.741(31) 0.602(71) 0.615(20) 0.787(75) 0.750(21) C — — 0.65(25) 0.822(77) 0.586(74) 0.63(17) 0.68(13) B 1.15(16) 1.077(60) 0.84(17) 0.82(22) 0.616(91) 0.44(11) 0.853(30) A 1.31(11) 1.398(46) — 0.42(25) 0.71(13) — — ID M 1.9 0.14435 32 L 1.9 0.14415 32 K 1.9 0.14387 32 J 1.9 0.14330 32 I 1.75 0.14950 32 H 1.75 0.14940 32 G 1.75 0.14930 24 F 1.75 0.14925 24 E 1.75 0.14920 24 D 1.75 0.14900 24 C 1.6 0.15750 24 B 1.6 0.15700 24 A 1.6 0.15500 24 ID some cases the data could not be extracted, because the fit intervals could not be determined reliably. Table 1 states in latticepresented units, in [ and mixing coefficient of the scalar channel. The ensembles have been JHEP04(2019)150 i ), c n by | the κ + n − (7.4) (7.1) (7.2) (7.3) + = ~v to the − (0) gb denotes κ 3.4(4) 0.7(1) ni 0.50(5) 1.01(1) am c ++ ) g ( ∗ kj and a meson c ) i (1) gg g ) ( ki g c ( , am 3.1(2) 1.1(2) φ 0.46(1) 0.66(5) | k i is the vacuum state. ) X . The quoted errors are ) m i ( n 0 g m ( nj η Ω 0 φ | v | (1) a- w ) . The subscript 0 g 0 + ( ∗ ni 3.1(2) w i am corresponding to the glueball 0.46(3) 0.68(5) 0.93(9) v ) g n ( n ~v ij φ i | X the restrictions of the ++ Ω . i | (1) 0 = = the pseudoscalar meson and gb ) n . i | i 0 ) ) m ) 2.8(3) 0.5(4) am ( i η 0.39(2) 0.59(6) g m m ( j mn ˆ ( i ) the corresponding physical states ( i O δ φ ) φ φ | | ) h m 3.2 = g ) is the matrix formed by the column vectors ( ni , respectively, and ( i v ) m (0) gg = φ ni ( ni h m c M v m ( ) i ) =1 n i am – 17 – g X m ∗ O mi 0.93(7) 0.17(1) 0.62(2) ( m nj ( ni 0.312(9) =1 v n v c ]. From the calculated correlation matrices we can i X + ) g can be calculated as the vectors dual to the i 3 i ( , + ∗ ni X 0 Ω i 2 i | η , where v and ) n ) 1 | g (0) a- g i ( i i − ( i ij φ φ are not normalised here. n X ˆ | | h of the GEVP ( O am M ) 0.98(6) 0.19(3) 0.36(4) ) ) i 0.151(5) g ) g g = . n ( i ( ( ni ni = m ~v i φ v v ( ) h g φ ni g g ( n | c =1 =1 ++ n n i i φ = X X (0) 0 | ) ) g = = g ( n 1.1(2) and ( ni am 0.17(3) 0.25(4) 0.37(7) i are the components of the eigenvectors φ c i : h n ) ) i | g ) ( m = ( ni m φ and the meson operators ( v | 2 ) /a , φ ) | g 0 g ] i ( n ( i w n and | N 19 O 5.86(8) 3.41(2) 1.859(4) ) g ( . Ground and excited state masses in lattice units extrapolated to the chiral limit ( ni v The inner products From the eigenvectors β 1.9 1.6 1.75 . Let us denote cont. n glueball and the mesonbe components, obtained respectively. as The normalisations of the vectors can means of [ So they are the row vectors of where operators Note that get insights aboutglueball the content. nature of the physical statescan with be respect reconstructed tocontribution and their decomposed mesonic and into a glueball contribution 7 Mixing between glueballs andThe mesons numerical results of theball and extended mesonic variational states. approach Such indicateof a mixing the mixing between has lowest glue- been chiral predicted supermultiplets in [ the literature on the structure Table 2 and the extrapolations tothe the continuum result limit in in unitspseudoscalar the of glueball. mixed the The channel, scale massesstatistical. gg in the the continuum gluino-glue, limit are a- given as ~v JHEP04(2019)150 (7.5) (7.6) (7.7) (7.8) 6(1) and . . Therefore, in general do t = 0 as the overlap )2 ) i g m n ( ( n | c c 82(3) and a meson of the ground state . ) . ) ) m and ( n g m = 0 ni ( c )2 c ) ) g ) g g (1)( ( n ( ( ni c c 0.8(3) 0.4(2) 0.6(1) m and the meson contribution c c 0.62(5) ( and ) ∗ ni i g ) ) ( v g g ∗ ni ( n ( ) do add up to 1 within errors as ) v c i φ m m | . Extrapolated to the continuum X ( )2 i ∗ kj 3 n m and perform a fit to a constant over X c (1)( ( ) 0.6(2) c t c 0.77(4) 0.83(2) 0.91(7) N n ) m ( ki 1 N m c ) ( 1 n g ) ( and n g k N . X N )2 ) g = (0)( and table ∗ kj ( c 0.8(1) m = i c c 0.88(9) 0.83(3) 0.80(6) ( nj 9 i n v – 18 – ki | ) c n ) | m ) ) m ( g ( k ∗ m ni ( X φ v φ h h (0)( nj n ij c 0.57(7) 0.70(5) 0.59(4) 0.62(6) n v X N ∗ N ni ) ) 1 v = 1 g m ( ( n n i ) ij N N X m ( n = 1 supersymmetric Yang-Mills theory. We have used the vari- . . β φ = = = 1.6 1.9 | 1.75 ) ) ) N i g m n m ( n 62(6), meaning that the ground state in the scalar channel shows | ( n ( n . c continuum c φ n h h = 0 = = . Interestingly the squares 2 2 ) n 9 ) m N ( m ( n c 91(6). While the ground state is more glueball like, the excited state is more N . . Meson and glueball contents of the ground state (0) and excited state(1) in the scalar are not necessarily orthogonal to each other and therefore = 0 i We have determined the glueball and meson contents ) ) m ( m ( φ 8 Conclusions In this work webound have state presented masses in in detailational method our in improved all techniques three for channels, the including estimations a of combined basis of glueball and mesonic opposite glueball andc meson contents than themeson ground like. state, This namely spacing, is figure especially visiblein in the orthogonal the case. chiral extrapolation at our smallest lattice and excited state inas the in scalar the channel using analysislimit, again of we two the meson find masses, and that seecontent two the figure glueball of ground operators state hassignificant a mixing glueball of content glueball of and meson contents. The excited state appears to have the | not add up to 1.we calculate The the formulae glueball above and arethe meson independent region contents of at where the each the time values separation form a plateau to extract the final results. We would like to point out that the glueball contribution Now we define the glueballof and this the state meson with contents the of the glueball physical and state meson contributions Table 3 channel, extrapolated to the chiral limit. JHEP04(2019)150 ++ 5 . 0 in the 0 ) 97(4). . m ]. ( 4 . c 0 26 = 0 and (0) ) 3 ) ) . 0 g m g 0 m ( 0 c (1)( (1)( a/w c c w 2 . 0 9). Right: extrapolation of the . ) ) g m 1 . = 1 0 (0)( (0)( β c c 0 . 0 0 2 4 6 8 0 ...... 0 0 0 0 0 1 . – 19 – 08 . 0 1. The masses of these states in lattice units are around . 3 06 . 0 ≈ 2 ) π − (1) a ) in the gluino-glue channel. The new techniques reduce excited state t 04 m m . a ( 0 ( 0 4 γ w C ) ) ) ) 02 for the extraction of the ground state masses. We have reanalysed the gauge . g m g m 0 (0)( (0)( (1)( (1)( c c c c min t . Left: chiral extrapolation of the glueball and meson contents 00 , i. e. around half of the cutoff scale. It is quite unexpected that states at these . 1 0 − 0 8 6 4 2 0 We have investigated the mixing between glueball and mesonic operators in the scalar Another very interesting result is the possible formation of a chiral supermultiplet of ...... a 1 0 0 0 0 0 5 . a complete analysis a moreincluding detailed also investigation of other the multipletsadding, bound and for state instance, a spectrum an is larger investigation required, set of the of baryonic quantum states numbers.and of pseudoscalar the channels. We theory are In [ the currently scalar channel we have found significant mixing between states. The average mass of the lightest supermultiplet isexcited around states at 0 high energies are not more affected by the supersymmetry breaking lattice artefacts. For basis and smearing techniques. Westates have and combined the first results excited for statesthe the from continuum masses several limit. of different the The lattice ground The first spacings ground interesting in state observation an is masses extrapolation of theimately to the formation degenerate scalar, of when pseudoscalar, supermultiplets. extrapolated andwith fermion to our channel the previous become results chiral approx- and and indicates continuum the limit. formation of This a is chiral in supermultiplet of line lightest contributions and therefore allow morevalues reliable of estimations of theconfigurations fit-intervals and of smaller our previousstate masses. investigations and Furthermore, improvedstates, we our which have have obtained estimates been a of out first of the reach estimation ground in of previous the studies masses without of the optimised excited variational operators in the scalar channel.of The the enlarged mixing basis allows between for theseJacobi the two smearing, first classes time which of an now includes investigation operators.tional presmearing Furthermore, correlator of we the have gauge optimised fields the and included an addi- Figure 9 ground state (0) andoverlaps excited at state different (1) lattice for spacings our to finest the lattices continuum ( limit. JHEP04(2019)150 – 1 -charge R ) SYM and QCD-like c ]. On the other hand, N ] that the lightest scalar 29 27 , 3 , 2 -charge 0) and meson ( R glueball operator, they conclude that ], which refines and extends the analysis + − 3 , equivalence of SU( 2 c N – 20 – ]. In [ glueball operator, is not heavier than the lightest 28 , ++ 27 ] the lightest states are conjectured to be of mesonic 28 appears to be much lighter than the scalar glueball, which 0 η ) for funding this project by providing computing time on the GCS ], the nature of the lowest two chiral supermultiplets has been investigated on the ]. Also, in contrast to the above arguments, we do not find the same mixing angle for Our numerical findings in the scalar channel, where the lightest state is dominantly On the other hand, in ref. [ Our results might help for a better understanding of the low energy effective action [ 27 28 ] and the conjectures presented in [ www.gauss-centre.eu supported by the Deutscheing Forschungsgemeinschaft Group (DFG) “GRK through 2149: theG. Strong Bergner Research and acknowledges support Train- Weak from Interactions the —No. Deutsche from Forschungsgemeinschaft BE (DFG) Hadrons Grant 5942/2-1. to Dark S.Austauschdienst Matter”. Ali (DAAD). acknowledges financial P. support Scior from acknowledges the support Deutsche Akademische by the Deutsche Forschungs- The authors( gratefully acknowledgeSupercomputers the JUQUEEN and Gauss JURECASuperMUC Centre at at Supercomputing Leibniz J¨ulich for Supercomputing Centreprovided Centre (JSC) Supercomputing on (LRZ). and the Further e. computing compute time cluster V. has PALMA been of the University of M¨unster. This work is in [ the scalar and the pseudoscalargluino channel. mass could A shed detailed more consideration light of on the these effects questions. ofAcknowledgments a small distinction between the scalar and pseudoscalar channels is, however,of not being glueball made. type,it are appears consistent that with thewith the pseudoscalar a predictions negligible ground mixing of state of is the [ glueball, dominated which by would correspond the to mesonic the contribution scenario advocated 2) multiplet. Generally,states the to be effective the Lagrangian lightestordering allows ones, of depending either states on is mesonic an thentheories. unknown or based parameter. In on glueball-type QCD The the the argument large- mesonic forleads the to the conjectures about the ordering of states and only a small mixing for SYM. A pseudoscalar state, which hasthe overlap multiplet with of the glueball 0 states must be lighter thantype the with multiplet a of small mesons. with mixing of an the arbitrary glueballs. mixing Their angle analysis between employs the an effective glueball Lagrangian ( basis of an effectivelightest action states including were mesonic conjectured andThe argument to glueball-type is degrees be based of on ofwhich leads freedom. the the to perturbation a The glueball splitting of of the type, thebecomes effective multiplets. lighter if theory In than the by mixing mesonic the a multiplet isstate, scalar small the pseudoscalar meson. not which gluino meson mass, has too Drawing overlap on strong. with the the proof [ 0 mixing. The pseudoscalar groundwhereas state the is scalar clearly ground dominated state by has the an mesonic apparent contribution, predominant3 glueball contribution. of [ glueball and meson contents. 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