PoS(LATTICE2016)222 1 = N http://pos.sissa.it/ , Metadata, citation and similar papers at core.ac.uk at papers similar and citation Metadata, y and light dynamical gluinos. ions of the four dimensional, y s, , Gernot Münster ∗ supersymmetric Yang-Mills 1 ive Commons = onal License (CC BY-NC-ND 4.0). N [email protected], [email protected] [email protected] [email protected] [email protected] Supersymmetric Yang-Mills theory with SU(3) gauge symmetr We report on our recent results regarding numerical simulat Speaker. ∗

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c Universität Münster, Institut für Theoretische Physik, Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany E-mail: [email protected], [email protected] Georg Bergner Universität Bern, Institut für Theoretische Physik, Sidlerstr. 5, CH-3012 Bern, Switzerland E-mail: Istvan Montvay Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22603 Hamburg, Germany E-mail: Stefano Piemonte Universität Regensburg, Institute for Theoretical Physic D-93040 Regensburg, Germany E-mail: Attribution-NonCommercial-NoDerivatives 4.0 Internati Sajid Ali, Henning Gerber, Pietro Giudice Simulations of

theory with three colours 34th annual International Symposium on Lattice24-30 Field July Theor 2016 University of Southampton, UK CORE Provided by Bern Open Repository and Information System (BORIS) System and Information Repository Bern Open by Provided PoS(LATTICE2016)222 (red π 2 a– ), where m c N Pietro Giudice , proportional to the 1 ) − defined as the value of S ) 7 ) Z c ( κ 9 [5, 6]. The coexistence S ( κ 3.7 3.8 3.9 am 0 , this model has been pro- . 1/(2 luino). We have confirmed e.g. 4 menological relevant SU(3) 139264 13860 y (SYM) with gauge group . . = 0 0 parameter in QCD. β here are important new aspects s. From the computational side = = Λ c c 3 colours. ne by our collaboration until now e [8] also for the SU(3) theory. es, see 32, κ κ (SUSY). Moreover, in [2] we have ve been presented in [1]: we have = 3.6 × c 3 n dark-matter sector that may explain n. ng result was the evidence that chiral N upermultiplet is recovered and we have Critical value of gluino, a-pion, Vol=16 3.5 where the square of the adjoint pion mass is the number of colours. 0.5 1.0 0.0 κ square) and the quantity renormalised value of thecircle), gluino for mass, vanish (blue Figure 1: c 1 N colors . . ) λ c x ( N sub- (2.1) λ c N 2 Z , is a scale parameter similar to the k i c Λ , the other one π N SYM with 2 ) 1 supersymmetric Yang-Mills theory with a gauge group SU( x e 1 ( 3 µ = Λ = A 1, and supersymmetric Yang-Mills theory with three colours N − . The result is a theory 1 N c 2 const Z = N symmetry is spontaneously is unbroken. As conjectured = c λ N i N 2 α ,..., vacua implies a first order phase transition (for a massless g Z 0 λ c α N vacua, where the gluino condensate = λ c h k 1 is the number of SUSY generators and N SU(2) SYM has been an interesting test case for the more pheno In this work we explore the supersymmetric Yang-Mills theor It is characterised by two fields: one Let us consider This result is supported by arguments in different approach = N where theory, which contains thein gluons SU(3) of SYM QCD. like On theSU(3) the new is other bound much hand, states more and t demandingposed than CP-violating as QCD phase an and attractive SU(2) candidate SYM.astrophysical for Moreover observations a [3]. supersymmetric hidde 2. Properties of SU(3). This work is asimulating natural the continuation gauge of group what SU(2).verified has that, been in Our do the continuum conclusive limit, resultsno the ha degeneracy sign of of the a s possiblestudied spontaneous the breaking of theory supersymmetry atsymmetry finite is temperature: restored near the the deconfinement most phase interesti transitio Simulations of 1. Introduction describing the gluon It is possible toanomalous show that and this that, symmetry is however, a The latter particlein is the a adjoint representation. Majoranafermion mass, fermion For SUSY non-zero iscause softly broken. there is Be- the only global one chiral symmetry Majorana is simply flavour, U(1) describing its superpartner, the gluino with can be usedWeyl as notation): an order parameter (using broken down to in [4] the group of U(1) of these this behaviour in the SU(2) theory [7] and we had some evidenc PoS(LATTICE2016)222 ), a . c ¯ λλ κ ∼ = 0 κ f Pietro Giudice re gluonic states 2 gluino-glueball: / rlo (RHMC) algorithm. gluinoball: a– + remnant of SUSY on the lat- eories on the lattice goes back ) algorithm [13, 14]. Some re- described by the Wilson action nt: the particle spectrum of the ntinuum limit, in the same way described by Wilson fermions in in our simulations is an improved Scatter plot expected for configurations have been obtained anslational invariance for non-zero effects can drastically increase the ctive action and derived a first su- e introduction of a non-zero gluino lar (0 bout 1.2 fm (in QCD units) or larger, uce the opposite order: clarifying this t, the particles are organised in mass- tions in spatial directions are compatible s we apply one or three levels of stout glueball, and again a gluino-glueball. Ac- r than the previous one; other authors [11], + Figure 2: 2 , ), and a Majorana fermion (spin 1 c λ glueball, a 0 κ 5 − ¯ λγ , 4.0 and ∼ ′ β η supersymmetric Yang-Mills theory with three colours 1 ). A second supermultiplet was introduced in [10] based on pu 32 volume, with two val- gluinoball: a–  = × λ − 3 µν N F lattices, have been obtained with a Rational Hybrid Monte Ca  4 . Tr κ µν σ Another feature conjectured for these theories is confineme On the lattice we can identify two The idea that it is possible to study supersymmetric gauge th ∼ χ Simulations of 4.3, and different values oframeter the hopping pa- pseudoscalar (0 theory consists of colourless bound states.degenerate multiplets. In the SUSY In limi [9]permultiplet the of authors the wrote low-lying down spectrum. an effe It consists of a sca Integrating out the Majorana fermions yields a Pfaffian. Itparticular can for have small a gluino negative masses sign, near in sources of explicitly SUSY breaking: the first one is due to th mass; the second one is alattice consequence spacing. of the Moreover breaking we ofmass have the splitting verified tr [15]; that in the finite case volume the of effect SU(2), is using negligible. a Note box that size periodic of boundary a condi in the effective action. It consists of a 0 ues of the inverse gauge coupling but the significance of the negativetions contribu- is reduced towards the continuum limit. If necessary, the signby is reweighting. taken into Part of account including the a work was clover done term:verified a great we improvement haveSU(2) in already the [1]. case of Wemainly on have a simulated 16 the theory cording to the authors, this last multipletusing should different be arguments, lighte and hints from ordinaryissue QCD, is ded one of the tasks of our project. 3. The theory on the lattice and simulations to [12]. Thetice, proposal one was should that, onlyas instead require it of that happens for trying SUSY chiral is toversion symmetry. recovered have of The in what some formulation we the was employ but first co proposed with in a [12]: tree-levelthe Symanzik the adjoint improvement; gauge representation. fields the are gluinossmearing To to are reduce the the link lattice fieldsmainly artifact in by the a two-step Wilson-Dirac operator. polynomialsults, hybrid on The Monte 6 Carlo (TS-PHMC PoS(LATTICE2016)222 c 6, κ . 5 . ritical = κ β 3 vacua. The Pietro Giudice = volume. c 4 is plotted against N 1 − malised mass of the S Z S se contribute to the dif- that they are compatible am ear linear dependence be- 587ona6 . 1 . As discussed in Sec. 2, we = i e gluino mass. This gives a solid sw ψ ric limit implies the supersymmet- c uantity s rise to two distinct condensates: 5 standard method to tune the SUSY ce spacing and a finite volume that ¯ e size effects. In future work a better ere we put on the abscissa the scalar ψγ h , see Figure 2. Scalar condensate distribution for nation of the adjoint pion mass is much 1658 and . 0 = 1 SYM theories. Figure 3: κ obtained using the RHMC algorithm with 3 = N Ward-Takahashi identity (WTI) (with the axial anomaly) λ . Given the ) κ 2 and a pseudoscalar condensate : fitting the data (the blue dashed line) we can determine the c ( ) , where the renor- / i c κ κ 2 ( is proportional to the ¯ ψψ / 1 SYM theory with gauge group SU(3) is characterised by supersymmetric Yang-Mills theory with three colours h π 1 = 2 a– = m and 1 N 1 N is called subtracted mass and can be identified with the renor is linear in 1 . Terms in the WTI proportional to the lattice spacing of cour − S S is a multiplicative renormalisation coefficient. We see a cl σ Z π . Comparing the results obtained using the two methods we see S 5 m S . 2 a– κ Z . m am ) As discussed in [12], and clarified in [16], the chiral symmet κ Eq. 2.1 when translated into the Dirac representation, give In practice, one has to tune the bare 2 ( / ric limit. More precisely, both the U(1) Simulations of with SUSY [4]. It is an interplay between having a finite latti leads to the measured larger mass splitting. 4. Tuning towards the SUSY limit malised mass of the gluino vanishes.direct A more tuning approach is,termine of the course, renormalised to gluino mass de- the WTI. using The procedure has already beenscribed de- in [18]; in Figure 1 we show directly the result: the q value of only in 7 ference between the results and therecontrol might of be systematic also errors some is finit needed.simpler Because and the numerically determi not expensive,limit it in will this be theory. used as our 5. The vacuum of the theory expect that our previous relation, we can determine theical crit- hopping parameter and the SUSY WTI are restoredtheoretical by basis a for single lattice fine-tuning formulations of of the bar a scalar condensate gluino; tween mass of the gluino.ently well satisfied, This as relation shown in is Figurered 1 appar- dashed (the line is a fit towhere the five data points), gluino mass so that themass renormalised vanishes. gluino As discussed in [9]OZI using approximation, the or intially [17] quenched using setup, a the par- joint square of pion the mass ad- three vacua lie in acondensate Cartesian and plane, on according the to ordinate Eq. the 2.1, pseudoscalar wh condensate 1 PoS(LATTICE2016)222 i- in 0 π 3 2 imple: i e y small vol- with Pietro Giudice (corresponding to the Pseudoscalar condensate c where the peak on the left κ c right one: this can be seen κ < pectation value of the scalar ≃ κ (Right) roportional to the square of the κ sian distributions. using our RHMC algorithm. For s of the bound states to the chiral eak, as in Figure 4 (Right). This 4 should remain. The reason for this three peaks should appear:responding one to the in two distributions o be occupied. So far we have never e only in a small volume; increasing n. We present here only the results c in Figure 2). This is exactly what ap- κ ess probable and above a certain value 2 in green). π 3 2 → t the two sides should appear. i e κ a second peak should emerge (corresponding c κ and 1 4 → π 3 2 i κ e 0 and only one peak should appear (corresponding to the distr > in Figure 2), but for we expect to see only one peak for i 0 i π 3 in Figure 2 ). The reason because they have the same height is s . Looking at the distribution of this quantity, in relativel 2 ¯ ψ c i ψψ 2 5 h e κ π in Figure 2 ). When 3 2 where the peak, corresponding to the distribution labelled ¯ i 0 ψγ = c e π h 3 2 κ κ i supersymmetric Yang-Mills theory with three colours e > 0. 1 and . 4 1 κ = π we expect 3 Scalar condensate distribution: this case corresponds to 2 = c i N e κ β < κ we should see two peaks, with the left one two times higher the c For A first order phase transition should show up as a jump in the ex As discussed in Sec. 4 the renormalised mass of the gluino is p κ When we look at = we are populating two vacua which haveobserved the the same probability double t peakhappen structure, even but for only one symmetric p gluino condensate at umes, one expects to see ait, two the peak tunnelling structure. between the This three ispractically ground impossible. possibl states becomes l (in black) is about twodistribution: times we higher see than only one the peak, one while on two the smaller right peaks ( a adjoint pion mass. Aslimit a consequence fitting we their extrapolate mass theobtained against masse for the square of the adjoint pio Figure 2, should disappear andphenomenon only is the still two under peaks investigation. at its sides 6. Mass spectrum: numerical results κ Figure 4: (Left) Simulations of distribution labelled with the center and two, withlabelled the with same height, at the two sides (cor in Figure 4 (Left) where data had to be fitted as a sum of two Gaus to the sum of thepears distributions in labelled Figure with 3. This result was obtained on a small volume 6 bution labelled with PoS(LATTICE2016)222 ′ , for their 0 η f ˜ gg a– =4.0 β . our case the gg Pietro Giudice 2 ) π ercomputing (GCS) for pro- (am S share of the supercomputer e techniques already discussed hi identities and of the adjoint ational supercomputing centres ercomputer SuperMUC at Leib- ue to discretisation effects, which ed the results with only one lattice estigations. We started to measure cua has been investigated and some and of the gluino-glueball ˜ olated to the chiral limit, are roughly lating the theory with a second value n one. The masses of both states are xtrapolate the results to the continuum ′ rder transition in the scalar condensate η . 0.0 0.1 0.2 0.3 0.4 for different values of the square of the adjoint ; the gluino-glueball is by far the state with

and of the gluino-glueball. Their error bars

′ 0.8 0.6 1.0 0.4 0.2 0.0 , showing both the glueball and the a– am -0.2 ++ ++ η 1 supersymmetric Yang-Mills theory with three ++ 5 = ++ 0 0 N f glb a– =4.0 β 3 and that we are currently simulating. . channel, taking the errors in that channel into account. But 4 and of the glueball 0 = 0 ++ f β . It is well known that this channel is particularly noisy; in 2 ) As for (Left) but for the mass of the a– π π 2 a– supersymmetric Yang-Mills theory with three colours 1 m (am = Mass of the a– (Right) . N π a– m 0.0 0.1 0.2 0.3 0.4

to estimate the discretization effects and, hopefully, to e We have presented our first results on The authors gratefully acknowledge the Gauss Centre for Sup The mass spectrum of this theory has been determined using th In Figure 5 (Right) we plot the mass of the a–

0.8 0.6 0.4 0.2 0.0 1.0 am -0.2 β 7. Conclusions and outlook colours. We have shown results on the use of the Ward-Takahas pion to tune the theorypreliminary to results supersymmetry. have been The obtained: structure we ofbut see va a not clear in first o thethe pseudoscalar particle one. spectrum of This the issuespacing theory. but is At the still the results under moment are weof promising. inv present We are currently simu the best results. The massescompatible of with these those two in states, the when 0 extrap different values of Figure 5: (Left) Simulations of pion mass are considerably smaller than those in the channel 0 masses are not compatible withshould be each already other. reduced at This is probably d viding computing time forJUQUEEN a at Jülich GCS Supercomputing Large-Scale Centre Project (JSC)niz and on Computing on the the Centre sup GC (LRZ). GCS is the alliance of the three n limit. Acknowledgements in [1]. In Figure 5 (Left) we plot the channel 0 glueball determination looks somewhat betterperfectly than compatible the when meso extrapolated to the chiral limit PoS(LATTICE2016)222 6 1209 (1987) 1100]. Pietro Giudice 1202.2598 (2014), 095016 66 90 (1999) 123 (2014) 049 (2016) 080 559 (2002) 719 1411 (2000) 661 23 1603 003.2073 [hep-lat]]. 83 (1999) 209 s been provided by the computer 446 . Tait, Phys. Rev. D of Education and Research (BMBF) Nucl. Phys. B (1998) 015009 emberg (MWK), Bayern (StMWFK) 1 [hep-lat]]. dbrink, JHEP ich), and LRZ (Bayerische Akademie onte, JHEP ünster, E. E. Scholz and J. Wuilloud, Eur. 58 ph]]. t]]. andbrink and I. Montvay, JHEP ]]. (1988) 445 [Sov. Phys. JETP (1982) 231. doi:10.1016/0370-2693(82)90828-0 (2004) 096004 doi:10.1103/PhysRevD.70.096004 296 113 70 6 (1987) 555. doi:10.1016/0550-3213(87)90660-2 (2005) 73 doi:10.1016/j.physletb.2005.07.050 292 623 (2014) 034 doi:10.1007/JHEP05(2014)034 [arXiv:1402.661 1405 (2012) 290 doi:10.1016/j.nuclphysb.2012.04.008 [arXiv: (1982) 253. doi:10.1016/0550-3213(82)90071-2 861 202 [DESY-Münster-Roma Collaboration], Eur. Phys. J. C [DESY-Münster Collaboration], Phys. Lett. B supersymmetric Yang-Mills theory with three colours 1 (2010) 147 doi:10.1140/epjc/s10052-010-1390-7 [arXiv:1 = [DESY-Münster Collaboration], Nucl. Phys. Proc. Suppl. et al. et al. 69 N et al. doi:10.1103/PhysRevD.90.095016 [arXiv:1408.6532 [hep- [hep-lat]]. (2012) 108 doi:10.1007/JHEP09(2012)108 [arXiv:1206.234 doi:10.1016/S0920-5632(00)91768-7 [hep-lat/9909070]. [hep-th/0408214]. [hep-lat/0506006]. [hep-th]]. doi:10.1007/s100520200898 [hep-lat/0111008]. doi:10.1103/PhysRevD.58.015009 [hep-th/9711166]. Phys. J. C doi:10.1016/S0370-2693(98)01523-8 [hep-lat/9810062]. doi:10.1016/0550-3213(88)90680-3 doi:10.1016/S0550-3213(99)00434-4 [hep-th/9905015]. doi:10.1007/JHEP11(2014)049 [arXiv:1405.3180 [hep-lat doi:10.1007/JHEP03(2016)080 [arXiv:1512.07014 [hep-la [9] G. Veneziano and S. Yankielowicz, Phys. Lett. B [8] A.Feo [3] K. K. Boddy, J. L. Feng, M. Kaplinghat, Y. Shadmi and T. M. P [1] G. Bergner, P. Giudice, G. Münster, I. Montvay and S. Piem [4] E. Witten, Nucl. Phys. B [5] M. A. Shifman and A. I. Vainshtein, Nucl. Phys. B [6] N. M. Davies, T. J. Hollowood, V. V. Khoze and M. P. Mattis, [2] G. Bergner, P. Giudice, G. Münster, S. Piemonte and D. San [7] R. Kirchner [15] G. Bergner, T. Berheide, G. Münster, U. D. Özugurel, D. S [16] H. Suzuki, Nucl. Phys. B [17] G. Münster and H. Stüwe, JHEP [13] I. Montvay and E. Scholz, Phys. Lett. B [12] G. Curci and G. Veneziano, Nucl. Phys. B [10] G. R. Farrar, G. Gabadadze and M. Schwetz, Phys. Rev. D [14] K. Demmouche, F. Farchioni, A. Ferling, I. Montvay, G. M [18] F. Farchioni [11] A. Feo, P. Merlatti and F. Sannino, Phys. Rev. D Simulations of HLRS (Universität Stuttgart), JSC (Forschungszentrum Jül der Wissenschaften), funded by the Germanand Federal the Ministry German State Ministries forand Research of Nordrhein-Westfalen Baden-Württ (MIWF). Furthercluster computing PALMA time of the ha University of Münster. References