Pos(LATTICE2018)209 1 Super Yang-Mills Theory, Most Recently with the = N ∗ [email protected] [email protected] Speaker

Pos(LATTICE2018)209 1 Super Yang-Mills Theory, Most Recently with the = N ∗ Stefano.Piemonte@Ur.De Georg.Bergner@Uni-Jena.De Speaker

PoS(LATTICE2018)209 https://pos.sissa.it/ 1 super Yang-Mills theory, most recently with the = N ∗ [email protected] [email protected] Speaker. This talk is an overview of ourtheories. recent investigations of We supersymmetric have and studied near conformal extensively gauge gauge group SU(3). Informal addition behaviour we at have an investigated infrared theories fixedfermion that point. action show setup We in indications have our for included studies. a a Isymmetric mixed will QCD con- fundamental explain on how the and this lattice adjoint is and related presentaddressed to some in the first the investigation studies investigation of of of super- the this main theory. obstacles that need to be ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). The 36th Annual International Symposium on22-28 Lattice July, 2018 Field Theory - LATTICE2018 Michigan State University, East Lansing, Michigan, USA. Stefano Piemonte Universität Regensburg, Institute for Theoretical Physics, D-93040 Regensburg, Germany E-mail: Georg Bergner Friedrich-Schiller-University Jena, Institute of Theoretical Physics, Max-Wien-Platz 1, D-07743 Jena, Germany E-mail: Supersymmetric and conformal theories on the lattice: from super Yang-Mills towards super QCD PoS(LATTICE2018)209 (2.1) Georg Bergner 2 supersymmetric = N ]. Both of these projects 6 , 5 , . 4 λ ) g m chiral matter superfield consisting of c + ¯ N / D ( ⊕ ¯ c λ N 1 2 ]. The second is the investigation of the strong 3 + , 1 2 , µν 1 F ]. The basic investigations of these theories are now 7 µν F 4 1 = , which are complex scalars. The Lagrangean is extended by 2 Φ SYM , leads to a soft supersymmetry breaking. The gluinos are Majorana 1 g and the usual gauge interactions, L Φ m λ 4 supersymmetric Yang-Mills theory, but also = 1 supersymmetric QCD are interesting from this perspective. N ]. = 8 and squarks N ψ 1 supersymmetric Yang-Mills theory (SYM) is the pure gauge part of supersymmetric = In separate contributions to this conference the status of our investigations of SYM for the The next natural step is the introduction of an In recent years our collaboration has done a number of studies of two interesting related sub- There are basically two alternative motivations for the investigation of supersymmetric strongly N In a quite similar way, there are two basic motivations for the studies of conformal and near gauge group SU(2) and SU(3) is reported [ fermions in the adjoint representation. A non vanishing gluino mass a Dirac quark dynamics of composite Higgs and walking Technicolor theories [ nearly completed. In a secondis contribution presented, the in status particular regarding of the ourthe measurement analysis Gradient of of the flow [ gluino phase condensate transitions with in the SYM methods of jects. One is the investigation of supersymmetricsymmetric Yang-Mills theory, extensions the of pure the gauge Standard sector Model of super- [ coupled theories on the lattice.sible On extensions the of one the hand Standard there Modelthere are based is the on a a investigations more in supersymmetric general the completion.better theoretical context On understanding of interest the of pos- in other the hand these non-perturbative theories.prominent dynamics example Supersymmetry of is is certain a theories tool based providing on a dualities. One QCD, with the gluino fields Supersymmetric and conformal theories 1. Investigations of supersymmetric Yang-Mills theory and near conformal theories are connected to the new studies of supersymmetric QCD that are reported in this contribution. Yang-Mills and conformal strongly interacting theories. The firststrong one dynamics is beyond the the phenomenological Standard possibilitythat of Model theories a that new with plays a the near rolebetween conformal, of the “walking”, the behavior scale Higgs can sector. of accommodatecompletion the It large inducing turned electroweak scale the out separation symmetry effective breaking operatorsfor and for the the fermion studies of mass scale conformal generation. and of neartheoretical conformal a The strongly questions. second strongly interacting theories motivation interacting are The again perturbative morenumber general beta of function fermion flavors. develops This a meansnumber fixed that of below point the fermions when loss there of increasing is asymptotican the a freedom infrared window at fixed an for point. even theories larger with Thismethods. a purely perturbative non-perturbative estimate conformal needs behavior to at be verified by non-perturbative 2. Supersymmetric Yang-Mills theory and supersymmetric QCD PoS(LATTICE2018)209 (2.3) (2.4) (2.2) 2 | 2 Georg Bergner a Φ | λ 2 ) supersymmetric c m † 2 there is no vacuum N Φ f + + 2 N | P 1 > + Φ c | 1 2 N Φ the theory is either close to m − f P + N ψ a ]. For ) T m 12 ¯ [ ψ + g f f 2 µ ]. N D √ 11 i µ γ and − ( c ¯ ψ ψ N . a 2 + 2 T 2 | † 2 † 2 − Φ P Φ ]. A perturbative setup for the tuning of the parameters a 2 µ T Φ 10 D 2 | Φ + + + − 2 | P 1 1 † 1 Φ Φ Φ a µ T supersymmetric Yang-Mills limit or in a Higgs phase resembling an a D the theory is expected to have an infrared fixed conformal point. In † 1 , the investigations of a conformal theory on the lattice has its own | ) ¯ λ c c c Φ g + N N N 2 3 ( 3 2 matter flavors. This theory has been considered in a perturbative setup on 2 √ g i f < SYM < SU f N , the theory has flat directions an a divergent scalar expectation value. Even f + + L f N N = N < < ] and the general form of possible supersymmetry breaking counterterms as well as but below the conformal window might in practice be hard to distinguish from the c c supersymmetric Yang-Mills theory except for some small regions of the parameter 9 f N N SQCD ) 2 3 f 2 3 N L N ≤ − c c N The third obstacle is related to the Yukawa couplings and the chiral symmetry of the theory. These are quite remarkable findings, in particular the prediction for the conformal window. It In practice, there are several difficult obstacles that one has to circumvent in order to simulate The second obstacle for the lattice simulations are the different phases of the theory sketched There are a number of interesting theoretical predictions for the low energy effective theory N ( In order to have a clear tuning of the scalar and pseudoscalar Yukawa interactions as well as the difficulties, but a number ofit tools is have interesting been to derived crosswith for check the the investigations lattice in simulationsconformal such with scenario. a the regime Supersymmetric analytic and QCD predictions.hard is to The hence control. theories always in a regime where the chiral limit is Further flavors can be added inpled terms in of the additional same matter way multipets to (quark theQCD and gluino (SQCD) squark and gluon fields), with fields. cou- The resulting theory is SU( the heavy mass pure SU the lattice in [ though there is arather clear difficult prediction towards for the this chiral phenomenon, limit. it makes It the seems simulations that of at thespace. small theory At would be interesting to checkThe whether simulations they of can be supersymmetricinvestigations, confirmed QCD like with the would dualities numerical also of lattice the open simulations. theory and the possible way supersymmetry breaking for scenarios. supersymmetric a QCD number on the of lattice. additional tion Supersymmetry of is the broken theory. in Therefore any thelimit. kind first In problem of supersymmetric is local Yang-Mills the lattice theory tuning realiza- there towardsbut is the as only supersymmetric the soon continuum tuning as of the a scalarself single interactions squark parameter and fields required, Yukawa of couplings the appear. matter multiplets are added, aabove. large number At of scalar small between these two cases therechiral are limit. theories with chiral symmetry breaking and confinement in the the symmetries have been discussed in [ is presented in another contribution to this conference [ and the phase structure of SQCD as a function of Supersymmetric and conformal theories Yukawa and four squark interactions, solution in the chiral limit andthe the region vacuum expectation value of the scalar condensate diverges. In PoS(LATTICE2018)209 ]. 13 9 . Georg Bergner 8 0 . = 0 1620 1600 . . a κ = 0 = 0 a a κ κ 7 0 . 6 0 . π am 5 0 . 4 0 . 3 0 . (b) fundamental meson mass ratios 2 0 . 0 9 8 7 6 5 4 3 2 1 1 9 ...... ]. Note that the extrapolation of the reference includes 1 1 1 1 1 1 1 1 1 0

π v /m m 16 ]. 13 8 . 7 0 . reference 6 0 . ]. 5 0 . 10 2 fundamental ) = 2 π 4 0 . f am ( N 1 and the results obtained in [ . 2 3 0 . = β 2 0 . at 0 (a) The comparison of the chiral extrapolation of the vector meson mass in units of the Gradient w 1 0 . (a) comparison to We consider SU(2) gauge theory with two Dirac fermions in the fundamental and one Ma- An additional obstacle for the numerical simulations is the sign problem thatWe appears are in super- addressing these problems first with a simplified setup. We are investigating the two In the end we discuss also some first results of our simulations of one flavor supersymmetric 0 0 Walking Technicolor 5 3 5 2 5 1 5 . . . .

3 2 1 0

0 v m w This theory has two Dirac fermionssentation. in the Investigations fundamental of and other one theories Dirac with fermion in fermions the in adjoint the repre- adjoint representation have shown jorana fermion in the adjointbesides representation. the mentioned The motivations relation for toscenario the SQCD, the consideration are Standard of based Model the on is theory, der the extended to walking with accommodate Technicolor the a scenario. fermion nearmeasurements. mass In conformal, generation this According or with to walking, the naive constraints gauge estimatessetup from theory with the electroweak in precision the Ultra or- least Minimal tension Walkingreview with Technicolor, of provides the other a electroweak interesting precision properties of data the for theory a like near the conformal possible dark theory, matter for candidate a see [ a chiral and continuum(b) extrapolation The up deviation to of the quadraticThe particle corrections mass spectrum which ratios once of the explains the Majorana the vector fermion remaining meson in and mismatch. the the adjoint pion representation tend is towards added. a constant value.l Figure 1: flow scale symmetric QCD. The integration of the Majorana fermions yields a Pfaffian. flavor SU(2) supersymmetric QCD withoutwe scalar are fields. already This in shouldright this help since case us it to close is find a to out candidatemodified a for whether version conformal a of composite behavior. the Higgs theory extension This proposed of in theory the [ is Standard Model. interesting It onQCD. is its We a are own slightly currently notapply able an to approach perform that the is based full on non-perturbative the tuning perturbation of matching the of theory, scalar3. but and fermion we Mixed propagators. representation setup for near conformal dynamics for Ultra Minimal Supersymmetric and conformal theories mass terms, a well definedwith chiral Ginsparg-Wilson symmetry fermions on on the the lattice lattice.is would possible In be as this helpful. described case in This even [ can an be offline achieved approach for the tuning PoS(LATTICE2018)209 ]. ] up to 18 17 , 16 Georg Bergner 5 10 . . ) mass parameters for 2 ad j 5 9 9 . κ 1350 . ]. The extension with the = 0 adj 17 /κ fund 1 κ , 5 8 8 16 . Fixed ) and adjoint ( ]. However, the theory with one Dirac f und κ 5 7 7 15 . ] and references therein. It might still be that 6 , 6 6 7 6 5 4 3 2 1 0 ...... 0 0 0 0 0 0 0 14

π fund ) m ( 2 (b) fundamental pseudoscalar mesontion mass of as the a adjoint func- mass parameter . 1(a) 5 10 . 4 has been found, see [ 1600 . . 0 = 0 5 9 9 . fund 1. (a) The dependence of the pion mass for the fundamental representation on the . adj − /κ κ 1 2 2 . = Fixed 0 β = ∗ γ 5 8 8 . Interplay between the tuning of the fundamental ( A second important condition for the walking scenario is the large mass anomalous dimen- These investigations are also an extension of the investigation of SU(2) gauge theory with two The ﬁrst step in our investigation has been to start from the heavy adjoint ﬂavor limit and The next step is to resolve the two mass parameter tuning that is required for the simulations in 7 7 6 4 2 1 8 6 4 2 0 ......

1 1 1 0 0 0 0

π adj ) m ( 2 sion. Minimal Walking Technicolor doesdimension of not fulfill this condition since a small mass anomalous the simulations at tuning of the adjoint massrepresentation on parameter. the (b) fermion The mass dependence parameter for of the the fundamental pion representation. mass for the fermions inthat the adjoint the near conformal behaviorby appears the already naive estimates. at a smaller number of fermions than indicated Dirac fermions in the fundamental representation presented in [ the complete theory coupled to thedifferent properties, Standard but Model according in to an the Extendeddates. standard Technicolor arguments One setup one of has has the slightly to largest considerwith mass alternative one candi- anomalous Dirac dimensions has fermion beenflavor in in found the the for adjoint adjoint the representation SU(2) representation hasdard gauge [ not Model. theory the Ultra right Minimal particle Walking content Technicolorallow is to the the be simplest coupling combined extension to with of the the the Standard Stan- close theory Model the in sector. order related to The theory aim with offundamental one representation our is Majorana investigations is to fermion a to in conformal quantify the behavior. how adjoint and two Dirac fermions in the crosscheck previous results for twoused fundamental a Dirac one-loop fermion clover improved flavorslimit Wilson with of fermion our a action decoupled setup. and adjoint flavor, a We we have plain are Wilson able gauge to action. reproduce the In results the presented in [ the mixed representation setup. It turnsthe out different that representations there towards is the chiral a limit.strongly strong affects interplay The the between tuning pion of of mass the the in tuning mass the of for one other representation representation, as shown in Figure adjoint fermions are complementary to the studies with additional fundamental fermions [ a reasonable accuracy as shown in Figure (a) adjoint pseudoscalar mesonthe mass fundamental as mass a parameter function of Figure 2: Supersymmetric and conformal theories PoS(LATTICE2018)209 . 1(b) Georg Bergner ). In the standard π m 5 ) over the pseudoscalar (pion) mass ( v m We have started our first studies of supersymmetric QCD. Besides the additional complication We have considered in our simulations supersymmetric QCD and related theories in numerical We have started our investigations with a related theory without scalar fields. This theory is We have also done out first simulations of supersymmetric QCD. Our current setup is moti- As a last step, we have investigated the influence of the fermions in the adjoint representation of introducing the additional scalar fields,adjoint it Majorana requires / fundamental the Dirac implementation Yukawa couplings. offor We the have very developed the the special RHMC simulation mixed algorithm ofthe these Pfaffian. types In of order couplings,considered to neglecting the be at one well the flavor enough case momentdiscretization separated of for the from the the a complex derivative gauge operator possible phase for group conformallike of the derivative SU(2). behavior, term squark including we and next Our the have to lattice quark nearestfields. fields, neighbor realization interactions which It uses and means a implies the a Wilson unbroken Wilson mass same supersymmetry forzero. the in scalar the So perturbative far limit, we settingstill the have very gauge preliminary. only coupling considered In to antheir our tree unimproved first level Wilson investigations relations and we fermion neglected have setup anyOne set further problem and tuning the that of our gauge we the results have and supersymmetry to breaking are value. Yukawa face operators. couplings In is to a the scan clear of signature the ofa adjoint the large and divergent fundamental values scalar mass for vacuum parameters, expectation the onemass scalar clearly limit. observes fields. regions with These In region theproper seems full way to and tuning be we approach close are to of currently working the the on expected parameters a small this better fermion algorithm problem for has this5. to regime. Conclusions be and controlled outlook in a lattice simulations. These theoriesrichness are of non-perturbative interesting phenomena from in these severalsimulations. theories perspectives: that can One The be of probed other them with isStandard numerical is Model. lattice a the possible deeper understanding of supersymmetric extensionsan of interesting the candidate forModel. a composite We were Higgs able (walking to Technicolor)flavors match and extension previous investigated results of the for the tuning the of Standard first SU(2) the preliminary gauge mass results theory parameters indicate with with a two transition twoscenario. fundamental form different a representations. chiral Our symmetry breaking towards a conformal vated by the perturbative limit andBesides a the much tuning larger to effort the is supersymmetric required limit, other for problems the have non-perturbative to tuning. be solved in the simulations 4. Numerical simulations of one flavor supersymmetric QCD on meson mass ratiosmass in ratio of the the fundamental vector meson one. mass ( The near conformal behavior predicts constant Supersymmetric and conformal theories chiral symmetry breaking scenario, which shouldMajorana apply fermion, for this the ratio case diverges withouttransition in the from the additional the chiral adjoint chiral limit. symmetry breaking Our scenario first towards very a preliminary conformal one, data see show Figure the PoS(LATTICE2018)209 78 48 for × , 237 3 34 Georg Bergner (2016) no.9, (2018) 119 94 (2017) no.3, 1801 96 (2016) 080 , 003 (2012) (2015) no.11, 114508 1603 1205 91 [hep-lat]]. , 1 (1996) [Subnucl. Ser. (2018, to appear). 45BC (2018, to appear). (2018) 08020 [arXiv:1710.07218 [hep-lat]]. 0903.2443 175 (2017) no.3, 034507 [arXiv:1706.05222 [hep-lat]]. 6 (2018) no.7, 074510 [arXiv:1709.07367 [hep-lat]]. 96 (2008) 115010 [arXiv:0809.0713 [hep-ph]]. LATTICE2018 97 78 LATTICE2018 (2009) 4045 [arXiv: 24 (2012) 025006 [arXiv:1204.4432 [hep-lat]]. 86 (2018, to appear). (2018) 113 [arXiv:1801.08062 [hep-lat]]. LATTICE2018 034504 [arXiv:1610.01576 [hep-lat]]. 1803 [arXiv:1512.07014 [hep-lat]]. (2018) no.5, 404 [arXiv:1802.07067 [hep-lat]]. [arXiv:1712.04692 [hep-lat]]. (1997)] [hep-th/9509066]. [arXiv:1412.5994 [hep-lat]]. 094507 [arXiv:1602.06559 [hep-lat]]. [arXiv:1111.4104 [hep-lat]]. Note that the simulations have to be in a regime where the supersymmetry breaking of the We thank the members of the DESY-Münster collaboration for helpful discussions, comments, [7] S. Ali, G. Bergner, H. Gerber,C.Lopez, I. Montvay, G.[8] Münster, S. PiemonteG. and Bergner, C. P.Scior, PoS Lopez and S. Piemonte, PoS [6] G. Bergner, P. Giudice, G. Münster, I. Montvay and S. Piemonte, Phys. Rev. D [9] M. Costa and H. Panagopoulos, Phys. Rev. D [3] S. Ali, H. Gerber, I. Montvay, G. Münster, S. Piemonte, P. Scior and G. Bergner, Eur. Phys. J. C [2] S. Ali, G. Bergner, H. Gerber, P. Giudice, I. Montvay, G. Münster, S. Piemonte and P. Scior, JHEP [4] G. Bergner, P. Giudice, G. Münster, P. Scior, I. Montvay[5] and S.G. Piemonte, Bergner JHEP and S. Piemonte, Phys. Rev. D [1] G. Bergner, P. Giudice, G. Münster, I. Montvay and S. Piemonte, JHEP [13] T. A. Ryttov and F. Sannino, Phys. Rev. D [16] R. Arthur, V. Drach, M. Hansen, A. Hietanen, C.[17] Pica andV. F. Drach, Sannino, T. Phys. Janowski Rev. and D C. Pica, EPJ Web Conf. [14] A. Patella, Phys. Rev. D [15] A. Athenodorou, E. Bennett, G. Bergner and B. Lucini, Phys. Rev. D [18] T. Karavirta, J. Rantaharju, K. Rummukainen and K. Tuominen, JHEP [10] J. Giedt, Int. J. Mod. Phys. A [11] B. H. Wellegehausen and A. Wipf, PoS [12] K. A. Intriligator and N. Seiberg, Nucl. Phys. Proc. Suppl. pure Yang-Mills sector can be under control, which means at least lattice sizes of 24 and advises. The(www.gauss-centre.eu) authors for funding gratefully this acknowledge projectcomputer the by JUQUEEN Gauss providing and computing Centre time JURECA forLeibniz on Supercomputing at Supercomputing Centre the (LRZ). Jülich e. GCS G. B. Supercomputing Super- V. acknowledges supportgemeinschaft Centre from (DFG) the (JSC) Grant Deutsche Forschungs- No. and BE SuperMUC 5942/2-1. at References Supersymmetric and conformal theories of the theory. One peculiar problem islimit. related to the divergent scalar field contributions in the chiral unimproved Wilson fermions. Consequently theregime current where simulations a non-perturbative for tuning SQCD can are beget still expected to to not the work. in required A regime the much where larger the effort supersymmetry is breaking needed can to be broughtAcknowledgments under control.