Continuum limit of SU(3) N=1 supersymmetric Yang-Mills theory and supersymmetric gauge theories on the lattice
Sajid Ali, Georg Bergner, Henning Gerber, Camilo López, Istvan Montvay, Gernot Münster, Stefano Piemonte, Philipp Scior
CContinuum limit of SU(3) N = supersymmetricYang-Mills theory and supersymmetric gaugetheories on the lattice Sajid Ali
Institut für Theoretische Physik, Universität Münster, Wilhelm-Klemm-Str. 9, D-48149 MünsterDepartment of Physics, Government College University Lahore, Lahore 54000, Pakistan
Georg Bergner * University of Jena, Institute for Theoretical PhysicsMax-Wien-Platz 1, D-07743 Jena, GermanyE-mail: [email protected]
Henning Gerber
Institut für Theoretische Physik, Universität MünsterWilhelm-Klemm-Str. 9, D-48149 Münster, Germany
Camilo López
University of Jena, Institute for Theoretical PhysicsMax-Wien-Platz 1, D-07743 Jena, Germany
Istvan Montvay
Deutsches Elektronen-Synchrotron DESYNotkestrasse 85, D-22607 Hamburg, Germany
Gernot Münster
Institut für Theoretische Physik, Universität MünsterWilhelm-Klemm-Str. 9, D-48149 Münster, Germany
Stefano Piemonte
University of Regensburg, Institute for Theoretical PhysicsUniversitätstr. 31, D-93040 Regensburg, Germany
Philipp Scior
Fakultät für Physik, Universität BielefeldUniversitätsstr. 25, D-33615 Bielefeld, Germany
We summarize our investigations of several aspects of N = N = R × S . * Speaker. © Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] J a n U(3) SYM and SUSY gauge theories on the lattice
Georg Bergner
1. Continuum limit of SU(3) N = super Yang-Mills theory with Wilson fermions Supersymmetric gauge theories play a central role for several theoretical developments to-wards an analytical understanding of strong interactions, such as gauge-gravity duality. However,their numerical investigation on the lattice leads to several challenges and unsolved problems. Itis the central goal of our investigations to perform numerical simulations of supersymmetric gaugetheories.The first target of our lattice simulations is the spectrum of bound states of N = λ ). The Euclidan Lagrangian is L = F µν F µν +
12 ¯ λ ( / D + m g ) λ . (1.1)The gluinos are Majorana fermions in the adjoint representation of the gauge group. The La-grangian L is invariant under the supersymmetry transformations: δ A µ = −
2i ¯ λ γ µ ε , δ λ = − σ µν F µν ε , in case of a vanishing gluino mass m g = f ( λ λ ) and the a– η (cid:48) ( λ γ λ ) meson, while thestates of glueball type are the 0 ++ and 0 − + glueballs [3, 4]. In our most recent investigationswe have considered in detail the relevant mixing of these two multiplets, and we obtained morereliable results for the lightest states [5]. We have also started to investigate other operators andbound states of the theory such as baryonic operators [6]. Our recent optimizations of the operatorbasis have been presented in a separate contribution to this conference.In the most generic case, supersymmetry can only be obtained by a fine-tuning of the pa-rameters of the lattice action. In the special case of SYM the fine-tuning can be avoided if chiralsymmetry is realized on the lattice using Ginsparg-Wilson fermions. In our first investigations wehave, however, relied on the simulations with Wilson fermions. These require the fine-tuning of asingle parameter, the fermion mass m g (or equivalently the hopping parameter κ ) [7]. Our tuningapproach relies on the signal for chiral symmetry restoration provided by a vanishing adjoint pionmass. This particle is not a physical state of the theory, but can be defined in a partially quenchedsetup [8].Our previous simulations of SU(2) gauge theory have been performed with a Wilson actionusing stout-smeared gauge links, and tree-level improved gauge action. For our new investigations1 U(3) SYM and SUSY gauge theories on the lattice
Georg Bergner of SU(3) SYM, we have employed one-loop clover-improvement of the Wilson fermion actionand demonstrated that it significantly reduces the lattice artefacts. We started by estimating thebest range of simulation parameters from investigations of finite volume effects, the sign problem,and topological freezing that slows down the simulations. The sign problem in SYM with Wilsonfermions is due to the Pfaffian of the Dirac operator that results from the functional integration ofMajorana fermions. Negative signs are rare in our simulations and can be taken into account byreweighting. In this way, we have estimated a reliable parameter range for our simulations [9]. Wehave also confirmed consistency with the supersymmetric Ward identities [10]. .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 . ( w , χ m π ) . . . . . . w , χ m gg ( ) a − η (cid:48) ( ) ++( ) . . . . . . . ( w , χ m π ) . . . . . . w , χ m gg ( ) a − η (cid:48) ( ) ++( ) Figure 1:
Extrapolation to the chiral and continuum limit shown in the plane of fixed lattice spacing. Thefigure on the left hand side shows our coarsest lattice spacing β = .
4, while the figure on the right hand sidecorresponds to the finest considered lattice spacing β = .
6. The lightest state of the gluino-glue ( gg ( ) ), thescalar (0 ++( ) ), and the pseudoscalar (a– η (cid:48) ( ) ) channel are shown as a function of the adjoint pion mass m π in units of the gradient flow scale w , χ . We have recently completed our simulations and we have been able to extrapolate the particlespectrum to the continuum and to the chiral limit. In the previous SU(2) SYM project, we haveperformed the two extrapolations independently. Our most recent data for SU(3) SYM allowedto apply a more reliable approach in terms of a simultaneous two-dimensional fit of all data, see[2] for further details. The data and fits projected in the plane of two lattice spacings are shownin Figure 1. The supersymmetric point is reached in the chiral plane at vanishing lattice spacing(Figure 2). Since we have used a one-loop improved lattice action, a functional dependence ofdiscretization effects according to ag is expected, where a is the lattice spacing and g the barecoupling constant. We have found that the our data are also in good agreement with a quadraticdependence on the lattice spacing.The final parameter ranges are: for the pion mass 0 . < am a– π < .
7, for the lattice spacing0 .
053 fm < a < .
082 fm, and for the lattice sizes from 12 ×
24 to 24 ×
48. The masses of boundstates in units of the gradient flow scale w are summarized in the following table:Fit w m g ˜ g w m ++ w m a − η (cid:48) linear fit 0.917(91) 1.15(30) 1.05(10)quadratic fit 0.991(55) 0.97(18) 0.950(63)SU(2) SYM 0.93(6) 1.3(2) 0.98(6)2 U(3) SYM and SUSY gauge theories on the lattice
Georg Bergner . . . . . . . . a / ( β w , χ ) . . . . . . . . w , χ m g ˜ ga − η (cid:48) ++ .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . ( a / w , χ ) . . . . . . . w , χ m g ˜ ga − η (cid:48) ++ Figure 2:
Extrapolation to the chiral and continuum limit shown in the plane of zero adjoint pion mass.The left hand side uses a linear extrapolation in the lattice spacing a , taking into account the one-loopimprovement with the inverse coupling constant β . The right hand side is a quadratic extrapolation in a . Allquantities are in units of the gradient flow scale w , χ . To conclude, our findings for SU(3) SYM show the formation of supermultiplets of boundstates and consistency with the supersymmetric Ward identities, confirming the validity of ournumerical approach and providing the starting point for further investigations in two different di-rections.The first direction is a more detailed study of the properties of SYM, such as the phase tran-sitions of this theory. At non-zero temperatures, we have found an interesting interplay betweenthe chiral and deconfinement transitions [11]. Deconfinement is absent in compactified SYM on R × S , and there is a continuity towards the semiclassical regime at a small compactification ra-dius [12]. The theory at zero temperature in the chiral limit provides further interesting featuresthat require a more detailed investigation. It is expected that spontaneous chiral symmetry breakingleads to N c different values of the gluino condensate for SU( N c ) SYM.The second line of investigations is the extension of numerical studies towards more generalsupersymmetric gauge models, such as super-QCD. The scalar fields of these theories lead to alarger number of relevant supersymmetry breaking operators, and a more delicate tuning procedureis required.
2. Exploratory studies of supersymmetric Yang-Mills theory with overlap fermions
Ginsparg-Wilson fermions allow to define an action with an intact (modified) chiral symmetryeven at non-vanishing lattice spacing [13]. The formulation of the Dirac operator fulfilling theGinsparg-Wilson relation is not unique. Domain-Wall fermions provide an exact solution in thelimit of an infinitely extended fifth dimension, and have been employed in some studies of thegluino condensate for gauge group SU(2) [14]. Several alternative approximation schemes havebeen suggested in the literature. In one exploratory investigation, also the overlap operator hasbeen considered [15].We have started to explore overlap gluinos. In our approach we implement the sign functionin terms of a polynomial approximation, allowing even in the massless limit to provide a smoothand regular force for the integration of the equation of motion required by the hybrid Monte Carlo3
U(3) SYM and SUSY gauge theories on the lattice
Georg Bergner method. The simulation cost of overlap gluinos is huge, as two expensive approximations arerequired to compute the sign function and the square root of the determinant corresponding to thePfaffian. In this context, a polynomial approximation avoids an additional inner inverter to the onealready required to compute the rational approximation of the square root employed by RHMC,and we can reach a stable precision of the rational approximation in this way.Several drawbacks have to be faced considering the overlap operator, mainly related to thezero modes. Effectively, each configuration with non-trivial topology has a zero mode, such thatits determinant and its Monte Carlo probabilistic weight is zero. This fact would effectively in-duce a fixed topology in the simulations. However, zero modes are also responsible for gluinocondensation, and therefore we are facing a “zero over zero” problem. A controlled approximationof the overlap formula can avoid zero modes as they are effectively smoothend by the polynomialapproximation.The huge simulation cost represents a challenge for the complete investigation of the boundstate spectrum of SYM with Ginsparg-Wilson fermions. However, several of our follow-up projectswould benefit from preserving chiral symmetry on the lattice. Wilson fermions lead to an additiverenormalization of the gluino condensate, and it is more difficult to investigate spontaneous chiralsymmetry breaking and the different phases of the gluino condensate. The additive renormalizationproblem can be avoided using the gradient flow, but the effects of chiral symmetry breaking preventfrom observing the N c phases of the gluino condensate even in the chiral limit. Therefore a detailedstudy of the gluino condensate requires a careful consideration of this fermion formulation.In super-QCD and other supersymmetric gauge theories, the large number of fine-tuning pa-rameters is reduced by the constraints of chiral symmetry. Even if fine-tuning can be handled bychecking the Ward identities, the cost of a Ginsparg-Wilson implementation will be balanced bythe reduced tuning costs. Figure 3:
The polynomial approximation of order N of the overlap formula. Left hand side: the deviationfrom the exact overlap operator, whose eigenvalues lie on a circle. The eigenvalues for polynomial orders160, 250 and 400 are shown. Right hand side: The sensitivity of the gluino condensate to the order of theapproximation. We have performed preliminary investigations of SU(2) SYM at β = . volume.The precision of our approximation is shown in Figure 3. The location of zeroes converges to thecircle in the complex plane, corresponding to the exact overlap operator. The zero mode problem issolved by a spectral gap, which is closing for more precise approximations, as expected. Remark-4 U(3) SYM and SUSY gauge theories on the lattice
Georg Bergner ably, the gluino condensate seems to be quite stable as a function of the order of the approximation,at least for the range considered in these first tests.
3. Phase transitions and compactified supersymmetric Yang-Mills theory
We have intensified our investigations of the phase transitions in SYM, in an attempt to un-derstand the relation between confinement and chiral symmetry breaking in this theory. We haveconfirmed the coincidence of the two transitions, considering also the gauge group SU(3), usingthe gradient flow to eliminate the difficulties with the additive renormalization of the condensate[16, 17].Another interesting aspect of the phase diagram in SYM is the absence of the deconfinementtransition if periodic boundary conditions are chosen instead of the usual thermal ones. This prop-erty of SYM on R × S has led to the conjecture of continuity down to a semiclassical regime at asmall radius of the compactified direction [18].In our first numerical investigations of compactified SYM, we have verified the absence ofthe deconfinement transition [12]. However, in the regime of small compactification radius, wehave observed deviations from the predicted behavior. The confinement region extends towardslarger masses when the compactification radius shrinks, whereas it is expected that this region getssmaller at smaller radii.In our most recent analysis [19] we were able to identify the difference between the observedand predicted phase boundaries as a lattice artefact stemming from the Wilson fermions. At smallradius, effectively a larger number of fermion fields contribute to the dynamics driven by latticeartefacts.
4. Conclusions
We have obtained our final results for the low-lying bound states of SU(3) SYM using one-loop clover-improved Wilson fermions, finding evidence for restoration of supersymmetry in thecontinuum limit from the bound state spectrum and from the SUSY Ward identities. We haveseveral uncertainties safely under control, like finite volume effects and the sign problem.We have started first exploratory studies with overlap fermions based on a polynomial approx-imation of the sign function. This appears to be a controlled and feasible algorithm for practicalsimulations, which would lead to a cleaner approach for the investigations of the gluino condensate,and which reduces the fine-tuning problem for super-QCD.Another aspect of our investigations are the phase transitions of SYM. The coincidence of thechiral and deconfinement transition has been confirmed in our most recent simulations. The numer-ical results for the compactified theory support the continuity down to the semiclassical regime atsmall compactification radius. At a very small radius, lattice artefacts have a significant influenceon the transition line. Nevertheless, at an intermediate radius, the theory should already mimic thesemiclassical expectations and is an interesting candidate for further investigations.5
U(3) SYM and SUSY gauge theories on the lattice
Georg Bergner
Acknowledgements
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