PoS(LATTICE2018)210 https://pos.sissa.it/ Gauge Theory - ) 3 ( SU ∗ 1 Supersymmetric Yang-Mills theory, supersymmetry is only broken by a = N Supersymmetric 1 supersymmetric QCD (SQCD) is a possible building block of theories beyond the standard 1 = [email protected] [email protected] = Speaker. model. It describes the interactionand between squarks. gluons and Since quarks supersymmetry withfine-tuning their is of superpartners, explicitly operators gluinos is broken necessary bygauge to the sector, obtain lattice a regularization, supersymmetric a continuum careful limit. For the pure N non-vanishing gluino mass. If wethe add scalar matter squark fields, sector this is have tocalculation no longer in be true lattice considered and perturbation for more theory, fine-tuning we operatorssector the show in is theory. that nevertheless Guided maintaining an by chiral important a step. symmetryon one-loop in the Furthermore, the fine-tuning we and light present Ward-identities first of preliminary SQCD. lattice results ∗ 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. Friedrich Schiller University Jena, 07743 Jena,E-mail: Germany Björn Wellegehausen Andreas Wipf Friedrich Schiller University Jena, 07743 Jena,E-mail: Germany N Towards simulations of Super-QCD PoS(LATTICE2018)210 (2.2) (2.1) (2.3) λ . The Pauli are the chiral 5 )  ) N αγ i φ − ( . e † ] P , ]. The mass spec- φ λ SU Björn Wellegehausen 1 + , → Z m P µ λ A − [ ψ = ( g µ i , P ¯ D a two-component complex ψψ ]. + ψ µ 9 5  ) λ (baryon number conservation) − αγ m ¯ µ i ψγ B φ ∂ i flavours is given by − ) , + f e 1 + 2 + = ( N
φ φ → λ U φ µ 3 µ ψ = ( D σ D with † ]. The next step towards a supersymmet- T φ 4 ) φ † . The : , µ N R φ 3 ( ) R [ D ) 1 and 2 ) ( 1 − g SU ( 3 U ( 1 2 1 ,  U 1 supersymmetric Yang-Mills theory (SYM), that ) ⊗ R λ − SU ) φ B µ
= , , 1 ) D ( 1 gauge transformations as well as under supersymmetry φ ψ ( µ 3 λφ ( U ) N ¯ U
] and P ασ ¯ N λγ i 2 ¯ µ ( ψ ⊗ [ i e 2 A ) gA − ) i SU 2 + → 1 ( ψ ( + φ µν P represents the generators of the gauge group µ U SU ¯ F λ ∂ † , ⊗ T µν φ R ψ F ) is a Majorana spinor in the adjoint representation of the gauge group, 5 f 4 1 g ) = N and αγ 2 λ i ]. In the present work we review the continuum formulation of SQCD and φ ( − e , 8 0 √  i γ , ψ SU 7 → P ( a Dirac spinor and the squark field − Tr ⊗ 0 µ ψ γ = L ψ D ) f = ], but non-perburbative results, especially for the mass spectrum of the theory, are not symmetry and R-symmetry L : N 5 ¯ ( P act on the two components of the squark field. The covariant derivatives are A A ) ) ]. A possible approach to control all relevant operators is a fine-tuning guided by lattice 1 σ SU 1 ( 6 ( U U is unbroken even in thechiral quantum theory and restricts operators to vanishing baryon number. The transformations and a discrete parityaction transformation. is The global symmetry group of the classical scalar field, both in the fundamental representation. The components of matrices The action is invariant under local where the gluino field the quark field projectors, discuss our lattice formulation andpotential fine-tuning for of the operators. squark Thenpersymmetry field we in to derive the a improve continuum one-loop our limit.sign effective lattice problem action Finally of with we the respect theory presentdiscussion and to first and present lattice further the our results results restoration result will regarding of be for a published supersymmetric su- in possible Ward-identities. a A follow-up paper detailed [ trum and chiral properties of SYMfermions theory for have the been gauge extensively studied groups on the lattice with Wilson perturbation theory. This method has beenric successfully gauge applied theories to [ two-dimensional supersymmet- 2. Continuum formulation of supersymmetric QCD The Lagrange function of SQCD for gauge group describes the interaction ofplicitly gluons broken with by their the lattice superpartners,to formulation, the restore but gluinos. chiral for this symmetry theory Supersymmetry in it the is has continuum ex- been limit shown to that recover it supersymmetry is [ sufficient ric standard model is thesupersymmetric inclusion QCD of (SQCD). matter SQCD describes fieldstheir the in superpartners, gluinos interaction the and between squarks. fundamental gluons So representation,approaches far and leading it [ quarks has to been with studied with classicalavailable. and semi-classical One reason islations the [ large number of operators, that has to be fine-tuned in lattice simu- Simulations of Super-QCD 1. Introduction Supersymmetric gauge theories are andard important building model. block of The many models most beyond simplest the one stan- is PoS(LATTICE2018)210 ). = 8 2.1 (3.3) (3.1) (3.2) break 1 and ⊗ 5 V ] = φ and . ¯ 8 and 8 3 of the gauge 2 . V ] = [ ⊕ −} } A , 1 1 [ Björn Wellegehausen is not difficult. σ + , , = f 1 ¯ 3 N 1. On a kinematic level ∈ { { 4 have to be fine-tuned. −− symmetry and therefore, ⊗ s . = Φ , = = B r ) f ) σ N D 1 ++ ( −− , Φ ≤ s Φ U , ] φ = } with † r + O 5 3 [ ) φ γ V 3 , ( ++ = 1 transformations while { Φ , rs SU A + )( = Φ ) is still preserved in the quantum theory. 2 − + 1 Γ ( − R with fermion number Φ ) gauge transformations restricts fundamental ) Φ U 1 + ( ) N + with − 3 ( U ( 2 2 + − with ⊗ + Φ SU ) SU 1 A Φ ) = σ λ 1 2 Γ ( Φ = ( V ( ¯ λ U 5 Fierz identities. Tr V ) , N = ( , 2 −− 2 and + Φ M SU − + Φ ψ are invariant under the − Γ The first term is not invariant under the Invariance under subgroup of 4 2 ++ + breaks the axial chiral symmetry while the first is invariant under chiral . and ¯ V ψ 3 2 Φ Φ The five marginal quartic operators, that have to be fine-tuned are AF ) M = = ⊗ Invariance under gauge symmetry restricts the Yukawa interactions to the ]. The mass dimensions of the fundamental fields are , 1 Φ and 1 4 8 9 ( V V 3 , Tr ⊗ U 6 V σ φ ¯ 3 , † = 1 φ 1 2. Only relevant operators with mass dimension V / M 3 or 3 ⊗ 3 ] = . The quarks and squarks transform in the fundamental representations 3 and λ ⊗ ... ] = [ ⊕ ψ chiral symmetry. The squark potential intion the of continuum this action five can operators be using written as a linear combina- 3. Lattice formulation and fine-tuning The lattice formulation with Wilson fermionssymmetry employed and in R-symmetry this explicitly. work The breaks remaining supersymmetry, intactthe chiral symmetries number of of the lattice operators theory that restrict needin the to continuum be limit. fine-tuned A in detailed acan discussion lattice of be simulation the to found important restore operators in and all the [ symmetries lattice formulation 1 group while the gluino transforms undercombinations the of adjoint component representation fields 8. are Therefore, the only invariant Its renormalization can always be absorbed byThe a Yukawa interaction wave-function renormalization should of be the a scalar marginal fields. operator in fourQuartic dimensions. interactions scalar and fermionic bilinears to two different tensor products, namely 3 The operators Simulations of Super-QCD are broken in the quantum theory byan the anomalous axial free anomaly, similar to QCD and SYM theory. However, [ Therefore, we have to consider onlysimplify the quadratic, discussion, cubic we and consider quartic only the interactionsthe gauge generalization in group to the an following. arbitrary To gauge group Quadratic interactions Quark and squark masses are compatiblequarks and with squarks supersymmetry has and to only be tuned. thetry relative An since mass explicit its gluino between superpartner, mass the on gluon, thescalar is other mass hand always terms breaks massless can supersymme- due be to written gauge as invariance. The two possible the only term invariant under all symmetries is the one already appearing in the Lagrangian ( The second mass term transformations. Cubic interactions form 3 PoS(LATTICE2018)210 , the (4.2) (4.1) (4.3) (3.4) (3.5) i m ) while . 0 3.5 . . 4 .
) = V  ) 5 ) is the quadratic 2 G λ φ y 3.5 λ . To reduce the i ( ¯ ∆ F P ] λ , = Ψ C ¯ φ 2 φ ψ [ Björn Wellegehausen 2 . δ ∆ 2 S ) λ ( − 2 p x 4 ( f δ / = ψ 1 contain only logarithmic Ψ 2 given by ( P + i δ f m = 3 ¯ λ ,  † V c G , ) i 1 φ ∆ 0 and different lattice volumes. λ 2 G , the two squark masses − g  ∆ 2 Trln i = , m = ]. Here, we compute an effective V √ 2 φ 1 2 i i m 4 ∆ λ 10 λ ( − + with 2 φ 3 , for f Z ! ψ c 1 φ 1 2
N + + and = q i + 1 i ϕ q m = c V

M ) 2 m ) . We observe, that at one-loop in perturbation 3 2 i − y g , 2 G depend on the squark and gluon propagator, the λ m ( m g µ ] i ∆ , f + φ 3 , m ϕ D c 1 [ 2 φ Z , δϕ µ i and gluon propagator S ) c ∆ γ λ − 2 , that are already present in the tree-level action, ap- + x ( ) N ( 4 c 1 2 δ f φ ¯ = and the five squark quartic couplings V ψ N m µ i ) is given by δϕ φ + λ + D + = Z 1 † 2 and

2.1 λ 1 φ p M ,
λ 3 ( µ g 2
V , / c D φ m , 1 1 Trln 2 1 φ ∆ g 2 1 − V 8 Z = = and the functions 1. This are the bare gluino mass , µ φ ) = φ 1 − + c = D ∆ Z ) + 2 2 G M f N µ 2 , φ µν m ( ∆ γ N ( F 6 m V ¯ SU λ , Tr + = and the quark / squark mass 1 φ 1 4 1 c Tr c ) = ∆ 2  m 2 1 N φ 2 g ( 2 for the 7 counterterms are given in Table 1 = + g i F + q c 2 C = m 1-loop m eff , V L ) = = 0 p 2 1 ( = m V g m with scalar field propagator Casimir eigenvalue of number of colors Before momentum integration we obtain the one-loop correction theory, only the operators pear. Furthermore, the quadratic divergencesdivergences cancel as and expected the from functions supersymmetry. On the lattice, the Wilsonreplace the mass continuum and propagators discrete by the latticepossible corresponding momenta 7 lattice squark propagators. break operators As supersymmetry. are a generated consequence, We atthat all one-loop, have the and to lattice we action have to at introducethe one-loop counterterms, couplings corresponds such to the continuum action at one-loop. The values of It turns out, that thelast counterterms line are of finite the table. and In theirIn our infinite the simulations volume following, we extrapolation the use is unimproved the lattice valuesthe given at action improved in the action corresponds the given is to lattice given the size by and parameters free squark given parameters mass. in ( The relation to the continuum action ( Simulations of Super-QCD After a rescaling of the fieldsdensity: and a Wick rotation we obtain the most general Euclidean Lagrange number of operators, we willsquark effective fix potential as in our many first as simulations possible of SQCD. by a perturbative4. calculation One-loop of effective the potential lattice Very recently, SQCD has beenand investigated squark with masses lattice have perturbation beensquark theory potential computed and at to critical one-loop one-loop in fermion contribution, order that lattice [ come perturbation with theory. a different It sign contains a bosonic and a fermionic squark wave function renormalization To restore supersymmetry in thefine-tune continuum 9 limit operators of for the corresponding lattice action, we have to PoS(LATTICE2018)210 3, . 32. 0 ) 5 0 and × = V 3 ( = 7 m 16 c m 0.077108 0.083166 0.084368 0.084646 0.084714 0.084731 0.084736 = 9 at . V 9 ) 4 = V ( β Björn Wellegehausen 97 at 6 . c 0 0.21088 0.22182 0.22434 0.22496 0.22511 0.22515 0.22516 − 3 to . 6 94 ) . V 14 18 22 3 = . . . . The sign problem increases 1 1 1 V 2 1 ( − − − g 5 / = = = c c g g g 0.252934 0.258885 0.259836 0.260038 0.260086 0.260098 0.260102 m m m N 16 and 0 . The result for the Pfaffian phase = g × m β 3 for different lattice volumes (left panel) and 8 ) g 2 0 200 400 600 800 1000 1200 1400 m = V ( 4 V c 1.0001 1.0000 0.9999 0.9998 0.9997 0.9996 0.9995 0.9994 0.9993 0.9992 0.9991 0.9990

-0.05689831 -0.05722842 -0.05723039 -0.05723041 -0.05723041 -0.05723041 -0.05723041

4

1 SYM theory. The simulations are performed on

Re e

φ i

998 at = . . 0 L ) 2 N 1 − V × ( 3 3 L 996 c . = 0.3108112 0.3109568 0.3109917 0.3110003 0.3110030 0.3057303 0.3101264 V 4 4 4 4 is shown in the left panel of Figure 3 4 5 6 i = = = = ) } 2 V V V V φ i M g 16 with inverse gauge couplings { ( m 2 × c 4. In this regime, we expect that the explicit breaking of chiral symmetry . 3 of the counterterms obtained from the one-loop effective potential for 0 -0.6366606 -0.6366606 -0.6366606 -0.6366606 -0.6366606 -0.6361834 -0.6366576 i c − Re exp h 1 SYM theory we determine the parameters for vanishing gluino mass by fine- = ) q 1 = m M ( 1 N c Pfaffian phase as function of the bare gluino mass Couplings -1.28 -1.26 -1.24 -1.22 -1.20 -1.18 -1.16 -1.14 -1.12 -1.378453 -1.378453 -1.379358 -1.378523 -1.378457 -1.378453 -1.378453 5 and .

3 and different lattive volumes 1.0000 0.9998 0.9996 0.9994 0.9992 0.9990 0.9988 0.9986 0.9984 0.9982 0.9980 0

Re

= e

8 L = ∞ φ i

16 32 64 c 256 128 5. Lattice results In a first test of our fine-tuningwe procedure expect we the consider theory the to limit of behave heavy similar quarks to and squarks, where Figure 1: lattice volumes up to 8 as function of the lattice volume for three different barewith gluino masses the (right lattice panel). volume andvery small depends lattices only where slightly we on canright compute the panel the distance we exact to show Pfaffian, the the theThe phase critical sign solid is point. lines as very are function close On linear to ofexpectation these fits one. the value to In of lattice the approximately the volume data 0 for points. three Extrapolated different to gluino larger volumes masses. we obtain a phase N m Table 1: dominates the spontaneous breaking.the Yukawa First interaction we between gluinos, investigate quarks thethe and Pfaffian squarks. Pfaffian sign On exactly rather problem as small induced lattices function by we of compute the bare gluino mass tuning the theory to vanishing adjoint pion mass. The adjoint pion is not a physical particle but its Simulations of Super-QCD expectation value Even at this large lattice thetaking sign into problem account is the almost Pfaffian absent phase. Similar and we to can simulate the theory without PoS(LATTICE2018)210 . g β m on a (5.1) (5.2) 2 6. The f -meson . is shown N 6 η c 16 lattice as N = W improved 2 β unimproved × 3 β 3 1 SYM theory. + 1 = . In the right panel − ) Björn Wellegehausen 2 c 1 N ( 3 and the improved or N . SYM 0 12 SQCD = . 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 = 1

β φ 0.065 0.060 0.055 0.050 0.045 0.040 0.035 0.030 0.025 0.020 − m † φ = for the unimproved and improved W 2 c g c g 5 or m . m , it is less broken for the improved m 16 lattice. In Figure 0 β + × = φ 3 µ m and . D κ g † and inverse gauge coupling β m φ 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 a g µ g m m D + -0.60 -0.70 -0.80 -0.90 -1.00 -1.10 -1.20 -1.30

∞ g 5 + 6 and critical line (right panel) on a 8 W . at the critical gluino mass to the function m 6  compared to the critical line of ) = β ) = β µν , β c g β F ( m -1.30 -1.20 -1.10 -1.00 -0.90 -0.80 -0.70 -0.60 -0.50 µν ( W F 0.07 0.06 0.05 0.04 0.03 0.02 . W . At fixed values of Tr β W for the unimproved (left panel) and improved (center panel) action as β 1 4 5 . − W 0 3 5 . .  as function of = 0 0 2 3 m c g = = g m m m = m c W g m improved, unimproved, improved, , 0 = 1 − Adjoint pion mass (left panel) for Bosonic Ward-identity c -1.16-1.14-1.12-1.10-1.08-1.06-1.04-1.02-1.00-0.98 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 14 W -1.30 -1.20 -1.10 -1.00 -0.90 -0.80 -0.70 -0.60 -0.50 m 0.07 0.06 0.05 0.04 0.03 0.02 16 lattice as function of the bare gluino mass = W × W 3 8 results is rather insensitive to the choiceunimproved of action, mass indicating parameter that indeed insquark this sector limit decouple. the We light obtain gluino a sectorwe critical and show bare the the gluino heavy critical mass quark mass of / As expected, the critical lines agreeNext in we the investigate, limit how of well heavy the quarks the and bosonic squarks. Ward-identity Figure 3: for the unimproved (left panel) andcoupling improved (center and panel) bare action as gluino function mass.while of it the decreases The inverse with gauge breaking increasing of the Ward-identity increases with increasing Figure 2: function of the gluino mass. The different colors encode different (from left to right increasing) values of Simulations of Super-QCD function of the inverse gauge coupling In the right panel, the bosonicaction Ward-identity at is the shown. critical The gluino solid mass lines represent fits to the data points. correlation function is calculated in analogycorrelation to function. QCD as The the extracted connected adjoint part pion of mass the adjoint is shown in the left panel of Figure is fulfilled for the unimproved and improved actions on the 8 action. We then fit the Ward-identity PoS(LATTICE2018)210 = ]. , Gauge Phys. ) super B292 , 3 unimproved ( ∞ ) . 2 W , 2 1801.08062 (2009) 4045 [ = ( Björn Wellegehausen A24 Nucl. Phys. N ]. , the fit values (2018) 113 The light bound states of ∞ Supersymmetric SU 03 1 → ]. with improved action is always β = -dimensional The light bound states of 2 W 1512.07014 [ JHEP N , Int. J. Mod. Phys. , PoS(LATTICE2018)211 (2016) 080 03 . 6 Mass spectrum of ,. hep-lat/0507019 [ JHEP these proceedings ]. , , . Both results are compatible with a restored bosonic Ward- Dynamical Supersymmetry Breaking in Supersymmetric QCD ) (2005) 082 1802.07797 11 Supersymmetric QCD on the Lattice: An Exploratory Study In preparation , ( 10 . Two-dimensional N = (2,2) super Yang-Mills theory on the lattice via Supersymmetry and the Lattice: A Reconciliation? (right panel). We obtain in the limit 003 . 3 1706.05222 0 [ JHEP − , ]. 9 = (1984) 493 We thank Stefano Piemonte, Marios Costa, Haralambos Panagopoulos, Andre 1 SYM theory on the lattice, supersymmetric QCD requires much more opera- ]. = B241 improved ∞ . (2017) 034507 W N Progress in four-dimensional lattice supersymmetry supersymmetric SU(3) Yang-Mills theory on the lattice 1 and D96 = ) 3 0903.2443 ( Yang-Mills theory on the lattice Rev. Nucl. Phys. [ dimensional reduction (1987) 555 supersymmetric SU(2) Yang-Mills theory N Theory – Pure Gauge sector with a twist 008 . [1] G. Curci and G. Veneziano, [2] G. Bergner, P. Giudice, G. Münster, I. Montvay and S. Piemonte, [3] S. Ali, G. Bergner, H. Gerber, P. Giudice, I. Montvay, G. Münster[4] et al., B. Wellegehausen, A. Wipf, M. Steinhauser and A. Sternbeck, [5] I. Affleck, M. Dine and N. Seiberg, [6] J. Giedt, [7] D. August, B. H. Wellegehausen and A. Wipf, [8] H. Suzuki and Y. Taniguchi, [9] B. Wellegehausen and A. Wipf, Sternbeck and Marc Steinhauser for discussions and helpful comments. References [10] M. Costa and H. Panagopoulos, Acknowledgments: closer to its continuum value 0 than for the unimproved action. 6. Conclusions Compared to tors to be fine-tuned torelevant perform operators a in supersymmetric a lattice continuum formulation limit. withfor Wilson In the fermions this and squark work calculated field we a to one-loop discussed guidetry. potential the the fine-tuning We towards simulated a supersymmetric continuumthat limit QCD the with in fermion restored sign the supersymme- problem limitour induced one-loop of by improved heavy the lattice quarks action Yukawa interaction leads andfulfilled is to in squarks almost an the and improved continuum absent. bosonic showed, limit. Furthermore, Ward-identityand So that the far, seems theory simulations to behaves have be similar been to donediscussion SYM for of theory. heavy our quarks Simulations lattice and for formulation squarks, lighter andwill matter be the fields made calculation and public of a in the [ detailed one-loop effective squark potential identity in this limit. At finite lattice spacing, the Ward-identity Simulations of Super-QCD The result is shown in Figure 0