EPJ Web of Conferences 175, 14001 (2018) https://doi.org/10.1051/epjconf/201817514001 Lattice 2017

Renormalization of Supersymmetric QCD on the Lattice

Marios Costa1,⋆ and Haralambos Panagopoulos1 1Department of Physics, University of Cyprus, CY-1678 Nicosia, Cyprus

Abstract. We perform a pilot study of the perturbative renormalization of a Supersym- metric gauge theory with matter fields on the lattice. As a specific example, we consider Supersymmetric =1 QCD (SQCD). We study the self-energies of all particles which N appear in this theory, as well as the renormalization of the coupling constant. To this end we compute, perturbatively to one-loop, the relevant two-point and three-point Green’s functions using both dimensional and lattice regularizations. Our lattice formulation in- volves the Wilson discretization for the gluino and quark fields; for gluons we employ the Wilson gauge action; for scalar fields (squarks) we use naive discretization. The gauge group that we consider is SU(Nc), while the number of colors, Nc, the number of flavors, N f , and the gauge parameter, α, are left unspecified. We obtain analytic expressions for the renormalization factors of the coupling constant (Zg) and of the quark (Zψ), gluon (Zu), gluino (Zλ), squark (ZA ), and ghost (Zc) fields on the lattice. We also compute the critical values of the gluino,± quark and squark masses. Finally, we address the mixing which occurs among squark degrees of freedom beyond tree level: we calculate the corresponding mixing matrix which is necessary in order to disentangle the components of the squark field via an additional finite renormalization.

1 Introduction

The current intensive searches for Physics Beyond the Standard Model (BSM) are becoming a very timely endeavor, given the precision experiments at LHC and elsewhere; at the same time, numeri- cal studies of BSM Physics are more viable due to the advent of lattice formulations which preserve chiral symmetry. Furthermore, the lattice formulation of various supersymmetric models is currently under active study [1, 2]. As a forerunner to a long-term prospect of addressing numerically super- symmetric extensions of the Standard Model, we have undertaken an investigation of SQCD, in order to address some of the fundamental difficulties which must be resolved before further progress can α α be made. Within the SQCD formulation we compute the quark (ψ), gluino (λ ), gluon (uµ), squark (A) propagators and the gluon-antighost-ghost Green’s function. Our computations are performed to one loop and to lowest order in the lattice spacing, a. We extract the renormalization factors for the coupling constant (Zg) and of the quark (Zψ), gluon (Zu), gluino (Zλ), squark (ZA ), and ghost (Zc) fields and the critical masses of quark, gluino and squark fields. ± The details of our work, along with a longer list of references, can be found in Ref.[3].

⋆Speaker, e-mail: [email protected]

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 175, 14001 (2018) https://doi.org/10.1051/epjconf/201817514001 Lattice 2017

2 Lattice Action Even though the lattice breaks supersymmetry explicitly [4], it is the only regulator which describes many aspects of strong interactions nonperturbatively. We will extend Wilson’s formulation of the QCD action, to encompass SUSY partner fields as well. In this standard discretization, quarks, squarks igaT αuα(x+aµ/ˆ 2) and gluinos live on the lattice sites, and gluons live on the links of the lattice: Uµ(x) = e µ . This formulation leaves no SUSY generators intact, and it also breaks chiral symmetry; it thus repre- sents a “worst case” scenario, which is worth investigating in order to address the complications [5] which will arise in numerical simulations of SUSY theories. In our ongoing investigation we plan to address also improved actions, so that we can check to what extent some of the SUSY breaking effects can be alleviated. For Wilson-type quarks (ψ) and gluinos (λ), the Euclidean action L on the lattice becomes SSQCD (A : squark field components): ±

L 4 Nc 1 r 2 = a 1 TrUµν + Tr λ¯ γµ µλ a Tr λ¯ λ SSQCD g2  − N  M D M − 2 MD M x  µ, ν c µ     r 2 + µA+† µA+ + µA µA† + ψ¯ Dγµ µψD a ψ¯ D ψD D D D −D − D − 2 D µ   + √ ¯ α α ¯ α α + ¯ α α ¯ α α i 2g A+† λMT P+ψD ψDP λMT A+ A λMT P ψD ψDP+λMT A† − − − − − − 1 2  α α 2  + g (A+† T A+ A T A† ) m(ψ¯ DψD mA+† A+ mA A† ) , (1) 2 − − − − − − − −  where: Uµν(x) = Uµ(x)Uν(x + aµˆ)Uµ†(x + aνˆ)Uν†(x), and a summation over flavors is understood in the last three lines of Eq. (1). The 4-vector x is restricted to the values x = na, with n being an integer 4-vector. Thus the momentum integration, after a Fourier transformation, is restricted to the first Brillouin zone (BZ) [ π/a, π/a]4 and the sum over x leads to momentum conservation in each − vertex. The terms proportional to the Wilson parameter, r, eliminate the problem of fermion doubling, at the expense of breaking chiral invariance. The definitions of the covariant derivatives are as follows: 1 λ (x) U (x)λ (x + aµˆ)U†(x) U†(x aµˆ)λ (x aµˆ)U (x aµˆ) (2) µ M 2a µ M µ µ M µ D ≡  − − − −  2 1 λ (x) Uµ(x)λ (x + aµˆ)U†(x) 2λ (x) D M ≡ a2 M µ − M µ 

+U†(x aµˆ)λ (x aµˆ)U (x aµˆ) (3) µ − M − µ − 1  ψ (x) U (x)ψ (x + aµˆ) U†(x aµˆ)ψ (x aµˆ) (4) µ D 2a µ D µ D D ≡  − − −  2 1 ψ (x) Uµ(x)ψ (x + aµˆ) 2ψ (x) + U†(x aµˆ)ψ (x aµˆ) (5) D D ≡ a2 D − D µ − D − µ   1 A+(x) U (x)A+(x + aµˆ) A+(x) (6) µ a µ D ≡  −  1 A+† (x) A+† (x + aµˆ)U†(x) A+† (x) (7) µ a µ D ≡  −  1 + µA (x) A (x aµˆ)Uµ†(x) A (x) (8) − a − − D ≡  −  1 µA† (x) Uµ(x)A† (x + aµˆ) A† (x) (9) − a − − D ≡  − 

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2 Lattice Action A gauge-fixing term, together with the compensating ghost field term, must be added to the action, in order to avoid divergences from the integration over gauge orbits; these terms are the same as in the Even though the lattice breaks supersymmetry explicitly [4], it is the only regulator which describes non-supersymmetric case. Similarly, a standard “measure” term must be added to the action, in order many aspects of strong interactions nonperturbatively. We will extend Wilson’s formulation of the to account for the Jacobian in the change of integration variables: Uµ uµ . QCD action, to encompass SUSY partner fields as well. In this standard discretization, quarks, squarks → igaT αuα(x+aµ/ˆ 2) and gluinos live on the lattice sites, and gluons live on the links of the lattice: Uµ(x) = e µ . This formulation leaves no SUSY generators intact, and it also breaks chiral symmetry; it thus repre- 3 The one-loop Feynman diagrams on the lattice sents a “worst case” scenario, which is worth investigating in order to address the complications [5] We calculate perturbatively 2-pt and 3-pt Green’s functions up to one loop, both in the continuum which will arise in numerical simulations of SUSY theories. In our ongoing investigation we plan to and on the lattice. The quantities that we study are the self-energies of the quark (ψ), gluon (u ), address also improved actions, so that we can check to what extent some of the SUSY breaking effects µ squark (A), gluino (λ), and ghost (c) fields, using both dimensional regularization (DR) and lattice can be alleviated. regularization (L). In addition we calculate the gluon-antighost-ghost Green’s function in order to For Wilson-type quarks (ψ) and gluinos (λ), the Euclidean action L on the lattice becomes SSQCD renormalize the coupling constant (g). The Green’s functions leading to self-energies of squarks (A : squark field components): ± exhibit also mixing among A+ and A† ; we calculate the elements of the corresponding 2 2 mixing − × L 4 Nc 1 r 2 matrix. = a 1 TrUµν + Tr λ¯ γµ µλ a Tr λ¯ λ SSQCD g2  − N  M D M − 2 MD M The one-loop Feynman diagrams (one-particle irreducible (1PI)) contributing to the quark propa- x  µ, ν c µ     gator, ψ(x)ψ¯(y) , are shown in Fig. 1, those contributing to the squark propagator, A+(x)A+† (y) , in r 2 � � � � + µA+† µA+ + µA µA† + ψ¯ Dγµ µψD a ψ¯ D ψD Fig. 2. Identical results are obtained for A+(x)A+† (y) and A† (x)A (y) . The last diagram in Fig. 2 D D D −D − D − 2 D � � � − − � µ   is responsible for mixing between A+ and A . The one-loop Feynman diagrams contributing to the α β − α ¯ β + √ ¯ α α ¯ α α + ¯ α α ¯ α α gluon propagator, uµ(x)uν(y) , and gluino propagator, λ (x)λ (y) , are shown in Fig. 3 and Fig. 4, i 2g A+† λMT P+ψD ψDP λMT A+ A λMT P ψD ψDP+λMT A† � � � � − − − − − − respectively. Lastly, the 1PI Feynman diagram which contributes to the ghost propagator, c(x)¯c(y) , 1 2  α α 2  � � + g (A+† T A+ A T A† ) m(ψ¯ DψD mA+† A+ mA A† ) , (1) is shown in Fig. 5. In this work we also calculate the gluon-antighost-ghost Green’s function in order 2 − − − − − − − −  to renormalize the coupling constant. In Fig. 6 we have drawn the corresponding lattice 1PI Feynman where: Uµν(x) = Uµ(x)Uν(x + aµˆ)Uµ†(x + aνˆ)Uν†(x), and a summation over flavors is understood in diagrams for the 3-pt function. the last three lines of Eq. (1). The 4-vector x is restricted to the values x = na, with n being an As is usually done, we will work in a mass-independent scheme, and thus all of our calculations, integer 4-vector. Thus the momentum integration, after a Fourier transformation, is restricted to the in the continuum as well as on the lattice, will be done at zero renormalized masses for all particles. first Brillouin zone (BZ) [ π/a, π/a]4 and the sum over x leads to momentum conservation in each − vertex. The terms proportional to the Wilson parameter, r, eliminate the problem of fermion doubling, + − at the expense of breaking chiral invariance. The definitions of the covariant derivatives are as follows: 1 ¯ λ (x) U (x)λ (x + aµˆ)U†(x) U†(x aµˆ)λ (x aµˆ)U (x aµˆ) (2) Figure 1. One-loop Feynman diagrams contributing to the 2-pt Green’s function ψ(x)ψ(y) . A wavy (solid) line µ M 2a µ M µ µ M µ � � D ≡  − − − −  represents gluons (quarks). A dotted (dashed) line corresponds to squarks (gluinos). Squark lines are further + 2 1 + marked with a ( ) sign, to denote an A+ (A ) field. A squark line arrow entering (exiting) a vertex denotes a A+ λM(x) Uµ(x)λM(x aµˆ)Uµ†(x) 2λM(x) − − D ≡ a2 − (A+† ) field; the opposite is true for A (A† ) fields. µ  − − + Uµ†(x aµˆ)λM(x aµˆ)Uµ(x aµˆ) (3) − − −  1 + − + µψD(x) Uµ(x)ψD(x aµˆ) Uµ†(x aµˆ)ψD(x aµˆ) (4) + − D ≡ 2a − − − + +   + + + + + + + + 2 1 ψ (x) Uµ(x)ψ (x + aµˆ) 2ψ (x) + U†(x aµˆ)ψ (x aµˆ) (5) D D ≡ a2 D − D µ − D − µ   Figure 2. The first 5 Feynman diagrams contribute to the 2-pt Green’s function A+(x)A+† (y) . The case of � � 1 + A† (x)A (y) is completely analogous. The last diagram is responsible for mixing between A+ and A† . µA+(x) Uµ(x)A+(x aµˆ) A+(x) (6) � − − � − a D ≡  −  1 A+† (x) A+† (x + aµˆ)U†(x) A+† (x) (7) µ a µ D ≡  −  1 + 3.1 MS-Renormalized Green’s Functions µA (x) A (x aµˆ)Uµ†(x) A (x) (8) D − ≡ a − − −   The first step in our perturbative procedure is to calculate the 2-pt and 3-pt Green’s functions in the 1 + µA† (x) Uµ(x)A† (x aµˆ) A† (x) (9) continuum, where we regularize the theory in D Euclidean dimensions (D = 4 2 ǫ)[6]. From this − a − − D ≡  −  −

3 EPJ Web of Conferences 175, 14001 (2018) https://doi.org/10.1051/epjconf/201817514001 Lattice 2017

+

− + −

Figure 3. One-loop Feynman diagrams contributing to the 2-pt Green’s function uα(x)uβ(y) . The “double � µ ν � dashed” line is the ghost field and the solid box in the bottom right vertex comes from the measure part of the lattice action.

+ −

Figure 4. One-loop Feynman diagrams contributing to the 2-pt Green’s function λα(x)λ¯ β(y) . � �

Figure 5. One-loop Feynman diagram contributing to the 2-pt Green’s function c(x)¯c(y) . � �

Figure 6. One-loop Feynman diagrams contributing to cα(x)¯cβ(y)uγ(z) . � µ �

computation we obtain the MS-renormalized Green’s functions by elimination of the pole part of the continuum bare Green’s functions. The renormalized Green’s functions are relevant for the ensuing calculation of the corresponding Green’s functions using lattice regularization and MS renormaliza- tion. Depending on the prescription used to define γ5 in D-dimensions, mixing between squarks may appear also in dimensional regularization. For the continnum calculations we adopt the t’Hooft- Veltman (HV) scheme [7]. Other prescriptions are related among themselves via finite conversion factors. Here we collect all MS-renormalized results for the 2-pt and 3-pt Green’s functions; the first result which we present, is the inverse quark renormalized propagator in momentum space:

2 2 R R MS 4 g CF µ¯ ψ˜ (q)ψ¯˜ (q′) = (2π) δ(q q′)iq 1 + 4 + α + (2 + α) log . (10) � �inv − � 16 π2   q2 

4 EPJ Web of Conferences 175, 14001 (2018) https://doi.org/10.1051/epjconf/201817514001 Lattice 2017

+ where C = (N2 1)/(2 N ) is the quadratic Casimir operator in the fundamental representation, q is F c − c the external momentum in the Feynman diagrams, andµ ¯ is the MS renormalization scale. Note also

− that a Kronecker delta for color indices is understood in Eqs.(10)-(14). + − In matrix notation, our results for the renormalized 2-pt Green’s functions with external squark legs are:

2 2 R R MS 4 2 10 2 g CF 16 µ¯ 10 A˜ (q)A˜ †(q′) = (2π) δ(q q′) q + q + (1 + α) log � �inv − 01 16 π2  3  q2  01

β 2 Figure 3. One-loop Feynman diagrams contributing to the 2-pt Green’s function uα(x)u (y) . The “double 2 g CF 4 01 � µ ν � +q , (11) dashed” line is the ghost field and the solid box in the bottom right vertex comes from the measure part of the 16 π2 3 10 lattice action. R A+ where AR is a 2-component column which contains the renormalized squark fields: AR = .  AR†  − + − We now turn to the gluon renormalized propagator. The contributions from the diagrams of Fig. 3, taken separately, are not transverse. But, their sum has this property, and it is found to take the following form:

2 2 α ¯ β R R MS 4 1 2 g N f µ¯ Figure 4. One-loop Feynman diagrams contributing to the 2-pt Green’s function λ (x)λ (y) . u˜ (q)˜u (q′) = (2π) δ(q + q′) q q + q δ q q 1 + 2 + log � � µ ν inv α µ ν µν µ ν π2 2 � �   −  16   q  g2 N 1 19 α2 µ¯ 2 c + α + + (3 α) log . (12) − 16 π2 2  6 2 −  q2   The result for the inverse gluino renormalized propagator to one-loop order is:

g2 2 2 2 Figure 5. One-loop Feynman diagram contributing to the 2-pt Green’s function c(x)¯c(y) . R ˜ R MS 4 i N f µ¯ g Nc µ¯ λ˜ (q)λ¯ (q′) = (2π) δ(q q′) q 1 + 2 + log + α + α log . (13) � � � �inv − 2 � 16 π2   q2  16 π2   q2  

The ghost propagator is the same as in the non-supersymmetric case:

2 2 R R MS 4 2 g Nc 1 µ¯ c˜ (q)c¯˜ (q′) = (2π) δ(q q′)q 1 1 + (3 α) log . (14) � �inv − − 16 π2  4 −  q2  Figure 6. One-loop Feynman diagrams contributing to cα(x)¯cβ(y)uγ(z) . � µ � Lastly, the 3-pt amputated Green’s function, at zero antighost momentum, in MS renormalization scheme, gives:

2 2 R α R β R γ MS 4 αβγ g Nc α µ¯ c˜ (q)c¯˜ (0)˜u (q′) = (2π) δ(q + q′) f (igqµ) 1 + 1 + log . (15) computation we obtain the MS-renormalized Green’s functions by elimination of the pole part of the � µ �amp 16 π2 2   q2   continuum bare Green’s functions. The renormalized Green’s functions are relevant for the ensuing calculation of the corresponding Green’s functions using lattice regularization and MS renormaliza- tion. Depending on the prescription used to define γ5 in D-dimensions, mixing between squarks 3.2 Green’s Functions on the lattice may appear also in dimensional regularization. For the continnum calculations we adopt the t’Hooft- The first result presented here, Eq. (16), is the lattice inverse quark propagator up to one loop. In all Veltman (HV) scheme [7]. Other prescriptions are related among themselves via finite conversion lattice expressions the systematic errors, coming from numerical loop integration, are smaller than the factors. last digit we present. Here we collect all MS-renormalized results for the 2-pt and 3-pt Green’s functions; the first result 2 which we present, is the inverse quark renormalized propagator in momentum space: B B L 4 g CF 2 2 ψ˜ (q)ψ¯˜ (q′) = (2π) δ(q q′) iq 1 12.8025 4.7920α + (2 + α) log a q inv π2 � � −  � − 16  −   2 2 2 R R MS 4 g CF µ¯ g CF 1 ψ˜ (q)ψ¯˜ (q′) = (2π) δ(q q′)iq 1 + 4 + α + (2 + α) log . (10) + 51.4347 r . (16) � �inv − � 16 π2   q2  16 π2 a 

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The inverse squark propagator, is: 2 B B L 2 g CF 2 2 2 10 A A † = q 11 + q 11.0173 3.7920α + (1 + α) log(a q ) inv π2 01 � � 16  −    01 1 65.3930 75.4031 +1.0087q2 + . (17) 10 a2 75.4031 65.3930 The gluino inverse propagator is: g2 B ˜ B L 4 i N f 2 2 λ˜ (q)λ¯ (q′) = (2π) δ(q q′) q 1 + 1.9209 log a q inv 2 π2 � � − �  16  −   g2 N c 16.6444 4.7920 α + α log a2 q2 π2 − 16  −    g2 N 1 + c 51.4347 r . (18) 16 π2 2 a  The gluon inverse propagator is given by:

B B L 4 1 2 u˜ (q)˜u (q′) = (2π) δ(q + q′) q q + q δ q q 1 µ ν inv α µ ν µν µ ν � �  −   g2 1 19.7392 + N 2.9622 + log a2 q2 π2 N f −16  − c −   α2 α 3 N 20.1472 0.8863 α + + log a2 q2 . (19) c 4 2 2 −  −  −     The ghost field renormalization, Zc, which enters the evaluation of Zg can be extracted from the ghost propagator: 2 B B L 4 2 g Nc 1 2 2 c˜ (q)c¯˜ (q′) = (2π) δ(q q′)q 1 + 4.6086 1.2029α (3 α) log a q . (20) inv π2 4 � � −  16  − − −   Lastly, the amputated gluon-antighost-ghost Green’s function is: 2 B α B β B γ L 4 αβγ g Nc 1 2 2 c˜ (q)c¯˜ (0)˜u (q′) = (2π) δ(q + q′) f igq 1 + 2.3960α α log a q . (21) µ amp µ π2 2 � �   16  −    The critical masses for the quark, squark and gluino can be read off (up to a minus sign) from the (q0) parts of Eqs. (16), (17), (18), respectively. O

3.3 Renormalizarion Factors Renormalization factors relate bare quantities (B) on the lattice to their renormalized (R) continuum counterparts: R B ψ = Zψ ψ , (22) R  B A = ZA A , (23) ± ± ± R =  B uµ Zu uµ , (24) R  B λ = Zλ λ , (25) R  B c = Zc c , (26) R ǫ B g = Zg µ− g . (27)

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The inverse squark propagator, is: Combining the Green’s functions on the lattice with the corresponding results from the continuum, L,MS L,MS L,MS L,MS L,MS L,MS 2 we extract Z , Z , Z , Z , Z and Zg in the MS scheme and on the lattice. B B L 2 g CF 2 2 2 10 ψ u λ A c A A † = q 11 + q 11.0173 3.7920α + (1 + α) log(a q ) ± inv π2 01 � � 16 −   g2   L,MS = + CF + + 2 2 Zψ 1 16.8025 3.7920α (2 α) log a µ¯ . (28) 2 01 1 65.3930 75.4031 π2 +1.0087q + . (17) 16 − −  10 a2 75.4031 65.3930 The gluino inverse propagator is: g2 ZL,MS = 1 N 16.6444 3.7920α + α log a2 µ¯2 + N 0.07907 + log a2 µ¯2 . (29) g2 λ 2 c f B ˜ B L 4 i N f 2 2 − 16 π − λ˜ (q)λ¯ (q′) = (2π) δ(q q′) q 1 + 1.9209 log a q    inv 2 π2 � � − �  16  −   2 g Nc 2 2 2 16.6444 4.7920 α + α log a q 1/2 L,MS g CF 1 2 2 10 01 2 Z = 11 8.1753 1.8960α + (1 + α) log a µ¯ 0.1623 . (30) − 16 π −  A π2 2 01 10     − 16  −    −   g2 N 1 + c 51.4347 r . (18) 16 π2 2 a  g2 1 3 α The gluon inverse propagator is given by: ZL,MS = 1 + 19.7392 N 18.5638 1.3863α + + log a2 µ¯2 u π2 N c 2 2 16  c −  − −   B B L 4 1 2 2 2 u˜ (q)˜u (q′) = (2π) δ(q + q′) q q + q δ q q 1 +N 0.9622 log a µ¯ . (31) µ ν inv α µ ν µν µ ν f − � �  −     g2 1 . + . + 2 2 2 2 19 7392 N f 2 9622 log a q L,MS g Nc 1 2 2 −16 π  − Nc − Z = 1 3.6086 1.2029α (3 α) log a µ¯ . (32)    c − 16π2 − − 4 − α2 α 3   N 20.1472 0.8863 α + + log a2 q2 . (19) − c  − 4  2 − 2     g2 1 3 1 The ghost field renormalization, Z , which enters the evaluation of Z can be extracted from the ghost ZL,MS = 1 + 9.8696 + N 12.8904 log a2 µ¯2 N 0.4811 log(a2 µ¯2) . (33) c g g 16 π2 − N c  − 2  − f  − 2  propagator:  c   2 L,MS g Nc 1 From the calculation of Z one can extract the Callan-Symanzik beta-function for SQCD. On the B ˜B L = π 4δ 2 + . . α α 2 2 . g c˜ (q)c¯ (q′) inv (2 ) (q q′)q 1 2 4 6086 1 2029 (3 ) log a q (20) B B � � −  16 π  − − 4 −  lattice, the bare beta-function is defined as: βL(g ) = adg /da gR, µ. The first term in this expansion   − | ¯ is: Lastly, the amputated gluon-antighost-ghost Green’s function is: g3 β (g) = 3N + N + (g5). (34) g2 L 16 π2 − c f O B α B β B γ L = 4 + αβγ + Nc 1 2 2  c˜ (q)c¯˜ (0)˜uµ (q′) amp (2π) δ(q q′) f igqµ 1 2.3960α α log a q . (21) � � 16 π2  − 2  For N < 3N , the (g3) term is negative, in other words, the theory is asymptotically free. Our finding      f c O The critical masses for the quark, squark and gluino can be read off (up to a minus sign) from the for the beta function agrees with what is obtained in the supersymmetric Yang-Mills theory [8]. (q0) parts of Eqs. (16), (17), (18), respectively. O 4 Future Plans – Conclusion 3.3 Renormalizarion Factors In this work we have performed a pilot investigation of issues related to the formulation of a super- Renormalization factors relate bare quantities (B) on the lattice to their renormalized (R) continuum symmetric theory on the lattice. As a prototype model, we have studied = 1 supersymmetric QCD. N counterparts: This model bears all major characteristics of potential extensions of the standard model, including R B ψ = Zψ ψ , (22) superpartners for gauge and matter fields; it is thus appropriate for a feasibility study on the lattice. R  B There are several well-known problems arising from the complete (or even partial) breaking of A = ZA A , (23) ± ± ± supersymmetry in a regularized theory, including the necessity for fine tuning of the theory’s bare R =  B uµ Zu uµ , (24) Lagrangian, and a rich mixing pattern of composite operators at the quantum level. We address these R  B problems via perturbative calculations at one loop. In order to provide the necessary ingredients λ = Zλ λ , (25) for performing numerical studies of supersymmetric theories, we have calculated the self energies R =  B c Zc c , (26) of all particles which appear in SQCD. We determined the renormalization factors for these fields; R ǫ B g = Zg µ− g . (27) in addition, for the squark propagator we found the mixing coefficients among its different degrees

7 EPJ Web of Conferences 175, 14001 (2018) https://doi.org/10.1051/epjconf/201817514001 Lattice 2017

of freedom. Furthermore, we have computed the gluon-antighost-ghost Green’s function in order to renormalize the coupling constant. Our results are also relevant to the investigation of relationships between different Green’s functions involved in SUSY Ward identities [9, 10]. There are several directions in which this work could be extended. A natural extension would be the computation of the Green’s functions for composite operators made of quark, squark, gluon and gluino fields; studies of such operators in the continuum can be found in, e.g., Refs. [11–14]. A serious complication in the supersymmetric case regards the mixing of quark bilinear operators with other composite operators. A whole host of operators with equal or lower dimensionality, having the same quantum numbers and same transformation properties can mix at the quantum level; on the lattice, the number of operators which mix among themselves is considerably greater than in the continuum regularization. We are planning to study their renormalization and mixing perturbatively. The perturbative computation of all relevant Green’s functions of these operators, will be followed by the construction of the mixing matrix, which may also involve nongauge invariant (but BRST invariant) operators or operators which vanish by the equations of motion. Finally, it would be important to extend our computations to further improved actions with reduced lattice artifacts and reduced symmetry breaking, e.g. the overlap fermion action, as a forerunner to numerical studies using these actions.

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