Supersymmetric QCD: Renormalization and Mixing of Composite Operators

# Supersymmetric QCD: Renormalization and Mixing of Composite Operators

Marios Costa    Haralambos Panagopoulos
###### Abstract

We study -dimensional SQCD with gauge group and flavors of chiral super-multiplets on the lattice. We perform extensive calculations of matrix elements and renormalization factors of composite operators in Perturbation Theory. In particular, we compute the renormalization factors of quark and squark bilinears, as well as their mixing at the quantum level with gluino and gluon bilinear operators. From these results we construct correctly renormalized composite operators, which are free of mixing effects and may be employed in non-perturbative studies of Supersymmetry. All our calculations have been performed with massive matter fields, in order to regulate the infrared singularities which are inherent in renormalizing squark bilinears. Furthermore, the quark and squark propagators are computed in momentum space with nonzero masses.

This work is a feasibility study for lattice computations relevant to a number of observables, such as spectra and distribution functions of hadrons, but in the context of supersymmetric QCD, as a forerunner to lattice investigations of SUSY extensions of the Standard Model.

## I Introduction

Current intensive searches for Physics Beyond the Standard Model (BSM) are becoming a very timely endeavor Tanabashi:2018 (), given the precision experiments at the Large Hadron Collider and elsewhere; at the same time, numerical studies of BSM Physics are more viable due to the advent of lattice formulations which preserve chiral symmetry Kaplan:1992bt (); Neuberger:1997fp (); Luscher:1998 (). In particular, the study of supersymmetric models on the lattice, which began a long time ago Curci:1986sm (), has been an object of intense research activity in recent years Kaplan:2009 (); Catterall:2014vga (); Joseph:2015xwa (); Ali:2018fbq (); Endrodi:2018ikq (), and applications to supersymmetric extensions of the Standard Model are gradually becoming within reach. Studies of hadronic properties using the lattice formulation of Supersymmetric Quantum Chromodynamics (SQCD) rely on the computation of matrix elements and correlation functions of composite operators, made out of quark (), gluino (), gluon (), squark () fields. These operators are of great phenomenological interest in the non-supersymmetric case, since they are employed in the calculation of certain transition amplitudes among bound states of particles and in the extraction of meson and baryon form factors. Correlation functions of such operators calculated in lattice SQCD therefore provide interesting probes of physical properties of the theory. A proper renormalization of these operators is most often indispensable for the extraction of results from the lattice. The main objective of this work is the calculation of the quantum corrections to a complete basis of “ultra local” bilinear currents, using both dimensional regularization and lattice regularization. We consider both flavor singlet and nonsinglet operators.

Within the SQCD formulation we compute the quark, squark propagators and all 2-pt Green’s functions of bilinear operators, made out of quark and squark fields. Our computations are performed to one loop and to lowest order in the lattice spacing, ; also, in order to avoid potential infrared singularities in Green’s functions of squark bilinears, we have employed massive chiral supermultiplets throughout. We extract from the above quantities the renormalization factors of the quark and squark fields and masses. Quantum corrections induce mixing of some of the bilinear operators which we study, both among themselves and with gluon and gluino bilinears having the same quantum numbers; we compute all the corresponding mixing coefficients, in renormalization scheme; the values of these coefficients can also be readily derived from the renormalized Green’s functions which we provide. Our calculations also reproduce the Adler-Bell-Jackiw (ABJ) anomaly of the axial vector quark current to one-loop order.

This work is a continuation to our recent paper MC:2018 (), in which we presented our perturbative results for the renormalization factors of the coupling constant () and of the quark (), gluon (), gluino (), squark (), and ghost () fields in the continuum and on the lattice; they were the first one-loop computations of these quantities using a lattice discretization of the action of SQCD.

The paper is organized as follows: Sec. II contains all relevant definitions and the calculational setup. In Sec. III we present our computation and results both for dimensional and lattice regularizations. Finally, we conclude in Sec. IV with a summary and a discussion of our results and possible future extensions of our work.

## Ii Formulation and Computational Setup

### ii.1 Lattice Action

In our lattice calculation, we extend Wilson’s formulation of the QCD action, to encompass SUSY partner fields as well. In this standard discretization quarks, squarks and gluinos live on the lattice sites, and gluons live on the links of the lattice: ; is a color index in the adjoint representation of the gauge group. This formulation leaves no SUSY generators intact, and it also breaks chiral symmetry; it thus represents a “worst case” scenario, which is worth investigating in order to address the complications Giedt () 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 () and gluinos (), the Euclidean action on the lattice becomes ( are the squark field components):

 SLSQCD = a4∑x[Ncg2∑μ,ν(1−1NcTrUμν)+∑μTr(¯λMγμDμλM)−ar2Tr(¯λMD2λM) (1) + ∑μ(DμA†+DμA++DμA−DμA†−+¯ψDγμDμψD)−ar2¯ψDD2ψD + i√2g(A†+¯λαMTαP+ψD−¯ψDP−λαMTαA++A−¯λαMTαP−ψD−¯ψDP+λαMTαA†−) + 12g2(A†+TαA+−A−TαA†−)2−m(¯ψDψD−mA†+A+−mA−A†−)],

where: , and a summation over flavors is understood in the last three lines of Eq. (1). The 4-vector is restricted to the values , with being an integer 4-vector. The terms proportional to the Wilson parameter, , eliminate the problem of fermion doubling, at the expense of breaking chiral invariance. In the limit the lattice action reproduces the continuum one.

The definitions of the covariant derivatives are as follows:

 DμλM(x) ≡ 12a[Uμ(x)λM(x+a^μ)U†μ(x)−U†μ(x−a^μ)λM(x−a^μ)Uμ(x−a^μ)] (2) D2λM(x) ≡ 1a2∑μ[Uμ(x)λM(x+a^μ)U†μ(x)−2λM(x)+U†μ(x−a^μ)λM(x−a^μ)Uμ(x−a^μ)] (3) DμψD(x) ≡ 12a[Uμ(x)ψD(x+a^μ)−U†μ(x−a^μ)ψD(x−a^μ)] (4) D2ψD(x) ≡ 1a2∑μ[Uμ(x)ψD(x+a^μ)−2ψD(x)+U†μ(x−a^μ)ψD(x−a^μ)] (5) DμA+(x) ≡ 1a[Uμ(x)A+(x+a^μ)−A+(x)] (6) DμA†+(x) ≡ 1a[A†+(x+a^μ)U†μ(x)−A†+(x)] (7) DμA−(x) ≡ 1a[A−(x+a^μ)U†μ(x)−A−(x)] (8) DμA†−(x) ≡ 1a[Uμ(x)A†−(x+a^μ)−A†−(x)] (9)

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 non-supersymmetric case. Similarly, a standard “measure” term must be added to the action, in order to accound for the Jacobian in the change of integration variables:  . All the details of the continuum and the lattice actions can be found in Ref.MC:2018 ().

### ii.2 Bilinear operators and their mixing

In studying the properties of physical states, the main observables are Green’s functions of operators made of quark fields, having the form , where denotes all possible distinct products of Dirac matrices, as well as operators made of squark fields , along with operators of higher dimensionality. The matter fields are considered to be massive, and for completeness’ sake we calculate the quark and squark propagators with nonzero masses; in this way, we have control over infrared (IR) divergences. Ultraviolet (UV) divergences are treated by a standard regularization, either the lattice (L) or dimensional regularization (DR).

The first new quantities of this paper are the renormalization factors for the squark and quark masses in the and schemes. After these computations, we focus on the matrix elements of composite bilinear operators. As we will see below, some of the operators carrying the same quantum numbers mix together beyond tree level. In order to determine their mixing coefficients, we calculate certain 2-pt Green’s functions of these operators. More specifically, we calculate the 2-pt Green’s function of squark bilinears with external squarks and gluons, as well as the 2-pt Green’s functions of quark bilinears with one external quark-antiquark pair, or one gluino-antigluino pair, or two gluons, or two squarks. All of our results are computed as functions of the coupling constant, the number of colors, the gauge fixing parameter and the external momentum. The renormalization conditions which we will impose involve the renormalization factors of the fields, that we have computed in Ref.MC:2018 ().

We identified all operators which can possibly mix with and all Green’s functions, e.g. , which must be calculated in order to compute those elements of the mixing matrix which are relevant for the renormalization of and .

The bilinear operators could in principle mix with four types of operators having the same quantum numbers. The four types are as follows: Type I are gauge invariant operators. Type II are operators which are not gauge invariant but are the BRST variation Suzuki () of some other operators. Type III operators vanish by the equations of motion. Type IV are operators which are not linear combinations of type I, II and III. By general renormalization theorems (see, e.g., Ref. Collins:1984xc ()), type I operators will not mix with type IV operators. We list the type I operators in Table 1 for the flavor non-singlet case () and in Table 2 additional operators which show up in the flavor singlet case (). Different values of the index “” lead to the following possibilities for the Dirac matrices: (scalar) , (pseudoscalar) , (vector) , (axial vector) and tensor . In Tables 1 and 2 we also include operators with lower dimensionalities, even though they do not mix with quark bilinears in dimensional regularization; they do however show up in the lattice formulation. Indeed, on the lattice, the number of operators which mix among themselves is considerably greater than in the continuum regularization. The perturbative computations of all relevant Green’s functions of operators and will be followed by the construction of the mixing matrix, which may also involve non gauge invariant (but BRST invariant) operators or operators which vanish by the equations of motion.

In this work we concentrate on extracting the mixing coefficients between quark, squark, gluino and gluon bilinears, which are relevant for physical external states and thus we do not take into account the ghost bilinears. As it turns out, flavor-singlet quark bilinears mix with gluino bilinears: . A particularly rich mixing pattern emerging from Tables 1 and 2 regards the case of scalar, pseudoscalar and vector quark bilinears, since they can mix with a variety of gluon and squark bilinear operators. Note also that the scalar operators mix with the identity at the quantum level.

In order to make the BRST symmetry explicit and elucidate the mixing with non-gauge-invariant operators, we write the Faddeev-Popov action, using a new auxiliary field :

 SFP=∫d4x[α2(Bα)2−Bα∂μuαμ−¯cα∂μDαβμcβ]. (10)

Under BRST transformations, the fields appearing in behave as follows:

 uαμ → uαμ+Dαβμcβ ξ, cα → cα−g2fαβγcβcγ ξ, ¯cα → ¯cα+Bα ξ, Bα → Bα, (11)

where is an infinitesimal anticommuting parameter. Under these transformations, is indeed invariant. Given that the effect of a BRST transformation on gauge and matter fields is that of a gauge transformation (with anticommuting parameter), all other parts of the action, being gauge invariant, will automatically also be BRST invariant.

From Eqs. (11), we see that type II operators may mix in the flavor-singlet case, e.g. :

 δBRST(uαμ¯cα)=(uαμBα+¯cαDαβμcβ)⇒δBRST(uαμBα+¯cαDαβμcβ)=0 (12)

This potential mixing of the non-gauge-invariant operator (which is also present in the non-supersymmetric case), is inconsequential if one is interested in physical external states with transverse polarization. Other flavor non-singlet operators, such as , etc., are not of types I, II or III and therefore cannot mix with gauge invariant operators. The presence of a global symmetry, which is preserved by the SQCD action, both in the continuum and on the lattice, forbids 3-squark operators from mixing with quark bilinears.

By investigating the behavior of dimension-2 and -3 bilinear operators under parity, , and charge conjugation, , we group them to the categories , , , and in Tables 1 and 2, based on the trasformation properties of the quark bilinears. These are symmetries of the action and their definitions are presented below.

 (13)

where .

 (14)

where means transpose. For bilinear operators, it is convenient to define a new trasformation , which is a combination of with an exchange in the flavors of the two fields and in their respective masses; this transformation is a (spurionic) symmetry of the action. The operators shown in Table 1 are eigenstates of both and . Mixing with further operators, such as and , is not allowed, due to incompatible eigenvalues under .

In order to calculate the one-loop mixing coefficients relevant to the squark- and quark-bilinear operators of lowest dimensionality, we must evaluate Feynman diagrams as shown in Figs. 1 and 2, respectively. In Figure 1 we have also included diagrams with external gluons; actually, since there are no BRST-invariant dimension-2 gluon operators, no mixing is expected to appear in this case, and we use this fact as a check on our perturbative results on the lattice. The diagrams in Figure 2 lead to the renormalization of dimension-3 quark bilinear operators and to the potential mixing coefficients with gluino, squark and gluon bilinears.

## Iii Details of the calculation and results

### iii.1 Renormalization of quark and squark propagators

The purpose of this section is to renormalize the quark and squark fields, as a prerequisite for the renormalization of bilinear composite fields. As a by-product, we also obtain the renormalization factors for the corresponding masses. We use both the dimensional and lattice regularizations of SQCD, in order to calculate the massive quark and squark propagators. Mass and field renormalizations as dictated by renormalization conditions in fact suffice to render finite all the terms in the inverse quark and squark propagators (Eqs. (19) and (32), respectively). The one-loop Feynman diagrams (one-particle irreducible (1PI)) contributing to the quark propagator, , are shown in Fig. 3, those contributing to the squark propagators, , , , , in Fig. 4. For our continuum results, we use renormalization in the HV (’t Hooft-Veltman) scheme Hooft:1972 () and for completeness we present also the conversion factors to the RI scheme. In our calculation of the quark propagator the indices carried by all gamma matrices are eventually contracted with the indices of external momenta; thus given that the latter only have 4 (rather than ) components, all prescriptions of in -dimensions Larin (); Siegel:1979 (), give the same 1-loop results.

It is convenient to express the squark field components as a doublet: ; the mass term then assumes the form , where the matrix is hermitian with non-negative eigenvalues, but not necessarily diagonal.

The definitions of the renormalization factors for the matter fields and their masses are:

 ψR = √ZψψB, (15) AR = √ZA±AB, (16) mRψ = ZmψmBψ, (17) mRA†mRA = Z†mAmBA†mBAZmA, (18)

where stands for the bare and for renormalized quantities and , are matrices corresponding to the doublet A.

After summing all the continuum Feynman diagrams of Fig. 3 the massive inverse quark propagator in dimensional regularization (DR) becomes:

 ⟨~ψB(q)~¯ψB(q′)⟩DRinv = (2π)4δ(q−q′){(i⧸q−mBψ)+g2CF16π2[i⧸q(4+α+2+αϵ+(2−α)m2q2+(2+α)log(¯μ2q2+m2) (19) −((2−α)m4q4+4m2q2)log(1+q2m2)) = (2π)4δ(q−q′)((i⧸q−m)−ΣBψ),

where is the quadratic Casimir operator in the fundamental representation, is the external momentum in the Feynman diagrams, and is the renormalization scale. For one-loop calculations, the distinction between and is inessential in many cases; we will simply use in those cases. We have also imposed that the renormalized masses of one flavor for quark and squark be the same. Note also that a Kronecker delta for color indices is understood in Eqs. (19) and (32). The results for the DR renormalization factors in the scheme are:

 ZDR,¯¯¯¯¯¯¯MSψ = 1+g2CF16π21ϵ(2+α) (20) ZDR,¯¯¯¯¯¯¯MSmψ = 1+g2CF16π21ϵ (21)

Using our results for the quark propagator, we can compute also the multiplicative renormalization function of the quark field and mass in the RI renormalization scheme ( and ). In order to find , we use the renormalization condition:

 [(ZDR,RI′ψ)−1i⧸q−ΣBψ∣∣terms∝⧸q]qρ=¯μρ=i⧸q|qρ=¯μρ, (22)

where is the renormalization scale 4-vector, is the quark self energy that we compute up to . We note that the renormalization scale appearing in need not coincide with the scale used in the scale. The RI counterpart for the multiplicative renormalization of the mass, , can be extracted from:

 [−(ZDR,RI′ψ)−1(ZDR,RI′mψ)−1mR−ΣBψ∣∣terms∝1]qρ=¯μρ=−mR. (23)

The expressions for the aforementioned renormalization factors are:

 ZDR,RI′ψ = 1+g2CF16π2[1ϵ(2+α)+4+α+(2+α)log(¯μ2¯μ2+m2)+m2¯μ2(2−α−4log(1+¯μ2m2)) −m4¯μ4(2−α)log(1+¯μ2m2)] ZDR,RI′mψ = 1+g2CF16π2[1ϵ+α+log(¯μ2¯μ2+m2)−m2¯μ2(2−α−(1−α)log(1+¯μ2m2)) +m4¯μ4(2−α)log(1+¯μ2m2)]

Notice that the expression for is not gauge independent; this was expected given that the renormalization condition relies on a gauge-variant Green’s function. The ratio between the and RI renormalization factors give the corresponding conversions factors.

 C¯¯¯¯¯¯¯MS,RI′ψ = ZDR,¯¯¯¯¯¯¯MSψ/ZDR,RI′mψ (26) C¯¯¯¯¯¯¯MS,RI′mψ = ZDR,¯¯¯¯¯¯¯MSmψ/ZDR,RI′mψ (27)

Being regularization independent, these same conversion factors can then be also used in the lattice regularization (L). Note also that continuum regularizations forbid the additive renormalization of the mass, so it is renormalized only multiplicatively, while in the lattice regularization the Lagrangian mass undergoes additive and multiplicative renormalization.

We next compute the lattice expression of the massive quark propagator to one loop. We take into account the gluon tadpole diagram which has no analog in the continuum (see Fig. 5). Our result is given by:

 ⟨~ψB(q)~¯ψB(q′)⟩Linv = (2π)4δ(q−q′){(i⧸q−mLψ)+g2CF16π2[i⧸q(−12.80254+4.79201α+(2−α)m2q2 (28) −(2+α)log(a2(m2+q2))−((2−α)m4q4+4m2q2)log(1+q2m2)) −m((0.30799+5.7920α)−(3+α)log(a2(m2+q2))−(3+α)m2q2log(1+q2m2)) + 1a51.4347r]} = (2π)4δ(q−q′)((i⧸q−m)−ΣB,Lψ)

In Eq. (28), just as in the corresponding equation in the continuum, terms with cancel out at one-loop level. This means that the Majorana components of , corresponding to and , do not mix under renormalization, unlike the case of the squark fields themselves, see Eqs. (32) and (46). The renormalization factors of the quark fields as well as of the quark mass in the scheme and on the lattice are extracted from the subtraction of the renormalized self-energy contributions, which were computed in the continuum, from the bare quark lattice self-energy, as required by the renormalization condition. The critical mass, , can also be read off Eq. (28); thus, we find:

 ZL,¯¯¯¯¯¯¯MSψ = 1+g2CF16π2(−16.8025+3.79201α−(2+α)log(a2¯μ2)) (29) ZL,¯¯¯¯¯¯¯MSmψ = 1+g2CF16π2(13.1105−log(a2¯μ2)) (30) mquarkcrit. = g2CF16π21a51.4347r (31)

We now turn to the one-loop corrections to the squark propagator. Given the mixing of squarks in the HV scheme, our results are written in matrix notation:

 ⟨~AB(q)~AB†(q′)⟩HVinv = (32) +m2g2CF16π2(18+7+αϵ+(7+α)log(¯μ2m2)−8log(1+q2m2)−(7−α)m2q2log(1+q2m2))(1001) −g2CF16π2(43q2+4m2)(0110)] ≡ (2π)4δ(q−q′)[(q2+m2)\openone−ΣBA].

where is a 2-component column which contains the bare squark fields.

Starting from Eq. (32), one requires the elimination of the pole part and determines the renormalized 2-pt Green’s function. Thus, one arrives at the expressions below for the squark field and mass renormalization respectively.

 ZDR,¯¯¯¯¯¯¯MSA = \openone[1+g2CF16π21ϵ(1+α)] (33) ZDR,¯¯¯¯¯¯¯MSmA = \openone[1+g2CF16π23ϵ] (34)

In the naive Dimensional Regularization (NDR) presecription, in which anticommutes with all matrices () Furman:2003 (), the nondiagonal elements in Eq. (32) vanish and the expression for the bare squark propagator in NDR scheme up to one-loop is:

 ⟨~AB(q)~AB†(q′)⟩NDRinv = + m2g2CF16π2(14+7+αϵ+(7+α)log(¯μ2m2)−8log(1+q2m2)−(7−α)m2q2log(1+q2m2))(1001)].

In computing the conversion factors between and NDR schemes, we cannot simply set because the two regularization actually lead to different renormalization schemes. Instead, we use the definitions of the conversion factors:

 A¯¯¯¯¯¯¯MS = C¯¯¯¯¯¯¯MS,¯¯¯¯¯¯¯MSNDRAA¯¯¯¯¯¯¯MSNDR (36) m2¯¯¯¯¯¯MSA = C¯¯¯¯¯¯¯MS,¯¯¯¯¯¯¯MSNDRmAm2¯¯¯¯¯¯MSNDRA(C¯¯¯¯¯¯¯MS,¯¯¯¯¯¯¯MSNDRmA)† (37)

which lead to the following values:

 C¯¯¯¯¯¯¯MS,¯¯¯¯¯¯¯MSNDRA = (38) C¯¯¯¯¯¯¯MS,¯¯¯¯¯¯¯MSNDRmA = (39)

Turning now to renormalization, there is a certain amount of freedom in defining it; an essential property which one would like to require is amenability to nonperturbative treatment. As will become clear below, when we discuss the lattice regularization, a natural definition satisfying this requirement is as follows:

 ⟨~ARI′(q)~ARI′†(q′)⟩inv∣∣q2=0 = mR2\openone (40) ⟨~ARI′(q)~ARI′†(q′)⟩inv∣∣q2=¯μ2 = (q2+mR2)|q2=¯μ2\openone (41)

where is the RI renormalized inverse squark propagator which is connected to the bare one through:

 ⟨~ARI′(q)~ARI′†(q′)⟩inv=(ZDR,RI′A)1/2⟨~AB(q)~AB†(q′)⟩DRinv(ZDR,RI′A)1/2, (42)

and similarly the renormalized mass is related to the bare mass through:

 mRI′2=ZL,RI′m†mB2ZL,RI′m (43)

Finally, is the RI renormalization scale 4-vector. In Eq. (41) the rhs is the tree level inverse squark propagator.

Using the renormalization conditions of Eq. (40)-(41) the multiplicative renomalization in the RI scheme for the squark field and mass can be determined:

 ZDR,RI′A = \openone+g2CF16π2[(1+αϵ+163+(7−α)m2¯μ2+(1+α)log(¯μ2m2)−(1+α)log(1+¯μ