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Advances in High Energy Physics
Volume 2013 (2013), Article ID 607807, 64 pages
Precision Calculations in Supersymmetric Theories
Institute for Theoretical Particle Physics, Karlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany
Received 28 January 2013; Accepted 8 May 2013
Academic Editor: Hong-Jian He
Copyright © 2013 L. Mihaila. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper we report on the newest developments in precision calculations in supersymmetric theories. An important issue related to this topic is the construction of a regularization scheme preserving simultaneously gauge invariance and supersymmetry. In this context, we discuss in detail dimensional reduction in component field formalism as it is currently the preferred framework employed in the literature. Furthermore, we set special emphasis on the application of multi-loop calculations to the analysis of gauge coupling unification, the prediction of the lightest Higgs boson mass, and the computation of the hadronic Higgs production and decay rates in supersymmetric models. Such precise theoretical calculations up to the fourth order in perturbation theory are required in order to cope with the expected experimental accuracy on the one hand and to enable us to distinguish between the predictions of the Standard Model and those of supersymmetric theories on the other hand.
Today we know that the Standard Model (SM) of particle physics [1####^~^~^~^~^~^####x2013;7], which is a renormalizable gauge theory for the group , is extremely successful at short distances of the order of ####^~^~^~^~^~^####x2009;cm. Up to now, all experiments verify it without any conclusive hint towards new physics. On the other hand, Einstein####^~^~^~^~^~^####x2019;s gravitational theory based on the same concept of gauging the symmetries gives a very good classical theory for long distances. However, the classical theory of gravity could not be quantized due to its abundant number of singularities. There seems to be a deep conflict between the classical theory of gravity and the quantum field theory. Thus, the question whether gauging is the only organizing principle or there is a deeper connection between space time and internal space symmetries arises naturally. In a long series of ####^~^~^~^~^~^####x201c;no-go theorems####^~^~^~^~^~^####x201d; among which the Coleman-Mandula theorem  is the most important one, it was shown that the only possible symmetry group of a consistent four-dimensional quantum field theory is the direct product of the internal symmetry group and the Poincar####^~^~^~^~^~^####xe9; group. Precisely, it states that internal symmetries cannot interact nontrivially with space time symmetry. Surprisingly, there is a unique way of combining nontrivially space time and inner space symmetries, namely, supersymmetry (SUSY). It was shown by Haag et al.  that weakening the assumptions of the Coleman-Mandula theorem by allowing both commuting and anticommuting symmetry generators, there is a nontrivial extension of the Poincar####^~^~^~^~^~^####xe9; algebra, namely, the supersymmetry algebra. The supersymmetry generators transform bosonic particles into fermionic ones and vice versa, but the commutator of two such transformations yields a translation in space time. In case of four-dimensional space time, the algebra generated by the SUSY generators will contain the algebra of Einstein####^~^~^~^~^~^####x2019;s general relativity.
The first attempts to construct physical models respecting SUSY can be traced back in the early seventies to the works by Golfand and Likhtman  and Volkov and Akulov . However, the first known example of a renormalizable supersymmetric four-dimensional quantum field theory is the Wess-Zumino model . Within SUSY it is very natural to extend the concept of space time to the concept of superspace . Along with the four-dimensional Minkowski space there are also two new ####^~^~^~^~^~^####x201c;anticommuting####^~^~^~^~^~^####x201d; coordinates and , that are labeled in Grassmann numbers rather than real numbers: The ordinary space dimensions correspond to bosonic degrees of freedom, the anticommuting dimensions to fermionic degrees of freedom. The fields are now functions of the superspace variables and they are organized into supersymmetric multiplets in a natural way . Expanding the multiplets in Taylor series over the Grassmannian variables, one obtains the components of the superfield as the coefficients of the expansion. They are ordinary functions of the space time coordinates and can be identified with the usual fields. Furthermore, in the superfield notation the manifestly supersymmetric Lagrangians are polynomials of the superfields. In the same way, as the ordinary action is the integral over the space time of the Lagrangian density, in the supersymmetric case the action may be expressed as an integral over the whole superspace.
As quantum field theories the supersymmetric theories are less divergent as they would be in the absence of SUSY. These properties can be traced back to the cancellation of diagrams containing bosonic or fermionic particles, as, for example, the cancellation of quadratic divergences present in the radiative corrections to the Higgs boson mass. Even more, it was shown [14####^~^~^~^~^~^####x2013;16] that there are parameters of the theory that do not get any radiative corrections; that is a very special feature in quantum field theories. The most important consequence for the particle phenomenology is the fact that in a supersymmetric theory there should be an equal number of bosons and fermions with equal masses. In other words, for every SM particle there should exist a supersymmetric partner with an equal mass. But in Nature we do not observe such a situation. An elegant solution to break SUSY in such a way that its renormalization properties remain valid (in particular the nonrenormalization theorems and the cancellation of quadratic divergences) is to introduce the so-called soft terms . In this way, the mass difference between supersymmetric partners can become of the order of SUSY breaking scale. Moreover, there will also be parameters that receive only finite radiative corrections of the order of magnitude of SUSY breaking parameters. This is the case of the Higgs masses and Higgs couplings. Accordingly, the SUSY partners of the SM particles should not be very heavy in order to account for the smallness of the Higgs mass and couplings. For example, requiring for consistency of the perturbation theory that the radiative corrections to the Higgs boson mass do not exceed the mass itself gives  where denotes the mass scale of SM superpartners. Thus, for ####^~^~^~^~^~^####x2009;GeV####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;and one obtains ####^~^~^~^~^~^####x2009;GeV. This feature is one of the great achievements of supersymmetric theories, namely, the solution to the hierarchy problem in particle physics.
The very old concept of the existence of an organizing principle that allows the unification of all interactions present in Nature is nowadays embedded in the so-called Grand Unified Theories (GUT). The predictions of such theories can be even precisely tested with the help of the experiments conducted at modern particle colliders. The most prominent example concerns, for sure, the prediction of gauge coupling unification. Once the gauge couplings for the electroweak and strong interactions had been precisely measured at the Large Electron-Proton Collider (LEP) , we could verify this hypothesis with high precision. The amazing result of evolving the low-energy values of the gauge couplings according to the SM predictions [20####^~^~^~^~^~^####x2013;22] is that unification is excluded by more than eight standard deviations. This means that unification can be achieved only if new physics occurs between the electroweak and the Planck scales. If one considers that a supersymmetric theory describes the new physics, one obtains that unification at an energy scale of about 1016####^~^~^~^~^~^####x2009;GeV can be realized if the typical supersymmetric mass scale is of the order of 103####^~^~^~^~^~^####x2009;GeV. This observation was interpreted as first ####^~^~^~^~^~^####x201c;evidence####^~^~^~^~^~^####x201d; for SUSY, especially because the supersymmetric mass scale was in the same range as that derived from the solution to the hierarchy problem.
Another virtue of SUSY is that it provides a candidate for the cold dark matter. Nowadays, it is well established that the visible matter amounts to only about 4####^~^~^~^~^~^####x25; of the matter in the Universe. A considerable fraction of the energy is made up from the so-called dark matter. The direct evidence for the existence of dark matter is the flat rotation curves of spiral galaxies (see, e.g.,  and references cited therein), the gravitational lensing caused by invisible gravitating matter in the sky [24, 25], and the formation of large structures like clusters of galaxies. The dark matter is classified in terms of the mass of the constituent particle(s) and its (their) typical velocity: The hot dark matter, consisting of light relativistic particles and the cold one, made of massive weakly interacting particles (WIMPs) . The hot dark matter might consist of neutrinos; however, this hypothesis cannot explain galaxy formation. For the cold dark matter, there is obviously no candidate within the SM. Nevertheless, SUSY provides an excellent candidate for WIMP, namely, the neutralino as the lightest supersymmetric particle.
These three fundamental predictions of SUSY make it one of the preferred candidates for physics beyond the SM. This explains the enormous efforts devoted to searches for SUSY in particle physics experiments at accelerators, in the deep sky with the help of telescopes, and with the help of underground facilities, that last already for four decades. The exclusion bounds on the supersymmetric mass spectrum are in general model dependent. In the case of the constrained MSSM (CMSSM), the current status is as follows: if one combines the excluded regions from the direct searches at the LHC , the stringent lower bound on the mass of the pseudoscalar Higgs from XENON100 , the constraints from the relic density from WMAP , and those from muon anomalous magnetic moment , one can set a lower limit on the WIMP mass of 230####^~^~^~^~^~^####x2009;GeV and on strongly interacting supersymmetric particles of about 1300####^~^~^~^~^~^####x2009;GeV. If in addition, the mass of the lightest Higgs boson of 125####^~^~^~^~^~^####x2009;GeV in agreement with the recent measurement at the LHC [31, 32] is imposed; one can exclude strongly interacting superpartners below 2####^~^~^~^~^~^####x2009;TeV. Nevertheless, such exclusion bounds concern the gluinos and mainly the first two generation of squarks. On the other hand, for the third generations of squarks, masses of the order of few hundred GeV are still allowed.
In this context, the question whether low-energy SUSY is still a valid candidate for physics beyond the SM arises naturally. Despite the slight tension that appears in particular models, as, for example, the constrained MSSM, (the constrained MSSM model is based on the universality hypothesis and is described by a set of five free parameters defining the mass scale for the Higgs potential and the scalar and fermion masses) the supersymmetric parameter space is large enough to accommodate all the experimental data known at present. However, the main prediction of low-energy SUSY, that is, the existence of supersymmetric particles at the TeV scale, is falsifiable at the LHC at the full energy run of 14####^~^~^~^~^~^####x2009;TeV. If no supersymmetric particle will be found at the TeV scale, we have to give up the main arguments in favor of SUSY, namely, the gauge coupling unification and the solution to the hierarchy problem.
To draw such powerful conclusions, one definitely needs an accurate comparison of the experimental data with the theory predictions based on SUSY models. There are various possibilities to perform such comparisons; one of them is high precision analyses, that requires precision data both at the experimental and theoretical level. On the theory side, the observables for which precise theoretical predictions up to the next-to-next-to leading order in perturbation theory are required are the electroweak precision observables (EWPO) , the muon anomalous magnetic moment , the lightest Higgs boson mass , the decay rate for the rare decay of a bottom quark into a strange quark and a photon , and, of course, the production and decay rates of the Higgs boson at hadron colliders . Details about the various topics can be found in the excellent review articles cited above. In this paper we report on the newest developments in precision calculations within SUSY models and set special emphasis on the recent calculations at the three-loop order involving several different mass scales. The latter constitute in many cases essential ingredients for the state of the art analyses of the experimental data taken currently at the LHC.
This paper is organized as follows. In the next section we briefly review the main results concerning the renormalizability of supersymmetric theories that can be derived from their holomorphic properties. In Sections 3 and 4 we describe the regularization method based on dimensional reduction applied to nonsupersymmetric and supersymmetric theories up to the fourth order in perturbation theory. In the second part of the paper we present the phenomenological applications of such precision calculations. Namely, in Section 5 we concentrate on computation of the three-loop gauge beta functions within the SM that allows us to predict the gauge couplings at high energies with very high accuracy. Furthermore, in Section 6 we report on the gauge coupling unification within SUSY models taking into account the most precise theoretical predictions and experimental measurements. Section 7 is devoted to the computation of the lightest Higgs boson mass within SUSY models with three-loop accuracy. In Section 8, the hadronic Higgs production and decay in SUSY models are reviewed and the required computations up to the third order in perturbation theory are presented. Finally, we draw our conclusions and present our perspective on precision calculations in SUSY models in Section 9. In the Appendix A we give details about the computation of the group invariants required in multiloop calculations. Appendix B contains the main renormalization constants needed for three-loop calculations in supersymmetric quantum chromodynamics (SUSY-QCD) within the modified minimal subtraction, that has been employed for the computations reviewed in Sections 7 and 8.
2. Holomorphy and Exact Beta Functions in Supersymmetric Theories
In the last decades, enormous progress has been made in understanding the dynamics of supersymmetric gauge theories. For many models even exact renormalization group equations (RGEs) for the gauge couplings have been derived. However, the connections between the exact results and those obtained in perturbation theory are still not completely elucidated. Shifman and Vainshtein  were the first to propose a solution to this puzzle. They based their argumentation on the difference between the quantities involved in the exact beta functions derived within the Wilsonian renormalization approach and those adopted in the common perturbative framework. A different derivation of the exact beta functions was presented in , where only the Wilsonian renormalization approach was used but the authors distinguished between the holomorphic and canonical normalization of the gauge kinetic term in the bare Lagrangian.
Within the Wilsonian framework  any field theory is defined by the fundamental Lagrangian, the bare couplings, and the cutoff parameter. Varying the cutoff parameter and the bare couplings in a concerted way so that the low-energy physics remains fixed, one finds the dependence of the bare couplings on the cutoff parameter which is encoded in the Wilsonian renormalization group equations (WRGEs). The transition from a fundamental Lagrangian to an effective Lagrangian involves integrating out the high momentum modes of the quantum fields (i.e., degrees of freedom with momenta between some large cutoff scale and some renormalization scale ). The coefficients of the resulting operators play the role of renormalized couplings and we will call them Wilsonian effective couplings. The virtue of this approach is the lack of any infrared effects, since none of the calculations involves infrared divergences.
Let us consider as an example supersymmetric electrodynamics (SQED). The vector superfield in the Wess-Zumino gauge has the following Grassmannian expansion: where the physical degrees of freedom correspond to the vector gauge field and the Majorana spinor field , known also as gaugino field. The field is an auxiliary field without any physical meaning and can be eliminated with the help of equations of motion for the physical fields.
The Lagrangian of the model at an energy scale can be written as follows: where the superfield strength tensor is defined through the following relation: with Here and are the supercovariant derivatives. The superfields and are chiral matter superfields with charges and , respectively. stands for the gauge coupling and denotes the superfield renormalization constant.
The maximal value of is equal to , the ultraviolet cutoff parameter. At this point the Lagrangian (4) is just the original SQED Lagrangian and the coefficients and are bare parameters.
Because the momentum integrals are performed in dimension and the regularization is introduced through the cutoff parameter, the Wilsonian renormalization procedure preserves SUSY. Thus, if one calculates the Wilsonian effective Lagrangian, it is manifestly supersymmetric. As a consequence, the resulting effective superpotential (the part of the Lagrangian density that does not contain any derivative) must be a holomorphic function of the couplings [14####^~^~^~^~^~^####x2013;16]. This constraint restricts the running of the Wilsonian couplings to just the one-loop order.
For example, let us assume that we integrate out the matter superfields passing to the low-energy limit of the theory. The low-energy effective coupling at the low-energy is given through the following relation: where denotes the renormalized or the Wilsonian low-energy effective coupling constant and is the cutoff-dependent bare coupling constant. is the coefficient of the one-loop beta function of the underlying theory, where the beta function is defined through Let us emphasize that (7) is exact at all orders. The two- and higher-loop RGEs involve at least which is a nonholomorphic function of the bare coupling and thus cannot contribute to (7). In , it was proved through a direct calculation using the supergraphs method that the two-loop contributions to the running of the effective coupling vanish. The generalization of this assertion to higher loops is based on the extension of the nonrenormalization theorem for -terms in supersymmetric theories [14####^~^~^~^~^~^####x2013;16].
As mentioned above, one has to distinguish between the holomorphic Wilsonian gauge couplings and the physically measurable momentum-dependent effective gauge couplings present in the one-particle irreducible generating functional. Unlike the Wilsonian couplings, the physical couplings do not depend on the ultraviolet cutoff scale but on momenta of the particles involved. The dependence of the physical couplings on the overall momentum scale is governed by the Gell-Mann-Low equations , which have different physical meaning as the WRGEs and have different -functions [41, 42] beyond one loop. Going from the effective Lagrangian in the Wilsonian approach to the classical effective action means to integrate out all of the degrees of freedom down to zero momentum, that will generate nonholomorphic corrections. is often interpreted as a sort of effective Lagrangian, but in general it does not have the form of a supersymmetric Lagrangian with holomorphic coefficients.
The connection between the Wilsonian gauge coupling and a physical gauge coupling was derived in the so-called Novikov-Shifman-Vainshtein-Zakharov renormalization scheme (NSVZ) . This scheme requires a manifestly supersymmetric regularization procedure. In addition, the definition of the physical couplings is close to that in the momentum subtraction scheme (MOM). The conversion relation reads where is the renormalization constant of the matter superfield and the coefficient is the Dynkin index of the representation of the matter superfield. The factor is related to the mass renormalization constants of the matter superfield through the nonrenormalization theorems, provided SUSY is preserved. However, in general the factors are not restricted by any holomorphic constraints and thus are not known analytically. They have to be computed order by order in perturbation theory. Combining (9) and (7) we get Using (8) we obtain for the beta function of the physical coupling in the NSVZ scheme the following relation: where we have specified the value of the coefficient for the SQED case and the superfield anomalous dimension is defined through Because (7) is exact at all orders, also the relation between the beta function of and the anomalous dimension of the matter superfields is valid at all orders. Let us remark, however, that this relation holds only in the NSVZ scheme. Unfortunately, it is highly nontrivial to fulfill the requirements of the NSVZ scheme in practice.
In supersymmetric nonabelian models with several matter supermultiplets, (9) becomes where is the quadratic Casimir operator of the adjoint representation and is the Dynkin index of the representation of the matter field . The second term stands for the gaugino contribution, while the third one for contributions generated by the matter superfields. A simple calculation provides us with the exact relation between the gauge beta function and the anomalous dimension of the matter superfields: From (14) it is easy to see that for the derivation of the -loop beta functions in the NSVZ scheme one needs the matter anomalous dimensions at the -loop order. As will be shown below this feature was intensively exploited in the literature.
In the case of SUSY-Yang-Mills theories the matter superfields are absent, so , and an exact formula for the gauge coupling beta function can be derived: Similar relations can also be derived for models with softly broken SUSY. The line of reasoning is as follows: the powerful supergraph method is also applicable for models with softly broken SUSY by using the ####^~^~^~^~^~^####x201c;spurion####^~^~^~^~^~^####x201d; external field method [17, 43]. Perhaps, one of the most prominent example is the relation that can be established between the gaugino mass and the gauge beta function. In the presence of the SUSY breaking gaugino mass term, the coefficient of the gauge kinetic term in the Wilsonian action becomes where is the Grassmann variable.
Using the same arguments based on holomorphy, it was shown [44, 45] that a renormalization group invariant (RGI) relation for the gaugino mass can be derived within NSVZ scheme: Moreover, it was shown with the help of the spurion formalism that the renormalization constants of softly broken SUSY gauge theory can be related to the renormalization constants of the underlying exact supersymmetric model [46####^~^~^~^~^~^####x2013;48]. Even more, the connecting formulas are valid at all orders in perturbation theory. The only necessary assumption for their derivation is the existence of a gauge and SUSY invariant regularization scheme. Thus, such relations are valid only in NSVZ-like regularization schemes.
At this point, a few remarks are in order to comment on the results discussed above. The authors of  state that in dimensions the only known regularization to conserve SUSY is the Pauli-Villars scheme for matter superfields and the higher derivative scheme for the gauge superfields. Technically this construction is rather complicated and hardly applicable to multiloop computations. In , an attempt was made to apply the ####^~^~^~^~^~^####x201c;supersymmetric dimensional regularization####^~^~^~^~^~^####x201d; or ####^~^~^~^~^~^####x201c;regularization by dimensional reduction####^~^~^~^~^~^####x201d; (DRED)  within the supergraph formalism. However, as pointed out by Siegel himself , this scheme is mathematically inconsistent in its original formulation and a consistent formulation will break supersymmetry in higher orders of perturbation theory. A similar situation occurs also for the application of DRED in component field formalism [59, 60] (A detailed analysis of this issue will be done in the next section). Thus, the exact formulas of the NSVZ scheme are not valid, in general, for calculations based on DRED since they do not involve a regularization scheme supersymmetric at all orders. For particle phenomenology, it means that the powerful predictions of (14) cannot be tested through experiments, since the beta functions are scheme dependent beyond two loops.
The breakthrough regarding this situation was obtained in [61####^~^~^~^~^~^####x2013;65], where it is stated that if the NSVZ scheme exists it can be perturbatively related to schemes based on DRED. Such arguments follow from the equivalence of different renormalization schemes in perturbation theory . Precisely, the computation of the three-loop mass anomalous dimension for the chiral matter superfield in a general nonabelian supersymmetric theory and of the three-loop gauge beta function in the abelian case allowed the derivation of the three- and four-loop gauge beta function for a general supersymmetric theory. Remarkably enough, the derivation (up to a numerical coefficient) of the four-loop gauge beta function was based on a three-loop calculation and theoretical considerations about special relations valid in supersymmetric theories and one-loop finite supersymmetric theories.
Let us mention at this point also the calculation of the three-loop gauge beta function for supersymmetric Yang-Mills (SYM) theories of . For this calculation, DRED was employed in component field formalism rather than superfield formalism, and hence a manifestly not supersymmetric gauge was used. The computations of [64, 65, 67] coincide as a consequence of gauge invariance of the gauge beta function.
Moreover, the authors of [61, 62] noticed that the differential operators relating the beta functions for soft SUSY breaking parameters to the beta functions of the gauge and Yukawa couplings are form invariant under change of scheme (i.e., from NSVZ to DRED scheme). Thus, similar relations for the soft SUSY breaking parameter valid to all orders of perturbation theory hold also in a DRED-like scheme (Actually, the scheme. proposed by the authors of [61, 62] is the so-called , for which beta functions of SUSY breaking parameters do not depend on the unphysical -scalar mass parameter. For more details about the scheme see Section 3.)
In the next section we will discuss in detail the application of DRED in component field formalism and give some example of important calculations that can be done within this approach. Nevertheless, already now we want to mention the coincidence of all results obtained with DRED in component field formalism and those derived via DRED in supergraphs formalism.
3. Dimensional Reduction in the Component Field Formalism
The precision of many present or forthcoming experiments in particle physics requires inevitably higher order perturbative calculations in the SM or its extensions like the Minimal Supersymmetric Standard Model (MSSM). Regularization of the divergent loop diagrams arising in the higher order calculations is commonly performed employing Dimensional Regularization (DREG) or its variants, due to its nice feature to respect gauge invariance. Higher order calculations within the SM predominantly use DREG in its original form [68, 69], while for calculations within supersymmetric theories DRED as defined in  is commonly employed. It is not a priori known whether SUSY as a symmetry of a given Lagrangian is still a symmetry of the full quantum theory in any particular case. Nevertheless, a detailed formal renormalization program has been pursued in  including a proof that SUSY is not anomalous. If the regularized theory does not respect SUSY, the finite amplitude will not satisfy the Ward identities required by SUSY, giving rise to an apparent anomaly. If SUSY is not anomalous, it is possible to restore the invariance by introducing finite counterterms.
In practice, the choice of regularization scheme is of considerable significance for the extraction of physical predictions. This is the case for the NSVZ scheme we alluded in the previous section, that rarely found direct practical applicability. It rather provides important checks for results predicted within DRED. In this section we discuss in detail the application of DRED in the component field formalism and its application to practical calculations.
DRED consists of continuing the number of space dimensions from to , where is less than , but keeping the dimension of all the fields fixed. In component field language, this means that the vector bosons and fermions preserve their four-dimensional character. Furthermore, it is assumed that all fields depend on rather than space time coordinates, so that the derivatives and momenta become -dimensional. It is the four-dimensional nature of the fields that is supposed to restore the supersymmetric Ward-Takahashi [71, 72] or Slavnov-Taylor  identities, while the -dimensional space time coordinates cure, as in DREG, the singularities of the loop integrals.
However, potential inconsistencies of DRED, arising from the use of purely four-dimensional relations between the Levi-Civita tensor and the metric tensor, have been pointed out by Siegel himself . Even more, inconsistencies of DRED arising without the direct use of Levi-Civita tensors have been revealed in . The authors have correlated them with the impossibility of decomposing the finite four-dimensional space into a direct sum of infinite-dimensional spaces. The solution proposed by the same authors is to introduce a formal space, called quasi-four-dimensional space (), with ####^~^~^~^~^~^####x201c;noninteger valued####^~^~^~^~^~^####x201d; vector and spinor indices (thus, the two types of indices range over an infinite set of values), obeying certain algebraic identities inspired from the properties of the four-dimensional Minkowski space. The existence of such a space was demonstrated by construction  starting from similar arguments as those used to prove the existence of the formal -dimensional space of DREG . In this way the consistency of the calculation rules is guaranteed. By construction, is represented as the direct sum of two infinite-dimensional spaces: which is formally -dimensional and is identical with the one of DREG and which is formally -dimensional. (One needs to perform twice the construction of -dimensional integrals and metric tensors for and . The -dimensional integral is the momentum integral in DRED, while integral is involved only in the definition of the -dimensional metric tensor.) According to the properties of the three formal spaces at hand , , one can derive the following relations for the corresponding metric tensors , , [59, 74]: Furthermore, any quasi-four-dimensional vector can be decomposed with the help of the projectors , : Imposing the Dirac algebra for the -matrices defined in we can derive similar commutation relations for the components in and : These relations together with the trace condition are sufficient for computing Feynman diagrams. Equation (23) is particularly useful in supersymmetric theories, because it ensures that the numbers of degrees of freedom for fermions and bosons are equal.
For practical computations, it is useful to note that the fermion traces that contain both types of -matrices can be factored out as follows: This relation can be derived from (23), (22), and the algebra of Dirac matrices in dimensions. Thus, the Dirac algebra can be performed separately in and in dimensions.
Once we introduced ####^~^~^~^~^~^####x201c;noninteger valued####^~^~^~^~^~^####x201d; spinor indices, we need infinite-dimensional -matrices to represent the Dirac algebra. Thus, the Fierz identities valid in the genuine four-dimensional space do not hold anymore in . Their use was identified with one of the sources of DRED inconsistencies. Moreover, within the invariance of the original Lagrangian under SUSY transformations might be broken. This feature can be directly correlated with the lack of Fierz identities that would ensure the cancellation of Lagrangian variation under SUSY transformations in the genuine four-dimensional space. However, it has been shown [60, 74] that such inconsistencies become active only in the higher orders of perturbation theory, when, for example, traces over at least ten -matrices and antisymmetrization over five indices are involved. Thus, DRED also breaks SUSY, but starting from higher orders of perturbation theory. This explains, why one- and even two-loop calculations of QCD corrections within DRED [76####^~^~^~^~^~^####x2013;80] based on genuine four-dimensional Dirac algebra and even Fierz rearrangement provided correct results. Even the supersymmetric character of DRED at low orders has been exploited in the context of QCD with massless quarks in . However, beyond the one-loop level the distinction between resulting from contractions of the quasi-four-dimensional vector fields and resulting from momentum integrals is difficult to follow. It turned out  that for higher order computations it is useful to decompose the quasi-four-dimensional vector fields according to (20). As we shall see in the next section, in the case of gauge theories the -dimensional components behave as vectors under the gauge transformations whereas the components as scalars, usually called -scalars.
Representing the underlying space of DRED as a formal infinite-dimensional space renders the extension of as subtle as in DREG. The consistent procedure proposed by ####^~^~^~^~^~^####x2019;t Hooft-Veltman (HV)  for defining as in four dimensions has in the context of SUSY theories two drawbacks. On the one hand, it is the fact that the mathematically consistent treatment of in DREG requires , whereas for DRED is needed. However, it has been shown up to two loops [82, 83] that the Adler-Bardeen theorem  could still be satisfied in DRED with HV scheme, if relations like which follow in are assumed to hold also for . On the other hand, the use of a not anticommuting leads to the breakdown of symmetries, for example, chiral symmetry of the SM or supersymmetry in case of the MSSM already at the one-loop level. These ####^~^~^~^~^~^####x201c;spurious anomalies####^~^~^~^~^~^####x201d; would spoil the renormalizability and they have to be cured by introducing appropriate counterterms to restore Ward-Takahashi and Slavnov-Taylor identities order by order in perturbation theory (see ). This approach was successfully applied for SM predictions within DREG up to three-loops [86, 87]. However, for the MSSM it becomes much more involved due to the complexity introduced by supersymmetric conditions and it rarely has been employed in practice .
The implementation of in DRED commonly used in practice is inspired by the naive scheme (NS) of DREG. Namely, it is treated rather like a formal object which is not well defined mathematically but anticommutes with all -matrices Nevertheless, one has to correct the false result that arises from (26), that the trace of and four or more -matrices vanishes. Paying attention that now two types of -matrices occur, the additional constraints read The tensor has some similarities with the four-dimensional Levi-Civita tensor: (i) it is completely antisymmetric in all indices; (ii) when contracted with a second one of its kind gives the following result: depending on the nature of Dirac matrices in (27). Here the square brackets denote complete antisymmetrization. When taking the limit , converts into the four-dimensional Levi-Civita tensor and (27) and (28) ensure that the correct four-dimensional results are reproduced. This last constraint is needed in order to correctly compute fermion triangle diagrams containing an axial vector current, that is, to cope with the Adler-Bardeen-Jackiw anomaly [88####^~^~^~^~^~^####x2013;90].
At this point a comment on (27) is in order. When we combine it with the cyclic property of traces, it necessarily follows that other traces are not well defined in dimensions. It turns out that there is an unavoidable ambiguity of order when fixing the trace condition in (27). Even if one does not use the cyclic property of the trace, an ambiguity in the distribution of the anomaly between the vector and the axial vector currents shows up . The occurrence of the ambiguity is a characteristic of the extension of away from dimensions. ####^~^~^~^~^~^####x2019;t Hooft and Veltman have pointed out in their original paper  that an ambiguity related to the location of shows up in HV scheme, too.
The use of an anticommuting in dimensions was applied for the first time to the evaluation of fermion traces with an even number of ####^~^~^~^~^~^####x2019;s in , and a few years later extended also to odd fermion traces in . The method (for more details see  and references cited therein) proved to be effective for SM calculations involving chiral fermions up to two-loop order [94####^~^~^~^~^~^####x2013;97]. The consistency of this prescription has been verified even in three-loop QCD-electroweak calculations [98, 99]. Within DRED, it has been successfully employed in MSSM calculations at the two- and three-loop order [100####^~^~^~^~^~^####x2013;103]. However, let us mention at this point that for these calculations at most the finite parts of two-loop and the divergent parts of three-loop diagrams are required. For the calculation of finite parts of three-loop diagrams containing two fermion triangle subdiagrams, the HV scheme has to be applied as the naive scheme does not provide correct results.
Through the consistent formulation of DRED we gain a regularization scheme which proves to be supersymmetric only in the lower orders of perturbation theory. Due to the violation of Fierz identities, SUSY invariance will be broken at higher orders. The first consequence of SUSY breaking is that the all-order relations between different anomalous dimensions valid in the NSVZ scheme do not hold in DRED. However, although DRED consistently formulated is not a supersymmetric scheme at all orders, it provides so far the best option for computations within SUSY theories.
3.2. Minimal Subtraction Schemes and
The common renormalization schemes used for multiloop calculations are the minimal subtraction (MS), momentum subtraction and on-shell schemes. Minimal subtraction, scheme has the advantage of involving the simplest computations, but it is nonphysical in the sense that it does not take into account mass threshold effects for heavy particles. Nevertheless, it is the main scheme used in renormalization group (RG) analyses relating the predictions of a given theory at different energy scales. The other two options are computationally much more involved but indispensable for the determination of the parameters of a theory from the quantities measured experimentally. We focus in this section on the minimal subtraction methods.
Minimal subtraction scheme with DREG as regulator  or the modified scheme  and its variant for DRED####^~^~^~^~^~^####x2014;the scheme####^~^~^~^~^~^####x2014;are in particular well suited for higher order calculations in perturbation theory. The advantage of these schemes is that all ultraviolet (UV) counterterms are polynomial both in external momenta and masses [106, 107]. This allows for setting to zero certain masses or external momenta, provided no spurious infrared divergences are introduced. This simplifies substantially the calculations of the Feynman integrals. It has been shown  by means of the infrared rearrangement (IRR) procedure [108####^~^~^~^~^~^####x2013;110] that the renormalization constants within the scheme can be reduced to the calculation of only massless propagator diagrams. This method was used for the first three-loop calculation of the QCD -function , applying it to each individual diagram. But the most effective approach is its use in combination with multiplicative renormalization. This amounts in general to solve recursively the equation where stands for the singular part of the Laurent expansion of in around . denotes the renormalized Green function with only one external momentum kept nonzero. denotes the renormalization constant associated with the Green function . In this case, the renormalization of through -loop order requires the renormalization of the Lagrangian parameters like couplings, masses, gauge parameters, mixing angles, and so forth. up to -loop order. The method was successfully applied to the three-loop calculations of anomalous dimensions within or schemes [49, 86, 101####^~^~^~^~^~^####x2013;103, 112####^~^~^~^~^~^####x2013;114] using the package MINCER  written in FORM , which computes analytically massless propagator diagrams up to three loops.
Apart from that, a second method was proposed in , which has been used for the calculation of the three- and even four-loop anomalous dimensions of QCD [118####^~^~^~^~^~^####x2013;121] and the beta function of the quartic coupling of the Higgs boson in the SM [114, 122, 123]. It deals with the IRR by introducing an artificial mass for all propagators. Expanding in all particles masses and external momenta, one can reduce the evaluation of the Feynman integrals to massive tadpoles. The analytic evaluation of the massive tadpoles up to three-loop order can be obtained with the help of the package MATAD .
A third method was introduced for the evaluation of the renormalization constants for the quark mass  and the vector  and quark scalar current correlators  through four loops. It is based on global IRR properties and amounts essentially to set to zero the external momentum and let an arbitrary subset of the internal lines to be massive. After nontrivial manipulations, the four-loop integrals can be reduced to three-loop massless, two-point integrals, and one-loop massive vacuum integrals.
The three-loop accuracy for the anomalous dimensions of theories involving not only vector but also Yukawa and quartic scalar interactions (e.g., the SM [49, 114]) was achieved only very recently. Remarkably, for supersymmetric and softly broken supersymmetric theories like the MSSM the three-loop anomalous dimensions were computed long before [63, 64, 127]. Their derivations used intensively the exact relations established between the various anomalous dimensions in the NSVZ scheme (for more details see Section 2) as well as the observation that the NSVZ scheme and DRED can be perturbatively connected.
3.3. DRED Applied to Nonsupersymmetric Theories
Although DRED was originally proposed as a candidate for an invariant regularization in supersymmetric theories, it proved to be useful also in nonsupersymmetric theories. Its use in SM calculations up to three-loop orders was motivated either by the possibility to apply four-dimensional algebra and even Fierz rearrangements [77, 80]####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;(the mathematical inconsistencies alluded to above do not occur at the two-loop level in this calculations), or by the possibility to easily convert a nonsupersymmetric gauge theory into a SUSY-Yang-Mills theory and use nontrivial Ward identities as checks of complicated calculations [78, 98, 128]. Apart from the computational advantages, DRED applied to nonsupersymmetric theories, in particular to QCD, provides us with a powerful tool to verify its consistency up to three-loop order via the connection that can be established with DREG (DRED and DREG are also perturbatively connected). ####^~^~^~^~^~^####x2009;Finally, it is motivated by the MSSM, as a softly broken supersymmetric theory or by various models derived from the MSSM which feature lower symmetries (e.g., the intermediate energy theory obtained by integrating out the squarks and sleptons). DRED applied to effective field theories, such that QCD extended to include the Higgs-top Yukawa coupling, was useful for the calculation of the production rate for the Higgs boson in gluon-fusion channel within MSSM [54, 129].
In the following, we consider a nonabelian gauge theory with Dirac fermions transforming according to a representation of the gauge group . For the moment we do not take into account any genuine scalar field.
The Lagrangian density (in terms of bare fields) reads where the field strength tensor is defined through is the covariant derivative. is the gauge field, is the Fadeev-Popov-ghost field, are the structure constants of the gauge group , is the gauge parameter, and is the gauge coupling.
For the case when the theory admits a gauge invariant fermion mass term we will have , where DRED amounts to imposing that all field variables depend only on a subset of the total number of space time dimensions; in this case out of where . We can then make the decomposition where It is then easy to show that  where where and denote the projection of the field strength and covariant derivative given in (31) onto , obtained with the help of the operator . Conventional dimensional regularization (DREG) amounts to using (36) and discarding (37).
Note that under the gauge transformationseach term in (37) is separately invariant. The fields behave exactly like scalar fields and are hence known as -scalars. There is therefore no reason to expect the vertex to renormalize in the same way as the vertex (except in the case of supersymmetric theories). The couplings associated with the vertex or with the quartic -scalar interaction are called evanescent couplings. They were first described in  and later independently discovered by van Damme and ####^~^~^~^~^~^####x2019;t Hooft . The vertices and , on the other hand, are renormalized in the same way as , and so forth because of the gauge invariance . Thus we can conclude that is the gauge particle, while acts as matter field transforming according to the adjoint representation. In order to avoid confusion, we denote in the following the gauge particles with and the -scalars with : Since -scalars are present only on internal lines we could, in fact, choose the wave function renormalization of and to be the same. However, such a renormalization prescription will break unitarity . The crucial point is the correct renormalization of subdivergences, which requires that vertices involving -scalars renormalize in a different way as their gauge counterparts. Thus, to renormalize the -scalars one has to treat them as new fields present in the theory.
For the renormalization of the theory we distinguish two new types of couplings: a Yukawa like coupling associated with the vertex and a set of quartic couplings associated with vertices containing four -scalars. The number is given by the number of independent rank four tensors which are nonvanishing when symmetrized with respect to and interchange. We address the issue of the quartic vertex renormalization in more detail in the next section.
The renormalization constants for the couplings, masses, and fields and vertices are defined as where is the renormalization scale and the bare quantities are marked by the superscript ####^~^~^~^~^~^####x201c;0.####^~^~^~^~^~^####x201d; stands for one-particle irreducible Green functions involving the external particles , , , . Equation (36) takes under renormalization the usual expression in terms of renormalized parameters as in DREG scheme. The renormalized Lagrangians is the new term that distinguishes DRED from DREG and it is given by Strictly speaking, (41) should also have a mass term for the -scalars; but since this mass term does not affect renormalization of the couplings and fermion masses we omit it here. We discuss this issue in more detail in Section 3.3.5.
The charge renormalization constants are obtained from the Slavnov-Taylor identities. For example, if one computes the -point Green function with external fields and denotes its coupling constant by , one obtains where the are the wave function renormalization constants for the , is the corresponding vertex renormalization constant, and the charge renormalization. Within the minimal subtraction scheme, one is free to choose any masses and external momenta, as long as infrared divergences are avoided. One can set all masses to zero, as well as one of the two independent external momenta in three-point functions. In this case, one arrives at three-loop integrals with one nonvanishing external momentum which can be calculated with the help of MINCER. One can also calculate the three-point functions setting a common mass to all particles and expanding the Feynman integrals in the limit with the help of asymptotic expansions . This approach is much more tedious, but possible infrared singularities would manifest in terms. If such terms are absent in the final expression, the limit can be taken and the result should coincide with the one obtained with the massless setup (for a comprehensive overview about the multiloop techniques within DREG see the review article ).
Precisely, the charge renormalization of the gauge coupling can be derived from the ghost-gauge boson, fermion-gauge boson, -scalar-gauge boson vertices, or the gauge boson self-interaction as a consequence of gauge invariance.
Similarly, for the charge renormalization constants of the evanescent couplings, the following relations hold: In general, even at one-loop order. However, in supersymmetric theories should hold at all orders because of SUSY. This can be understood following the same line of reasoning as for the derivation of the equality of the charge renormalization constants for the interactions involving gluons and those involving gluinos.
3.3.1. The -Scalar Self-Couplings
Let us discuss the structure of the quartic -scalar couplings for an arbitrary gauge group. These interactions are invariant under the symmetry , where only is gauged. The number of independent quartic -scalar couplings is given by the number of independent rank tensors invariant with respect to and exchange, because of the invariance. It has been shown that for , , with there are four such tensors . For the case only three independent tensors can be built , while for their number reduces to two . The answer to the general question concerning rank tensors is not yet known. For the explicit construction of the set of tensors we consider first the group and then generalize the results for the other two groups.
A natural choice for a basis for rank tensors when is given by . (An alternative way to define a basis which has the virtue of being immediately generalizable to any group  is in terms of traces of products of the generators in the defining representation, thus , , etc.) one has Here stands for the completely symmetric rank tensors. The dimension of the basis reduces to in the case of . This is achieved via the relation [137, 138] which is not valid for .
To describe the -scalar quartic interactions one needs to construct rank tensors invariant with respect to exchange of pairs of indices. Thus, one has to take linear combinations of the basis tensors and symmetrize them with respect to the pair of indices and . A possible choice for is given by Note that the absence of a type term from (47) follows from the identity  However, for practical purposes a basis constructed with the help of the structure constants and avoiding the use of the -tensors is more suited. For example, it would allow to explore more easily the supersymmetric case and to generalize to other groups. It is natural to consider the alternative choice [113, 131] Let us introduce the coupling constants Then we can write the last term in (41) where denote the quartic -scalar couplings in the basis . The renormalization constants , , and so forth have been computed through one loop in the scheme for a general gauge group in [131, 135] and in  for . The calculation performed in  has employed the method of  to introduce an artificial mass for all propagators in order to avoid spurious infrared divergences. For the calculation of the results in terms of group invariants the package color  has been used. For completeness, we reproduce here the one-loop results for the couplings : with the group invariants defined in Appendix A and the abbreviation , where denotes the number of active fermions. Let us notice at this point the presence of negative power of couplings in the expressions of the renormalization constants. This results in beta functions that are not proportional to the coupling itself. This feature is specific to scalar couplings and it implies that, even if we set such a coupling to zero at a given scale, it will receive nonvanishing radiative corrections due to the other couplings present in the theory.
The above results have been computed using an gauge group. However, they are parametrized in terms of group invariants. Thus they are also valid for other physically interesting groups like and . The explicit values of the group invariants for the three groups can be found in Appendix A.
In the case of group, the invariant becomes a linear combination of , , because of relation (46). The same is also true for the coupling that can be expressed in terms of the other three couplings. Thus in this case one can ignore .
Actually, the one- and two-loop renormalization constants for totally symmetric quartic scalar couplings with scalars in an arbitrary representation have been known for long time . However, these results cannot be directly applied to -scalar self-interactions, due to their particular symmetry with respect to exchange between pairs of indices.
3.3.2. Three-Loop Renormalization Constants for a Nonsupersymmetric Theory
In this section we report on the explicit computation of the charge , and mass , renormalization constants to three-loop order within scheme. This requires the calculation of divergent parts of logarithmically divergent integrals. One can exploit the fact that such contributions are independent of the masses and external momenta. Precisely, one sets all internal masses to zero and keeps only one external momentum different from zero and then solve recursively (29). In practice, use of the automated programs QGRAF , q2e and exp [142, 143] and MINCER are essential due to the large number of diagrams that occur.
The analytical form of up to two-loop order is identical to the corresponding result in the scheme. This has been shown by an explicit calculation for the first time in  and is a consequence of the minimal renormalization. The three- and four-loop results for a general theory have been derived in [113, 135, 144]. For completeness we present in the following the three-loop results: The one-loop result for can be found in . For the particular case of QCD, that is, and , the two-, three-, and four-loop results have been computed in [113, 144]. The two-, three-, and four-loop results for a general theory have been derived in . Because of the complexity of the results, we reproduce below only the two-loop contributions that are, however, enough for most of the practical applications: The group invariants , , , occurring in the above equations are defined in Appendix A and we used the abbreviation .
There is also an indirect way to derive the three-loop gauge beta function in the scheme starting from the knowledge of the three-loop gauge beta function in the scheme and the fact that the gauge couplings defined in the two schemes can be perturbatively related to each other. This method will be discussed in more detail in the next section. Let us mention, however, that using the expression for the three-loop gauge beta function in the scheme and the two-loop conversion relation of given in (57) one obtains exactly the same results for as given in (53). This is a powerful consistency check for the calculation reviewed in this section. It is interesting to mention that the equality of the two results can be obtain only if one keeps during the calculation and renormalize them differently. The identification of and leads to inconsistent results. In case of the error is a finite, gauge parameter independent term . For quark mass renormalization, this identification (precisely the identification of the renormalization constants for the two couplings) generates much more severe problems. Namely, the renormalization constant for the quark mass will contain nonlocal terms at three-loop order and the mass anomalous dimension will erroneously become divergent at this loop order.
The renormalization constant for the fermion masses has been computed in  to three- and in [135, 144] even to four-loop order. Whereas in [113, 135, 144] only the anomalous dimensions were given we want to present the explicit three-loop result for the renormalization constant, that reads where is Riemann####^~^~^~^~^~^####x2019;s zeta function with ####^~^~^~^~^~^####x2026;.
Again, the consistency of the above results can be proved using the indirect method alluded above. To derive the three-loop quark mass anomalous dimension in the scheme , one needs the three-loop result for and the two-loop conversion relation for the quark mass as given in (58). Full agreement has been found between the two methods , that provides a further consistency check of the calculation.
3.3.3. The General Four-Loop Order Results in the Scheme
The direct way to compute the renormalization constants in minimal subtraction schemes as or requires the calculation of divergent parts of logarithmically divergent integrals. Up to three loops there are well established methods and automated programs exist to perform such calculations (see, e.g., [115, 124]). Also at four-loop order a similar approach is applicable. Nevertheless, it is technically much more involved [118, 119, 121, 145, 146]. There is, however, an indirect method discussed in [113, 128] to derive the renormalization constants in the scheme starting from their expressions. It relies on the perturbative relation that can be established between the couplings and masses defined in the two schemes and takes into account that the four-loop results in the scheme are known [118, 119, 121]. For example, to derive the beta function for the gauge coupling to four-loop order in scheme one needs the relation between the gauge couplings defined in the and schemes up to three-loop order. The latter can be determined using the following arguments.
To compute the relations between running parameters defined in two different renormalization schemes, one has to relate them to physical observables which cannot depend on the choice of scheme. For example, the relationship between the strong coupling constant defined in the and schemes can be obtained from the -matrix amplitude of a physical process involving the gauge coupling computed in the two schemes. However, beyond one loop the computation of the physical amplitudes becomes very much involved and requires the computation of multiloop and multiscale on-shell Feynman integrals that is a highly nontrivial task. Nevertheless, one can avoid the use of on-shell kinematics introducing a physical renormalization scheme defined through convenient kinematics, for which the renormalization constants can be computed applying the ####^~^~^~^~^~^####x201c;large-momentum####^~^~^~^~^~^####x201d; or the ####^~^~^~^~^~^####x201c;hard-mass####^~^~^~^~^~^####x201d; procedures. Up to three loops, there are well established methods (for details see previous sections) to compute the divergent as well as finite pieces of the Feynman integrals and automated programs exist to perform such calculations. Once the renormalization constants in the physical renormalization scheme are determined, one uses the constraint that the effective gauge coupling constant defined in such a scheme is unique and thus independent of the regularization procedure. Furthermore, one relates the running gauge couplings defined in the two regularization schemes through the following relations: where are the charge renormalization constants using minimal subtraction in DREG/DRED, as defined above. Note that the various in (56) depend on differently renormalized , so that the equations have to be used iteratively at higher orders of perturbation theory. Working out these considerations for the gauge coupling and for the fermion mass up to the three-loop order, one obtains where we have suppressed the explicit dependence on the renormalization scale . and denote the three-loop terms and they are obtained from the finite parts of three-loop diagrams (see  for details). They read [131, 135] Inserting (57) and (58) into the definition of the beta function for the gauge coupling equation (8) and the mass anomalous dimension equation (12), one can show that where the first equality is due to the definition of and and the second one is a consequence of the chain rule. Let us briefly discuss the order in perturbation theory up to which the individual building blocks are needed. Of course, the quantities are needed to four-loop order; they can be found in [118####^~^~^~^~^~^####x2013;121]. The dependence of and on starts at two- and one-loop order , respectively. Thus, is needed up to the three-loop level (cf. (60)). On the other hand, both and