IFT-UAM/CSIC-19-020 LAPTH-004/19 Reduction of Couplings and its applicationin Particle Physics

# Ift-Uam/csic-19-020 Lapth-004/19 Reduction of Couplings and its application in Particle Physics

S. Heinemeyer email: Sven.Heinemeyer@cern.ch M. Mondragón email: myriam@fisica.unam.mx N. Tracas and G. Zoupanos
Campus of International Excellence UAM+CSIC, Cantoblanco, 28049, Madrid, Spain
Instituto de Física de Cantabria (CSIC-UC), E-39005 Santander, Spain
Instituto de Física, Universidad Nacional Autónoma de México, A.P. 20-364, CDMX 01000, México
Physics Department, Nat. Technical University, 157 80 Zografou, Athens, Greece
Institute of Theoretical Physics, D-69120 Heidelberg, Germany
Max-Planck Institut für Physik, Föhringer Ring 6, D-80805 München, Germany
Laboratoire d’Annecy-le-Vieux de Physique Théorique, Annecy, France
Dedicated to the memory of Wolfhart Zimmermann,
the brilliant theoretical physicist,
who initiated the subject of reduction of couplings
email: ntrac@central.ntua.gremail: George.Zoupanos@cern.ch
###### Abstract

The idea of reduction of couplings in renormalizable theories will be presented and then will be applied in Particle Physics models. Reduced couplings appeared as functions of a primary one, compatible with the renormalization group equation and thus solutions of a specific set of ordinary differential equations. If these functions have the form of power series the respective theories resemble standard renormalizable ones and thus widen considerably the area covered until then by symmetries as a tool for constraining the number of couplings consistently. Still on the more abstract level reducing couplings enabled one to construct theories with beta-functions vanishing to all orders of perturbation theory. Reduction of couplings became physics-wise truly interesting and phenomenologically important when applied to the standard model and its possible extensions. In particular in the context of supersymmetric theories it became the most powerful tool known today once it was learned how to apply it also to couplings having dimension of mass and to mass parameters. Technically this all relies on the basic property that reducing couplings is a renormalization scheme independent procedure. Predictions of top and Higgs mass prior to their experimental finding highlight the fundamental physical significance of this notion.

Prologue and Synopsis

In spite of their limitations, perturbative local field theories are still of prominent practical value.

It is remarkable that the intrinsic ambiguities connected with locality and causality - most of the time associated with ultraviolet infinities - can be summarized in terms of a formal group which acts in the space of the coupling constants or coupling functions attached to each type of local interaction.

It is therefore natural to look systematically for stable submanifolds. Some such have been known for a long time: e.g., spaces of renormalizable interactions and subspaces characterized by system of Ward identities mostly related to symmetries.

A systematic search for such stable submanifolds has been initiated by W. Zimmermann in the early eighties.

Disappointing for some time, this program has attracted several other active researchers and recently produced physically interesting results.

It looks at the moment as the only theoretically founded algorithm potentially able to decrease the number of parameters within the physically favoured perturbative models aaa The above text has appeared, as Geleitwort (preface), in the book “Reduction of Couplings and its Application in Particle Physics Finite Theories Higgs and Top Mass Predictions”, Ed. Klaus Sibold, Authors: Jisuke Kubo, Sven Heinemeyer, Myriam Mondragon, Olivier Piguet, Klaus Sibold, Wolfhart Zimmermann, George Zoupanos. Published in PoS (Higgs & top)001. .

Raymond Stora, CERN (Switzerland), December 16, 2013

## Chapter 1 Introduction: The Basic Ideas

In the recent years the theoretical endeavours that attempt to achieve a deeper understanding of Nature have presented a series of successes in developing frameworks such as String Theories and Noncommutativity that aim to describe the fundamental theory at the Planck scale. However, the essence of all theoretical efforts in Elementary Particle Physics (EPP) is to understand the present day free parameters of the Standard Model (SM) in terms of few fundamental ones, i.e. to achieve reductions of couplings[1]. Unfortunately, despite the several successes in the above frameworks they do not offer anything in the understanding of the free paramaters of the SM. The pathology of the plethora of free parameters is deeply connected to the presence of infinities at the quantum level. The renormalization program can remove the infinities by introducing counterterms, but only at the cost of leaving the corresponding terms as free parameters.

Although the Standard Model (SM) has been very successful in describing elementary particles and its interactions, it has been known for some time that it must be the low energy limit of a more fundamental theory. This quest for a theory beyond the Standard Model (BSM) has expanded in various directions. The usual, and very efficient, way of reducing the number of free parameters of a theory to render it more predictive, is to introduce a symmetry. Grand Unified Theories (GUTs) are very good examples of such a procedure [2, 3, 4, 5, 6, 7]. First in the case of minimal , because of the (approximate) gauge coupling unification, it was possible to reduce the gauge couplings of the SM and give a prediction for one of them. By adding a further symmetry, namely global supersymmetry [8, 9, 10] it was possible to make the prediction viable. GUTs can also relate the Yukawa couplings among themselves, again provided an example of this by predicting the ratio [11] in the SM. Unfortunately, requiring more gauge symmetry does not seem to help, since additional complications are introduced due to new degrees of freedom, for instance in the ways and channels of breaking the symmetry.

A possible way to look for relations among unrelated parameters is the method of reduction of couplings [12, 13, 14]; see also refs [15, 16, 17]. This method, as its name proclaims, reduces the number of couplings in a theory by relating either all or a number of couplings to a single coupling denoted as the “primary coupling”. This method might help to identify hidden symmetries in a system, but it is also possible to have reduction of couplings in systems where there is no apparent symmetry. The reduction of couplings is based on the assumption that both the original and the reduced theory are renormalizable and that there exist renormalization group invariant (RGI) relations among parameters.

A natural extension of the GUT idea and successful application of the method of reduction of couplings is to find a way to relate the gauge and Yukawa sectors of a theory, that is to achieve gauge-Yukawa Unification (GYU). This will be presented in Chapter 5. Following the original suggestion for reducing the couplings within the framework of GUTs we were hunting for renormalization group invariant (RGI) relations holding below the Planck scale, which in turn are preserved down to the GUT scale. It is indeed an impressive observation that one can guarantee the validity of the RGI relations to all-orders in perturbation theory by studying the uniqueness of the resulting relations at one-loop. Even more remarkable is the fact that it is possible to find RGI relations among couplings that guarantee finiteness to all-orders in perturbation theory. The above principles have only been applied in supersymmetric GUTs for reasons that will be transparent in the following sections, here we should only note that the use of supersymmetric GUTs comprises the demand of the cancellation of quadratic divergencies in the SM. The above GYU program applied in the dimensionless couplings of supersymmetric GUTs had a great success by predicting correctly, among others, the top quark mass in the finite [18, 19] and in the minimal supersymmetric [20] before its discovery [21].

Although supersymmetry seems to be an essential feature for a successful realization of the above program, its breaking has to be understood too, since it has the ambition to supply the SM with predictions for several of its free parameters. Indeed, the search for RGI relations has been extended to the soft supersymmetry breaking sector (SSB) of these theories, which involves parameters of dimension one and two. In addition, there was important progress concerning the renormalization properties of the SSB parameters, based on the powerful supergraph method for studying supersymmetric theories, and it was applied to the softly broken ones by using the “spurion” external space-time independent superfields. According to this method a softly broken supersymmetric gauge theory is considered as a supersymmetric one in which the various parameters, such as couplings and masses, have been promoted to external superfields. Then, relations among the soft term renormalization and that of an unbroken supersymmetric theory have been derived. In particular the -functions of the parameters of the softly broken theory are expressed in terms of partial differential operators involving the dimensionless parameters of the unbroken theory. The key point in solving the set of coupled differential equations so as to be able to express all parameters in a RGI way, was to transform the partial differential operators involved to total derivative operators. It is indeed possible to do this by choosing a suitable RGI surface.

On the phenomenological side the application on the reduction of coupling method to supersymmetric theories has led to very interesting developments too. Previously an appealing “universal” set of soft scalar masses was assumed in the SSB sector of supersymmetric theories, given that apart from economy and simplicity (1) they are part of the constraints that preserve finiteness up to two-loops, (2) they appear in the attractive dilaton dominated supersymmetry breaking superstring scenarios. However, further studies have exhibited a number of problems, all due to the restrictive nature of the “universality” assumption for the soft scalar masses. Therefore, there were attempts to relax this constraint without loosing its attractive features. Indeed an interesting observation on GYU theories is that there exists a RGI sum rule for the soft scalar masses at lower orders in perturbation theory, which was later extended to all-orders, and manages to overcome all the unpleasant phenomenological consequences. Armed with the above tools and results we were in a position to study the spectrum of the full finite models in terms of few free parameters, with emphasis on the predictions of supersymmetric particles and the lightest Higgs mass.

The result was indeed very impressive since it led to a prediction of the Higgs mass which coincided with the results of the LHC for the Higgs mass by ATLAS [22, 23] and CMS [24, 25], and predicted a relatively heavy spectrum consistent with the non-observation of supersymmetric particles at the LHC. The coloured supersymmetric particles are predicted to be above 2.7 TeV, while the electroweak supersymmetric spectrum starts below 1 TeV. These successes will be presented in Chapter 6.

Last but certainly not least, the above machinery has been recently applied in the MSSM with impressive results concerning the predictivity of the top, bottom and Higgs masses, being at the same time consistent with the non-observation of supersymmeric particles at the LHC. More specifically the electroweak supersymmetric spectrum starts at 1.3 TeV and the coloured at  TeV. These results will be presented too in Chapter 6.

## Chapter 2 Theoretical Basis

### 2.1 Reduction of Dimensionless Parameters

In this section we outline the idea of reduction of couplings. Any RGI relation among couplings (i.e. which does not depend on the renormalization scale explicitly) can be expressed, in the implicit form , which has to satisfy the partial differential equation (PDE)

 μdΦdμ=→∇Φ⋅→β = A∑a=1βa∂Φ∂ga = 0 , (2.1)

where is the -function of . This PDE is equivalent to a set of ordinary differential equations, the so-called reduction equations (REs) [12, 13, 14],

where and are the primary coupling and its -function, and the counting on does not include . Since maximally () independent RGI “constraints” in the -dimensional space of couplings can be imposed by the ’s, one could in principle express all the couplings in terms of a single coupling . However, a closer look to the set of Eqs. (2.2) reveals that their general solutions contain as many integration constants as the number of equations themselves. Thus, using such integration constants we have just traded an integration constant for each ordinary renormalized coupling, and consequently, these general solutions cannot be considered as reduced ones. The crucial requirement in the search for RGE relations is to demand power series solutions to the REs,

 ga=∑nρ(n)ag2n+1 , (2.3)

which preserve perturbative renormalizability. Such an ansatz fixes the corresponding integration constant in each of the REs and picks up a special solution out of the general one. Remarkably, the uniqueness of such power series solutions can be decided already at the one-loop level [12, 13, 14]. To illustrate this, let us assume that the -functions have the form

 βa=116π2⎡⎣∑b,c,d≠gβ(1)bcdagbgcgd+∑b≠gβ(1)bagbg2⎤⎦+⋯ ,βg=116π2β(1)gg3+⋯ , (2.4)

where stands for higher order terms, and ’s are symmetric in . We then assume that the ’s with have been uniquely determined. To obtain ’s, we insert the power series (2.3) into the REs (2.2) and collect terms of and find

 ∑d≠gM(r)daρ(r+1)d=lower order quantities ,

where the r.h.s. is known by assumption, and

 M(r)da =3∑b,c≠gβ(1)bcdaρ(1)bρ(1)c+β(1)da−(2r+1)β(1)gδda , (2.5) 0 =∑b,c,d≠gβ(1)bcdaρ(1)bρ(1)cρ(1)d+∑d≠gβ(1)daρ(1)d−β(1)gρ(1)a . (2.6)

Therefore, the ’s for all for a given set of ’s can be uniquely determined if for all .

As it will be clear later by examining specific examples, the various couplings in supersymmetric theories have the same asymptotic behaviour. Therefore searching for a power series solution of the form (2.3) to the REs (2.2) is justified.

The possibility of coupling unification described in this section is without any doubt attractive because the “completely reduced” theory contains only one independent coupling, but it can be unrealistic. Therefore, one often would like to impose fewer RGI constraints, and this is the idea of partial reduction [26, 27].

The above facts lead us to suspect that there is and intimate connection among the requirement of reduction of couplings and supersymmetry which still waits to be uncovered. The connection becomes more clear by examining the following example.

Consider an gauge theory with the following matter content: and are complex scalars, and are left-handed Weyl spinor, and is a right-handed Weyl spinor in the adjoint representation of .

The Lagrangian, omitting kinetic terms, includes:

 L⊃i√2{ gY¯¯¯¯ψλaTaϕ−^gY¯¯¯¯^ψλaTa^ϕ+h.c. }−V(ϕ,¯¯¯ϕ), (2.7)

where

 V(ϕ,¯¯¯ϕ)=14λ1(ϕiϕ∗i)2+14λ2(^ϕi^ϕ∗ i)2+λ3(ϕiϕ∗i)(^ϕj^ϕ∗ j)+λ4(ϕiϕ∗j)(^ϕi^ϕ∗ j), (2.8)

which is the most general renormalizable form of dimension four, consistent with the global symmetry.

Searching for a solution of the form of Eq. (2.3) for the REs (2.2,) we find in lowest order the following one ( is the gauge coupling):

 gY=^gY=g ,λ1=λ2=N−1Ng2 ,λ3=12Ng2 , λ4=−12g2 , (2.9)

which corresponds to an supersymmetric gauge theory. Clearly the above remarks do not answer the question of the relation among reduction of couplings and supersymmetry but rather try to trigger the interest for further investigation.

### 2.2 Reduction of Couplings in N = 1 Supersymmetric Gauge Theories. Partial Reduction

Let us consider a chiral, anomaly free, globally supersymmetric gauge theory based on a group G with gauge coupling constant . The superpotential of the theory is given by

 W = 12mijϕiϕj+16Cijkϕiϕjϕk , (2.10)

where and are gauge invariant tensors and the matter field (chiral superfield) transforms according to the irreducible representation of the gauge group . The renormalization constants associated with the superpotential (2.10), assuming that supersymmetry is preserved, are

 ϕ0i =(Zji)(1/2)ϕj , (2.11) m0ij =Zi′j′ijmi′j′ , (2.12) C0ijk =Zi′j′k′ijkCi′j′k′ . (2.13)

The non-renormalization theorem [28, 29, 30, 31] ensures that there are no mass and cubic-interaction-term infinities and therefore

 Zi′j′ij(Zi′′i′)(1/2)(Zj′′j′)(1/2)=δi′′(iδj′′j) ,Zi′j′k′ijk(Zi′′i′)(1/2)(Zj′′j′)(1/2)(Zk′′k′)(1/2)=δi′′(iδj′′jδk′′k) . (2.14)

As a result the only surviving possible infinities are the wave-function renormalization constants , i.e., one infinity for each field. The one-loop -function of the gauge coupling is given by [32, 33, 34, 35, 36]

 β(1)g=dgdt=g316π2[∑iT(Ri)−3C2(G)] , (2.15)

where, as usual, is the logarithm of the ratio of the energy scale over a reference scale, is the quadratic Casimir of the adjoint representation of the associated gauge group and is given by the relation while is the generators of the group in the appropriate representation. The -functions of , by virtue of the non-renormalization theorem, are related to the anomalous dimension matrix of the matter fields as:

 βijk=dCijkdt = Cijlγlk+Ciklγlj+Cjklγli . (2.16)

At one-loop level is given by [32]

 γ(1)ji=132π2[CiklCjkl−2g2C2(Ri)δij], (2.17)

where is the quadratic Casimir of the representation , and . Since dimensional coupling parameters such as masses and couplings of scalar field cubic terms do not influence the asymptotic properties of a theory on which we are interested here, it is sufficient to take into account only the dimensionless supersymmetric couplings such as and . So we neglect the existence of dimensional parameters, and assume furthermore that are real so that always are positive numbers. For our purposes, it is convenient to work with the square of the couplings and to arrange in such a way that they are covered by a single index :

 α=g24π , αi = g2i4π . (2.18)

The evolution equations of ’s in perturbation theory then take the form

 dαdt=β = −β(1)α2+⋯ ,dαidt=βi = −β(1)iαiα+∑j,kβ(1)i,jkαjαk+⋯ , (2.19)

where denotes the contributions from higher orders, and .

Given the set of the evolution equations (2.19), we investigate the asymptotic properties, as follows. First we define [12, 14, 37, 38, 16]

 ~αi≡αiα , i=1,⋯,n , (2.20)

and derive from Eq. (2.19)

 αd~αidα=−~αi+βiβ=⎛⎝−1+β(1)iβ(1)⎞⎠~αi−∑j,kβ(1)i,jkβ(1)~αj~αk+∑r=2(απ)r−1~β(r)i(~α) , (2.21)

where are power series of ’s and can be computed from the -th loop -functions. Next we search for fixed points of Eq. (2.20) at . To this end, we have to solve

 ⎛⎝−1+β(1)iβ(1)⎞⎠ρi−∑j,kβ(1)i,jkβ(1)ρjρk=0 , (2.22)

and assume that the fixed points have the form

 ρi=0 for i=1,⋯,n′ ; ρi >0 % for i=n′+1,⋯,n . (2.23)

We then regard with as small perturbations to the undisturbed system which is defined by setting with equal to zero. As we have seen, it is possible to verify at the one-loop level [12, 13, 14, 37] the existence of the unique power series solution

 ~αi=ρi+∑r=2ρ(r)iαr−1 , i=n′+1,⋯,n (2.24)

of the reduction equations (2.21) to all orders in the undisturbed system. These are RGI relations among couplings and keep formally perturbative renormalizability of the undisturbed system. So in the undisturbed system there is only one independent coupling, the primary coupling .

The small perturbations caused by nonvanishing with enter in such a way that the reduced couplings, i.e. with , become functions not only of but also of with . It turned out that, to investigate such partially reduced systems, it is most convenient to work with the partial differential equations

 {~β∂∂α+n′∑a=1~βa∂∂~αa} ~αi(α,~α)=~βi(α,~α) ,~βi(a) = βi(a)α2−βα2 ~αi(a),~β ≡ βα , (2.25)

which are equivalent to the reduction equations (2.21), where we let run from to and from to in order to avoid confusion. We then look for solutions of the form

 ~αi=ρi+∑r=2(απ)r−1f(r)i(~αa) , i=n′+1,⋯,n , (2.26)

where are supposed to be power series of . This particular type of solution can be motivated by requiring that in the limit of vanishing perturbations we obtain the undisturbed solutions (2.24) [27, 39]. Again it is possible to obtain the sufficient conditions for the uniqueness of in terms of the lowest order coefficients.

### 2.3 Reduction of Dimension-1 and -2 Parameters

The reduction of couplings was originally formulated for massless theories on the basis of the Callan-Symanzik equation [12, 13]. The extension to theories with massive parameters is not straightforward if one wants to keep the generality and the rigor on the same level as for the massless case; one has to fulfill a set of requirements coming from the renormalization group equations, the Callan-Symanzik equations, etc. along with the normalization conditions imposed on irreducible Green’s functions [40]. There has been a lot of progress in this direction starting from ref. [41], as it is already mentioned in the Introduction, where it was assumed that a mass-independent renormalization scheme could be employed so that all the RG functions have only trivial dependencies on dimensional parameters and then the mass parameters were introduced similarly to couplings (i.e. as a power series in the couplings). This choice was justified later in [42, 43] where the scheme independence of the reduction principle has been proven generally, i.e it was shown that apart from dimensionless couplings, pole masses and gauge parameters, the model may also involve coupling parameters carrying a dimension and masses. Therefore here, to simplify the analysis, we follow Ref. [41] and make use also of a mass-independent renormalization scheme.

We start by considering a renormalizable theory which contain a set of dimension-zero couplings, , a set of parameters with mass-dimension one, , and a set of parameters with mass-dimension two, . The renormalized irreducible vertex function satisfies the RG equation

 DΓ[Φ′s;^g0,^g1,...,^gN;^h1,...,^hL;^m21,...,^m2M;μ]=0 , (2.27)

where

 D=μ∂∂μ+N∑i=0βi∂∂^gi+L∑a=1γha∂∂^ha+M∑α=1γm2α∂∂^m2α+∑JΦIγϕIJδδΦJ , (2.28)

where is the energy scale, while are the -functions of the various dimensionless couplings , are the various matter fields and , and are the mass, trilinear coupling and wave function anomalous dimensions, respectively (where enumerates the matter fields). In a mass independent renormalization scheme, the ’s are given by

 (2.29)

where , and are power series of the ’s (which are dimensionless) in perturbation theory.

We look for a reduced theory where

 g≡g0,ha≡^hafor 1≤a≤P,m2α≡^m2αfor 1≤α≤Q

are independent parameters and the reduction of the remaining parameters

 ^gi=^gi(g),(i=1,...,N),^ha=P∑b=1fba(g)hb,(a=P+1,...,L),^m2α=Q∑β=1eβα(g)m2β+P∑a,b=1kabα(g)hahb,(α=Q+1,...,M) (2.30)

is consistent with the RG equations (2.27,2.28). It turns out that the following relations should be satisfied

 βg∂^gi∂g=βi,(i=1,...,N),βg∂^ha∂g+P∑b=1γhb∂^ha∂hb=γha,(a=P+1,...,L),βg∂^m2α∂g+P∑a=1γha∂^m2α∂ha+Q∑β=1γm2β∂^m2α∂m2β=γm2α,(α=Q+1,...,M). (2.31)

Using Eqs. (2.29) and (2.30), the above relations reduce to

The above relations ensure that the irreducible vertex function of the reduced theory

 ΓR[Φ's;g;h1,...,hP;m21,...,m2Q;μ]≡Γ[Φ's;g,^g1(g)...,^gN(g);h1,...,hP,^hP+1(g,h),...,^hL(g,h);m21,...,m2Q,^m2Q+1(g,h,m2),...,^m2M(g,h,m2);μ] (2.33)

has the same renormalization group flow as the original one.

The assumption that the reduced theory is perturbatively renormalizable means that the functions , , and , defined in Eq. (2.30), should be expressed as a power series in the primary coupling :

 ^gi=g∞∑n=0ρ(n)ign,fba=g∞∑n=0ηb(n)agneβα=∞∑n=0ξβ(n)αgn,kabα=∞∑n=0χab(n)αgn. (2.34)

The above expansion coefficients can be found by inserting these power series into Eqs. (2.31), (2.32) and requiring the equations to be satisfied at each order of . It should be noted that the existence of a unique power series solution is a non-trivial matter: It depends on the theory as well as on the choice of the set of independent parameters.

It should also be noted that in the case that there are no independent mass-dimension 1 parameters () the reduction of these terms take naturally the form

 ^ha=L∑b=1fba(g)M,

where is a mass-dimension 1 parameter which could be a gaugino mass that corresponds to the independent (gauge) coupling. Furthermore, if there are no independent mass-dimension 2 parameters (), the corresponding reduction takes the analogous form

 ^m2a=M∑b=1eba(g)M2.

### 2.4 Reduction of Couplings of Soft Breaking Terms in N=1 Suspersymmetric Theories

The method of reducing the dimensionless couplings was extended[41, 44], as we have discussed in the introduction, to the soft supersymmetry breaking (SSB) dimensionful parameters of supersymmetric theories. In addition it was found [45, 46] that RGI SSB scalar masses in Gauge-Yukawa unified models satisfy a universal sum rule.

Consider the superpotential given by

 W=12μijΦiΦj+16CijkΦiΦjΦk , (2.35)

along with the Lagrangian for SSB terms

 −LSSB=16hijkϕiϕjϕk+12bijϕiϕj+12(m2)jiϕ∗iϕj+12Mλλ+H.c., (2.36)

where the are the scalar parts of the chiral superfields , are the gauginos and their unified mass.

Let us recall (see Eqs.(2.15-2.17)) that the one-loop -function of the gauge coupling is given by [32, 33, 34, 35, 36]

 β(1)g=dgdt=g316π2[∑iT(Ri)−3C2(G)] , (2.37)

the -function of is given by

 βijkC=dCijkdt = Cijlγlk+Ciklγlj+Cjklγli , (2.38)

and, at one-loop level, the anomalous dimension of the chiral superfield is

 γ(1)ij=132π2[CiklCjkl−2g2C2(Ri)δij]. (2.39)

Then, the non-renormalization theorem [28, 29, 31] ensures there are no extra mass and cubic-interaction-term renormalizations, implying that the -functions of can be expressed as linear combinations of the anomalous dimensions .

Here we assume that the reduction equations admit power series solutions of the form

 Cijk=g∑n=0ρijk(n)g2n . (2.40)

In order to obtain higher-loop results instead of knowledge of explicit -functions, which anyway are known only up to two-loops, relations among -functions are required.

Judicious use of the spurion technique, [31, 47, 48, 49, 50] leads to the following all-loop relations among SSB -functions (in an obvious notation), [51, 111, 55, 54, 53, 56, 57]

 βM =2O(βgg) , (2.41) βijkh =γilhljk+γjlhilk+γklhijl −2(γ1)ilCljk−2(γ1)jlCilk−2(γ1)klCijl , (2.42) (βm2)ij =[Δ+X∂∂g]γij , (2.43)

where

 O =(Mg2∂∂g2−hlmn∂∂Clmn) , (2.44) Δ =2OO∗+2|M|2g2∂∂g2+~Clmn∂∂Clmn+~Clmn∂∂Clmn , (2.45) (γ1)ij =Oγij, (2.46) ~Cijk =(m2)ilCljk+(m2)jlCilk+(m2)klCijl . (2.47)

The assumption, following [55], that the relation among couplings

 hijk=−M(Cijk)′≡−MdCijk(g)dlng , (2.48)

is RGI and furthermore, the use of the all-loop gauge -function of Novikov et al. [58, 59, 60] given by

 βNSVZg=g316π2[∑lT(Rl)(1−γl/2)−3C2(G)1−g2C2(G)/8π2] , (2.49)

lead to the all-loop RGI sum rule [61] (assuming ),

 m2i+m2j+m2k=|M|2{ 11−g2C2(G)/(8π2)dlnCijkdlng+12d2lnCijkd(lng)2 }+∑lm2lT(Rl)C2(G)−8π2/g2dlnCijkdlng . (2.50)

Surprisingly enough, the all-loop result of Eq.(2.50) coincides with the superstring result for the finite case in a certain class of orbifold models [62, 63, 46] if

 dlnCijkdlng=1 ,

as discussed in ref. [19].

Let us now see how the all-loop results on the SSB -functions, Eqs.(2.41)-(2.47), lead to all-loop RGI relations. We assume:
(a) the existence of a RGI surfaces on which , or equivalently that the expression

 dCijkdg=βijkCβg (2.51)

holds, i.e. reduction of couplings is possible, and
(b) the existence of a RGI surface on which

 hijk=−MdC(g)ijkdlng (2.52)

holds too in all-orders.
Then one can prove [64, 65], that the following relations are RGI to all-loops (note that in both (a) and (b) assumptions above we do not rely on specific solutions of these equations)

 M =M0 βgg, (2.53) hijk =−M0 βijkC, (2.54) bij =−M0 βijμ, (2.55) (m2)ij =12 |M0|2 μdγijdμ, (2.56)

where is an arbitrary reference mass scale to be specified shortly. The assumption that

 Ca∂∂Ca=C∗a∂∂C∗a