1 Introduction

LAPTH-009/16

LPSC16022

One-loop renormalisation of the NMSSM in SloopS : 1. the neutralino-chargino and sfermion sectors.

G. Bélanger111Email: belanger@lapth.cnrs.fr, V. Bizouard222Email: bizouard@lapth.cnrs.fr, F. Boudjema333Email: boudjema@lapth.cnrs.fr , G. Chalons444Email: chalons@lpsc.in2p3.fr,

LAPTh, Université Savoie Mont Blanc, CNRS, B.P.110, F-74941 Annecy-le-Vieux Cedex, France

[2mm] Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3, 53 Rue des Martyrs, 38026 Grenoble, France

[2mm]

We have completed the one-loop renormalisation of the Next-to-Minimal Supersymmetric Standard Model (NMSSM) allowing for and comparing between different renormalisation schemes. A special attention is paid to on-shell schemes. We study a variety of these schemes based on alternative choices of the physical input parameters. In this paper we present our approach to the renormalisation of the NMSSM and report on our results for the neutralino-chargino and sfermion sectors. We will borrow some results from our study of the Higgs sector whose full discussion is left for a separate publication. We have implemented the set up for all the sectors of the NMSSM within SloopS, a code for the automatic computation of one-loop corrections initially developed for the standard model and the MSSM. Among the many applications that allows the code, we present here the one-loop corrections to neutralino masses and to partial widths of neutralinos and charginos into final states with one gauge boson. One-loop electroweak and QCD corrections to the partial widths of third generation sfermions into a fermion and a chargino or a neutralino are also computed.

1 Introduction

Supersymmetry has long been considered as the most natural extension of the standard model that can address the hierarchy problem while providing a dark matter candidate. The discovery of a Higgs boson with a mass of 125 GeV whose properties are compatible with those of the Standard Model is a great achievement of the first Run of the LHC  [1, 2] and in some sense supports supersymmetry. Indeed, one can argue that a Higgs with a mass below 130 GeV is a prediction of the minimal supersymmetric standard model (MSSM). However, the fact that the observed Higgs mass is so close to the largest value that can be achieved in the MSSM, a value obtained by requiring a rather heavy supersymmetric spectrum, raises the issue of naturalness [3, 4]. Another issue with the MSSM is the problem [5]. Namely why , a supersymmetry preserving mass parameter as it appears in the superpotential through the operator mixing the two (superfield) Higgs doublets , should be, for a viable phenomenology, small i.e. of the order the electroweak scale, whereas one expects its value to be rather of order the cut-off scale. Both these problems are solved in the singlet extension of the MSSM, the Next-to-Minimal Supersymmetric Standard Model (NMSSM) where the parameter is generated dynamically through the vacuum expectation value of the scalar component of the additional singlet superfield. Moreover, as a bonus new terms in the superpotential are now present and give a contribution to the quartic Higgs couplings beside the gauge induced quartic coupling of the MSSM. These new contributions can lead to an increase of the tree-level mass of the lightest Higgs, thus more easily explaining the observed value of the Higgs mass [6, 7] without relying on very large corrections from the stop/top sector. Although fine-tuning issues remain [8, 9, 10, 11] they are not as severe as in the MSSM.

The Higgs discovery has thus led to a renewed interest in the NMSSM both at the theoretical and experimental level with new studies of specific signatures of the NMSSM Higgs sector  [12, 13] and/or of the neutralino and sfermion sectors [14, 15, 16] being pursued at the LHC. With the exciting possibility of discovering new particles at the second Run of the LHC, it becomes even more important for a correct interpretation of a future new particle signal to know precisely the particle spectrum as well as to make precise predictions for the relevant production and decay processes.

The importance of loop corrections to the Higgs mass in supersymmetry cannot be stressed enough. After all, it is because of radiative corrections that the MSSM has survived. The large radiative corrections from the top and stop sector are necessary to raise the Higgs mass beyond the bounds imposed by LEP and to bring it in the range compatible with the LHC. Higher-order corrections are also of relevance for supersymmetric particles, higher-order SUSY-QCD and electroweak corrections to the full SUSY spectrum have been computed for some time in the MSSM and are incorporated in several public codes [17, 18, 19, 20]. More recently higher-order corrections to Higgs and sparticle masses have been extended to the NMSSM  [21, 22]. Several public codes incorporate these corrections with different scopes and approximations, NMSSMTools [23, 24], SPheno [25, 26], SoftSUSY [27], NMSSMCalc [28] and FlexibleSUSY [29]. See also the recent work [30] on the corrections to the Higgs masses in the NMSSM. Moreover, higher-order corrections to decays have also been computed with some of these codes  [31, 32, 33, 34].

The code SloopS was developed for the MSSM with the objective of computing one-loop corrections for collider and dark matter observables in supersymmetry. The complete renormalisation of the model was performed in  [35, 36] and several renormalisation schemes were implemented. This code relies on an improved version of LanHEP [37, 38, 39] for the generation of Feynman rules and counter terms. The model file generated is then interfaced to FeynArts [40], FormCalC [41] and LoopTools for the automatic computation of one-loop processes [42]. One-loop corrections to masses, two-body decays and production cross sections at colliders were realized together with one-loop corrections for various dark matter annihilation  [43, 44, 45, 46, 47] and coannihilation processes [44]. SloopS has first been extended to include the NMSSM for one-loop processes not requiring renormalisation, such as the rates for gamma-ray lines relevant for Dark Matter indirect detection [48, 49] and Higgs decays to photons at the LHC [50, 51].

The present paper is the first in a serie that describes the implementation of the one-loop corrections for all sectors of the NMSSM. We will concentrate in this first paper on the details and issues having to do mainly with the neutralino/chargino sector since the addition of a singlet brings new features compared to the MSSM. We will be brief on the set-up of the renormalisation in the sfermion sector since the particle content is the same as within the MSSM, for this sector we therefore adhere to the approach given in [36] for the MSSM. The chargino-neutralino sector, in particular through the singlet superfield, is quite tied up with the Higgs sector. We will therefore have to borrow some elements from our study of the Higgs sector which we will go over in more detail in a follow-up paper [52]. For the neutralino/chargino sector, different renormalisation schemes are defined. In particular we have aimed at studying different on-shell, OS, schemes. The latter are based on choosing a minimal set of observables, namely masses of physical particles in the NMSSM spectrum to define the set of input parameters and necessary counterterms which will allow to get rid of all ultra-violet divergences in all calculated observables. Finding the minimal set of necessary counterterms requires solving a system of coupled equations. For the case of the NMSSM where mixing between different components occurs and where the same parameters appears in different sectors, the system of equations can be large. Moreover some choices of the minimal set (and therefore the relevant coupled equations) will lead to solutions that are extremely sensitive to a particular choice of a parameter which may, in some process, induce large radiative corrections. It is also possible, when a renormalisation scale has been chosen, to follow a simpler implementation of the counterterms, à la , where these counterterms are pure divergent terms. In some instances these can also lead to splitting a large system of coupled equations to a smaller and more manageable system of equations. The renormalisation of the ubiquitous which, at tree-level, represents the ratio of the vacuum expectation values, vev, of the 2 Higgs doublets is a case in point. We will also study mixed schemes where some parameters are while others are OS. The study of different renormalisation schemes is very important. First it can provide an estimate on the theoretical uncertainty due to the truncation to one-loop of the perturbative prediction and may also point at a bad choice of a renormalisation scheme. Second, for the NMSSM where a large part of the spectrum has not been seen it is difficult to predict which, from the point of view of an OS scheme, are the input parameters that one can use or which are the masses that will be discovered and measured (precisely) first. It is therefore wise to be open and prepare for different possibilities. In particular, our discussion will touch on some important issues regarding the relationship between the underlying parameters at the level of the Lagrangian and the physical parameters. This will bring up the issue of the reconstruction of the underlying parameters which is very much tied up to the renormalisation scheme and the differences in how we define the counterterms.

One of our goals has been to implement our approach in a code for the automatic generation of one-loop corrected observables and for an easy implementation of the counterterms. We have relied on SloopS. Therefore this work is also a natural extension of the work performed in  [35, 36] for the MSSM. Taking advantage of this automation we are able to provide and discuss a series of applications, pertaining to corrections to masses and various decays involving charginos, neutralinos and sfermions.

The paper is organized as follows. Section 2 contains a brief description of the NMSSM. Our general approach to the renormalisation of the NMSSM and its implementation in SloopS as well as how we handle infra-red divergences is explained in Section 3. The renormalisation of the neutralino and chargino sector is detailed in Section 4. We also give a rather extensive presentation of the different choices for the on-shell schemes and the problematic of the choice of the input parameters. The renormalisation of the sfermion sector follows the one of the MSSM. It is briefly reviewed in Section 5. We are then ready to apply the general approach and principles to specific observables. We start in Section 6 by defining a set of 5 benchmark points. In Section 7 we first start by giving results for different schemes for the one-loop corrected masses of the neutralinos before presenting results for the one loop corrected two-body decays of charginos and neutralinos into gauge bosons. This is performed for all 5 benchmark points and for different schemes. We then turn in Section 8 to the one-loop two-body decays of third generation sfermions into a fermion and chargino or neutralino. Section 9 contains our conclusions.

2 Description of the NMSSM

The NMSSM contains all the superfields of the MSSM as well as one additional gauge singlet superfield . Thus the Higgs sector consists of two SU(2) Higgs doublets superfields , and the singlet superfield,

(1)

The interaction Lagrangian can be decomposed in terms derived from the superpotential and from the soft SUSY breaking Lagrangian. In the -invariant NMSSM that we consider here, the superpotential can be split into two parts [5]. The first one depends only on the Higgs superfields , via two dimensionless couplings and ,

(2)

where and is the two dimensional Levi-Civita symbol with . The second part corresponds to the Yukawa couplings between Higgs and quarks or leptons superfields,

(3)

where

(4)

are respectively the superfields associated with the left-handed (LH) quark doublets, LH lepton doublets, right-handed (RH) quark and lepton singlets. The index i=1..3 indicates the generation. In what follows, this index will be omitted and a sum over the three generations will be implicit. No generation mixing is assumed in our study. These supersymmetric scalar partners will be denoted as and for the LH states and and for the partners of the RH states. In an abuse of language we will also refer to these partners as LH and RH. Let us keep in mind, at this point already, that parameters from the superpotential will find their way into the Lagrangian of the particle and the superparticles. For example, the same enter both the Higgs sector and the neutralino (higgsino) sector, thus offering ways to extract these parameters from different sectors. The soft SUSY breaking Lagrangian reads,

(5)
  • The first two lines belong to the Higgs sector with the first line representing the soft mass terms for the Higgs bosons while the second line, not present in the MSSM, represents the NMSSM trilinear Higgs couplings .

  • The third and fourth lines belong to the sfermion sector with a structure and a content exactly the same as in the MSSM with first the soft sfermion masses ( for the doublet squark/slepton, and for the RH singlets) followed by the MSSM-like tri-linear -terms for squarks and sleptons and . We have only written the terms for one generic generation since we are not considering inter-generation mixing.

  • The last line contains the soft mass terms for, respectively, the , and gauginos, also called bino, winos and gluinos.

We consider the NMSSM with CP conservation so that all parameters are taken to be real.

The neutral components of the Higgs doublets, and , contain both a CP even and a CP odd part. After expanding around their vacuum expectation values, their scalar neutral component reads

(6)

The vacuum expectation values, are chosen to be real and positive. As in the MSSM we define and such that the mass comes out to be .

The so-called higgsino mass parameter in the MSSM is now a derived parameter. is generated dynamically from the vev of the singlet field,

(7)

It is convenient to keep as an independent parameter, comparison with the MSSM will then be easier. With , we take and as independent parameter while is kept as a shorthand notation for in the same way as we use as a short-hand notation for .

The particle content of the NMSSM has extra particles in the neutralino and Higgs sector than what constitutes the MSSM. The physical scalar fields consist of 3 neutral CP-even Higgs bosons, , 2 CP-odd Higgs bosons, and a charged Higgs, . The fermionic component of is a neutralino called singlino. It mixes with the two higgsinos. With the two gauginos (U(1) and SU(2)) the NMSSM has five neutralinos.

To summarise, the parameters that will be relevant for the present paper which covers the neutralino, chargino and sfermion sector and which need to be renormalised (apart from the SM parameters) are

(8)

The first six of these parameters enter the chargino/neutralino sector. also enter the Higgs sector. In fact and are also present in the sfermion sector. The second group corresponds to the squark sector while the last group corresponds to the sleptons.

Other parameters not listed in Eq. 8 such as and enter only the Higgs sector. They will be studied in a separate publication detailing the treatment of the Higgs sector. Because of the supersymmetric nature of the model, in particular the origin of the parameter, the neutralino/chargino sector and the Higgs share parameters in common as was presented in Eq. 8. Since it may be advantageous to use inputs from the Higgs sector to extract one or all of the parameters in Eq. 8, their extraction and definition will be influenced by how all the parameters of the Higgs sector are extracted. Let us therefore list the 9 parameters of the Higgs sector:

(9)

Finally since we concentrate on electroweak corrections and do not consider gluino production or decay, the renormalisation of is not needed.

3 Full one-loop corrections: general approach

3.1 Renormalisation: our general approach

The renormalisation procedure follows the same approach as the one adopted in SloopS for the SM and the MSSM. Namely we aim primarily at an on-shell renormalisation of all parameters [35, 36]. Other realisations of on-shell renormalisation schemes for the chargino/neutralino sector have also been performed both in the MSSM [53, 31], the complex MSSM [54] and the NMSSM [31].

OS schemes mean that one uses as inputs physical observables which are therefore defined when particles taking part in these observables are physical and on their mass shell. Technically, the easiest and most obvious set of this type of observables are the masses of the particles themselves. In this case one only exploits the pole structure of two-point self-energy functions and require that the residue at the pole be unity. One difficulty occurs when we have mixing between particles sharing the same quantum numbers and therefore transitions from one to the other are possible. This will occur for Higgses, charginos, neutralinos and sfermions. The OS conditions mean also that when these physical particles are on their mass shell these (non-diagonal) transitions vanish. From another technical point of view this means that in the calculation of scattering amplitudes and decays we should not worry about corrections on the external legs, the wave functions will be automatically normalized. Recall that at tree-level one starts with the underlying parameters of a Lagrangian in terms of current/gauge fields where mixing between these fields is present. We then move to the physical basis where the physical fields are defined. This is achieved by some diagonalising matrices. At one-loop each underlying parameter is shifted by the addition of a counterterm. There is then a minimum set of conditions to restrict the form and the value of the counterterm. This shifting of parameters will at one-loop mix some particles. To perform a full definition of a physical particle at one-loop, in our approach, we introduce a matrix of wave functions with the conditions that when these transitions (containing one-loop plus counterterms) are evaluated OS, all transitions vanish. It is important to stress that it is unnecessary to introduce shifts in the diagonalising matrix that was used at tree-level.

Related to mixing also is the fact that one physical parameter, for example the mass of one neutralino in the NMSSM, depends on a large number of independent underlying parameters contained, in this case, in the mixing matrix. For instance, besides the SM parameters, 6 parameters (the first set in Eq. 8), contribute to the neutralino mass matrix. In this particular case one needs to solve a system of 6 coupled equations. This is the reason why the reconstruction of the parameters, or in other words the necessary counterterms, requires finding the solution to a (large) system of coupled equations. Finding the solutions can be extremely difficult and sometimes impossible from a partial or even total knowledge of the physical parameters. For example, the chargino masses can furnish but with a degeneracy. If the system of coupled equations can be split into different independent subsystems of equations, the extraction of the parameters will be much easier and their evaluations less subject to uncertainties in the sense of being less sensitive to small variations in the input parameters. Therefore, by combining different sectors one can work with smaller, independent blocks which are easier or more efficiently solved. For example, take the set in Eq. 8, originates from the Higgs sector and finds it way in all sectors of the NMSSM. As we will illustrate, it is much easier to get the counterterm for from the Higgs sector for which we could revert to a scheme. In this case this involves a one-to-one mapping between the required counterterm for and some simple evaluation of 2-point functions involving the Higgs. Reverting to the Higgs sector for this particular parameter is therefore technically much easier than trying to extract all the 6 parameters in the first set of Eq. 8 solely from the neutralino/chargino sector. Moreover by extracting from the Higgs sector we can choose a scheme where one further decomposes the remaining system of the coupled equations into 2 blocks: 2 equations from the chargino sector that will then furnish and the rest can be determined from the neutralino sector. Another advantage is that we have a much better handle on the extraction of , indeed as we stressed and as we will see explicitly the effect of on the neutralino/chargino is quite small. In a nutshell, a physical mass of a neutralino/chargino is essentially given by a soft mass with a small correction which is proportional to , such that , then . Although we will propose to use the Higgs sector for a definition of , we will in this first paper be very brief about the renormalisation of the Higgs, the full renormalisation of the Higgs sector will be detailed in a forthcoming publication [52]. In order to facilitate the comparison with other computations, we will also use a scheme in which the six parameters of the neutralino/chargino sector are taken as while on-shell conditions are used for the SM parameters.

Leaving aside the issue of (where it is defined from), the chargino/neutralino sector through the masses of the 7 particles it contains could furnish enough input to constrain the set of 6 parameters. There are various choices for the minimal set of inputs. We will propose a few. The most appropriate choice of input may depend on the observable considered. For example imagine a scenario where is much larger than all other masses. The scheme with the 3 lightest neutralinos will be quite insensitive to and its counterterm. As long as we concentrate on correcting observables that are not sensitive to the bino component, this should be fine but clearly within this scheme we should not expect to make a good prediction to any observable where the bino component plays a role. Similar issues occur with the singlino. The mention of the bino and singlino component, or any other component for that matter, raises the issue of how can one weigh any of these components from a knowledge of masses only. In general this is not possible. This is one of the shortcomings of the OS approach based solely on masses that we will present here. Schemes where one can use a particular decay of a neutralino which is sensitive to a particular coupling and hence component, in lieu of a mass, are possible but they are technically challenging (use of three-point function) and we will not implement this approach in this first publication.

To be complete, let us recall that the fermion and gauge sector of the SM is renormalised on-shell which means that the gauge boson masses are defined from the pole masses and that the electromagnetic coupling, , is defined in the Thomson limit. One should keep in mind that the scale of the latter, , is far smaller than the electroweak scale or the masses of the various supersymmetric particles we are dealing with. A running of , from to brings in about a correction.

If a complete and proper renormalisation procedure has been achieved, all observables should be ultra-violet finite. We always perform this stringent test and check for the absence of ultraviolet divergences. Such divergences arise in loop integrals and are encoded in the parameter defined in dimensional reduction as where , being the number of dimensions and is the Euler constant111In SloopS, we apply the constrained differential renormalisation scheme which has been shown to be equivalent to the SUSY conserving dimensional reduction scheme [41].. Since physical processes must be finite, we simply check that the numerical results, for one-loop corrections to masses or to decay processes, are independent of by varying the numerical value of from 0 to . We require that the numerical results agree up to five or seven digits (recall that SloopS uses double precision). Such tests have proven extremely useful in testing the code at each step of its implementation.
In schemes where at least one parameter is taken to be , a dependence on the renormalisation scale also appears. For all decay processes we have set this scale to the mass of the decaying particle and in calculating corrected masses this scale is set at the tree-level mass of the particle.

3.2 Infrared and real corrections

A second test concerns infrared finiteness. Infrared divergences arise in processes involving charged particles in external legs. The regularization of the divergence from the pure loop contribution is done in FormCalc by adding a fictitious mass to the photon (). After adding the real photon emission, the divergence associated with the soft photon emission will exactly cancel that of the pure loop contribution ()

(10)

where is a cut on the energy of the photon introduced to separate the soft and hard part when performing the phase space integral for the real emission,

(11)

To check the convergence we modify the value of . Note that the dependence should disappear when calculating the sum of the soft and hard part. This check is not automatized in SloopS, one has to calculate the sum of soft and hard emission for different values of until a plateau is reached.

In our calculations of the decays of squarks we have also considered QCD corrections. In all examples we have considered in the present paper the genuine non-abelian structure of QCD is not present. For all these cases we adopt the same procedure for taming the infrared divergences concerning gluons as the one we apply to infrared photons. For these applications we give the gluon a mass.

4 Renormalisation of the Chargino and Neutralino sector

4.1 Implementing our general considerations

Before entering into the details of the chargino/neutralino sector let us review our set-up for the renormalisation of the fermions as fit for the neutralinos and charginos. We will follow almost verbatim the implementation in the MSSM carried out in [35, 36]. We reproduce the different steps so the reader can follow exactly how we impose our conditions on the different counterterms.

For a Dirac fermionic field with a bare mass , the kinetic and mass terms of the Lagrangian can be written at tree level as :

(12)

When several fermions mix, the mass term simply becomes a matrix. can involve a large number of underlying parameters. The mass eigenstates are obtained after diagonalizing the mass matrix with two unitary matrices and ,

(13)

such that

(14)

We now shift by shifting the parameters of its elements and proceed to shift fields through wave function normalization,

(15)
(16)

and are the renormalised matrix and fields respectively and where are projection operators. For a Majorana fermion, as will be the case for the neutralinos, , only one counterterm matrix is required, likewise one unitary matrix is needed for the diagonalisation of the mass matrix.
Following [36] the renormalised two-point function describing the transition can be written in a compact notation,

(17)

including the one-loop self-energy and the counter-terms that represent the correction to the element , i.e. . We stress again that we are using the same diagonalising matrices as those used at tree level. This formula makes it clear that the mass and wave-function counterterms can be obtained separately from on-shell (OS) conditions.

Using one of the masses one can impose one of the OS conditions on the physical pole mass

(18)

means that the imaginary dispersive part of the loop function is discarded so as to maintain hermiticity at one-loop. is the tree-level mass. Using a mass as an input means that the tree-level mass that is used in Eq. 18 receives no correction at one-loop. This gives a direct constraint on the element which will be used as one condition to solve for the system of equations that define the full set of counterterms. When this full set of counterterms is solved equation Eq. 18 is used to calculate the pole mass for the particles that were not used as input, see [36] for the algebraic details. Considering the number of coupled equations, finiteness of the mass(es) derived at one-loop is a highly non trivial test and shows the robustness of our code. We always perform this finiteness test.

Wave-function renormalisation constants are derived by requiring that

  • the diagonal renormalised 2-point self-energies for transitions have residue of at the pole mass. This pole mass may get a one-loop correction. For our treatment at one-loop it is sufficient to impose the residue condition by taking the tree-level mass. This translates into

    (19)
  • To avoid any , transition we impose

    (20)

4.2 Specialising to the case of the charginos and neutralinos

The new fermions in the electroweak sector of the NMSSM are the two charginos, combination of charged winos and higgsinos as in the MSSM, and the five neutralinos, combination of bino, wino, neutral higgsinos and the singlino. In the basis

(21)

the mass matrix for the charginos reads,

(22)

while for the neutralinos in the basis

(23)

the mass matrix reads

(24)

The charginos and neutralinos eigenstates are obtained with the help of two unitary matrices and for charginos and one unitary matrix for neutralinos ( are particular manifestations of the matrices introduced in Eq. 13:

(25)

leading to the mass eigenstates

(26)

Following our program we proceed to shift the underlying parameters. This results in introducing counterterms to the mass matrices

(27)

with, in the chargino case,

(28)

and for the neutralino counterterms

(29)

with the constraint and ( is also defined from , a constraint which is implemented explicitly in Eq. 29 ).
As we have shown in the general presentation, the renormalised self energies lead to corrections, , to the tree-level masses. Imposing that some of these corrections vanish will put constraints on or else will give finite one-loop correction to the mass. Note again that since after the shifts on the parameters are made we still keep the same diagonalising matrices, we have for the corrections on the physical masses

(30)

4.3 Issues in the reconstruction of the counterterms of the chargino and neutralino sector

To fully define the chargino/neutralino sector one needs, besides the SM parameters and , to reconstruct and define the 6 parameters listed in Eq. 8 namely . This set defines the matrices , see Eq. 22,24. Three of these parameters are common to both the neutralino sector and the chargino sector, these are while are present only in the neutralino sector. Clearly the sole knowledge of two chargino masses is not sufficient to constrain and . However, if is provided from some other source then input from the two chargino masses can reconstruct . In this case three neutralino masses are sufficient to define for this one needs to solve a system of three equations.
In principle, the chargino/neutralino sector by providing 7 physical masses can furnish enough constraint to define the set of the 6 counterterms. However apart from assuming that one is in the lucky situation that as many as 6 (or 7) masses in the chargino/neutralino sector, have been measured, a cursory look at the tree-level mass matrices Eq. 22 and Eq. 24 already reveals the problems encountered in reconstructing the fundamental parameters of these mass matrices from the masses of the charginos and neutralinos only. First of all, we see that in the chargino sector, the contribution is quite small. In the neutralino sector the situation as concerns this parameter is not much better since either its contribution vanishes in the gaugeless limit ( or ), as in the chargino case or it is very much tangled up with the parameter . Moreover both represent mixing effects that may be difficult to extract from masses only. This is different from the extraction of for example where if an almost bino-like neutralino mass, , is used as input we would have an almost one-to-one mapping . This said one must not forget that the problematic are also present in the Higgs sector and in view of the observations we have made it is worth studying whether some input from the Higgs sector may not be a better way of extracting . However other parameters enter the Higgs sector but not the chargino/neutralino sector, see Eq. 9. Hence combining the Higgs and the chargino/neutralino sectors as many as 11 parameters should be reconstructed and we would therefore need as many inputs.
We would also like to point at an important conceptual issue having to do with the reconstruction of the underlying parameters from the sole knowledge of the physical masses, in particular from the chargino and neutralino sector. As is clear from the chargino mass matrix Eq. 22 there is a symmetry. Although the system can be solved by giving the two physical chargino masses it is impossible to unambiguously assign the value of or to the correct “position” in the mass matrix. In other words the higgsino/wino content is not unambiguously assigned. This would however be important to know when we want to solve for the other remaining parameters in the neutralino sector. Even without this caveat a similar problem occurs if one wants to unambiguously extract for example. A good reconstruction would require knowing not only the mass but the bino or singlino content of that mass. This is a challenging problem even in the (simpler) MSSM,  [55, 53, 56]. We will assume that some knowledge of the content is available from a measurement of some decay or cross section and from comparing the chargino and neutralino mass spectrum, see [36] for a discussion on this issue.
Setting aside these issues and remarks, let us return to the problem of defining and reconstructing the underlying parameters and counterterms. Since, for the chargino/neutralino system, we need to define and solve for 6 counterterms, we need a trade-off that supplies 6 inputs or conditions, . Different choices of the inputs correspond to a renormalisation scheme. We have also discussed that we may have to revert to a larger set that includes the Higgs sector, in this case solving for both the Higgs and chargino/neutralino we may have to extend the 6 needed inputs to as many as , see Eq. 9.
Therefore, in all generality, one needs to invert a system such as

(31)

contains other counterterms, such as gauge couplings, that are defined separately. Using the physical mass of one of the neutralinos/charginos as an input, see Eq. 18, is a possible choice in an OS scheme. Not all inputs need to be OS. In fact it is perfectly legitimate to adopt a fully scheme. In this particular case, the counterterms can be simply read off from an external code such as NMSSMTools or any code based on the solution of the Renormalisation Group Equation (RGE), at one-loop. In passing let us add that we have checked systematically that the part of our counterterms are the same, independently of how we extract them and we checked that they are consistent with the values extracted from NMSSMTools.

To make the system Eq. 31 manageable one should strive to reduce the rank of the matrix by breaking it into independent blocks, such that

(32)

We will compare a few schemes and implementations. In what we will call the mixed on-shell schemes, we work to reconstruct the 6 parameters of the chargino/neutralino sector, therefore . will be extracted from a condition on (from the Higgs sector), from the charginos and the rest of the three parameters solely from the neutralinos. In this case we have

(33)

As with all resolutions of a system of equations, the inversion of the matrix could introduce the inverse of a small determinant. We have already encountered such an example with and the division by the small in section 3.1. Another case concerns that can only be reconstructed precisely using the neutralino that is dominantly bino. This can easily be seen from the first term in eq. 30, . If the mass of the dominantly bino neutralino is not chosen as an input parameter, then the extraction of involves a division by a small number since is suppressed, hence can induce numerical instabilities. This is the reason we have brought up the issue of the content of the particle when its mass is used as input.

A second set of schemes, full OS-scheme, is a full where all inputs are masses from the chargino/neutralino sector. We have pointed at some of the shortcomings of this approach, lack of sensitivity to and to to some extent. To achieve a better determination of the parameters in particular the problematic , we get help from the Higgs sector but this time all parameters are defined OS. In this case among the inputs we will take some Higgs masses. This will be done at the expense of having a larger system, , the extra two parameters that come into play are

4.4 Mixed on-shell schemes

This set up is done along the decomposition where gets its source in the Higgs sector, implementing a condition for .

4.4.1 from the Higgs sector

The renormalisation of the Higgs sector is done within the same spirit as the one followed for the neutralino sector by the introduction of wave function renormalisation constants, details will be given in a separate paper. The condition calls for the wave function renormalisation constants of the Higgs doublets. It is an extension of the DCPR scheme[57, 58] used in the context of the MSSM to the NMSSM[59],

(34)

where and are the wave function renormalisation constants of the and doublets. The infinity symbol indicates that we take the divergent part of the expression. and are related to the wave function renormalisation constants of the physical CP-even eigenstates , and . The latter are obtained from the CP-even neutral elements of and through the diagonalising matrix

(35)

Explicitly,

(36)

with

(37)

where is the fully antisymmetric rank 3 tensor with and is the derivative of the self-energy of the Higgs (with respect to its external momentum), evaluated at its mass , this condition is such that the residue of the Higgs propagator is unity. The same requirement was imposed on the charginos and neutralinos.

In a scheme only the divergent part of the countertem is defined i.e, any finite term is set to . Nonetheless, the scheme and the one-loop result is still not fully defined unless one specifies the renormalisation scale . The latter is the remnant scale introduced by the regularization procedure, dimensional reduction. Varying can give some estimate on the theoretical uncertainty of the calculation due to the truncation at one-loop. In the numerical results obtained using a scheme, the default value of is fixed to be equal to the mass of the decaying particle or to the (tree-level) mass of the particle whose one-loop correction is calculated.

4.4.2 The charginos

Having solved for , the chargino system, provides the simplest set up for defining from the masses of both charginos as input. Exactly the same approach and the same expressions are found for the MSSM

(38)

The explicit solutions shown in Eq. 38 gives us the opportunity to go over the ambiguity on the true reconstruction of . In fact Eq. 38 corresponds to 4 solutions, since are given up to a sign and since we have a ambiguity. This issue was discussed at some length and some suggestions were given on how to lift the degeneracy [36]. By looking at the values of some decays (or cross sections) involving a chargino, for example, we can check that only one of the solutions is compatible with the value of the decay rate. This is a limitation on using only the value of the physical masses as input. Having chosen the correct we can now pass them to the neutralino sector 222Numerical problems may arise in the limit , see  [36] for a more thorough discussion..

4.4.3 Three neutralino masses as input

We are now left with determining , (or ) and using three neutralino masses, this is the . Out of the five possible neutralino masses, assuming they have all been measured, one must pick up 3 masses that give the best reconstruction of the remaining parameters. As we pointed out, technically we should avoid having . Obviously the best extraction of would, ideally, need the bino like neutralino, whereas (or ) is most directly tied up with the singlino component. A wino-like neutralino as a third input will not do since this is essentially sensitive to with only feeble mixing with the contribution. The third neutralino to use as input is necessarily a higgsino like neutralino, again this is evident since in the NMSSM is intimately related to , the higgsino parameter. One can also look at the mass matrix (Eq. 24) to see that enters only in the singlino - higgsino off-diagonal element. Therefore the subset to choose calls for and . We see again that a judicious choice calls for a knowledge of the identity of the particle apart from knowing the value of the corresponding mass exactly.
Having implemented the approach this way, one can calculate the one-loop corrections to two neutralinos, the remaining wino-like and the remaining higgsino-like neutralinos.

4.5 Full OS-schemes

4.5.1 The neutralino/chargino sector

Since all 6 parameters are necessary to describe the chargino/neutralino sector which provides physical masses one could entertain defining all these parameters from this sector. We have pointed out at the shortcomings of this extraction which has to do with the fact that the dependence on is very weak and that the dependence on is complicated. From the technical point of view the reconstruction is also involved as it requires inverting a system, . The best choice for builds up on the remarks we have just made in picking up the three most appropriate neutralinos in the previous paragraph. Based on those arguments the OS scheme uses the following set of inputs