One-loop effects on MSSM parameter determination via chargino production at the LC

Abstract

At a future linear collider very precise measurements, typically with errors of <1 %, are expected to be achievable. Such an accuracy gives sensitivity to the quantum corrections, which therefore must be incorporated in theoretical calculations in order to determine the underlying new physics parameters from prospective linear collider measurements. In the context of the chargino-neutralino sector of the minimal supersymmetric standard model, this involves fitting one-loop predictions to prospective measurements of the cross sections, forward-backward asymmetries and of the accessible chargino and neutralino masses. Taking recent results from LHC SUSY and Higgs searches into account we consider three phenomenological scenarios, each with characteristic features. Our analysis shows how an accurate determination of the desired parameters is possible, providing in addition access to the stop masses and mixing angle.

Eur. Phys. J. C (2013) 73:2446DOI 10.1140/epjc/s10052-013-2446-2

Regular Article - Theoretical Physics

One-loop effects on MSSM parameter determination via chargino production at the LC

Aoife Bharucha1,a, Jan Kalinowski2,b, Gudrid Moortgat-Pick1,3,c, Krzysztof Rolbiecki3,4,d, Georg Weiglein3,e

1II. Institut fr Theoretische Physik, University of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany

2Faculty of Physics, University of Warsaw, 00681 Warsaw, Poland

3DESY, Deutsches Elektronen-Synchrotron, Notkestr. 85, 22607 Hamburg, Germany

4Instituto de Fsica Terica, IFT-UAM/CSIC, 28049, Madrid, Spain

Received: 21 March 2013 / Published online: 8 June 2013 Springer-Verlag Berlin Heidelberg and Societ Italiana di Fisica 2013

Abstract At a future linear collider very precise measurements, typically with errors of <1 %, are expected to be achievable. Such an accuracy gives sensitivity to the quantum corrections, which therefore must be incorporated in theoretical calculations in order to determine the underlying new physics parameters from prospective linear collider measurements. In the context of the charginoneutralino sector of the minimal supersymmetric standard model, this involves tting one-loop predictions to prospective measurements of the cross sections, forwardbackward asymmetries and of the accessible chargino and neutralino masses. Taking recent results from LHC SUSY and Higgs searches into account we consider three phenomenological scenarios, each with characteristic features. Our analysis shows how an accurate determination of the desired parameters is possible, providing in addition access to the stop masses and mixing angle.

1 Introduction

A linear collider (LC) [15] will be an ideal environment for high precision studies of physics beyond the standard model (BSM). A particularly well-motivated BSM theory is the minimal supersymmetric standard model (MSSM). This provides the lightest neutralino as a candidate to explain the evidence for dark matter in the universe [6, 7]. Further, naturalness arguments (see e.g. Ref. [8]) support light higgsino-like charginos and neutralinos, as also predicted by

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b e-mail: mailto:[email protected]

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d e-mail: mailto:[email protected]

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e e-mail: mailto:[email protected]

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GUT motivated SUSY models [9]. Due to the challenges involved in detecting electroweakinos at the LHC, current bounds coming from the ATLAS and CMS exclude only small regions of parameter space, see e.g. Refs. [10, 11]. The charginos and neutralinos could therefore be within reach of a rst stage linear collider.

One approach to determine the fundamental MSSM parameters is to consider constrained models such as the constrained minimal supersymmetric standard model (CMSSM), and perform a global t of this reduced set of parameters to all relevant experimental results available, see e.g. Ref. [12]. Here on the other hand, in order to precisely determine the nature of the underlying SUSY model, we wish to determine the fundamental parameters in the most model-independent way possible. The determination of the U(1) parameter M1, the SU(2) parameter M2, the higgsino parameter and tan , the ratio of the vacuum expectation values of the two neutral Higgs doublet elds, at the percent level via chargino and neutralino pair-production has been shown to be possible at LO (see Ref. [13] and references therein). Due to the expected high precision of mass and coupling measurements at the LC, as well as the fact that one-loop effects in the MSSM may be sizeable, higher order effects have to be considered. Taking these corrections into account additional MSSM parameters become relevant, such as the masses of the stops and sleptons, which are also so far weakly constrained by the LHC.

In this paper we show how it would be possible to determine the fundamental parameters of the chargino and neutralino sector at the LC, including the complications arising due to higher order effects. Specically, we calculate the next-to-leading order (NLO) corrections to the cross-section ( ) and forwardbackward asymmetry (AFB) for chargino production, and also to the chargino and neutralino masses. A number of next-to-leading order (NLO) calculations of

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chargino and neutralino pair production at the LC can be found in the literature [1418]. We perform our calculations in the on-shell (OS) scheme such that, as far as possible, the mass parameters can be interpreted as the physical masses.Recent work on the OS renormalization of the chargino neutralino sector can be found in Refs. [1824].

By tting loop corrected predictions to these experimental results we show that it is possible to extract the fundamental parameters of the MSSM Lagrangian. However, due to the greater number of parameters, performing the t is more involved than for the LO analysis. Choosing three potential MSSM scenarios, we assess the impact of the loop corrections and the feasibility of such an extraction in each.We further investigate the impact of obtaining masses of the charginos and neutralinos from threshold scans rather than the continuum (see Ref. [1]) on the resulting accuracy of the parameters obtained from the t.

The paper is organized as follows. In Sect. 2 we introduce the process studied and dene necessary notation. We then provide details of the calculation of the loop corrections in Sect. 3, including details of the renormalization scheme used. In Sect. 4 we further discuss the method employed in order to t to the MSSM parameters, dene the scenarios considered, and present our results. Finally in Sect. 5 we discuss the implications of the results of the ts.

2 Process studied and tree-level relations

In this paper we study the determination of the fundamental parameters in the charginoneutralino sector of the MSSM, via chargino production at a LC. The charginos,

, and

( B, W, H1, H2) basis is given by

Y =

M1 0 MZcsW MZssW

0 M2 MZccW MZscW MZcsW MZccW 0

MZssW MZscW 0

,

(3)

where sW (cW ) is the sin(cos) of the weak mixing angle W . Since Y is complex symmetric, its diagonalization requires only one unitary matrix N, via M

0 = NY N.

As described in detail in Sect. 4, the parameter determination relies on the measurement of the masses of the charginos and neutralinos, the polarized cross-section for the pair production of charginos,

1,

1 , (4)

and the forwardbackward asymmetry dened by,

AFB =

(cos > 0) (cos < 0)

(cos > 0) + (cos < 0)

e+e +1

, (5)

for the unpolarized cross-section, where is the angle of the momentum of the chargino

1 with respect to the momentum of the incoming electron e.

Neglecting the electronHiggs couplings, this process occurs at leading order via three diagrams, as seen in Fig. 1. The transition matrix element can be written as [26]

M

e+e +i j

=

0, are the mass eigenstates of the gauginos and higgsinos, as seen from the relevant part of the MSSM Lagrangian [25],

L

neutralinos,

es Q v e+ Pu e

j

Pv

i /

pij PL UXV ij PR V XUT ij

j

+i , (6)

where Q denotes the bilinear charges, = L, R refers to

the chirality of the e+e current and = L, R to that of

the

+i

j current. The summation over and is implied. The bilinear charges are comprised of the propagators and couplings

QLL = CL+i

j DZGLCL+i

jZ,

=

+

1

2

0j,(1)

where PL/R = 1/2(1 5). The mass matrix for the

charginos is given by

X =

M2 2MW s2MW c

0i /pij PL NY N ij PR NY

NT ij

QRL = CL+i

j DZGRCL+i

jZ,

, (2)

where s/c sin / cos , and MW is the mass of the W

boson. This matrix is diagonalized via the bi-unitary transformation M

+ = UXV , where U and V are complex

unitary matrices. The mass matrix for the neutralinos in the

QLR = CR+i

j + DZGL CR

+i

jZ

(7)

i2e D CR

ee+

i CR

ee+

j,

QRR = CR+i

+ j + DZGR CR

+i

jZ ,

Eur. Phys. J. C (2013) 73:2446 Page 3 of 11

Fig. 1 Tree-level diagrams for the production of charginos

+1

and

1 at the LC

for which the required MSSM couplings for the

+i

j ,

+i jZ and ee +i vertices are given by

CL/R+i

j = ieij ,

CL+i

,

jZ =

ie cW sW

s2W ij Uj1Ui1

1

2Uj2Ui2

CR+i

jZ = CL+i

jZ U V ,

CRee+

i =

iesW Vi1,

and GL, GR, DZ, and D are dened via

GL =

s2W 12sW cW , GR =

sW cW ,

(9)

In the equations above, e denotes the electric charge, me and MZ are the masses of the electron and Z boson. DZ and D

refer to the propagators of the Z boson and sneutrino (of mass m), in terms of the Mandelstam variables s and t.

One can therefore express the transition matrix element in terms of M2, and tan , in addition to the known SM

parameters. However, the expected accuracy of the measurements at the linear collider is such that one-loop corrections become relevant, and we shall see in the following section how the higher order expressions depend on many additional MSSM parameters.

(8) 3 NLO contributions and renormalization

We have calculated the full one-loop corrections to the forwardbackward asymmetry for process e+e +1

1,

DZ =

ss M2Z

, D =

s

t m2

within the complex MSSM; the corresponding corrections to the cross section were calculated in Ref. [18]. Examples for the contributing self-energy, vertex and box diagrams are shown in Fig. 2. As in Ref. [18], for the calculation we have used the program FeynArts [2731], which allowed an automated generation of the Feynman diagrams and amplitudes. Together with the packages FormCalc [3234] and LoopTools [32] we derived the nal matrix elements and loop integrals. We assume a unit CKM matrix. We regularize using dimensional reduction [3537], which ensures that SUSY is preserved, via the implementation described in Refs. [32, 38].

A number of one-loop calculations in the gaugino higgsino sector can be found in the literature, mainly in the

.

Fig. 2 Examples of one-loop self-energy (upper), vertex (middle) and box (lower) diagrams for the production of charginos

+1 and

1 at the LC (where m, g denote the generation)

Page 4 of 11 Eur. Phys. J. C (2013) 73:2446

CP-conserving MSSM [14, 3947], but some of which apply a renormalization scheme that is also applicable for complex parameters [14, 45]. CP-odd observables have also been calculated at the one-loop level, for instance in Refs. [4850], but no dedicated renormalization scheme was required in these cases as the observables studied were UV-nite. Since we intend to extend the current study to the case of complex parameters, here we follow the approach of Refs. [18, 20] closely, where a dedicated on-shell renormalization scheme for the chargino and neutralino sector of the MSSM with complex parameters was developed. In the following we will therefore only discuss the parameter renormalization of the chargino and neutralino sector, relevant for the denitions of the parameters at loop level, briey and for further details about the chargino eld renormalization and the renormalization of other sectors we refer the reader to Refs. [18, 20, 21, 24].

The mass matrix in the chargino sector, Eq. (2), is renormalized via

X X + X, (10) where X is dened by

X =

M2 2(MW s)2(MW c)

and dene the left and right handed vector and scalar coefcients of the renormalized self-energy analogously via

L/Rij(p2) and SL/SRij(p2), respectively.

As stated earlier, we consider the parameter renormalization as for the complex MSSM, such that our setup is easily adaptable for future extensions. In Ref. [18, 21], it was shown that in the CP violating case, the 1-loop corrections to the phases of M1 and , i.e. M1 and respectively1 are

UV nite. Therefore we take the approach that these phases can be left unrenormalized. We can then determine the necessary conditions to obtain the absolute values |M1|, |M2|

and ||, depending on which three physical masses are cho

sen to be on-shell. As we have two external charginos, and in order to easily extend our setup to the case of

+1

2 pro-

duction, we assume the NCC scheme with

01,

1 and

2

on-shell [18, 2022], such that i = 1 and i = 1 and 2. Note

that in choosing the scheme, it is desirable that the on-shell particles should contain signicant bino, wino, and higgsino components, in order that the M1, M2 and parameters are accessible [18, 2022]. For the above choice, these conditions are satised for all the scenarios dened in Sect. 4, in which the lightest neutralino always has a sizeable bino-like component. The parameters in question of the chargino mass matrix can then be renormalized via expressions given in Refs. [18, 21, 24], which we list here for completeness,

|M1| =

, (11)

containing the renormalization constants M2 and , as well as renormalization constants (RCs) from other sectors, c, s (which can be expressed in terms of tan ), and MW , dened in Ref. [18]. The neutralino mass matrix, Eq. (3), is similarly renormalized via

Y Y + Y, (12) where Y is dened analogously to X in Eq. (11) and contains the additional RC M1, cf. Eq. (3).

Following e.g. Ref. [20], M1, M2, and are determined by choosing three out of the total six physical masses of the charginos and neutralinos to be on-shell, i.e. the tree-level masses, m

i , coincide with the one-loop renormalized masses, M

i = m i + m i ,

m

i

m

1 Re(eiM1 N2i1)F

2 Re eiNi3Ni4 Re(Uj1Vj1)

+Re N2i2 Re eiUj2Vj2 Ck

+ Re(Uj1Vj1) Re eiUk2Vk2

Re eiUj2Vj2 Re(Uk1Vk1) Ni

Re N2i2 Re eiUk2Vk2

+2 Re eiNi3Ni4 Re(Uk1Vk1) Cj , (15)

|M2| =

1F Re eiUj2Vj2 Ck

Re eiUk2Vk2 Cj , (16)

|| =

2 Re

Lii m2

i

i

+

Rii m2

i

1F Re(Uj1Vj1)Ck Re(Uk1Vk1)Cj . (17)

F , Ci, and Ni are dened by

F = 2 Re(Uk1Vk1) Re eiUj2Vj2

Re(Uj1Vj1) Re eiUk2Vk2 , (18)

1We adopt the convention that M2 is real.

1

2Re

SLii m2

i

+

SRii m2

i

= 0. (13)

We dene the coefcients L/Rij(p2) and SL/SRij(p2) of the self energy via

ij p2 = /

pPLLij p2 + /

pPRRij p2

+PLSLij p2 + PRSRij p2 , (14)

Eur. Phys. J. C (2013) 73:2446 Page 5 of 11

Ci = Re m

+i L

,ii

m2+i

+R,ii m2+i + SL,ii m2+i + SR,ii m2 +i

j=1,2

k=1,2

2Xjk Re(Uij Vik), (19)

where the analogous right-handed parts are obtained by the replacement L R, and

CRee+

i = CRee+

i

Ze

sW sW +

1

2 Z

e + ZLe

ee+ 1ZR,1i + CRee+ 2ZR,2i . (22)

Note that the renormalization constants of the SM elds,i.e. ZV V (V = , Z) and ZLe for the vector bosons and

electron, and parameters, i.e. Ze and cW (sW ) for the electric charge and cos(sin) of the weak mixing angle respectively, can be found in Ref. [18]. The renormalization for the chargino elds is performed in the most general manner, making use of separate RCs for the incoming and outgoing elds, i.e. coefcients ZL/R,ij and ZL/R,ij, respectively, for

left- and right-handed charginos as given in Ref. [18]. Finally, the counterterm for the sneutrino self energy takes the form

Ci

j = iij

+

1

2 CR

Ni = Re m

0i L0,ii m2

0i

+R0,ii m2

0i + SL0,ii m20i + SR0,ii m2 0i

4Yjk Re(NijNik), (20)

and the subscripts and 0 identify the coefcients of the

chargino and neutralino self-energy, respectively.2

Finite results for the process of interest at one-loop are obtained by adding the counterterm diagrams shown in Fig. 3. Although FeynArts generates these diagrams, expressions for the counterterms which renormalize the couplings dened at tree-level in Eq. (6), calculated in Ref. [18], are required as input, and therefore, again for completeness, we provide expressions for these explicitly. For the

+i

j,

j=1,2

k=3,4

2 Z

1 i + Zi p2 m2i

m2i

2 Z

i + Zi

, (23)

for

i = e,

,

Z

+i

j and e

e

+i vertices, these can be expressed as fol-

lows:

CL+i

j = CL+i

j

Ze +

Z 2

+CL+i jZ

ZZ

2 +

ie

2 ZL,ij + ZL,ij ,

CL+i

jZ = CL+i

jZ

Ze

cW cW

sW sW +

, where the sneutrino eld and mass RCs, Zi and mi , are also dened following Ref. [18].

Initial and nal state soft radiation must also be included to obtain an infra-red nite result as the incoming and outgoing particles are charged, and this is done as described in detail in Ref. [18], using the phase-space slicing method to dene the singular soft and collinear contributions in the regions E < E and < , respectively. In the soft and collinear limit, the results are regularized using electron and photon masses, respectively, and factorized into analytically integrable expressions proportional to the tree-level cross-section tree(e+e +1

ZZZ 2

(21)

+CL+i j

Z Z

2 2ie

sW cW ij

+

1

2

n=1,2

CL+i

nZZL,nj + CL+n

1). However, the result is cut-off dependent (i.e. on E and ), and removing this dependence requires a calculation of the cross section for the three body nal state, excluding the soft and collinear regions, which we perform using FeynArts and FormCalc. We further require that soft photon radiation is included in the

jZ ZL,in ,

2Here Ni should not be confused with the neutralino mass matrix Nij .

Fig. 3 Counterterm diagrams in the MSSM for the production of charginos

+1 and

1 at the LC

Page 6 of 11 Eur. Phys. J. C (2013) 73:2446

cross-section obtained from FormCalc. Finally we obtain a complete IR nite and cut-off independent result by adding the collinear contribution, which is calculated following the procedure outlined in Ref. [14].

4 Fit strategy and numerical results

4.1 Obtaining MSSM parameters from the t

With the loop corrections calculated as in Sect. 3, we can determine the fundamental parameters of the MSSM at NLO.From now on, we will restrict our study to the case of real parameters. In the chargino and neutralino sectors there are four real parameters, see Sect. 2, which we t to,

M1, M2, , tan . (24)

We additionally t to the sneutrino mass, as this enters at tree level and will therefore signicantly affect cross sections and forwardbackward asymmetries. However, in those scenarios where the sneutrino would already have been ob-served at the LC, its mass is assumed to be known. At the loop level, a large number of MSSM parameters will contribute. Depending on the scenario, only limited knowledge about some of these may be available. In particular LHC data may only provide limited information about the parameters of the stop sector, and direct production at the LC might not be possible. However, our analysis also offers good sensitivity to these parameters at the LC, as stops could signicantly contribute to chargino/neutralino observables at NLO.

At the LC, masses are expected to be measured with high precision using different methods [1]. In the following we adopt the experimental precision which could be achieved using the threshold scan method, however, we also investigate how the t precision would change if the masses were obtained from the continuum. In case of the cross sections, the experimental uncertainty is dominated by the statistical uncertainty [51],

=

S + B

S , (25)

where S and B are the signal and background contributions, respectively. In addition, we assume that the statistical uncertainties for the cross sections correspond to an integrated luminosity of L = 200 fb1 per polarization assuming the

efciency of = 15 %, which includes branching ratios

for semileptonic nal states and a selection efciency of 50 % [51]. Similarly, for the forwardbackward asymmetry we have

AstatFB =

1 A2FB

N , (26)

and the total number of events N = N+ + N [51].

In order to estimate the theoretical uncertainty on the masses, cross-sections and forwardbackward asymmetries, we consider the size of possible effects due to neglected higher order corrections as well as unknown MSSM parameters not included in the t. NNLO corrections are an important source of theoretical uncertainty, however, at present, corrections of this kind are only known for chargino and neutralino masses, for which the leading SUSY-QCD NNLO corrections were calculated in Ref. [46, 47]. Based on these results we estimate the uncertainty on the masses due to NNLO corrections to be of the order of 0.5 GeV,i.e. comparable to the expected experimental uncertainty. Note that the masses chosen on-shell are assigned no theoretical uncertainty. We further neglect the currently unknown uncertainties arising due to NNLO corrections to the cross-sections and forwardbackward asymmetries, assuming that in the future NNLO results for these could be incorporated. However, we do include the additional uncertainty arising due to any unknown MSSM parameters which are not included in the t, dominated by the contribution from the heavy pseudoscalar Higgs boson mass MA0 . We perform a multi-dimensional 2 t using Minuit [52, 53]

2 =

i

OiLi

Oi

2 , (27)

where the sum runs over the input observables Oi, depend

ing on the scenario, with their corresponding experimental uncertainties Oi.

4.2 Scenarios studied and motivation

We carry out the t for three scenarios, S1, S2 and S3, shown in Table 1, chosen in order to realistically assess the sensitivity to the desired parameters in a number of possible situations. Due to the current status of direct LHC searches [54, 55], in all scenarios we require heavy rst and second generation squarks and gluinos, while the stop sector is assumed to be relatively light.3 In S1 and S2 we take the masses of the stops, mt1 and mt2, to be 400 GeV and

800 GeV respectively, and the mixing angle to be cos t = 0.

The sbottom sector can then be obtained by dening m

b1 =

400 GeV and cos b = 0. On the other hand in S3, in or

der to obtain mh = 125 GeV, calculated using FeynHiggs

2.9.1 [5659], such that it is compatible with the recent Higgs results from the LHC [60, 61], the stop sector parameters are chosen to be mu3 = 450 GeV, mq3 = 1500 GeV and

At = 1850 GeV, ensuring large mixing between the stops,

3Note that in light of current LHC limits, the value M3 = 700 GeV

in S1 and S2 means that the gluino mass is rather low, however, our results are largely independent of this choice as M3 only enters our calculations via two loop corrections to mh.

Eur. Phys. J. C (2013) 73:2446 Page 7 of 11

Table 1 Parameters for scenarios 1/2 and 3 (S1/S2 and S3), in GeV with the exception of tan . Here M(l/q)i (M(e/u/d)i ) represents the left (right) handed mass parameter for a slepton/squark of generation i respectively (jointly referred to as Mfi ), and Af is the trilinear coupling for a sfermion f. See text for stop and sbottom parameter denitions

Scenario 1/2

M1 125 M2 250/2000 180 MA0 1000

M3 700 tan 10 Mq1,2 1500 Aq1,2 650

Ml/e1,2 1500 Ali 650 Ml3 800 Me3 400

Scenario 3

M1 106 M2 212 180 MA0 500

M3 1500 tan 12 Mq1,2 1500 Aq1,2 1850

Mli 180 Ali 1850

Me1,2 125 Me3 106

As a result of indirect limits (checked using micr-Omegas 2.4.1 [62, 63]), we have chosen mixed gaugino higgsino scenarios favored by the relic density measurements [64] and relatively high pseudoscalar Higgs masses in light of avor physics constraints, e.g. the branching ratio of B(Bs +) [65]. We also check that our scenar

ios agree with the experimental results for branching ratio

B(b s ) and the anomalous magnetic moment of the

muon (g 2)/2. Further, in S2 we study the sensitiv

ity of the t to large values of M2, such that the wino-like chargino and neutralino are heavy and decoupled from the bino- and higgsino-like particles. Finally, in S1/S2 we consider the case that the sleptons (with the exception of the light stau) and pseudoscalar Higgs bosons are at the TeV scale, and in S3 the case that they are relatively light. Therefore, while S1/S2 are not in keeping with the 125 GeV Higgs boson, they provide illustrative examples of the potential of the LC in scenarios complementary to S3.4

4.3 Results for scenario 1

In this scenario, only the charginos and three neutralinos will be accessible at the LC. As input for the t we therefore use:

the masses of the charginos (

1,

2) and three lightest

neutralinos (

such that cos t = 0.148. The sbottom sector is then obtained

by dening m

b1 = 450 GeV and cos b = 0. In Fig. 4, for

each of these scenarios, the mass corrections for neutralinos

02 and 03 are seen to be sensitive to the stop mixing angle.

03)

the light chargino production cross section (

+1

1)

01,

02,

with polarized beams at s = 350 and 500 GeV the forwardbackward asymmetry AFB at s = 350 and

500 GeV

the branching ratio B(b s ) calculated using mi

crOmegas 2.4.1 [62, 63].

The input variables, together with errors, namely the assumed experimental precision of the prospective LC measurements as well as the theoretical uncertainties, are listed in Table 2. It is interesting to observe the large NLO corrections to AFB, which even result in a change of sign. Note that

B(b s ) is included in order to increase sensitivity to the

third generation squark sector, and the estimated experimental precision of 0.3 104, taken from Ref. [66], is adopted.

We found that the impact of the muon anomalous magnetic moment is negligible in this scenario, mainly due to the heavy smuon sector. It should be possible to probe the supersymmetric QCD sector, with squark masses of 1.5 TeV

and the gluino mass of 700 GeV, at the LHC, such that

the theoretical uncertainty arising due to these parameters is small in comparison to that due to the unknown MA0 . We therefore include the small dependence on the A0 mass as an additional source of error, having explicitly checked that the impact of all other parameters is negligible. Note that there

4Note that in S1(S2) a Higgs mass of mh = 125 GeV can also be

achieved by adopting cos t = 0.4(0.5).

Fig. 4 One-loop corrections to the masses of neutralinos

02 (upper)

03 (lower) as a function of the stop mixing angle cos t , for scenarios S1 (blue), S2 (red, dashed) and S3 (green, dotted)

and

Page 8 of 11 Eur. Phys. J. C (2013) 73:2446

Table 2 Observables (masses in GeV, cross sections in fb) used as input for the t in S1, tree-level values and loop corrections are specied. Here the superscript on and AFB denotes s in GeV, and the subscript on denotes the beam polarization (P(e), P(e+)).

The central value of the theoretical prediction,

B(b s ) = 3.3 104 GeV,

calculated using state-of-the-art tools, is also included in the t.Errors in brackets are for masses obtained from the continuum.See text for details of error estimation

Observable Tree value Loop corr. Error exp. Error th.

m

1 149.6 0.1 (0.2) m

2 292.3 0.5 (2.0) m

01 106.9 0.2 m

02 164.0 2.0 0.5 (1.0) 0.5 m

03 188.6 1.5 0.5 (1.0) 0.5 (

+1

1)350(0.8,0.6) 2347.5 291.3 8.7 2.0 (

+1

1)350(0.8,0.6) 224.4 7.6 2.7 0.5 (

+1

1)500(0.8,0.6) 1450.6 24.4 8.7 2.0 (

+1

1)500(0.8,0.6) 154.8 12.7 2.0 0.5 A350FB (%) 2.2 6.8 0.8 0.1

A500FB (%) 2.6 5.3 1.0 0.1

are no theoretical errors for masses chosen to be on-shell.

Even at one loop, these masses are related to the fundamental parameters via the tree level relations, and are included in the t.

In S1 we t eight MSSM parameters: M1, M2, , tan , m, cos t, mt1, and mt2. The results of the t are given in

Table 3. We nd that the gaugino and higgsino mass parameters are determined with an accuracy better than 1 %, while tan is determined with an accuracy of 5 %. Excellent precision of 23 % is obtained for the mass of the otherwise unobservable sneutrino. Including NLO effects even allows us to constrain the parameters of the stop sector. Although the precision shown in Table 3 is rather limited, this could lead to an important hint concerning the masses of the stops, which, if not already seen, might allow for a well-targeted search at the LHC. This could be another example of LCLHC interplay [67].

Finally, in Table 3 we compare the t results using masses of the charginos and neutralinos obtained from threshold scans and from the continuum. For the latter, the accuracy at which the parameters can be determined is seen to de-

Table 3 Fit results (masses in GeV) for S1, for masses obtained from threshold scans (threshold t) and from the continuum (continuum t). Numbers in brackets denote 2 errors

Parameter Threshold t Continuum t

M1 125 0.3 (0.7) 125 0.6 (1.2)

M2 250 0.6 (1.3) 250 1.6 (3)

180 0.4 (0.8) 180 0.7 (1.3)

tan 10 0.5 (1) 10 1.3 (2.6)

m 1500 24 ([phone] (40) cos t 0 0.15 (+0.40.3)

mt1 400+180120 (at limitatlimit) mt2 800+300170 ([phone]+350220 (at limitatlimit)

teriorate, with errors on the fundamental parameters almost doubling, clearly indicating the need to measure chargino and neutralino masses via threshold scans.

4.4 Results for scenario 2

In this scenario, where the M2 parameter is set to 2 TeV, only the light chargino and three lightest neutralinos will be accessible at the LC. As input for the t we therefore use:

the masses of the lighter chargino (

1) and neutralinos

(

01,

02,

03)

with polarized beams at s = 400 and 500 GeV the forwardbackward asymmetry AFB at s = 400 and

500 GeV

the branching ratio B(b s ).

As we again nd that the muon anomalous magnetic moment has a negligible impact, it is not used in the t. The input variables, together with errors, namely the assumed experimental precision of the prospective LC measurements as well as the theoretical uncertainties, are listed in Table 4. While AFB is negligible at LO, the NLO corrections to it are again found to be large.

We again t eight MSSM parameters: M1, M2, , tan , m, cos t, mt1, and mt2. The impact of other parameters,

except the heavy Higgs boson mass, can be neglected. The results from the t are given in Table 5. The higgsino and bino mass parameters are well constrained in this scenario since bino-like neutralino and all higgsinos are directly accessible. Even though the winos are not directly accessible, the wino mass parameter M2 can be constrained with 10 %

accuracy at 1 level. An accuracy of 20 % is achieved for tan , signicantly worse than in S1. This can be understood by the fact that the mixing in S2 between chargino states is weak due to M2 being heavy, and the constraint on tan is

the light chargino production cross section (

+1

1)

Eur. Phys. J. C (2013) 73:2446 Page 9 of 11

Table 4 Observables (masses in GeV, cross sections in fb) used as an input for the t in S2, as in Table 4. The central value of the theoretical prediction,

B(b s ) = 3.3 104 GeV,

calculated using state-of-the-art tools, is also included in the t.See text for details of error estimation

Observable Tree value Loop corr. Error exp. Error th.

m

1 179.1 0.1 m

01 111.1 0.2 m

02 183.6 0.07 0.5 0.5 m

03 194.2 1.9 0.5 0.5 (

+1

1)400(0.8,0.6) 1214.9 344.7 6.0 0.1 (

+1

1)400(0.8,0.6) 250.6 32.4 2.7 0.1 (

+1

1)500(0.8,0.6) 1079.2 194.8 6.0 0.1 (

+1

1)500(0.8,0.6) 229.6 8.7 2.7 0.1 A400FB (%) 0.0 3.0 1.0 0.1

A500FB (%) 0.0 5.0 1.0 0.1

Table 5 Fit results (in GeV with the exception of tan and cos t ) for S2, as in Table 3, where numbers in brackets denote 2 errors

Parameter Fit result

M1 125+0.90.6 (+2.11.2)

M2 2000 200 (+600400)

180 0.2 (+0.50.3) tan 10 2 (+54)

m unconstrained cos t 0+0.130.09 (+0.40.3)

mt1 400+25050 (+50080) mt2 800+300200 (+900400)

dependent on this mixing. No limits can be derived on the sneutrino mass, due to the Yukawa suppressed coupling of the higgsino-like

1 to the electron and sneutrino. We are, however, as shown in Table 5, still able to derive limits on the stop masses and mixing parameter.

4.5 Results for parameters in scenario 3

This nal scenario features the richest phenomenology of the studied benchmark scenarios. As input for the t we therefore use:

the masses of the charginos (

1,

2) and neutralinos

(

01,

02,

03)

the light chargino production cross section (

+1

1)

with polarized beams at s = 400 and 500 GeV the forwardbackward asymmetry AFB at s = 400 and

500 GeV

the Higgs boson mass, mh the branching ratio B(b s ) the anomalous muon magnetic moment.

Compared to the previous scenarios, these observables are supplemented by the Higgs boson mass, mh, calculated us-

ing FeynHiggs 2.9.1 [5659]. The estimated experimental precision at the LC for mh, taken from Ref. [1], is adopted. We further assume the future theoretical uncertainty on the Higgs boson mass to be 1 GeV [59]. As before, the remaining two observables, the branching ratio

B(b s ) and the anomalous muon magnetic moment are

calculated using micrOmegas 2.4.1 [62, 63], and a projected experimental error on the anomalous muon magnetic moment of 3.4 1010 is employed [68], which we assume

would dominate over the theoretical uncertainty. The input variables, together with errors, namely the assumed experimental precision of the prospective LC measurements and the theoretical uncertainties, are summarized in Table 6. Because the sneutrino is now directly accessible, we assume that its mass has been measured and it is therefore not included in the t. On the other hand, due to the stronger dependence of the NLO cross-section and forwardbackward asymmetry on MA0 , this is now used as an additional t parameter. We neglect the remaining theoretical uncertainty on the cross-sections, as it is found to be negligible in comparison to the experimental error.

This means that in scenario 3, we t to M1, M2, , tan , cos t, mt1, mt2 and MA

0 . The results of the t are collected in Table 7. The parameters of the electroweak gaugino higgsino sector are determined with high precision. Due to a signicant mixing in the stop sector, and the improvement in the t quality due to the inclusion of the Higgs mass, we nd that the t is now also sensitive to the mass of the heavy stop. The accuracy is better than 20 % for this particle even though it is far beyond the reach of the LC and also most likely of the LHC. In addition, in this scenario an upper limit on the mass of the heavy Higgs boson can be placed at 1000 GeV, at the 2 level. It is the particular sensitivity of the NLO corrections to MA0 which presents this unique opportunity to set such an upper bound.

Page 10 of 11 Eur. Phys. J. C (2013) 73:2446

Table 6 Observables (masses in GeV, cross sections in fb) used as an input for the t in S3, as in Table 2. The central values of the theoretical predictions

B(b s ) = 2.7 104,

(g 2)/2 = 2.4 109 and

mh = 125 GeV, calculated using

state-of-the-art tools, are also included in the t. See text for details of error estimation

Observable Tree value Loop corr. Error exp. Error th.

m

1 139.3 0.1 m

2 266.2 0.5 m

01 92.8 0.2 m

02 148.5 2.4 0.5 0.5 m

03 189.7 7.3 0.5 0.5 (

+1

1)400(0.8,0.6) 709.7 85.1 4.5 (

+1

1)400(0.8,0.6) 129.8 20.0 2.0 (

+1

1)500(0.8,0.6) 560.0 70.1 4.5 (

+1

1)500(0.8,0.6) 97.1 16.4 2.0 A400FB (%) 24.7 2.8 1.4 0.1

A500FB (%) 39.2 5.8 1.5 0.1

Table 7 Fit results (in GeV with the exception of tan and cos t ) for S3, including results for the masses of the heavier stop mass (mt2) and the

pseudoscalar Higgs boson (MA0 )

Parameter Fit result

M1 106 0.3 (0.5) M2 212 0.5 (1.0)

180 0.4 (0.9) tan 12 0.3 (0.7)

cos t 0.15+0.080.06 (+0.160.09) mt1 430+200130 (+300400)

mt2 1520+200300 (+300400) MA0 <650 (<1000)

5 Conclusions

The evidence for the Higgs boson and dark matter, when examined in the context of supersymmetry, suggests the possibility of a light and M1. We have extended previous analyses, which tted observables for chargino production at the LC to extract fundamental MSSM parameters, by incorporating NLO corrections. The loop corrections are calculated for all observables tted, namely the polarized cross-sections and forwardbackward asymmetry for chargino production as well as the

1,

2 and

01,

02,

03

masses, in an on-shell scheme which facilitates the extension to the complex case. We have tted these observables for three complementary scenarios. We found that on including NLO corrections, when M1, M2 and are light they can be determined to percent-level accuracy, and tan to <5 %. Further we showed that obtaining masses of the charginos and neutralinos from the continuum as opposed to threshold scans would result in the uncertainty on the fundamental parameters almost doubling, reinforcing the importance of threshold scans for mass measurements. As a heavy M2 is still a viable possibility, we also considered

M2 = 2000 GeV, and found that the sensitivity to M2 is ap

proximately 10 %. As the error on tan is dependent on the degree of mixing in the chargino sector, here it increases to 20 %. Note that the inclusion of B(b s ), as well

as the use of masses determined via threshold scanning, in the t was seen to improve the sensitivity to the stop sector. We nally considered a scenario compatible with the latest Higgs results. For this scenario we found that including B(b s ), (g 2)/2 and mh in the t, along with

the signicant mixing in the stop sector, helped to obtain an accuracy better than 20 % on the mass of the heavy stop, even though this particle is far beyond the reach of the LC and also most likely of the LHC. We also included MA0 in the t, and found that, due to the particular sensitivity of the

NLO corrections to MA0 , it would even be possible to place a 2 upper bound on this parameter of 1000 GeV. In summary, we have shown that incorporating NLO corrections is required for the precise determination of the fundamental parameters of the chargino and neutralino sector at the LC, and could further provide sensitivity to the parameters describing particles which contribute via loop corrections.

Acknowledgements The authors gratefully acknowledge support of the DFG through the grant SFB 676, Particles, Strings, and the Early Universe, as well as the Helmholtz Alliance, Physics at the Terascale. This work was also partially supported by the Polish National Science Centre under research grant DEC-2011/01/M/ST2/02466 and the MICINN, Spain, under contract FPA2010-17747; Consolider-Ingenio CPAN CSD2007- 00042. KR thanks as well the Comunidad de Madrid through Proyecto HEPHACOS S2009/ESP-1473 and the European Commission under contract PITN-GA-2009-237920.

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