Extracting Chargino/Neutralino Mass

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PM/98{26

hep-ph/9810214

EXTRACTING CHARGINO/NEUTRALINO MASS

PARAMETERS FROM PHYSICAL OBSERVABLES

G. MOULTAKA

Physique Mathematique et Theorique, UMR{CNRS,

Universite Montp ellier I I, F{34095 Montp ellier Cedex 5, France.

E-mail: [email protected]

Abstract

I rep ort on two pap ers, hep-ph/9806279 and hep-ph/9807336, where complemen-

tary strategies are prop osed for the determination of the chargino/neutralino sector

parameters, M ;M ; and tan , from the knowledge of some physical observables.

1 2

This determination and the o ccurrence of p ossible ambiguities are studied as far

as p ossible analytically within the context of the unconstrained MSSM, assuming

however no CP-violation.

Talk given at the International Conference on High Energy Physics,

Vancouver 1998

(to appear in the proceedings) 1

1 Intro duction

The gauge b osons and Higgs b osons sup erpartners have every chance to play, in the

minimal version of the sup ersymmetric standard mo del (MSSM), an imp ortant part in the

rst direct exp erimental evidence for sup ersymmetry, if the latter happ ens to b e linearly

realized in nature around the electroweak scale. This would go through the study of the

direct pro duction of the light states and their subsequent decays, eventually cascading

; ;

down to leptons (or jets) and missing energy [3] [4] [5].

The chargino/neutralino sector is an over-constrained system in the sense that only a

few basic parameters in the Lagrangian are needed to determine all the six physical masses

and the mixing angles of the various states. The latter determine the couplings to gauge

b osons, Higgs b osons and matter fermions, so that various phenomenological tests could

be in principle envisaged in the pro cess of exp erimental identi cation. Alternatively, one

might hop e that a partial exp erimental knowledge of this sector would be sucient to

allow a reasonably unequivo cal reconstruction of the full set of parameters; at stakes, on

one hand the determination of the magnitude of the fermion soft susy breaking parame-

ters, on the other, the existence of a heavy neutral stable particle, of prime imp ortance

to the cold dark matter issue [6]. Furthermore, the sensitivity to tan , the ratio of the

twovacuum exp ectation values of the Higgs elds, and to the sup ersymmetric parameter

, brings in a further correlation with the other sectors of the MSSM.

Hereafter we describ e two strategies: the rst deals with the extraction of M ; and

tan form the study of the lightest chargino pair pro duction and decayine e collisions

[1], the second with the extraction of M ;M and form the knowledge of any three ino

1 2

masses and tan [2]. We start by stating the common features to these complementary

approaches as well as their sp eci c assumptions. We then highlight the main ingredients

of each of them and illustrate some of their results. Finally we show in what sense they

eventually complement one another. [Obviously, the reader is referred to [1] and [2] for

more details and references. Still, we add some comments at various places of the ongoing

presentation, which di er slightly from, and hop efully complete, the latter references.]

The reconstruction of the basic parameters of the theory involves generically two steps

which can b e sketched as follows:

Exp erimental Observables

y (I )

Physical Parameters

(1.1)

y (II)

Lagrangian parameters

Each of these steps can su er from equivo cal reconstructions due to partial exp erimental

knowledge or to theoretical ambiguities. In the present rep ort we concentrate on the

theoretical asp ects of b oth steps. 2

2 CDDKZ and KM common features

The ino sector is considered in b oth [1] (referred to as CDDKZ) and [2] (KM) with the

following assumptions:

No reference to mo del-dep endent assumptions ab out physics at energies much higher

than the electroweak scale, like the GUT scale, and their p ossible implication on the

parameters of this sector. [Thus the study is mainly carried out in the unconstrained

MSSM, but any mo del-assumptions can b e easily overlaid.]

R-parity conservation;

CP-conservation in the ino sector; This assumption is here only for practical rea-

sons and should be eventually removed in future studies in order to cop e with the

p ossibility to deal with (complex) phases [7];

CDDKZ and KM cho ose M > 0. This is of course a mere convention due to the

partial phase freedom through rede nition of elds, the only physical signs b eing

the relative ones among M ;M and as one can easily see from the relevant terms

1 2

in the Lagrangian. (also tan is taken p ositive and the term convention is that

of ref.[[8 ]].)

Let us now recall brie y the basic ingredients of the ino mass matrices. The physical

charginos (resp. neutralinos) are mixtures of charged (resp. neutral) higgsino and gaugino

comp onents. The chargino mass matrix reads:

M 2m sin

2 W

M = (2.2)

2m cos

It has a sup ersymmetric contribution coming from the term in the sup erp otential, the

higgsino comp onent, a contribution from the soft susy breaking wino mass term, and o -

diagonal terms due to the electroweak symmetry breaking. Since M is not symmetric one

needs two indep endent unitary matrices for the diagonalization. This is but the re ection

of the fact that there are two indep endent mixings involving separately the two higgsino

SU (2) doublets. The eigenvalues are most easily obtained from the diagonalization of

M M giving the squares of the chargino masses:

C C

2 2 2 2

M = [M + +2m

2 W

1;2

(2.3)

2 2 2

(M + +2m ) 4(M m sin 2 ) ]

W W

On the other hand, the angles ; de ning the two indep endent left- and right-

L R

chiral mixings among the winos and higgsinos in the four comp onent Dirac representation

are given by 3

2 2 2

M 2m cos 2

cos 2 =

2 2

2(M m )M

2m (M cos + sin ) 2

sin 2 =

2 2

2(M m )M

(2.4)

2 2 2

M +2m cos 2

cos 2 =

2 2

2(M m )M

2 2m (M sin + cos )

sin 2 =

2 2

2(M m )M

where M is the lightest chargino mass given by eq.(2.3). This form of the mixing

angles is such that the eigenvalues of M are always p ositive de nite.

The neutralino mass matrix corresp onds to bilinear terms in the photino, zino and

neutral higgsino two-comp onent elds. It receives contributions from the term, the soft

mass terms of the gaugino SU (2) triplet (M ) and singlet (M ), while the mixing among

L 2 1

states is triggered by the electroweak symmetry breaking:

1 0

M 0 m s cos m s sin

1 Z w Z w

C B

0 M m c cos m c sin

C B

2 Z w Z w

C M = B

A @

m s cos m c cos 0

(2.5)

Z w Z w

m s sin m c sin 0

Z w Z w

In contrast with M , M is symmetric so that it can be diagonalized with one

C N

unitary matrix. On the other hand the eigenmasses are not p ositive de nite . Finally we

note that in general the diagonalization of M cannot be achieved through a similarity

transformation, unless all three parameters M ;M and are real. This will be a key

1 2

p oint in the algorithm we present for the reconstruction of the parameters in the neutralino

sector.

3 Sp eci c features

3.1 CDDKZ

The lightest chargino can b e pro duced in pairs in e e collisions, at LEPI I [4] or NLC

[5] energies, through and Z s-channel exchange as well as sneutrino t-channel exchange.

The pro duction cross section will thus dep end on the chargino mass m , the sneutrino

mass m and the mixing angles, eq.(2.4), which determine the couplings of the chargino

states to the Z and the sneutrino. The unp olarized total cross section is illustrated in

g.1 with three representative cases of higgsino, gaugino or mixed content of the lightest

For more details ab out the ino sector see for instance [8],[9] and references therein. 4 2 4 √ s=200 GeV 3 [pb] [pb] 1 2 tot tot σ σ 1

0 0 150 300 450 600 0 200 400 600 800 √ s [GeV] Sneutrino Mass

(a) (b)

Figure 1: Total cross section for the charginos pair pro duction for a representative set of

M ;, solid line gaugino case, dashed line higgsino case, dot-dashed line mixed case. In

(a) m = 200GeV . (taken from ref.[1])

chargino mass. The sharp rise near threshold should allow a precise determination of the

chargino mass. Also the sensitivity to the sneutrino mass with the typical destructive

interference in the gaugino and mixed cases necessitates the knowledge of this parameter.

Subsequently the chargino will decay directly to a pair of matter fermions (leptons or

quarks) and the (stable) lightest neutralino, through the exchange of a W b oson (charged

Higgs exchange is suppressed for light fermions) or scalar partners of leptons or quarks.

Of course the decay matrix elements will dep end on further parameters like the susy scalar

masses and couplings to the neutralino. However, CDDKZ prop ose that, lo oking at the

total pro duction cross section and some p olarization comp onents and spin-spin correla-

tions of the nal state charginos, one can de ne measurable combinations for which the

details of the chargino decay pro ducts cancel out. This allows to isolate to a large extent

the chargino system from the neutralino system and thus extract the mixing angles and

chargino mass from these observables (step (I) in eq.(1.1)). In fact, for step (I) to work

completely for the chargino system, one needs to know, b esides the sneutrino mass, also

the lightest neutralino mass, as will b ecome clear later on. Once the chargino mass m

and cos 2 ; cos 2 are known one can determine M ; and tan up to p ossible ambi-

L R 2

guities, [step (I I) of eq.(1.1).]

Before going further let us rst describ e in some detail the basic ingredients of step (I).

The presence of invisible neutralinos, in the nal state of the pro cess e e ! !

1 1

0 0

(f f )(f f ), makes it imp ossible to measure directly the chargino pro duction angle

1 2 3 4

1 1

in the lab oratory frame. From nowonwe will thus concentrate on observables where this

angle is integrated out. Integrating also over the invariant masses of the fermionic systems

(f f ) and (f f ) one can write the di erential cross section in the following form:

1 2 3 4

4 + 0 0

d (e e ! ! (f f )(f f ))

1 2 3 4

1 1

(3.6)

Br = Br ( ; ; ; )

0 + 0

! f f ! f f

1 2 3 4

124s

d cos d d cos d

where is the ne structure constant, the velo city of the chargino in the c.m. frame, 5 1.0

σ =0.37 pb P2/Q=−0.24 0.5 P2/Y=−3.66 R φ 0.0 cos2

−0.5

−1.0 −1.0 −0.5 0.0 0.5 1.0 φ

cos2 L

, Figure 2: Contours for the \measured values" of the total cross section (solid line),

and (dot-dashed line) for m =95GeV [m = 250GeV ]. (taken from ref.[1])

( ) denotes the p olar angle of the f f (f f ) system in the ( ) rest frame with

1 2 3 4

1 1

resp ect to the charginos ight direction in the lab oratory frame, and ( ) the corre-

sp onding azimuthal angle with resp ect to a canonical pro duction plane. ( ; ; ; )

is made out of combinations of helicity amplitudes which lead to an unp olarized term plus

fteen other contributions from p olarization comp onents and spin-spin correlations. We

repro duce here only those comp onents which are relevant to our discussion.

= +( ) cos P + cos cos Q

unpol

(3.7)

+ sin sin cos( + ) Y + :::

Actually, among the sixteen terms which contribute to only ten survive b ecause of

CP-invariance (when violation of CP from the Z-b oson width or radiative corrections is

neglected). Of these ten, three are redundant b eing CP eigenstates. Of the remaining

seven indep endent comp onents, only those which can be extracted from exp erimentally

measurable angular distributions are explicitly written in eq.(3.7). This means that the

others will b e integrated out through appropriate pro jections.

In eq.(3.7) corresp onds to the unp olarized cross section for the chargino pair

unp ol

pro duction and is given in terms of helicity amplitudes by

2 2

= d cos [jh ;++ij + jh ;+ij

unp ol

(3.8)

2 2

+jh ; +ij + jh ; ij ] 6

P is a p olarization comp onent coming separately from the (or ) system

2 2

P = d cos [jh ;++ij + jh ;+ij

(3.9)

2 2

jh ; +ij jh;ij ]

while Q and Y describ e the spin correlations between the two chargino systems and

have the following structure

2 2

Q = d cos [jh ;++ij jh;+ij

2 2

jh ; +ij + jh ; ij ]

(3.10)

Y = d cos Refh ; ih ;++i g

where is the initial state electron helicity. The strategy of CDDKZ is based on the

following two observations:

i) The three angular contributions, cos ; cos and sin sin cos ( + ) are fully

determined by the measurable parameters E; jP j (the energy and momentum of each

of the decay systems f f in the lab oratory frame), and the chargino mass m ;

i j

2 2

ii) The three quantities ; P =Q and P =Y lead to free observables, where (and

unpol

= )) measures the asymmetry b etween left- and right-chirality form factors in

the decay pro ducts of the chargino;

Here the kinematic con guration is similar to that of a lepton pair pro duction with

successive decays in light leptons or quarks plus missing energy. However in the present

context the invisible particle has a non negligible mass whose knowledge is necessary to

relate the energy of the f f system in the chargino rest frame to that in the lab oratory

i j

frame. Thus the neutralino mass is actually necessary in the reconstruction of the angular

contributions in i). The crucial p ointin ii) is that the dep endence on the nal state decay

fermions through the asymmetry in the left- and right- chiral structure cancels out. Thus

2 2

; P =Q and P =Y allow to study the chargino sector indep endently of the details

unpol

of the decay pro ducts. In the same time, their extraction from the exp erimental data,

via convolution with appropriate moments, requires the measurement of the energies and

momenta of the two f f systems, the chargino mass (ex. via threshold e ects, see g.1),

i j

as well as the neutralino mass (ex. from the energy distribution of the nal particles).

2 2

Once extracted, one can combine ; P =Q and P =Y which dep end on the c.m. en-

unpol

ergy s, the sneutrino mass m , and cos 2 ; cos 2 to determine the latter cosines. An

~ L R

illustration is given in g. 2 of a unique consistent solution corresp onding to the inter-

section p oint of the contour plots at (cos 2 = 0:8; cos 2 = 0:5). The requirement

L R

that the three curves should meet in one p oint o ers clearly a very stringent consistency

check of the mo del. On the other hand, while is a quadratic p olynomial in cos 2 ,

unpol L

cos 2 , the two other observables are quartic in these variables. A p otential ambiguityin

the determination of (cos 2 ; cos 2 ) will b e, however, very unlikely, esp ecially if m is

L R ~

xed indep endently and the c.m. energy varied. We do not dwell here on further asp ects 7

of step (I) which can b e found in [1].

Wenow go to step (I I) of eq.(1) and describ e brie y how to determine M ; and tan .

Starting from eq.(2.4) and m in eq.(2.3), CDDKZ give closed expressions for M ;;tan

in terms of the quantities p = cot( );q = cot( + ). They considered all p ossible

R L R L

cases and concluded to the existence of at most a two-fold ambiguity in the determination

of the Lagrangian parameters, traceable to a sign ambiguityin sin 2 , (see [1] for de-

L;R

tails). Here we only sketch an equivalent discussion which shows that, when it o ccurs, this

two-fold ambiguityis always asso ciated with opp osite sign solutions. This can b e most

easily seen as follows: from cos 2 ; cos 2 in eq.(2.4) one easily determines M uniquely

L R 2

(remember that M is p ositive in our convention) and with a global sign ambiguity,as

+ +

functions of m ; tan ; cos 2 and cos 2 . Plugging those functions in the m part

L R

1 1

of eq.(2.3) one gets, thanks to some cancellations, a simple quadratic equation in tan .

The two solutions encompass automatically the sign of sin 2 sin 2 . Furthermore, each

L R

of them is consistent only with (at most) one sign repro ducing the correct m , since

eq.(2.3) is not invariant under ! for a given M ; tan . Asanumerical illustration,

2 2

taking the input of g.2, =0:37pb, P =Q = 0:24; P =Y = 3:66, CDDKZ nd the

tot

following two-fold solution

(A) [1:06; 83GeV ; 59GeV ]

[tan ; M ;]= (3.11)

(B ) [3:33; 248GeV ; 123GeV ]

We see that the two-fold ambiguity comes with a sign change for in accord with the

general pattern just describ ed. To eliminate this discrete ambiguity one would clearly need

an indep endent information ab out any of the three parameters. Finally, the reconstruction

obviously dep ends on the quality of the exp erimental accuracy with which the needed

observables can b e determined. This requires among other things:

- running at di erent c.m. energies: at threshold for a good determination of m ,

away from threshold to increase the sensitivity to chargino p olarization;

- a good reconstruction of the nal state fermion systems for a good determination

of the neutralino mass;

- identi cation of the chargino electric charge, necessary for the extraction of P ;

- an indep endent knowledge of the sneutrino mass, to avoid a three parameter t to

the observables;

3.2 KM

In this section we describ e another strategy for extracting the Lagrangian parameters

[2]. It consists in assuming that only ino masses are known. Among other things, this

strategy will b e complementary to the one describ ed in the previous section in the sense 8

that it provides (within the CDDKZ strategy) an algorithm for the determination of M ,

the only parameter whichwas not reconstructed in ref.[1]. KM concerns mainly step (I I)

of eq.(1). The emphasis is put on the extent to which the reconstruction can be made

through a controllable analytical pro cedure including all p ossible ambiguities, if three ino

masses and tan were known . This is particularly relevant for the neutralino sector

where the analytical reconstruction is far from trivial.

The next aim in [2] is to provide a numerical co de which uses as much of the analytical

solutions as p ossible and allows a direct reconstruction of M ;M and from the physical

1 2

ino masses. We do not address here the more realistic issues when only mass di erences

are measured [3], however it is clear that the study provides a useful building blo ckeven

in this case, and practically allows to avoid parameter scanning numerical pro cedures as

well as mo del-dep endent assumptions. KM distinguish two cases:

S : The two charginos and one neutralino masses are input;

S : One chargino and two neutralino masses are input;

Although S is phenomenologically less comp elling than S as far as the generic pattern

1 2

of low lying states is concerned, it turns out that it leads to a full analytical reconstruction.

In contrast, S needs partly a purely numerical algorithm which is, however, minimized

through the use of the S solutions. The b ottom line is that the resulting algorithms are

very fast, the rst b eing fully analytical and the second needing seldom more than a few

iterations to reach numerical convergence (see [2] for more details).

Let us now describ e brie y the solutions for S .

Chargino sector:

+ +

Starting from eq.(2.3) one can determine analytically and M in terms of M ;M ;tan

1 2

and m . Without further information in the chargino sector, and M will be deter-

W 2

mined, but up to a jj$M ambiguity (that is, one cannot determine uniquely at this

level the Higgsino and Gaugino content of the charginos). On the other hand the global

sign ambiguityin , due to the fact that only is known, is actually lifted by the relation

+ +

M = m sin 2 M M (3.12)

1 2

since M is p ositiveby de nition. Nonetheless there remains a two-fold ambiguity coming

from the relative sign in eq.(3.12). On the other hand, some constraints will come

from the requirement of real-valuedness of M and . All these asp ects are analytically

delineated in [2] in terms of domains of tan and the sum and di erence of the input

chargino masses.

Neutralino sector:

Let us now assume that M ;;tan and one neutralino mass have b een determined, and

address the question of reconstructing M and thus the three remaining neutralino masses.

The fact that tan needs to b e an input is actually a marginal p oint here, as one can assume that this

parameter has b een determined from elsewhere, like for instance in [1] or from the study of yet another

sector of the MSSM. 9

It should b e clear that the answer to this question is not straightforward indep endently of

whether it can be phrased analytically or not. Indeed, with all parameters but M xed

in eq.(2.5), and the knowledge of the mass of just one neutralino state (say the lightest),

it could well be that multiple branch solutions exist which would be lifted only through

extra information ab out the couplings in this sector. It turns out, however, not to b e the

case (at least when phases are ignored): there is basically a unique solution, apart from

the fact that one should allow for negative and p ositive values for the input neutralino

mass since M can have negative eigenvalues. (This sign lib erty is actually the only

ambiguity which can be eventually xed through the study of the couplings and will not

be discussed further here.)

The trick is to write down the four indep endent combinations of the entries of M which

are invariant under similarity transformations, and thus relate them simply to the four

eigenvalues of M . One can then express the correlations between the eigenvalues and

the basic parameters in the following form:

2 2 2 2 2 2

P +P ( +m +M S S )+m M s sin 2

2 2

ij ij

Z Z

(3.13)

M =

P (S M )+(c m sin 2 M )

2 2

ij ij

2 2 2 2 2 2

S P +P m sin 2 (P +( +m )P +S m sin 2 )M

ij ij ij ij

w w

ij ij

(3.14) M =

2 2 2

P +P ( +s m )+S (s m sin 2 M )

ij ij

w w

Z Z

where

~ ~

S M + M

ij N N

i j

~ ~

P M M

ij N N

i j

and i 6= j index any neutralino mass parameter. (The tilde denotes the fact that the

mass parameters can be negative valued) The nice thing ab out the ab ove equations is

that if any of the neutralino masses is taken as input (say M ), then the other three

are determined analytically through a simple cubic equation. A unique value for M is

then determined from eq.(3.13) after plugging any of these solutions. Eqs.(3.13, 3.14)

express in a sp ecially convenient way the various correlations among the four eigenvalues

and the basic parameters. It is also noteworthy that the input set (M ;;tan ) plus

one neutralino mass is optimal for a fully analytical determination. In particular this is

precisely the input set required in ref.[1]. We illustrate here the complementarity of the

two approaches by taking the two sets of numb ers (A) and (B) in eq.(3.11) and a lightest

neutralino M =30GeV , to reconstruct M and the remaining neutralino masses from

N 1

eqs.(3.13, 3.14): 10

~ ~ ~

[ M ; M ; M ; M ]

1 N N N

2 3 4

(+); [30GeV ; 59GeV ; 107GeV ; 122GeV ]

(A)

(3.15)

(); [52GeV ; 58GeV ; 119GeV ; 120GeV ]

(+); [46GeV ; 110GeV ; 130GeV ; 284GeV ]

(B )

(); [25GeV ; 101GeV ; 132GeV ; 284GeV ]

Here () refer to the two p ossible signs of the 30GeV lightest eigenmass input. The

e ect of this sign tends to be more imp ortant for M than for the neutralino masses. Of

course a minus sign should b e accompanied with the appropriate sign change in the Feyn-

man rules involving neutralinos (see [8]). A further study of the left and right form factors

in the chargino decay system could then lift partially the four-fold ambiguity in eq.(3.15).

Lifting the remaining two-fold ambiguity will still necessitate further measurements from

the ino sector.

Back to the S strategy,we give in g.3 an illustration of the sensitivitytoachargino

mass, xing the other two masses of chargino and neutralino. The b ehavior of the recon-

structed (M ;M ;) turns out to be fairly simple, up to the two-fold ambiguity induced

1 2

by in the chargino sector. A simple b ehavior shows as well for the remaining three

neutralino masses (see [2]), when the input neutralino mass is varied. This b ehavior

which is fully controlled analytically can be used to discuss the generic gross features of

the sp ectrum even when one deviates from the present input strategy. For instance the

sensitivity to tan turns out to be rather mild, and the e ect of the sign change in the

input neutralino mass tends to be negligible apart from well lo calized regions (see ref.[2]

for further illustrations, including a reconstruction of the parameters at the GUT scale).

In g.4 we illustrate the S strategy. The input set in this case requires a partial numerical

algorithm since the output b ecomes controlled by high degree p olynomials. However using

eqs.(3.13, 3.14) in conjunction with the chargino sector relations allows an optimized iter-

ative algorithm. Fig.4 shows a rather intricate b ehavior of (M ;M ;) when one chargino

1 2

and two neutralino masses are taken as input, a re ection of the ab ove mentioned high

degree p olynomials, which nevertheless b oils down (at least in our numerical trials) to

at most a four-fold ambiguity. The regions of many-fold ambiguities or no ambiguity at

all are separated by domains where the output parameters b ecome complex valued (the

shaded areas). Furthermore the singular b ehavior in some small regions is generically

traced back to zeroing some parameters (see ref.[2] for more details). 11 1000

900 M 2 800 µ not 700 real 600 500 µ 400 M2 300

200 µ M 100 1 M 1 0 -100 µ -200

-300 µ -400 -500

-600 twofold twofold -700 ambiguity ambiguity -800 -900 -1000 0 100 200 300 400 500 600 700 800 900 1000

Mχ+ (GeV)

Figure 3: ; M and M (with the \higgsino-like" convention jj M ) as functions of

1 2 2

+ +

M for xed M ( = 400 GeV); M ( = 50 GeV), and tan ( = 2). (taken from ref.[2])

2 1

300

250 M 2 200

150

100

50 M 1 0

-50 µ

-100

-150

-200

-250

-300 0 50 100 150 200 250 300

M χ0 (GeV)

Figure 4: , M and M as function of M for xed M ( = 100GeV), M (= 80

2 1 N

GeV) and tan (= 2). (taken from ref.[2]) 12

4 Final comments

In this talk we presented two p ossible theoretical strategies for the extraction of the ino

sector parameters from physical observables. The rst relied on the study of the pro duc-

tion and decay of the lightest chargino in e e collisions, the second on the knowledge of

some ino masses. We also illustrated a full reconstruction of the ino sector when the two

complementary approaches are brought together. Generally sp eaking, these approaches

provide with ecient to ols for the study of the ino sector. In the same time, they suggest

the need in some cases for further exp erimental information due to the o ccurrence of

p ossible discrete ambiguities in the reconstruction.

Furthermore, although we only considered real-valued parameters, some of the material

presented here go es through unaltered if phases are allowed. This is the case in CDDKZ

for the chargino sector, even though extra information will still be needed to determine

those phases. The inclusion of phases is less obvious in KM, esp ecially in the neutralino

sector, and deserves a separate study by itself. One should, however, keep in mind that

the ab ove strategies can give indirect information ab out the need for phases, whenever

the exp erimental data place the parameters in the forbidden regions delineated in KM. In

any case, a by-pro duct of the analytical study would have b een the construction of fast

and exible algorithms which can be used in various ways when reconstructing the ino

parameters.

Finally, it should be stressed that the strategies we presented here are just at the

theoretical level. Obviously a more realistic examination of the exp erimental extraction

of observables and related errors is still needed to assess their degree of eciency. In

addition, these strategies should eventually be placed in a wider context involving the

other sectors of the MSSM, taking into account plausible discovery scenarios of the susy

partners. The inter-correlations between these sectors, endemic to sup ersymmetry, will

then hop efully allow a unique determination of all the parameters of the mo del.

Acknowledgements

I would like to thank Jean-Loc Kneur and Peter Zerwas for valuable discussions on

their contributed pap ers. Thanks go also to Seongyoul Choi and Ab delhak Djouadi for

providing me with useful material for the talk.

References

[1] S.Y. Choi, A. Djouadi, H. Dreiner, J. Kalinowski and P.M. Zerwas, hep-ph/9806279,

(to app ear in Eur.Phys.J. C)

[2] J.L. Kneur and G. Moultaka, hep-ph/9807336,

(to app ear in Phys.Rev. D) 13

[3] I. Hinchli e et al., Phys. Rev. D55 (1997) 5520; D. Denegri, W. Ma jerotto and L.

Rurua, CMS NOTE 1997/094, HEPHY PUB 678/97, hep-ph/9711357; CMS Collab-

oration (S. Ab dullin et al.), CMS-NOTE-1998-006, hep-ph/9806366;

[4] Physics at LEP II, Rep ort No. CERN-96-01, eds. G. Altarelli, T. Sjostrand, and

F. Zwirner;

[5] E. Accomando et al. Physics Rep orts 299 (1998) 1;

[6] see for instance, G. Jungman, M. Kamionkowski and K. Griest, Phys.Rep. 267 (1996)

195;

[7] M. Brhlik and G.L. Kane, hep-ph/9803391; T. Ibrahim and P. Nath, hep-ph/9807501;

S.Y. Choi, J.S. Shim, H.S. Song, W.Y. Song hep-ph/9808227;

[8] J.F. Gunion and H.E. Hab er Nucl. Phys. B272 (1986) and Erratum hep-ph/9301205;

[9] H.E. Hab er and G.L. Kane Phys.Rep. 117 (1985) 75; 14