Determining SUSY Parameters from Chargino Pair Production

Corfu Summer Institute on Elementary Particle Physics, 1998

PROCEEDINGS

Jan Kalinowski Instytut Fizyki Teoretycznej, Ho˙za 69, 00681 Warszawa, Poland E-mail: [email protected]

Abstract: A procedure to determine the chargino mixing angles and, subsequently, the fundamental SUSY parameters M2,µand tan β by measurements of the total cross section and the spin correlations + − + − in e e annihilation toχ ˜1 χ˜1 chargino pairs is discussed.

1. Introduction role in the ﬁrst direct experimental evidence for supersymmetry.

Despite the lack of direct experimental evidence The doubling of the spectrum of states in for supersymmetry (SUSY), the concept of sym- the MSSM together with our ignorance on the metry between bosons and fermions [1] has so dynamics of the supersymmetry breaking mech- many attractive features that the supersymmet- anisms gives rise to a large number of unknown ric extension of the Standard Model is widely parameters. Even with the R-parity conserving considered as a most natural scenario. SUSY en- and CP-invariant SUSY sector, which we will as- sures the cancellation of quadratically divergent sume in what follows, in total more than 100 new corrections from scalar and fermion loops and parameters are introduced. This number of pa- thus stabilizes the Higgs boson mass in the de- rameters can be reduced by additional physical 2 sired range of order 10 GeV, predicts the renor- assumptions. In the literature several theoreti- 2 θ malized electroweak mixing angle sin W in strik- cally motivated scenarios have been considered. ing agreement with the measured value, and pro- The most radical reduction is achieved by embed- vides the opportunity to generate the electroweak ding the low–energy supersymmetric theory into symmetry breaking radiatively. a grand unified (SUSY-GUT) framework called Supersymmetry predicts the quarks and lep- mSUGRA. The mSUGRA is fully specified at tons to have scalar partners, called squarks and the GUT scale by a common gaugino mass m1/2, sleptons, the gauge bosons to have fermionic part- a common scalar mass m0, a common trilinear ners, called gauginos. In the Minimal Supersym- scalar coupling AG, the ratio tan β = v2/v1 of metric Standard Model (MSSM) [2] two Higgs the vev’s of the two neutral Higgs fields, and the doublets with opposite hypercharges, and with sign of the Higgs mass parameter µ.Allthecou- their superpartners: higgsinos, are required to plings, masses and mixings at the electroweak give masses to the up and down type fermions scale are then determined by the RGEs [3]. It and to ensure anomaly cancellation. The hig- turns out, however, that the interpretation of ex- gsinos and electroweak gauginos mix; the mass perimental data and derived limits on sparticle eigenstates are called charginosχ ˜1,2 and neutrali- masses and their couplings strongly depends on 0 nosχ ˜1,2,3,4 for electrically charged and neutral the adopted scenario. Therefore it is important states, respectively. Actually in many supersym- to develop strategies to measure all low-energy metric models the lighter chargino statesχ ˜1 are SUSY parameters independently of any theoret- expected to be the lightest observable supersym- ical assumptions. The experimental program to metric particles and they may play an important search for and explore SUSY at present and fu- Corfu Summer Institute on Elementary Particle Physics, 1998 Jan Kalinowski ture colliders should thus include the following 2. Chargino masses and couplings points: The superpartners of the W boson and charged (a) discover supersymmetric particles and mea- Higgs boson, W˜ and H˜ , necessarily mix since sure their quantum numbers to prove that the mass matrix (in the {W˜ −, H˜ −} basis) [2] they are superpartners of standard parti- √ M2 2mW cβ cles, MC = √ (2.1) 2mW sβ µ (b) determine the low-energy Lagrangian pa- is nondiagonal (cβ =cosβ,sβ =sinβ). It is ex- rameters, pressed in terms of the fundamental supersym- (c) verify the relations among them in order to metric parameters: the gaugino mass M2,the distinguish between various SUSY models. Higgs mass parameter µ,andtanβ. Two dif- ferent matrices acting on the left– and right– If SUSY is realized in Nature, it will be a matter chiral (W,˜ H˜) states are needed to diagonalize + − of days for the future e e linear colliders (LC) the asymmetric mass matrix (2.1). The (posi- to discover the kinematically accessible super- tive) eigenvalues are given by symmetric particles. Once they are discovered, m2 1 M 2 µ2 m2 ∓ χ˜ = 2 2 + +2 W ∆ (2.2) the priority will be to measure the low-energy 1,2 SUSY parameters independently of theoretical prejudices and then check whether the correla- where tions among parameters, if any, support a given 2 2 2 2 ∆= [(M2 +µ +2mW) theoretical framework [4], like SUSY-GUT rela- − M µ − m2 β 2 1/2 tions. Particularly in this respect the e+e− lin- 4( 2 W sin 2 ) ] (2.3) ear colliders are indispensable tools, as has been The left– and right–chiral components of the cor- demonstrated in a number of dedicated work- − responding chargino mass eigenstateχ ˜1 are re- shops [5]. In my talk I will concentrate on the lated to the wino and higgsino components in the i.e. first phase of LC, when only a limited number following way of supersymmetric particles can kinematically be − − − produced. In contrast to earlier analyses [6, 7], I χ˜1L = W˜ L cos φL + H˜1L sin φL will not elaborate on global chargino/neutralino − − − χ˜1R = W˜ R cos φR + H˜2R sin φR (2.4) fits but rather attempt to explore the event char- acteristics to isolate the chargino sector. The with the rotation angles given by analysis will be based strictly on low–energy su- φ − M 2 − µ2 − m2 β / persymmetry. We will see that even if the light- cos 2 L = ( 2 2 W cos 2 ) ∆ 2 2 2 est chargino statesχ ˜1 are, before the collider cos 2φR = −(M2 − µ +2mW cos 2β)/∆ (2.5) energy upgrade, the only supersymmetric states β M that can be explored experimentally in detail, As usual, we take tan positive, 2 positive and µ some of the fundamental SUSY parameters can of either sign. φ φ χχZ be reconstructed from the measurement of char- The angles L and R determine the ˜˜ and theχ ˜νe˜ couplings: gino mass m , the total production cross sec- χ˜1 tion and the chargino polarization in the final − − e 2 hχ˜1L|Z|χ˜1Li = − (4sW − 3 − cos 2φL) state. Beam polarization is helpful but not nec- 4sW cW e essarily required. hχ− |Z|χ− i − s2 − − φ ˜1R ˜1R = s c (4 W 3 cos 2 R) The results presented here have been obtained 4 W W − − e in collaboration with S.Y. Choi, A. Djouadi, H. hχ˜1R|ν˜|eL i = − cos φR (2.6) sW Dreiner and P. Zerwas [8]. For an alternative way 2 2 2 of determining the SUSY parameters from the where sW =1−cW ≡sin θW . The coupling measurement of (some) chargino and neutralino to the higgsino component, being proportional masses, see [9]. to the electron mass, has been neglected in the

2 Corfu Summer Institute on Elementary Particle Physics, 1998 Jan Kalinowski sneutrino vertex. The photon–chargino vertex After a Fierz transformation of theν ˜–exchange is diagonal, i.e. it is independent of the mixing term, the amplitude angles: 2 e + − T = Qαβ[¯v(e )γµPαu(e )] − − s hχ˜1L,R|γ|χ˜1L,Ri = e (2.7) − µ + ×[¯u(˜χ1 )γ Pβv(˜χ1 )] (3.2) The chargino couplings, and therefore mixing angles, are physical observables and they can be can be expressed in terms of four bilinear charges, measured. Their knowledge together with the classified according to the chiralities of the asso- measurement of the chargino masses is sufficient ciated lepton (α = L, R) and chargino (β = L, R) to determine the fundamental supersymmetric currents M2 µ β parameters , and tan . DZ 2 2 QLβ =1+ 2 2(2sW − 1)(4sW − 3 − cos 2φβ) 8sWcW 3. Chargino production and decay Dν˜ + δβ,R 2 (1 + cos 2φR) 4sW + − Charginos are produced in e e collisions, either DZ 2 QRβ =1+ 2(4sW − 3 − cos 2φβ) (3.3) in diagonal or in mixed pairs. Here we will con- 4cW sider the diagonal pair production of the lightest + − Theν ˜ exchange affects only the LR charge while charginoχ ˜1 in e e collisions, all other charges are built up by γ and Z ex- + − + − D e e → χ˜1 χ˜1 (3.1) changes. ν˜ denotes the sneutrino propagator 2 Dν˜ = s/(t − mν˜), and DZ the Z propagator 2 assuming the second charginoχ ˜2 too heavy to DZ = s/(s − mZ + imZΓZ). Above the Z pole be produced in the first phase of e+e− linear col- the non–zero width can be neglected so that the liders. If the chargino production angle could be charges are real. − + measured unambiguously on an event-by-event Theχ ˜1 andχ ˜1 helicities are in general not basis, the chargino couplings could be extracted correlated due to the non–zero masses of the par- χ directly from the angular dependence of the cross ticles; amplitudes√ with equal ˜1 helicities vanish section. However, charginos are not stable. We only ∝ mχ˜ / s for asymptotic energies. De- 1 will assume that they decay into the lightest neu- noting the electron helicity by the first index 0 − + tralinoχ ˜1, which is taken to be stable, and a pair σ and theχ ˜1 andχ ˜1 helicities by the remain- of quarks and antiquarks or charged leptons and ing two indices λ and λ¯, the helicity amplitudes 0 neutrinos. Since two neutral particlesχ ˜1 escape T (σ; λλ¯)=2παAσ;λλ¯ can be expressed as func- undetected, it is not possible to reconstruct the tions of bilinear chirality charges [10] events unambiguously. In particular the produc- p tion angle Θ cannot be reconstructed for a given A+;++ = − β−β+(QRR + QRL)sinΘ event. The distribution of observed final state A+;+− = −(β+QRR + β−QRL)(1 + cos Θ) quark jets or leptons is given by a convolution A+;−+ =+(β−QRR + β+QRL)(1 − cos Θ) of the chargino production and decay processes. p A+;−− =+ β−β+(QRR + QRL)sinΘ Therefore we will consider the production of po-

larized charginos and their subsequent decays. It

~ ~ ~

e e

1 1 has to be stressed that the spin correlations be- 1

tween production and decay have to be properly

taken into account [7, 8].

+ + +

~ ~ ~

e e e

1 1 1 3.1 Polarized chargino production + − + − The process e e → χ˜1 χ˜1 is generated by the Figure 1: The three mechanisms contributing to the s γ + − + − three mechanisms shown in Fig. 1: –channel production of diagonal chargino pairs χ˜1 χ˜1 in e e and Z exchanges, and t–channelν ˜ exchange. annihilation.

3 Corfu Summer Institute on Elementary Particle Physics, 1998 Jan Kalinowski

p A−;++ = − β−β+(QLR + QLL)sinΘ GeV)

A−;+− =+(β+QLR + β−QLL)(1 − cos Θ) gaugino : (M2,µ)=(81,−215) A−;−+ = −(β−QLR + β+QLL)(1 + cos Θ) p higgsino : (M2,µ) = (215, −81) (3.7) A−;−− =+ β−β+(QLR + QLL) sin Θ (3.4) mixed : (M2,µ)=(92,−93)

β β β − m2 /s 1/2 for which the light chargino mass m is approx- where =1 ,and =(1 4 ) is χ˜1 χ˜1 imately 95 GeV. The sharp rise of the produc- theχ ˜1 velocity in the c.m. frame. tion cross section in Fig. 2 allows us to mea- The unpolarized differential cross section for + − + − sure the chargino mass mχ˜ very precisely. As e e → χ˜ χ˜ is obtained by averaging/sum- 1 1 1 is well-known, the sneutrino exchange leads to ming over the electron/chargino helicities. It can a strong destructive interference for the gaug- be written as ino and mixed regions, while the dependence of 2 dσ πα 2 2 the cross section on mν˜ decreases as the higgsino = β{(1 + β cos Θ)Q1 dcosΘ 2s component of the chargino increases. Prior or si- 2 +(1 − β )Q2 +2βcos ΘQ3} (3.5) multaneous determination of mν˜ is therefore necessary to determine the other SUSY parameters. where the quartic charges Qi are given in terms In general charginos are produced with non- of bilinear charges as follows [11] zero polarization. The key point in our analysis in Sect. 4 will be to exploit the partial informa- Q 1 |Q |2 |Q |2 |Q |2 |Q |2 1 = 4 ( RR + LL + RL + LR ) tion on the chargino polarizations that can be 1 ∗ ∗ Q2 = 2 Re(QRRQRL + QLLQLR) (3.6) obtained from the distribution of the chargino 1 2 2 2 2 χ− Q3 = 4 (|QRR| + |QLL| −|QRL| −|QLR| ) decay products. Therefore we define the ˜1 po- ~ − larization vector P =(PT,PN,PL)inthe˜χ1 The total production cross section as a func- rest frame. PL denotes the component parallel to χ− P tion of the√ c.m.energy and the angular distribu- the ˜1 flight direction in the c.m. frame, T the tion at s = 500 GeV are shown in Fig. 2 for transverse component in the production plane, a representative set of (M2,µ) parameters; the and PN the component normal to the production sneutrino mass has been taken mν˜ = 200 GeV. plane. The three components can be written in The parameters are chosen in the higgsino region terms the helicity amplitudes as M2 |µ|, the gaugino region M2 |µ|and in 1 X M ∼|µ| β P |A |2 |A |2 the mixed region 2 for tan =2as(in L = N ( σ;++ + σ;+− 4 σ= 2 2 −|Aσ;−+| −|Aσ;−−| ) (3.8) 2 1.0 X √ s=500 GeV P 1 A A∗ A A∗ T = N Re( σ;++ σ;−+ + σ;−− σ;+−) 2 σ= [pb] X Θ

[pb] 1 0.5 1 ∗ ∗

tot PN = Im(Aσ;−−Aσ;+− −Aσ;++Aσ;−+)

σ N /dcos 2 σ= σ d

0 0.0 with the normalization 150 300 450 600 −1.0 −0.5 0.0 0.5 1.0 Θ X √ s [GeV] cos 1 2 2 N = (|Aσ;++| + |Aσ;+−| 4 σ= 2 2 Figure 2: Left panel: the total cross section as a +|Aσ;−+| + |Aσ;−−| ) (3.9) function of the c.m.energy. Right panel: the angular The normal component PN can only be gener- √distribution as a function of the production angle for s = 500 GeV. In both panels: mν˜ = 200 GeV, solid ated by complex production amplitudes. Ne- lines for the gaugino, dashed lines for the higgsino glecting small higher order loop eﬀects and the − + and dot-dashed lines for the mixed cases, see eq. 3.7 small Z–width eﬀect, the normalχ ˜1 andχ ˜1 po- [taken from [8]]. larizations are zero since theχ ˜1χ˜1γ andχ ˜1χ˜1Z

4 Corfu Summer Institute on Elementary Particle Physics, 1998 Jan Kalinowski

0 3 0 0 0 0

u d

1 B,˜ W˜ ,H˜1,H˜2 to the mass eigenstatesχ ˜1, .., χ˜4.

~ The neutralino mass matrix N is given by

1 1

0 0

d L

L  

~ ~

d 1 1 M1 0 −mZ sW cβ mZ sW sβ

 M m c c −m c s 

u u d  0 2 Z W β Z W β    −mZ sW cβ mZ cW cβ 0 −µ Figure 3: Chargino decay mechanisms; the ex- mZ sW sβ −mZ cW sβ −µ 0 change of the charged Higgs boson is neglected.

Besides the parameters M2,µ and tan β, which vertices are real (even for non-zero phases in the already appear in the chargino mass matrix, the chargino mass matrix) and theν ˜–exchange am- only additional parameter in the neutralino mass 1 plitude is real too. The CP–violating phases will matrix is M1 . In general supersymmetric mod- change the chargino mass and the mixing angles els, however, the neutralino sector can poten- [8, 13] but they do not induce complex charges in tially be more complex than in the MSSM with the production amplitudes of the diagonal char- more additional parameters. gino pairs. In the chargino rest frame, the angular distribution of the neutralino and the f1f¯2 system 3.2 Decay of polarized charginos 0 from the polarized chargino decayχ ˜1 → χ˜1 + We will assume that charginos decay into the f1f¯2 (summed over the final state polarizations) 0 lightest neutralinoχ ˜1 and a pair of quarks and is determined by the energy and the polar an- 0 f f¯ antiquarks or leptons and neutrinos:χ ˜1 → χ˜1 + gle of the neutralino (or equivalently of the 1 2 ff¯0. The decay can proceed via the exchange of system) with respect to the chargino polarization ∗ ∗ the W , squarks or sleptons; the exchange of the vector. Let us define θ (θ¯ ) as the polar angle of ¯ ¯ − + charged Higgs bosons can be neglected for light the f1f2 (f3f4)systeminthe˜χ1 (˜χ1 ) rest frame fermions in the final state. The corresponding with respect to the original flight direction in the ∗ ∗ diagrams are shown in Fig. 3 for the decay into laboratory frame, and φ (φ¯ ) the corresponding quark pairs. After a Fierz transformation of the azimuthal angle with respect to the production − + q˜–exchange contributions, the decay amplitude plane. Taking theχ ˜1 (˜χ1 ) flight direction as − 0 χ˜1 → χ˜1du¯ may be written as: quantization axis and for the kinematical config- 2 uration defined above, the spin–density matrix e µ ∗ D = √ [¯u(d)γ PLv(¯u)] ρλλ0 ∼DλDλ0 can conveniently be written in the s2 2 2 W following form 0 − ×[¯u(˜χ1)γµ(FLPL +FRPR)u(˜χ )](3.10) 1 ∗ 1 1+κcos θ∗ κ sin θ∗eiφ ρλλ0 = ∗ (3.12) with 2 κ sin θ∗e−iφ 1 − κ cos θ∗ √ FL = DW (2N12 cos φL + 2N13 sin φL) − + for theχ ˜1 decay, and similarly for theχ ˜1 with + D ˜ cos φL(N12 − 2Yq tan θW N11) dL √ barred quantities. ∗ ∗ FR = DW (2N12 cos φR − 2N14 sin φR) The spin analysis–powers κ andκ ¯, depend on ∗ ∗ the final ud or lν pairs considered in the chargino + Du˜L cos φR(N12 +2Yqtan θW N11)(3.11) decays. Since left– and right–chiral form factors Y / where q =16 is the quark hypercharge and FL,FR contribute at the same time, their values 0 2 0 2 DW = s − mW + imW ΓW , D ˜ = t − m ˜ , dL dL are determined by the masses and couplings of D u0 − m2 u˜L = u˜L . Analogous expressions ap- all the particles involved in the decay. Charac- ply to decays into lepton pairs with Yl = −1/2. teristic examples for κ as a function of the in- 0 0 0 The Mandelstam variables s , t , u are defined variant mass of the fermion system are shown in 0 in terms of the 4–momenta q0,q andq ¯ ofχ ˜1,d Fig. 4; the squark masses are set to 300 GeV, 0 2 0 andu ¯, respectively, as s =(q+¯q),t =(q0+ 1In GUT models with the unification of gaugino q2 u0 q q2 N × ), =(0+¯), while ij is the 4 4ma- masses at a high scale, the M1 and M2 are related by M 5 2 θ M trix rotating the current neutralino eigenstates 1 = 3 tan W 2.

5 Corfu Summer Institute on Elementary Particle Physics, 1998 Jan Kalinowski

1.0 The quantity Σ can be decomposed into six- tanβ=2 teen independent angular parts

0.5 ∗ ∗ Σ=Σun +cosθ κP+cosθ¯ κ¯P¯ +cosθ∗ cos θ¯∗κκ¯Q

κ 0.0 +sinθ∗ sin θ¯∗ cos(φ∗ + φ¯∗)κκ¯Y +sinθ∗ sin θ¯∗ sin(φ∗ + φ¯∗)κκ¯Y¯ −0.5 gaugino ∗ ∗ ∗ ∗ higgsino +sinθ cos φ κU +sinθ sin φ κU¯ mixed +sinθ¯∗ cos φ¯∗κ¯V +sinθ¯∗sin φ¯∗κ¯V¯ −1.0 ∗ ∗ ∗ ∗ 0 102030405060 +sinθ cos θ¯ (cos φ κκ¯W +sinφ κκ¯W¯) √ s’ [GeV] ∗ ∗ ∗ ∗ +cosθ sin θ¯ (cos φ¯ κκ¯X +sinφ¯ κκ¯X¯) +sinθ∗ sin θ¯∗(cos(φ∗ − φ¯∗)κκ¯Z Figure 4: The polarization analysis–power κ as a √ ∗ ∗ ∗ ∗ function of the hadron invariant mass s0 for the +sinθ sin θ¯ (sin(φ − φ¯ )κκ¯Z¯ (3.16) same set of parameters as for the cross section (3.7). where the sixteen coefficients are combinations of helicity amplitudes, corresponding to the un- and the gaugino masses are assumed universal polarized cross section (Σun), 2 × 3 polarization at the unification scale. For our purposes, how- components (P, U, V, P¯, U¯, V¯)and3×3 spin–spin ever, it will suffice that they are finite. Let us correlations (the remaining ones) multiplied by also note that neglecting effects from non–zero the spin-analysis power factors κ andκ ¯.Inthe κ widths, loops and CP–noninvariant phases, and CP-invariant theory, and neglecting loops and κ κ −κ χ ¯ are real, and = ¯ for charginos ˜1 decaying the width of the Z–boson for high energies, the to charge-conjugated final states. six functions U¯, V¯, W¯ , X¯, Y¯, Z¯ can be discarded. Moreover, from CP–invariance 3.3 Physical observables λ−λ0−1 Charginos decay fast and only correlated produc- Aσ;λλ0 =(−1) Aσ;−λ0 −λ (3.17) tion and decay can be observed experimentally. it follows that P¯ = −P, U = −V and W = X . The matrix element for the physical process The overall topology is therefore determined by + − − + 0 0 un P Q U W e e → χ˜1 χ˜1 → (˜χ1f1f¯2)(˜χ1f¯3f4) (3.13) seven independent functions: Σ , , , , , Y and Z. summed/averaged over the final/initialP state po- The key point in our analysis is to exploit larizations, is given by M = Tσ;λλ¯ DλDλ¯ .In- the partial information on the chargino polariza- tegrating over the production angle Θ and also tions with which they are produced. Theχ ˜ po- over the invariant masses of the fermionic sys- larization vectors andχ ˜–˜χ spin–spin correlation tems (f1f¯2)and(f¯3f4), one can write the 4-fold tensors can be determined from the decay dis- differential cross section in the following form: tributions of the charginos independently of the

4 2 chargino decay dynamics. d σ α β ∗ ∗ ∗ ∗ = B B¯Σ (3.14) The decay angles (θ ,φ )and(θ¯,φ¯ ), which d cos θ∗dφ∗d cos θ¯∗dφ¯∗ 124πs are used to measure the χ1 chiralities, can not − 0 + be reconstructed completely since there are two where B = Br(˜χ1 → χ˜1f1f¯2)andB¯=Br(˜χ1 → 0 invisible neutralinos in the ﬁnal state. However, χ˜1f¯3f4). The scaled diﬀerential cross section Σ = Σ(θ∗,φ∗,θ¯∗,φ¯∗) is a product of the production the longitudinal components and the inner product of the transverse components can be recon- helicity amplitudes Aσ;λλ¯ and the spin density structed from the momenta measured in the lab- matrices for the chargino decays ρλλ0 XXX oratory frame ∗ Σ= Aσ;λλ¯A 0¯0 ρλλ0 ρ¯λ¯λ¯0 (3.15) σ;λλ θ∗ E−γE∗ /βγ|p~∗| λλ¯ λ0λ¯0 σ cos =( )

6 Corfu Summer Institute on Elementary Particle Physics, 1998 Jan Kalinowski

∗ ∗ ∗ cos θ¯ =(E¯−γE¯ )/βγ|~p¯ | mν˜ can be also extracted. The three quantities ∗ θ∗ θ¯∗ φ∗ φ¯∗ p~·~p/|p~∗||~p | m and cos 2φL,R can be determined from the sin sin cos( + )= ¯ ¯ χ˜1 ∗ ∗ 2 2 ∗ ∗ production cross section and the spin correlations +(γE − E )(γE¯ − E¯ )/β γ |p~ ||~p¯ | (3.18) as follows. √ ∗ ∗ The mass mχ˜ can be measured very pre- where γ = s/2m . E(E¯)andE(E¯) are 1 χ˜1 the energies of the two hadronic systems in the cisely near the threshold where the production σ e+e− → χ+χ− laboratory frame and in the rest frame of the cross section ( ˜1 ˜1 ) rises sharply with ∗ β charginos, respectively; ~p(~p¯)and~p(p¯~∗) are the the chargino velocity . Alternatively the masses corresponding momenta; the angle between the of chargino and neutralino can be determined by vectors in the transverse plane is given by ∆φ∗ = fitting the energy spectra of the jet-jet final state 2π−(φ∗+φ¯∗) for the reference frames defined ear- systems [12]. For the analysis below we assume lier.2 Therefore, by means of kinematical projec- that the mass of the light chargino has been measured and m =95GeV tions the terms in the first three lines of eq.(3.16) χ˜1 P can be extracted and three κ-independent phys- Since the polarization is odd under parity 2 2 ical observables, Σun, P /Q and P /Y, construc- and charge–conjugation, it is necessary to iden- ted. They are unambiguously related to the prop- tify the chargino electric charges in this case. erties of the chargino sector, not affected by the This can be accomplished by making use of the neutralinos, since they are given in terms of the mixed leptonic and hadronic decays of the chargi- Q helicity production amplitudes as follows no pairs. On the other hand, the observables Y R and are defined without charge identification Σun = dcosΘN so that the dominant hadronic decay modes of R 1 the charginos can be exploited. Suppose that the P NPL = 4 dcosΘ 2 2 R X quantities σt, P /Q and P /Y have been mea- Q 1 |A |2 −|A |2 = 4 dcosΘ ( σ;++ σ;+− sured and are taken to be σ= 2 2 σ e+e− → χ+χ− . −|Aσ;−+| +|Aσ;−−| ) t( ˜1 ˜1 )=037 pb R X 2 2 Y 1 A A∗ P P = 2 dcosΘ Re( σ;−− σ;++)(3.19) = −0.24, = −3.66 (4.1) σ= Q Y √ at s = 500 GeV. These measurements can be It is thus possible to study the chargino sec- interpreted as contour lines in the plane (cos 2φL, tor in isolation by measuring the mass of the cos 2φR) which intersect with large angles so that lightest chargino, the total production cross sec- a high precision in the resolution can be achieved. tion and the spin(–spin) correlations. A representative example for the determination of cos 2φL and cos 2φR based on the values in 4. Extraction of SUSY parameters eq. (4.1) is shown in Fig. 5. The three contour lines meet at a single point (cos 2φL, cos 2φR)= χ The pair production of the lightest chargino 1 is (−0.58, −0.48) for mν˜ = 250 GeV; note that the characterized by the chargino mass m and the χ˜1 sneutrino mass can be determined together with two mixing angles cos 2φL,R. For simplicity we the mixing angles from the “measured values” in assume that the sneutrino mass mν˜ is obtained eq. (4.1). from elsewhere, although combining the energy Finally the Lagrangian parameters M2,µand variation of the cross section with the measure- tan β can be obtained from m ,cos2φL and χ˜1 ment of the spin correlations, the sneutrino mass cos 2φR up to a two-fold ambiguity. It is most 2Actually to determine the kinematical variables, transparently achieved by introducing the two ∗ cos θ etc., the knowledge of mχ˜0 is also needed, which auxiliary quantities 1 can be extracted from the energy distributions of final state particles, see later. However, it must be stressed p =cot(φR−φL),q=cot(φR +φL) (4.2) that the above procedure does not depend on the details of decay dynamics nor on the structure of (potentially They are expressed in terms of the measured val- more complex) neutralino and sfermion sectors. ues cos 2φL and cos 2φR up to a discrete ambi-

7 Corfu Summer Institute on Elementary Particle Physics, 1998 Jan Kalinowski

correlations among the chargino decay products 2 2 (P /Q, P /Y), the physical parameters m , χ˜1 1.0 cos 2φL and cos 2φR are determined unambiguously. Then the fundamental parameters tan β, σ =0.37 pb P2/Q=−0.24 M2 and µ are extracted up to a two-fold discrete 0.5 2 P /Y=−3.66 ambiguity. If the collider energy is suﬃcient to produce R φ 0.0 the two chargino states in pairs, the above ambi-

cos2 guity can be removed [13] by the measurement of the heavier chargino mass. With polarized −0.5 beams available at the LC, the measurement of the left-right asymmetry ALR can provide [13] an −1.0 alternative way to extract the mixing angles (or −1.0 −0.5 0.0 0.5 1.0 φ serve as a consistency check). cos2 L

Figure 5: Contours for the “measured values” (4.1) 5. Conclusions of the total cross section (solid line), P 2/Q (dashed P2/Y m line), and (dot-dashed line) for χ =95GeV We have discussed how the parameters of the 1 [mν˜ = 250 GeV]. chargino system, the mass of the light chargino m and the two angles φL and φR,canbeex- χ˜1 tracted from pair production of the light chargino guity due to undetermined signs of sin 2φL and state in e+e− annihilation. In addition to the to- sin 2φR tal production cross section, the measurements 2 φ 2 φ p2 q2 2(sin 2 L +sin 2 R) of angular correlations among the chargino de- + = 2 (cos 2φL − cos 2φR) cay products give rise to two independent ob- cos 2φL +cos2φR servables which can be measured directly despite pq = cos 2φL − cos 2φR of the two invisible neutralinos in the final state. 2 2 4sin2φL sin 2φR From the chargino mass mχ˜ and the two p − q 1 = 2 (4.3) (cos 2φL − cos 2φR) mixing parameters cos 2φL,R, the fundamental supersymmetric parameters tan β, M2 and µ can Solving then eqs. (2.5) for tan β one finds at most be extracted up to at most a two-fold discrete two possible solutions, and using √ ambiguity. Moreover, from the energy distribu- M2 = mW [(p + q)sβ − (p − q)cβ]/ 2 tion of the final particles in the decay of the √ chargino, the mass of the lightest neutralino can µ = mW [(p − q)sβ − (p + q)cβ]/ 2 (4.4) be measured; this allows us to determine the pa- we arrive at tan β, M2 and µ up to a two-fold dis- rameter M1 so that also the neutralino mass ma- crete ambiguity. For example, taking the “mea- trix can be reconstructed in a model-independent sured values” from eq. (4.1), the following results way. are found [8] Although we only considered real-valued pa-  rameters, some of the material presented here [1.06; 83GeV, −59GeV] goes through unaltered if phases are allowed [8, [tan β; M2,µ]= (4.5)  13] even though extra information will still be [3.33; 248GeV, 123GeV] needed to determine those phases. Other sets of “measured values” can lead to a It should be stressed that the strategy pre- unique solution if the other “possible solution” sented here is just at the theoretical level. More has a negative tan β. realistic simulations of the experimental measure- To summarize, from the light chargino pair ments of physical observables and related errors, production, the measurements of the total pro- including radiative corrections, are needed to as- duction cross section and either of the angular sess fully its usefulness. Nevertheless, if the LC

8 Corfu Summer Institute on Elementary Particle Physics, 1998 Jan Kalinowski

and detectors are built and work as expected, G. Moortgat-Pick and H. Fraas, Phys. Rev. D no doubt that the actual measurements will be 59 (1999) 015016 [hep-ph/9708481]. better than anything presented here – provided [7] G. Moortgat-Pick, H. Fraas, A. Bartl and supersymmetry is discovered! W. Majerotto, Eur. Phys. J. C7 (1999) 113 [hep-ph/9804306]; Acknowledgments V. Lafage it et al., Spin and spin-spin correlations in chargino pair production at future linear e+e− colliders, hep-ph/9810504. I thank the organisers G. Zoupanos, N. Tracas and G. Koutsoumbas for their warm hospitality [8] S.Y. Choi, A. Djouadi, H. Dreiner, J. Kali- at Corfu. I would also like to thank my collabo- nowski and P.M. Zerwas, Eur. Phys. J. C7 hep-ph/9806279 rators S.Y. Choi, A. Djouadi, H. Dreiner and P. (1999) 123 [ ] Zerwas for many valuable discussions. This work [9] J.L. Kneur and G. Moultaka, Phys. Rev. D59 has been partially supported by the KBN grant (1999) 015005 [hep-ph/9807336]. 2 P03B 052 16. [10] K. Hagiwara and D. Zeppenfeld, Nucl. Phys. B 274 (1986) 1. References [11] L.M. Sehgal and P.M. Zerwas, Nucl. Phys. B 183 (1981) 417. [1] Yu.A. Gol’fand and E.P. Likhtman, JETP Lett. [12] H.U. Martyn, in DESY-ECFA Conceptual LC 13 (1971) 452; Design Report, DESY 1997-048, ECFA 1997- J. Wess and B. Zumino, Nucl. Phys. B70 182. For an updated analysis at high luminosity, (1974) 39; see H.U. Martyn, talk given at the 2nd ECFA- R. Haag, J.T.Lopusza´ nski and M.F. Sohnius, DESY Linear Collider Workshop, Oxford 1999. Nucl. Phys. B88(1975) 257. [13] S.Y. Choi, A. Djouadi, H.S. Song and P.M. [2] For reviews of supersymmetry and the Minimal Zerwas, Determining SUSY parameters in Supersymmetric Standard Model, see H. Nilles, chargino pair-production in e+e− collisions, Phys. Rep. 110 (1984) 1; hep-ph/9812236. H.E. Haber and G.L. Kane, Phys. Rep. 117 (1985) 75. [3] V. Barger, M.S. Berger and P. Ohmann, Phys. Rev. D49(1994) 4908. [4] J. Kalinowski, Supersymmetry searches at e+e− linear colliders, hep-ph/9904260. [5] Proceedings Physics and Experiments with Lin- ear Colliders: R. Orava, P. Eerola, M. Nordberg (Eds.), Sariselkä, Finland 1991, World Scientific (1992); F.A. Harris, S. Olsen, S. Pakvasa, X. Tata (Eds.), Waikoloa, Hawaii 1993, World Scientific (1994); A. Miyamoto, Y. Fujii, T. Matsui, S.Iwata (Eds.), Morioka, Japan 1995, World Scientific (1996); E. Accomando et al., LC CDR Report DESY 97-100 [hep-ph/9705442], and Phys. Rep. 299 (1998) 1. [6] A. Leike, Int. J. Mod. Phys. A3(1988) 2895; M.A. Diaz and S.F. King, Phys. Lett. B 349 (1995) 105 and B373 (1996) 100; J.L. Feng and M.J. Strassler, Phys. Rev. D51 (1995) 4461 and D55 (1997) 1326;