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Physics Letters B 315 ( 1993 ) 494-502
North-Holland
PHYSICS LETTERS B
An S matrix analysis of the Z resonance
L3 Collaboration
0 . A d ria n i0, M . Aguilar-Benitezx, S. A hlen1, J. A lcarazp, A. A loisioaa, G . Alversonj ,
M .G . A lvig giaa, G . Am brosiaf, Q. A n q, H. Anderhubat, A .L. Anderson“ , V .P. Andreev2-',
T. Angelescuk, L. A ntonovan, D . Antreasyan8, P. A rcex, A. A refievz, A. Atamanchuk^,
T. Azem oonc, T . A zizh, P .V .K .S . Bab aq, P. Bagnaia31, J.A . Bakkenah, R .C . B a ll0, S. Banerjee11,
J. B a o e, R . B arillèrep, L. Barone31, A. Baschirotto y, R . Battistonaf, A. B a y r, F. Becattini0,
J. Bechtluft3, R . Becker3, U . Becker“ >
3t, F. Behnerat, J. Behrens31, G y.L. Bencze*, J. Berdugo*,
P. Berges“ , B. Bertu cciaf, B .L . B e te van,3t, M . B ia sin iaf, A. B ila n d at, G .M . B ile iaf, R . Bizzarri31,
J.J. Blaisin g d, G .J. Bobbinkp-b, R . Bo ck3, A. Böhm 3, B . Borgia31, M . Bosettiy, D. Bo u rilko vac,
M . Bourquinr, D . Boutignyp, B . Bouwensb, E. Bram b illa33, J.G . B ran so n ^ I.C . Bro ckag,
M . Brooksv, A. Bujak iK), J.D . Burger", W .J. Burgerr, J. Busenitz3p, A. Buytenhuijsac, X .D . C a iq,
M . C apell“ , M . Caria af, G . Carlino
A .M . Cartacei0, R . Castello y, M . Cerrada*, F. Cesaron i al,
Y .H . Chang", U .K . C haturvediq, M . Chem arinw, A. Chenav, C. Chenf, G . Chenf, G .M . C henf,
H .F. Chens, H .S. C henf, M . C hen", W .Y . Chenav, G. C hiefari33, C .Y. C hiene, M .T. C h o iao,
S. Chungn, C. C iv in in i0,1. C lare“ , R . C lare", T .E . Coanv, H .O . Cohn3d, G. Coignetd,
N . Colino p, A. Contin s, F. Cotorobaik, X .T . C u iq, X .Y . C u iq, T.S. D a in, R. D ’Alessandro °,
R . de Asm undisaa, A. Degré d, K . D eitersar, E. Dénes*, P. Denesah, F. DeNotaristefani31,
M . D h in a3t, D . D iB ito n to ap, M . Diem ozai, H .R . D im itro van, C. D io n isiai, M . D ittm ar31,
L. Djam bazov3*, M .T . D o va q, E . Drago33, D . Duchesneaur, P. D uinkerb, I. D uranaf, S. Easoaf,
H . E l M am ouniw, A. Engler38, F .J. Eppling” , F.C . E rn é b, P. Exterm annr, R . Fabbrettiar,
M . Fab re3r, S. Falcian o 31, S .J. F a n 31“ , O. Fackler“ , J. F a y w, M . F e lcin ip, T. Ferguson38,
D . Fernandez*, G . Fernandez*, F. Ferro n i31, H . Fesefeldt3, E. Fian d rin i3f, J.H . F ie ld r,
F. Filth au tac, G . Finocchiaro“ , P.H . Fish ere, G . Fo rco n ir, L. Fre.djr, K . Freudenreich31,
W . Frieb el3S, M . Fukushim a” , M . G aillo u d 1, Y u . G alaktionovz,n, E. G allo 0, S.N. G angulip,h,
P. Garcia-Abiax, D . G elew, S. G en tile“ , N . Gheordanescuk, S. G iaguai, S. Goldfarbj ,
Z .F. Gong5, E. Gonzalez*, A. Gougase, D. G oujonr, G . G rattaae, M . Gruenewaldp, C. G u q,
M . G uanziroliq, J.K . G u o am, V .K . G upta311, A. G u rtu h, H .R . Gustafson0, L .J. G u tayaq,
K . Hangarter3, B. H artm ann3, A. H asanq, D . H auschildtb, C.F. H e 3m, J.T . H e f, T. Hebbekerp,
M . Hebert ^ A. H ervé p, K . H ilgers3, H . H o fer3*, H. H ooranir, G . H u q, G .Q . H u 3m, B. Ille w,
M .M . IIya sq, V . Innocente11, H . Janssenp, S. Jezequeld, B.N . Jin f, L.W . Jo n es0,
1. Josa-M utuberriap, A. Kasser1, R .A . K h an q, Yu. Kam yshkovad, P. Kapinosaj,as,
J.S . Kapustinskyv, Y . K aryotakisp, M . K a u rq, S. Khokharq, M .N . Kienzle-Focaccir,
J.K . K im 30, S.C. K im ao, Y .G . K im 30, W .W . K innisonv, A. K irk b y3e, D . K irk b yae, S. K irsch as,
W . K itte i30, A. K lim en to vn,z, R . Klöckner3, A.C. K ö n ig 30, E. Koffem anb, O. Kornadt3,
V. Koutsenko“ >
2, A. KoulbardisaJ, R .W . Kraem er38, T. K ram er“ , V .R . K rastevan,af, W . K ren z3,
A. Krivshich aj, H . K u ijte n 30, K .S. K u m arm, A. Kunin n-2, G . Lan d i0, D . Lanske3, S. Lanzano33,
A. Lebedev” , P . Lebrun w, P. Lecom teat, P. Lecoqp, P. Le Coultre31, D .M . L e e v, J.S . Le e ao,
K .Y . L e e 30, 1. Leedom J, C. L e g g e ttJ.M . Le G o ffp, R . Leiste3S, M . L e n ti0, E. Leonardiai,
C. L i s>q, H .T . L i f, P .J. L i am, J.Y . L ia o 3"1, W .T . L in 3V, Z .Y . L in s, F.L. Lin d e b, B. Lindem ann3,
L. L is ta 33, Y . L iu q, W . Lohm annas, E. Longoai, Y .S. L u f, J.M . Lubbersp, K . Lübelsm eyera,
494
\
Elsevier Science Publishers B.V.
Volume 315, number 3,4
P H Y SIC S L E T T E R S B
7 October Í993
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M . M acD erm ottat, R . M alik q, A. M alin in z, C. M aña*, M . M aolinbay3', P. M archesiniat,
F. M arion d, A. M a rin 1, J.P . M artin w, L. Martinez-Laso *, F. M arzano31, G .G .G . Massaro b,
K . M azum darr, P. M cB rid e 1", T. M cM ahonaq, D. M cN a llyat, M. M erkag, L. M erola33,
M . M eschini0, W .J. M etzgerac, Y . M i\ A. M ih u lk, G .B. M ills v, Y . M irq, G. M irab elliai,
J. M n ich 3, M . M ö lle r3, B. M onteleoni0, R. Morand d, S. M organti31, N .E. M o u laiq, R . M ountae,
S. M ü lle r3, A. N adtochy3->,E. N ag y\ M . N apolitano33, F. Ncssi-Tedaldia\ H . Newm anae,
C. N eyerat, M .A . N ia z q, A. N ip p e3, H . N ow ak3S, G . O rgantini31, D. Pandoulas3, S. Pao letti0,
P. Paolucci33, G . Pascale31, G. Passaleva0,af, S. P a trice lli3a, T. P a u l6, M . Pauluzziaf, C. Pau s3,
F. Paussa\ Y .J. P e i3, S. Pensottiy, D. Perret-Gallixd, J. Perrierr, A. Pevsnere, D. Piccoloaa,
M . Pieri p , P.A . Piro u éah, F. P la s ilad, V. Plyaskinz, M. P o h lat, V. Pojidaevz-°, H. Postem a11,
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M . Rattaggiy, G . Raven b, P. R azisab, K . R ead ad, D. R e n 3t, Z. R e n q, M . Rescigno31,
S. Reucroftj, A. R ick e r3, S. Riem ann35, B.C . Riem ers3q, K. R ile sc, O. R in d c, H .A. R iz v iq,
S. R o 30, F .J. Rodriguez*, B .P . R o e c, M . Röhner3, L. Rom ero*, S. Rosier-Leesd,
R . Rosm alenac, Ph. Rosselet \ W . van Rossum b, S. R o th 3, A. R ubbian, J.A . R u b io p,
H. Rykaczewski3t, M . Sachw itz35, J. SalicioP, J.M . Salicio*, G .S. Sanders v, A. Santocchia af,
M .S. Sarakinos", G. Sarto rellig,q, M . Sassowsky3, G. Sauvaged, V. Schegelskyaj, D. Schm itz3,
P. Schm itz3, M . Schneegansd, H . Schopper3U, D J. Schotanusac, S. Shotkin", H J. Schreiberas,
aa
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V. Soulim ov33, C. Spartiotise, T . Spickerm ann3, P. Sp illan tin i°, R . Starosta3, M. Steuer6,1*,
D .P. Stickland3h, F. Sticozzi", H . Stoneah, K . Strauch“ , B.C . Stringfellowaq, K . Sudhakarh,
G. Sultanovq, L .Z . Su n s,q, G .F. Susinnor, H . Suter31, J.D . Sw ainq, A.A. Syed3C, X .W . Tangf,
L. Taylor J, G. T erz iy, Samuel C.C. T in g ", S.M . T in g D, M . T o n u tti3, S.C. Tonw arh, J. Tóth*,
A. Tsaregorodtsev3-*, G . T sip o litis38, C. T u lly ab, K .L . Tungf, J. U lb rich t3*, L. Urbán
U . U w e r3, E. V alente31, R .T . Van de W a lle 3C, I. V etlitskyz, G . V iertel31, P. V ikasq, U . V ik a sq,
M . Vivargentd, H . Vogelag, H . Vogt35,1. V orobievz, A.A. Vorobyov«, L. Vuilleum ier1,
M . W adhw ad, W . W a llra ff3, C. W ang", C .R . W ang5, X .L . W angs, Y .F . W ang", Z.M . W angq-S,
C. W arn er3, A. W eb er3, J. W eb er31, R . W e ill \ T .J. W enaus“ , J. W enningerr, M . W h iten,
C. W illm ott *, F. Wittgenstein p, D. W rig h t3h, S.X . W u q, S. W ynhoff3, B. Wyslouch ",
Y .Y . X ie am, J.G . X u f, Z.Z. X u s, Z .L. X u e am, D .S. Y a n 31", B.Z. Yang5, C.G. Yangf, G. Yangq,
C.H . Y e q, J.B . Y e s, Q. Y e q, S.C. Y e h 3V, Z .W . Y in 3m, J.M . Y o u q, N. Yunusq, M . Yzerm anb,
C. Zaccardelli3e, N. Z aitsev33, P. Zem p3t, M . Zengq, Y . Zenga, D .H . Zhangb, Z.P. Zhangs,q,
B. Zhou1, G .J. Zhouf, J.F . Zh o u 3, R .Y . Z h u ae, A. Zichichi g-P>q and B.C.C. van der Zw aanb
â L Physikalisches Institut, R W T H , W-5100 Aachen, F R G 1
and III. Physikalisches Institut, RW TH , W-5100 Aachen, F R G 1
b National Institute for High Energy Physics, N IK H E F \ NL-1009 D B Amsterdam, The Netherlands
c University of Michigan, Ann Arbor, M l 48109, USA
d Laboratoire d’Annecy-le- Vieux de Physique des Particules, LAPP, IN2P3-CNRS, B P 110,
F-74941 Annecy-le-Vieux CEDEX\ France
e Johns Hopkins University; Baltimore, M D 21218, USA
f Institute of High Energy Physics, IH E P , 100039 Beijing, China
g IN F N - Sezione di Bologna, 1-40126 Bologna, Italy
h Tata Institute of Fundamental Research, Bombay 400 005, India
1 Boston University, Boston, MA 02215, USA
j Northeastern University, Boston, MA 02115, USA
495
Volume 315, number 3,4
P H Y SIC S L E T T E R S B
7 October 1993
k Institute of Atomic Physics and University of Bucharest, R-76900 Bucharest, Romania
1 Central Research Institute fo r Physics of the Hungarian Academy of Sciences, H-Ì525 Budapest i 14, Hungary2
m Harvard University, Cambridge, MA 02139, USA
n Massachusetts Institute of Technology, Cambridge, MA 02139, USA
0 IN FN Sezione di Firenze and University of Florence, 1-50125 Florence, Italy
p European Laboratory for Particle Physics, CERN, CH-1211 Geneva 23, Switzerland
q World Laboratory, F B L JA Project, CH-1211 Geneva 23, Switzerland
T University of Geneva, CH-1211 Geneva 4, Switzerland
s Chinese University o f Science and Technology, USTC, Hefei, Anhui 230 029, China
1 University of Lausanne, CH-1015 Lausanne, Switzerland
u Lawrence Livermore National Laboratory, Livermore, CA 94550, USA
v Los Alamos National Laboratory, Los Alamos, N M 87544, USA
w Institut de Physique Nucléaire de Lyon, IN2P3-CNRS, Université Claude Bernard, F-69622 Villeurbanne Cedex, France
x Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas, CIEM AT, E-28040 Madrid, Spain
y IN F N - Sezione di M ilano , 1-20133 M ilan , Italy
z Institute of Theoretical and Experimental Physics, IT EP, Moscow, Russia
aa IN FN - Sezione di Napoli and University of Naples, 1-80125 Naples, Italy
ab Department of Natural Sciences, University of Cyprus, Nicosia, Cyprus
ac University of Nymegen and N IK IIE F , NL-6525 ED Nymegen, The Netherlands
ad Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
ae California Institute o f Technology, Pasadena, CA 91125, USA
af IN F N - Sezione di Perugia and Università Degli Studi di Perugia, 1-06100 Perugia, Italy
ag Carnegie Mellon University, Pittsburgh, PA 15213, USA
ah Princeton University, Princeton, N J 08544, USA
ai IN FN - Sezione di Roma and University of Rome, “La Sapienza ”, LOO 185 Rome, Italy
aj Nuclear Physics Institute, St. Petersburg, Russia
ak University of California, San Diego, CA 92093, USA
Dept, de Fìsica de Partículas Elementales, Univ. de Santiago, E-15706 Santiago de Compostela, Spain
am Shanghai Institute of Ceramics, SIC , Shanghai, China
an Bulgarian Academy of Sciences, Institute of Mechatronics, BU-1113 Sofia, Bulgaria
ao Center for High Energy Physics, Korea Advanced Inst, o f Sciences and Technology, 305-701 Taejon, Republic of Korea
ap University of Alabama, Tuscaloosa, A L 35486, USA
aq Purdue University, West Lafayette>IN 47907, USA
ar Paul Sc herrer Institut, PSI, CH-5232 Villigen, Switzerland
as D ES Y - Institut für Hochenergiephysik, 0-1615 Zeuthen, FR G
at Eidgenössische Technische Hochschule, E T H Zürich, CH-8093 Zurich, Switzerland
au University of Hamburg, W-2000 Hamburg, FR G
av High Energy Physics Group, Taiwan, China
w
♦
Received 5 August 1993
Editor: K. Winter
The 5* matrix ansatz is a rigorously model independent approach to describe the cross-sections and asymmetries in
e+e“ annihilation. Using the cross-sections and asymmetries measured with the L3 detector during the 1990 and 1991
running period, we determine the mass and the width of the Z boson, the contributions of the Z exchange and of the yZ
interference. Including the polarization of the t lepton in the analysis, the leptonic helicity amplitudes of the scattering
process are determined assuming lepton universality. The results are compared with other model independent ansatzes
as realized in Z F IT T E R . A systematic bias of the Z mass due to the yZ interference term is detected, which leads to
an underestimation of the error on mz for model independent determinations.
1
Supported by the German Bundesministerium
Forschung und Technologie.
für
2
Supported by the Hungarian O TKA fund under contract
number 2970.
Volume 315, number 3,4
7 October 1993
PH Y SIC S L E T T E R S B
1. Introduction
2. The L3 detector
The successful operation of LEP has allowed a
precise measurement of the e+e_ annihilation crosssections and asymmetries near the Z resonance. The
mass, total and partial widths of the Z boson have
been determined with an excellent accuracy. The ex­
perimental results confirm the Standard Model with
percent precision [1-3].
In this paper we investigate to what extent a sat­
isfactory description of the experimental data on the
Z line shape can be reached with minimal assump­
tions. We base this study on an S matrix approach,
the details of which are explained elsewhere [4]. The
scattering process is described by the superposition of
massless and massive boson exchange, without mak­
ing detailed assumptions about the dynamics of the
process.
For the total cross-section the S matrix approach
is equivalent to the model independent approach
derived earlier [5]. Other model independent ap­
proaches to the Z line shape have been described in
the literature [6-8] and used by L3 in previous stud­
ies [ 1]. All of these studies have in common that the
interference between the massless and the massive
boson exchange for hadronic reactions is fixed to the
value it assumes in the Standard Model. This treat­
ment was shown to be sufficient at the previous level
of accuracy, since the interference term is suppressed
in the vicinity of the Z resonance. The present ac­
curacy of lineshape measurements allows, however,
to determine limits on the value of this interference
term for total cross-sections as well as asymmetries,
and for leptonic and hadronic final slates separately.
We also study the influence of this term on the value
of the Z mass, and the potential bias caused by fix­
ing it to its Standard Model value. A discussion of
the theoretical predictions for the interference term
in the Standard Model and its measurability can be
found in ref. [9].
We use our experimental measurements of the total
cross-sections, the forward-backward asymmetries for
all leptonic and hadronic Z decay channels as well
as the polarization of tau leptons from Z decay. The
total luminosity used is 17.2 pb*” 1 (corresponding to
about 40000 leptonic and 423 000 hadronic events)
collected with the L3 detector in 1990 and 1991.
The L3 detector at LEP covers 99% of the full solid
angle. It is designed to measure energy and position
of leptons, photons and jets with high precision. A de­
tailed description of the detector and its performance
can be found elsewhere [10].
The detector consists of a time expansion cham­
ber (T EC ) for the tracking and vertex reconstruc­
tion of charged particles, a high resolution electromag­
netic calorimeter made of about 11 000 bismuth ger­
manium oxide (BG O ) crystals, a hadron calorimeter
(H C A L) with uranium absorber and brass propor­
tional wire chambers and a high precision muon spec­
trometer, consisting of three layers of multi-wire drift
chambers. A cylindrical array of 30 scintillation coun­
ters is installed in the barrel region between the BGO
and the HCAL. The luminosity is measured by the
luminosity monitors, two electromagnetic calorime­
ters, situated symmetrically on either side of the inter­
action point. Each calorimeter is a finely segmented
and azimuthally symmetric array of 304 BGO crystals
covering the polar range 24.93 < 0 < 69,94 mrad. All
detectors are inside a 12 m inner diameter solenoidal
magnet which provides a uniform magnetic field of
0.5 T along the beam direction.
3. Z lineshape measurements
Operating the LEP storage ring in the vicinity of
the Z mass with high luminosity permits a detailed
study of the lineshape of the Z resonance. We have
performed measurements of the reactions
(1) e+e- -» hadrons,
(2) e+e~
ß +ju~(y),
(3 )e+e~ -► T+r~ (y ) ,
(4) e+e” —» e+e~ ( y ) .
The analysis methods used for these reactions are de­
scribed in detail elsewhere [1,11]. The cross-sections
and asymmetries determined with the data taken in
the 1990 and 1991 runs of LEP have been published
[1 ]. Additionally, we include the average r polariza497
ar
PHYSICS LETTERS B
Volume 315, number 3,4
tion [ 12 ] at x/? = 91.222 GeV, Vx = -0.132±0.026±
0 .021.
The matrix element for the exchange of a photon
and a Z boson in e+e" annihilation into massless
fermions can be written as
M u (s)
+
R
-f CTq + CTj
(1)
m\ - imzr z.
—
ÍJ2
”
*
0*3•
(4 cont’d)
^
+ (j - m\ ) j {A
( s - m l)2 + m lT 2
z
A = tot, fb, poi, fbpol,
lÆyl2 if A = tot,
0 if A ^ tot.
(5)
The pole position for the Z-boson is given by sz, R y
and Rz are the residuals for the photon and Z boson
respectively. R r is defined by
( 2)
Qf is the charge of the corresponding fermion and
a (s) the running QED coupling constant.
The coefficients R ¿ describe the four helicity am­
plitudes for the Z exchange:
I* 9
fr fR+),
(6 )
The Z exchange term, rA, and the yZ interference
term, j Ai are given by
rf
'A
+
R z {tL eg
a 2 + Cr3 ,
ijy
aA(s) = \na 1 ' A
/
n~ 1
with
Rz
crS, (¿ )
,y
/ = 0,3,
sz
-j- (7q -
where rl is the photon exchange term:
OG
fi
Sz
tf p o lO ) =
The cross-sections can be parametrized as follows:
4. The S matrix formalism
R
7 October 1993
R
/=i
ÎA
f/
+ 2 ^ l m C (A ,
mz
2 Re C\ ,
rA
f
(7)
/=1
u
where {±1} indicates that the sign of Rz * and of Rz
corresponds to the sign of cr, in eq. (4).
For the hadron channel one has to sum rA, rA and j A
over all colours and open flavours. The asymmetries
are defined by
A a (s ) =
<rA{s)
A zfz tot.
(8 )
Ö’tot (S )
,fl
Rz = ^z(eL eR
♦fR fL ) ,
The coefficients f)ll (s) of the power series in eq. ( 1)
describe non-resonant contributions to the scattering
process. As shown in refs. [4,5] these are numerically
small around the Z resonance and are neglected for
our analysis.
The cross-sections, a¡, arising from the correspond­
ing M u can be combined to four linearly independent
cross-sections, observable at LEP:
Photonic corrections are included by convolution, for
details see ref. [4],
Eqs. (l)- (8 ) completely define the framework of
an S matrix analysis of the Z lineshape. However,
when comparing Z parameters to other approaches,
the following clarification is necessary:
First, it should be noted that in the S matrix ap­
proach the total width, r z, is constant in contrast to
the parametrization of the Breit-Wigner resonance of
the Z lineshape, where p¿ is a result of quantum cor­
rections, which are ^-dependent. This leads to a trans­
formation of the Z mass, mz, to mz and of the total
Z width, 7z, to Fz [13]:
ö-tot fa) = +a0 + ÖÏ + <r2 + <?3
rñz — [1 + (/z/^z)2] “ l/2mz ~ wz - 34 MeV,
f2
Rz = -^z(eReL
ß
R'z = ^z(eReL
fR" fL+) ,
*
(3)
,
O fb (s ) =
498
+ # 0 “ G I + Ö2
0*3 Í
(4)
r
[1 + (r z/mz)2]- l,2r z * r z - l MeV.
(9)
Volume 315, number 3,4
PH Y SIC S L E T T E R S B
Thus, the mass, Tñz, obtained from S matrix fits
should be shifted by -34 MeV and the total width,
F z, by -1 MeV, with respect to the results obtained
from the standard procedure [ 1].
Second, with vector and axial vector couplings of
the Z to the fermions, eqs. (3) can be expressed as
R
r o
K(gv + ga)(gl + g l),
ri
R
K (g ï + g l) (gl - g l),
1
2
R
K(gv - g l) (gl - g l) ,
R
Í3
K ('gv ~ ga) (gl + g l),
with
K
\
V lln a
( 11)
Interpreting the couplings, gv and ga, as effective pa­
rameters, the weak corrections are absorbed in the
couplings.
For leptonic channels, the cross-sections (eqs, (4 )),
or respectively, the helicity amplitudes (eqs. (3) ), can
all be measured separately, assuming lepton universal­
ity. However, for hadronic reactions the contributing
flavours cannot all be separated. Therefore, one can
only measure the sum of all contributions according
to eq. (7), i.e. in terms of r^ad and ybad. In previous
model independent studies [ 1], terms relating to jxSi
were evaluated using the Standard Model relation
gl = g l( l - 4|Qf| sin20W) ,
photon exponentiation. Interference between initial
and final state is neglected for radiative corrections.
The data listed in ref [ 1] have systematic uncer­
tainties in addition to their statistical errors. These
are caused by selection bias, theoretical uncertainties,
limited Monte Carlo statistics etc. We consider a par­
tial error correlation calculating a #2,
X
( 10)
( 12 )
with sin 0w taken from the leptonic lineshape.
5. S matrix analysis
We used the program SM ATASY [14] together
with Z FIT T ER version 4,53 [7]. SM ATASY relies
on the S matrix ansatz for the total cross-section
and for the three asymmetries. It is a generalization
of the existing Z FIT T ER branch ZUSM AT, which
considers only the total cross-section. Initial and fi­
nal state QED corrections are taken into account by
convolution in 0 (a 2), higher order corrections for
initial state radiation are considered with common
7 October 1993
-l
 l V ~ xÀ,
(13)
where A is a column vector with elements such as
(<7th - crexp) and M th - ^ exp) and V is the N x N
error correlation matrix between measurements. The
diagonal elements of V are given by the quadratic
sum of statistical and systematic errors, while the offdiagonal elements are given by the product of the com­
mon systematic errors. This can be generalized also to
the common systematic error between different data
sets. The procedure to implement the LEP energy un­
certainty is described in detail elsewhere [15].
6. Results
To study the influence of the yZ interference on the
final results of mz and Fz the fits in the following
sections are performed in two steps:
(a) All parameters except the photon exchange, ry
A,
are left free.
(b) In addition to r\, the contributions to the yZ
interference, 7 th0atd, are fixed to the value expected by
the Standard Model.
In order to reduce the number of free parameters,
lepton universality is assumed. The photon exchange
term, rj0l (see eq. (2 )), is fixed using the running
coupling constant value at LE P energies, | a -1 (5 )
128,8, The quality of all fits is good. The x 2per degree
of freedom varies between 0,75 and 0.78. The results
of the S matrix approach are compared to those of
the model independent ansatzes of ZFITTER.
6.1. A fit to the total cross-section and
forward-backward asymmetry
We perform a fit to the leptonic and hadronic
cross-section data and the leptonic forward-backward
asymmetries according to eq. (5), Assuming lepton
universality, one gets for case (a) 8 and for (b) 6 free
parameters: mz, Fz, and rth0atd,
for hadrons and
499
Volume 315, number 3,4
PH Y SIC S L E T T E R S B
Table 1
Results of the S matrix fit to total cross-sections and
forward-backward asymmetries: (a) all parameters except
the photon exchange are left free; (b) in addition the yZ
interference terms are fixed to the Standard Model expec­
tation.
Case (a)
Case (b)
Parameter
mz (G eV)
Pz (G eV)
Jep
'tot
•lep
-'tot
Jep
91.152 ± 0.015
2.494 ±0.012
91.160 ± 0.010
2.492 ±0.012
0.141 ±0.002
0.140 ± 0.002
0.032 ± 0.064
fixed to 0.0058
mz (G eV)
Pz (G eV)
lepQ
R
iepl
R
0.004 ±0.001
0.004 ±0.001
0.674 ±0.087
2.859 ± 0.030
0.720 ±0.700
0.675 ± 0.087
2.855 ± 0.029
fixed to 0,219
Xd
tot
/had
-'tot
rton Jtou rfbP>^lbP f ° r lePtons- The results are shown
in table 1 . Comparing both fits, one notices that mz
increases by 8 MeV, when the yZ interference terms
are fixed in fit (b ), whereas the error on mz decreases
by 5 MeV. That means that fixing the yZ interference
introduces a systematic bias in the determination of
the Z mass. For all the other parameters the mean
values and errors remain unchanged. If one compares
the values for Tñz with the results for mz, determined
with Z FIT T ER (see table 3), one finds the expected
offset between mz and mz of -34 MeV, only when
the yZ interference term for the hadron channel j th0atd
is treated in the same way.
(5.2. Determining the helicity amplitudes with the S
matrix approach
In addition to the data set used in section 6.1, the
t polarization measured in 1991 is also taken into
account. Assuming lepton universality and C P con­
servation three independent helicity amplitudes /?zp0,
i?zPl and R f 1 can be determined corresponding to
eqs. (3). It should be noted here, that in the r channel
one has the possibility to determine all four ampli­
tudes, because one can measure the total cross-section
and all three asymmetries. The amplitudes i?zpl an(^
i? 2 p3 are equal by time reversal symmetry. The avail­
able information is not sufficient to express rtoad and
7totd by helicity amplitudes. Therefore, they still re­
main independent parameters. The number of free pa­
rameters is 7 in case (a) and 6 in case (b). For the
500
Table 2
Results of the S matrix fit to total cross-sections, forwardbackward asymmetries and x polarization: (a) all parame­
ters except the photon exchange are left free; (b) in addi­
tion the hadronic yZ interference terms for the total crosssection are fixed to the Standard Model expectation.
Parameter
%
7 October 1993
Case (a)
Case (b)
91.155 dh0.013
2,494
0 ,0 1 2
91.160 ± 0 .0 10
2.492 ± 0 .0 12
0.429 ± 0 .0 12
-0.370 ± 0.003
0.429 ± 0 .0 1 2
0.323 i 0.016
2.860 ± 0.030
0.620 ± 0.620
-0.370 ± 0.003
0.323 ± 0.016
2.856 ± 0.029
fixed to 0.219
mz (G eV)
r z (G eV)
91.189 ± 0.013
2.495 ± 0 .0 1 2
91.194 ± 0 .0 10
2.493 ± 0 .0 12
&,lcp
ÔlcP
&a
-0.037 ± 0 .0 10
-0.037 ± 0 .0 10
-0.4991± 0,0019
—0.4988± 0.0019
0.2317 ± 0.0037
0.2316 ± 0.0037
D¡ep2
AZ,
rhad
tot
jhad
Jtot
sin¿ 0w
fit we assume the helicity amplitudes to be real.
The first part of table 2 shows the results. The sec­
ond part of table 2 shows quantities which are derived
from the parameters in the upper part of the table,
mz, Pz and mz, P z are related by eq. (9). Refering to
eqs. ( 1 0 ) one can write the amplitudes R 1
^ 1as func­
tion of the couplings gjep and galep. Here we use i? 2 P0
and R l pi to determine the couplings and sin2 #w, de­
fined by eq. ( 1 2 ).
For the helicity fit one finds the same behaviour
for mz as in the previous section. The error for mz
increases when yth0ad is left free. The mean values and
errors for all other parameters are almost unchanged.
As a cross check we compare the results of the S
matrix approach with the Z FIT T ER results using the
same measurements as for the helicity fit. In Z F IT ­
T ER two alternative model independent ansatzes are
applied: the first is based on the assumption of real
vector and axial vector couplings of the Z boson to
fermions; the second relies on the assumption that
scattering through the Z boson may be considered as
subsequent formation and decay of a resonance de­
scribed by the widths into the initial and final state
fermions. The complications due to the handling of
the yZ interference contribution for the hadronic final
state, 7 th0atd, are solved by fixing it to the value expected
in the Standard Model. In order to check this proce-
P H Y SIC S L E T T E R S B
Volume 315, number 3,4
Table 3
Results of a model independent fit to total cross-sections,
forward-backward asymmetries and r polarization using
Z F IT T E R : (a) all parameters except the photon exchange
are left free; (b) in addition the hadronic yZ interference
terms for the total cross-section are fixed to the Standard
Model expectation.
Parameter
Case (a)
Case (b)
mz (G eV )
Tz (G eV)
91.187 ±0.013
2.492 ±0.012
91.194 ±0.009
2.490 ±0.012
?,lep
—0.040 ±0.006
-0.040 ±0.006
? Jep
-0.4989± 0.0016
1.751 ±0.011
1.0 0
± 0.86
-0.4986± 0.0016
1.750 ±0.011
fixed to 0.219
0.2300± 0.0030
0.2300± 0.0029
^had
/had
J tot
sin¿ 0 \y
dure a five parameter fit to the leptonic and hadronic
cross-section data and the leptonic forward-backward
asymmetry and the r polarization is performed.
In case (a) the term yjjjt1 is left free, and in case (b)
the standard Z FIT T ER code is used with the fixed
yth0ad. The results are shown in table 3* The second part
of table 3 shows sin2 0W derived from &jep and #Jep.
If one modifies the standard Z FIT T ER program to
allow for a fit to the hadronic interference term yth0ad,
as it is done for the results (a) in table 3, one also finds
that the mean value for mz decreases and that the
error increases with respect to the standard Z FIT T ER
results (b), by the same amount observed for the fits
using the S matrix formalism,
A comparison of the second part of table 2 with
table 3 shows that one gets the same results for the
S matrix approach as with Z FIT T ER . The 5 matrix
approach can, however, reproduce the mean value and
error for mz only if the treatment of yth0ad is identical
to the standard ZFIT T ER .
7. Conclusions
- The S matrix approach allows a general model inde­
pendent investigation of the cross-sections and asym­
metries measured in the vicinity of the Z resonance
and a determination of the mass and the width of the
Z boson.
- The S matrix approach can reproduce, with addi­
tional assumptions, the results of other model inde­
7 October 1993
pendent ansatzes realized in ZFITTER.
- For model independent determinations, the fixing
of the hadronic interference term for the total crosssection to the Standard Model expectation value leads
to a systematic bias in the value of the Z mass and un­
derestimates the systematical error on mz. This effect
is also observed for ZFITTER. All other parameter
are independent of yth0ad. Although the yZ interference
contribution to the hadronic final state, yth0ad, is sup­
pressed and its measurement is very poor, the influ­
ence ofythod on the value mz is not negligible. The pro­
cedure of fixing yth0ad by expressions predicted by the
Standard Model should be checked when performing
model independent fits to avoid misinterpretation of
the results. An improved measurement of
is ex­
pected by running with higher luminosity at energies
off the resonance (see [9 ]).
The results of the 5 matrix approach confirm the
present values for mz and 7z within their errors. The
good agreement of the 5 matrix approach with the
Standard Model fit values means that there is no ev­
idence for new physics in the data.
Acknowledgement
We express our gratitude to the CERN accelerator
divisions for the excellent performance of the LE P ma­
chine. We acknowledge the effort of all engineers and
technicians who have participated in the construction
and maintenance of this experiment.
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