M easurem ent ofthe E nergy Spectrum of
e
from M uon D ecay and
Im plications for the Lorentz Structure of the W eak Interaction
B.A rm bruster,1 I.M .Blair,2 B.A .Bodm ann,3 N .E.Booth,4 G .D rexlin,1 V .Eberhard,1
J.A .Edgington,2 C .Eichner,5 K .Eitel,1 E.Finckh,3 H .G em m eke,1 J.H o l,3 T .Jannakos,1
arXiv:hep-ex/9806024v1 24 Jun 1998
P.Junger,3 M .K leifges,1 J.K leinfeller,1 W .K retschm er,3 R .M aschuw ,5 C .O ehler,1
P.Plischke,1 J.R app,1 C .R uf,5 M .Steidl,1 O .Stum m ,3 J.W olf,1 B.Zeitnitz1
1
Institutfur K ernphysik I,Forschungszentrum K arlsruhe,Institutfur experim entelle
K ernphysik,U niversitatK arlsruhe,Postfach 3640,D -76021 K arlsruhe,G erm any
2
Physics D epartm ent,Q ueen M ary and W est eld C ollege,M ile End Road,London E1 4N S,
U nited K ingdom
3
Physikalisches Institut,U niversitatErlangen-N urnberg,Erwin-Rom m el-Stra e 1,D -91058
Erlangen,G erm any
4
D epartm ent ofPhysics,U niversity ofO xford,K eble Road,O xford O X 1 3R H ,U nited K ingdom
5Institutfur Strahlen- und
K ernphysik,U niversitatB onn,N u allee 14{16,D -53115 B onn,
G erm any
Presentaddress: D epartm ent ofPhysics and A stronom y,U niversity ofA labam a,Tuscaloosa A L
35487
A bstract
T he K A R M EN experim ent uses the reaction
theenergy distribution of
T he
e
e
12C
(
e ,e
)12N g:s: to m easure
em itted in m uon decay atrest
analog !L of the fam ous M ichel param eter
+
! e+ +
e+
.
has been derived
from a m axim um -likelihood analysis ofevents near the kinem atic end point,
3:8
E m ax = 52:8 M eV . T he result,!L = (2:7+3:
3
3:1)
10 2 ,is in good agree-
m ent w ith the standard m odel prediction !L = 0. W e deduce a 90%
con dence upper lim it of ! L
0:113, w hich corresponds to a lim it of
1
jgRS L + 2gRT L j 0:78 on the interference term between scalar and tensor
coupling constants.
Experim ental results from nuclear
decay and m uon decay form the basis of the V -A
hypothesis,w hich isan essentialfeature ofthe standard m odel(SM )ofelectroweak interactions. T he rate ofm uon decay,the purely leptonic process
+
! e+ +
e+
,hasbeen used
to determ ine the universalFerm icoupling constantG F . Precise m easurem ents ofthe shape
ofthe e+ energy spectrum ,the decay asym m etry between the
+
spin and e+ m om entum ,
and the polarisation vector ofthe e+ have led to bounds on the scalar,vector and tensor
coupling constants,w hich form theLorentz structure ofthecharged weak interaction.T hese
results com bined w ith the inverse process
+e !
+
e
underpin the SM assum ption
oflepton num berconservation,the V -A interaction and universality [1].A llexperim ents up
to now support the V -A structure ofthe weak interaction;however,substantialnon-(V -A )
com ponents are not ruled out.
C om plem entary to these experim ents,w hich are allbased on observation ofthe charged
leptons only,the K arlsruhe R utherford M edium Energy N eutrino experim ent (K A R M EN )
determ ines the energy spectrum of the
e
em itted in the decay
+
! e+ +
e+
of un-
polarized m uons to draw conclusions on the Lorentz structure. In the well-know n case of
e+ spectroscopy, it is the M ichel param eter
w hich governs the shape of the e+ energy
spectrum . In an analogous way,the shape ofthe
e
energy spectrum is determ ined by the
param eter !L , w hich also depends on vector, scalar, and tensor com ponents of the weak
interaction,but in a di erent com bination. In the SM allnon- (V -A ) com ponents vanish,
and !L is predicted to be 0. T hus an upper lim it on !L derived from the analysis ofthe
e
energy spectrum provides new lim its on nonstandard couplings.
A llfeaturesofm uon decay arem ostgenerally described by a local,derivative-free,leptonnum ber-conserving,four-lepton point interaction w ith the m atrix elem ent given by [2]
2
4
M = p GF
2
T he index
X
g he j j( e)n ih( )m j j i:
(1)
= S ;V ;T
; = R ;L
labels the type ofinteraction
(S = 4-scalar,V = 4-vector,T = 4-tensor)and
the indices and indicate the chirality (L = left-,R = right-handed)ofelectron and m uon
spinors,respectively. In this representation the chirality ofthe neutrino n or m is xed to
be equalto thatofthe associated charged lepton forthe V interaction,butopposite forthe
S and T interactions. A s G F sets the absolute strength ofthe interaction,the ten coupling
constants g are dim ensionless com plex quantities norm alized by
3jgRT L j2 + 3jgLT R j2 +
w ith gRT R = gLT L
1
( jgS j2 + jgV j2)= 1
; = R ;L 4
X
(2)
0. In the SM ,m uon decay is a pure V interaction m ediated between
left-handed particles,so allcoupling constants vanish except gLV L
1. A lthough this rep-
resentation is elegant from the theoreticalpoint ofview ,the individualcoupling constants
cannot be determ ined directly by experim ent. H owever,the m easurable param eters ( , ,
!L ,etc.) are expressable as positive sem ide nite bilinear com binations ofg
from w hich
upper or lower lim its for the coupling constants can be derived.
T he possibility ofm easuring !L w ith the K A R M EN experim entwas rstpointed outby
Fetscher [3]. M ore recently G reub et al.[4]have calculated the spectrum ofleft-handed
e
including radiative corrections and e ects of nite lepton m asses. Taking signi cant term s
only,the spectrum dN L =dx can be described by
G 2F m 5
dN L
=
Q fG 0(x)+ G 1(x)+ !L G 2(x)g
dx
16 3 L
w here m
is the m uon m ass,x = 2E =m
(3)
is the reduced neutrino energy,and Q L denotes
the probability ofem ission ofa left-handed
e. T he function
G 0(x) describes the pure V -A
interaction,G 1(x) takes into account radiative corrections,and !L G 2(x) includes the e ect
ofscalar and tensor com ponents according to
!L =
jgRS R j2 + 4jgLV R j2 + jgRS L + 2gRT L j2
3
:
4 jgRS L j2 + jgRS R j2 + 4jgLV L j2 + 4jgLV R j2 + 12jgRT L j2
3
(4)
T he calculated
energy spectra for di erent values of! L are show n in Fig.1(a). M o-
e
m entum conservation in the decay xes the em ission direction of
point to be opposite to that ofthe positron and the
near the kinem atic end
e
. Together w ith angular m om entum
conservation thisim plies suppression ofem ission ofleft-handed
e
in the case ofvector cou-
pling,w hile allother couplings enhance the decay rate at the end point. T he totaldecay
rate,and therefore the integralneutrino ux,is unchanged by nonstandard interactions.
T heK A R M EN experim entusesthepulsed spallation neutron facility ISIS attheR uther+
ford A ppleton Laboratory to investigateneutrinosfrom
decay.T he800 M eV proton beam
from ISIS is stopped in a Ta-D 2O target producing neutrons and pions. A llcharged pions
are stopped inside the target w ithin 10 10 s, the
m aterialw hile the
burst of
+
/
, e,and
+
decay chain
+
+
!
+
being absorbed by the heavy target
,
+
! e+ +
e+
producesan intense
,em itted isotropically w ith equalintensity. Since both
decay at rest,the energy spectra ofthe neutrinos are wellde ned. T he
m onoenergetic
w ith E
= 29:8 M eV ;the
e
and
from the
+
+
+
and
+
decay produces
decay have continuous
energy distributionsup to E m ax = 52:8 M eV .T he tim e structure ofISIS | two 100 nsw ide
proton bunches324 nsapartand recurring at50 H z| determ inestheproduction tim eofthe
di erent
avors: the short
+
lifetim e ( = 26 ns) leads to two
pulses w ithin the rst
500 nsafterbeam -on target.T hese pulsesare wellseparated in tim e from the production of
e
and
,w hich follow w ith the m uch longer lifetim e ofthe
+
(
+
= 2:2 s). T his leads
to a suppression factor ofabout 104 for cosm ic-ray background.
T he neutrinosare detected in a segm ented 56 ton liquid scintillation calorim eterconsisting of512 opticalm odules,each w ith a length of3.53 m and a crosssection of18 18 cm 2 [5].
T he detector is an alm ost com pletely (96% ) active calorim eter optim ized for the m easureq
m ent ofelectrons around 30 M eV and achieves resolutions of (E )=E = 11:5% = E (M eV )
for energy,and (X )
7 cm for position m easurem ent. A 7000 ton shielding steelblock-
house togetherw ith two layersofactive veto counterssuppressesbeam -correlated spallation
neutrons and cosm ic-ray m uons.
T he signature that unam biguously identi es a
4
e
is a delayed coincidence consisting of
an electron from the charged current reaction
e
12
C ( e ,e )12N g:s: in the tim e w indow of
production followed by a positron from the subsequent
decay of 12N g:s: ( = 15:9 m s)
atthe sam e location in the detector. Each event fully contained w ithin the centraldetector
w ith tim e 0.6{9.6 s after beam -on target and energy 10{36 M eV is identi ed as electron,
provided it is followed by a positron event w ithin 0.5{36 m s w ith energy 3.5{16.5 M eV .
W e dem and the sequence to be detected in the sam e or adjacent m odule w ithin a distance
X
35 cm along them oduleaxis.C utsused to reduce cosm ic background arethesam eas
used in previousdata evaluations[6].In data accum ulated between June1990 and D ecem ber
1995 | corresponding to 9122 C ofprotons or 2:51
we nd 513 e /e+ sequences. Subtracting 13:3
1021
+
decays in the ISIS target |
0:8 background events and taking into
account an overalldetection e ciency = 32:8% ,the ux-averaged cross section is
h iexp = (9:4
0:4(stat) 0:8(sys))
10 42 cm 2.
(5)
T his is in good agreem ent w ith di erent theoretical calculations of h ith in the range of
(9.1{9.4) 10 42 cm 2 [7,8].
A sthe recoilenergy ofthe 12N nucleusisnegligible,the
e
energy E isdeterm ined from
the m easurem ent ofthe electron energy E e via the kinem atic relation E = E e + Q ,w here
Q = 17:3 M eV isthe Q value ofthe detection reaction. T he energy dependence ofthe cross
section isdom inated by thephase-space factor(E
e
Q )2.T herefore,a low rateofadditional
atthe kinem atic end point E m ax = 52:8 M eV due to nonstandard couplings is translated
to the observation ofa signi cantly higher rate ofelectrons and thus to a distortion ofthe
visible energy spectrum ofFig.1(b).
T he K A R M EN calorim eterallow sa precise m easurem entofthe energy E e [see Fig.1(c)].
T heenergy spectrum of
e
from
+
decay isthen determ ined in two steps.First,weapply the
procedure ofregularized unfolding described by Blobel[9]to derive the true electron energy.
T his m ethod takes into account the detector response and m inim izes inherent instabilities
(oscillating solutions) by dem anding a priori a certain degree of sm oothness of the true
electron distribution depending on statisticalaccuracy. T he
5
e
energy distribution is then
calculated from the num ber of prim ary electrons, w ithin a given interval E from the
unfolding procedure, divided by the corresponding m ean cross section. T his yields a
e
energy spectrum w ith seven data points as show n in Fig.2 and com pared w ith the V -A
expectation. T his represents the rst m easurem ent ofthe neutrino energy spectrum from
m uon decay in addition to the well-know n e+ spectrum .
Because ofthe strong energy dependence ofthe detection crosssection the m ostdetailed
inform ation on !L and Q L isobtained from the experim entalelectron spectrum ofFig.1(c).
T he analysis is done by two independent m ethods: (1) the investigation ofthe m easured
decay rate on the basisofthe ux-averaged crosssection,and (2)the analysisofthe spectral
shape w ith a m axim um likelihood (M L) m ethod.
A s can be seen from Fig.1(b), !L > 0 would result in additional 12C ( e ,e )12N g:s:
events; on the other hand, Q L < 1 would reduce the num ber of events [see Eq.(3)]. In
order to nd allowed regions in the Q L -!L param eter space, we com pared m easured and
expected ux averaged cross sections. A s theoretical cross section h ith w ith a realistic
estim ate ofthe system atic error we use h ith = (9:2
0:5)
10 42 cm 2. T he experim ental
crosssection istaken from Eq.(5)w ith statisticaland system atic erroradded quadratically.
T he probability distribution ofthe ratio
R (Q L ;!L )=
h iexp
= Q L (1 + S
h ith
incorporatesa ux decrease by righthanded
e
!L )=
9:4
9:2
0:9
0:5
(6)
through Q L aswellasan increase by nonzero
!L values;S is the ratio ofadditionalevents in case of!L = 1 relative to the expectation
for!L = 0.W e have sam pled the probability density function ofthe ratio R from G aussian
distributionsofh iexp and h ith for3 di erentenergy ranges:(a)the range 10{36 M eV w ith
the highest statisticalaccuracy,but only m oderate sensitivity S = 0:81,(b) the range 28{
36 M eV ,w here w ith S = 3:48 we are very sensitive to !L ,and (c) the range 10{22.5 M eV ,
w here the expected event num ber is alm ost independent of!L [S = 0:002,see Fig.1(b)].
From range (c) we deduce a lower lim it Q L
0:796.T he shaded param eter space show n in
Fig.3 com binesregionsexcluded at90% con dence levelofall3 energy ranges.From inverse
6
m uon decay experim ents it is know n that Q L > 0:92 [10,11]. Including this inform ation in
our analysis ofrange (b) restricts the allowed area and sets a 90% con dence upper lim it
!L
0:12.
In the second m ethod we determ ine !L by analyzing the shape ofthe visible electron
spectrum independent ofQ L . In order to increase the energy resolution and to reduce the
background levelwe applied m ore stringent cuts on the electron position along the m odule
axis jX j
150 cm and on the electron tim e 0.6{7.2 s. T hese cuts reduce the background
to only 6.0 events in a sam ple of441 events,thus nearly doubling the signal-to-background
ratio.
T hetheoretical e energy spectrum ofEq.3 wasconverted into a visibleelectron spectrum
using the energy-dependent (E )taken from [8]folded w ith the detectorresponse by a M C
calculation. T he M L procedure was carried out on an event-by-event basis for several t
intervals allofw hich gave results com patible w ith !L = 0 w ithin a 1 -error.T he netresult
is
3:8
!L = (2:7+3:
3 (stat)
3:1(sys))
10 2 :
(7)
Including the system atic error(energy shiftof0.25 M eV or0.7% scaling error)we nd,w ith
the m ost conservative Bayesian approach,a 90% con dence upper lim it ! L
0:113. T his
excludes the region above the horizontalline in Fig.3. C om bining Eq.(2) and Eq.(4) the
follow ing relation between the shape param eter !L and nonstandard couplings is [10,12]
s
jgRS L
+
2gRT L j
16
!L .
3
T he lim it on !L thus results in an upper lim it ofjgRS L + 2gRT L j
(8)
0:78 forthe interference
term ofscalar and tensor am plitudes.
In conclusion, the K A R M EN experim ent nds no evidence for nonstandard coupling
constantsin
+
decay atrest,eitherby a determ ination ofthe absolute
e
ux orby analysis
ofthe spectralshape. O uranalysis excludes m ost ofthe Q L -!L param eter space and yields
for the rst tim e an upper lim it on the neutrino M ichelparam eter ! L .
7
D uring 1996 the experim ent was upgraded by an additionalactive veto counter in order
to increase the sensitivity ofthe search forneutrino oscillations in the channel
!
e
[13].
Since 1997 K A R M EN has been taking data again. U p to the end of1999 we expect about
400 further charged current events,w hich w illreduce the statisticalerror by abouta factor
of 1.4. C onsidering also a reduction of the system atic error, this m ay result in a lim it
com petitive w ith the present best lim it jgRS L + 2gRT L j
0:45 deduced from m easurem ents
ofthe positron polarization [10].
W e gratefully acknow ledge the nancialsupport from the G erm an Bundesm inisterium
fur Bildung, W issenschaft, Forschung und Technologie (BM BF),the Particle Physics and
A stronom y R esearch C ouncil(PPA RC ),and theC entralLaboratory oftheR esearch C ouncil
(C LRC ).In particular,we thank W .Fetscher for num erous discussions.
8
R EFER EN C ES
[1]W . Fetscher, Z.Phys. C 56, 109 (1992); W . Fetscher and H .-J.G erber in Precision
Tests ofthe Standard Electroweak M odel,W orld Scienti c,Singapore (1993).
[2]W .Fetscher,H .-J.G erber and K .F.Johnson,Phys.Lett.B 173,102 (1986).
[3]W .Fetscher,Phys.R ev.Lett.69,2758 (1992);71,2511(E) (1993).
[4]C .G reub,D .W yler and W .Fetscher,Phys.Lett.B 324,109 (1994).
[5]G .D rexlin etal.,N ucl.Instrum .and M ethods Phys.R es.,Sect.A 289,490 (1990).
[6]B.Bodm ann etal.,Phys.Lett.B 332,251 (1994).
[7]M . Fukugita et al., Phys. Lett. B 212, 139 (1988); S.L. M intz and M . Pourkaviani,
Phys.R ev.C 40,2458 (1989);E.K olbe etal.,Phys.R ev.C 49,1122 (1994);J.Engel
etal.,Phys.R ev.C 54,2740 (1996).
[8]T .W .D onnelly,Phys.Lett.B 43,93 (1973).
[9]V .Blobel,in Proceedings ofthe 1984 C ER N SchoolofC om puting (C ER N -R eport N o.
85/09,1985).
[10]W .Fetscher,Phys.R ev.D 49,5945 (1994).
[11]Particle D ata G roup,R .M .Barnett etal.,Phys.R ev.D 54,250 (1996).
[12]Lim its on jgRS R j and jgLV R j are possible as well, but are not as stringent as in other
experim ents.
[13]G .D rexlin etal.,Prog.Part.N ucl.Phys.40,Vol.1,193 (1998).
9
10
8
ωL= 0.30
ωL= 0.15
ωL= 0.0
25
20
b)
Events / 1.5 MeV
12
a)
Event Rate [ rel. units ]
ωL= 0.30
ωL= 0.15
ωL= 0.0
14
60
c)
50
40
15
30
6
10
20
4
5
10
2
0
0
20
0
40
Neutrino Energy Eν [MeV]
10
20
0
30
10
trum in
12C
(
e ,e
+
20
30
Exp. visible Electron Energy [MeV]
Visible Electron Energy [MeV]
FIG . 1. In uence of di erent values of ! L = 0:0, 0.15, 0.3 on (a) the
e
energy spec-
decay and on (b) the visible electron energy spectrum m easured w ith the reaction
)12N g:s: (c) Experim entalelectron energy distribution together w ith M C expectation
(solid line) and the subtracted background (shaded).
dN/dE [rel. units]
Neutrino Flux [ rel. units ]
FIG U R ES
5
4
3
2
1
0
20
FIG .2. Energy spectrum of
25
e
from
30
+
35
40
45
50
Neutrino Energy Eν [MeV]
decay determ ined by an unfolding m ethod com pared
w ith the standard m odelexpectation (solid line).
10
ωL [10-2]
ν
Me
V
QL =0.92
36
10-
25
10-22.5 MeV
30
20
28-
36
15
Me
V
ωL =0.113
10
excl. region
5
0
0.7
0.75
0.8
0.85
0.9
0.95
1
QνL
FIG .3. T he Q L -!L param eterspace: T he shaded regions are excluded at90% con dence from
the di erentanalysesofthe absolute ux in severalenergy ranges. T he horizontalline isthe result
ofthe spectralshape analysis !L
lim it Q L
0:113 at 90% con dence. T he verticalline is the current best
0:92.
11