Scattering of neutrinos on a polarized electron target as a test for new physics beyond the Standard Model

0.27 MB·English

by  S. Ciechanowicz

Scattering of neutrinos on a polarized electron target as a test for new physics beyond
the Standard Model
S. Ciechanowicz∗ and W. Sobk´ow†
Institute of Theoretical Physics,
University of Wrocl aw, Pl. M. Born 9,
PL-50-204 Wrocl aw, Poland
M. Misiaszek‡
M. Smoluchowski Institute of Physics,
Jagiellonian University, ul. Reymonta 4,
PL-30-059 Krak´ow, Poland
(Dated: February 1, 2008)
5 In this paper, we analyze the scattering of the neutrino beam on the polarized electron target,
0 and predict theeffects of two theoretically possible scenarios beyond theStandard Model. In both
0 scenarios, Dirac neutrinos are assumed to be massive.
2 First, we consider how the existence of CP violation phase between the complex vector V and
axial A couplings of the Left-handed neutrinos affects the azimuthal dependenceof the differential
n
cross section. Thisasymmetry doesnot vanishin themassless neutrinolimit. Theazimuthal angle
a
J φe′ of outgoing electron momentum is measured with respect to the transverse component of the
initial electron polarization ηe⊥. We indicate the possibility of using the polarized electron target
0
1 to measure the CP violation in the νµe− scattering. The future superbeam and neutrino factory
experiments will provide the unique opportunity for the leptonic CP violation studies, if the large

magnetized sampling calorimeters with good event reconstruction capabilities are build.
2
Next, we take into account a scenario with the participation of the exotic scalar S coupling of
v
theRight-handedneutrinosinaddition tothestandardvectorVand axialAcouplingsoftheLeft-
6
8 handed neutrinos. The main goal is to show how the presence of the R-handed neutrinos, in the
2 above process changes the spectrum of recoil electrons in relation to the expected Standard Model
9 prediction,usingthecurrentlimits on thenon-standardcouplings. Theinterference termsbetween
0 the standard and exotic couplings in the differential cross section depend on the angle α between
3 thetransverseincomingneutrinopolarizationandthetransverseelectronpolarizationofthetarget,
0 and do not vanish in thelimit of massless neutrino. The detection of the dependenceon this angle
/ in the energy spectrum of recoil electrons would be a signature of the presence of the R-handed
h
neutrinos in the neutrino-electron scattering. To make this test feasible, the polarized artificial
p
neutrinosource needs to beidentified.
-
p
e PACSnumbers: 13.15.+g,13.88.+e,14.60.St
h
:
v I. INTRODUCTION However,inthegeneralcaseofcomplexgL andgLcou-
i V A
X plings, we have one additional free parameter: the rela-
tivephasebetweenthese couplingsdenotedasβ . The
r Standard Model (SM) of electro-weak interactions VA
a CP-odd interference contribution enters the differential
[1,2,3]hasavector-axial(V-A)Lorentzstructure[4],i.e.
cross section for the scattering of left-handed neutrinos
onlyleft-handed(L-handedchiralitystates)andmassless
onthepolarizedelectrontarget(PET),if sin(β ) =0.
Dirac neutrinos may take part in the charged and neu- VA
| |6
The experimental measurement of the azimuthal angle
tralweakinteractions. The observedCPviolationinthe
decaysof neutralkaonsand B-mesons[5] is described by φe′ ofoutgoingelectronmomentumcouldbeusedtotest
theCPsymmetryinleptonsectorofelectroweakinterac-
a single phase of the CKM matrix.
ThevectorgL andaxial-vectorgL neutralcurrentcou- tions. The observation of asymmetry in the angular dis-
V A tribution of recoil electrons, caused by the interference
pling constants are assumed to be real numbers, which
means that Im(gL) = Im(gL) = 0. The values of these terms between the standard complex couplings would
V A giveadditionalinformationaboutthecouplingconstants.
two couplings are derived from neutrino electron scat-
tering and from e+e− l+l− annihilation studies, but
→ The magnetized sampling calorimeters (e.g. MINOS
in the fitting procedure the imaginary parts are fixed to
far detector) are composed of many steel leyers, which
their Standard Model values.
are magnetized using a coil through a hole in the center
oftheplanestoanaveragefieldofabout1.5T.Inapiece
of magnetized iron, there are lots of unpaired electrons
all pointing the same direction. The Moeller polarime-
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] ters determine the polarization of the electron beam by
‡Electronicaddress: misiaszek@zefir.if.uj.edu.pl measuring the cross section asymmetry in the scattering
2
dard couplings gL, gL. The main goal of the other part
TABLE I: Current limits on thenon-standard couplings V A
(SectionIII)istoshowhowthepresenceoftheR-handed
neutrinos, in the (ν e−) scattering process, changes the
Coupling constants SM Current limits µ
energy spectrum of recoil electrons in relation to the ex-
|gV | 1 >0.960
LL pected SM prediction, using the current limits on the
|gV | 0 <0.060
LR non-standardcouplings[7]. We concernthe scatteringof
|gV | 0 <0.110
RL transversely polarized muon neutrino beam on the PET
|gRVR| 0 <0.039 to probe ERWI effects and include a theoretical discus-
|gS | 0 <0.550 sion of the possibility of developing such a beam.
LL
|gS | 0 <0.125 Our analysis is model-independent and is made in
LR
|gS | 0 <0.424 the massless neutrino limit. All the calculations are
RL
|gS | 0 <0.066 made in the limit of vanishing neutrino mass with the
RR
|gT | 0 0 Michel-Wightman density matrices [8] for the polarized
LL incoming neutrino beam (Appendix C) and for the
|gT | 0 <0.036
LR polarized electron target, respectively. We use the
|gT | 0 <0.122
RL system of natural units with h¯ = c = 1, Dirac-Pauli
|gRTR| 0 0 representation of the γ-matrices and the (+, , , )
− − −
metric [9].
of polarized electrons by polarized electrons. Polarized
electrons are scattered off a polarized ferromagnetic foil,
and the foil polarizationis determined by measurements II. LEFT HANDED NEUTRINO SCATTERING
ON A POLARIZED ELECTRON TARGET
of the magnetization of the foil and its thickness. So,
there is the well-known and commonly used in accelera-
tor physics technique for developing PETs. The Standard Model (SM) of electroweak interactions
Although the SM agrees well with all experimental isbasedonthe gaugegroupSU(2) (1). Theleft-handed
×
data up to available energies, the experimental precision fermion fields ψ = νi and ui of the ith fermion
of present measurements still does not rule out the pos- i l−i d′i
family transform as d(cid:16)oub(cid:17)lets un(cid:16)der(cid:17)SU(2), where d′
sible participation of the exotic scalar S, tensor T and i ≡
V d andV istheCabibbo-Kobayashi-Maskawamix-
pseudoscalar P couplings of the right-handed(R-handed j ij j
ing matrix. The vectorand axial-vectorcouplings in SM
chirality states)Dirac neutrinos beyondthe SM [6]. The P
are
current upper limits on the all non-standard couplings,
obtained from the normal and inverse muon decay, are
gL(i) tL(i) 2q(i)sin2θ
presented in the Table I [7]. In the SM, only gV is non- V ≡ 3 − W
zero value. LL gAL(i) ≡ tL3(i) (1)
New effects due to the exotic right-handedweakinter-
where tL(i) is the weak isospin of fermion i (+1/2 for
actions(ERWI)couldbedetectedbythemeasurementof 3
u and ν ; 1/2 for d and l ), q is the charge of ψ
neutrinoobservables(NO)whichconsistonlyoftheinter- i i − i i i i
in units of e and θ is the weak angle. Because of the
ferencesbetweenthe standardV AandERWI,anddo W
− model-dependentinterpretationofthecouplingconstants
notdependontheneutrinomass. TheNOincludethein-
values, they are assumed to be real numbers. For exam-
formationonthe transversecomponentsofneutrinospin
ple,thetotalcrosssectionforhighenergyneutral-current
polarization (TCNSP), both T-even and T-odd. These
(ν e−) scattering is
quantities vanish in the SM, so the detection of the non- µ
zero values of the TCNSP wouldbe a direct signatureof
σ (ν +e− ν +e−)
theR-handedneutrinopresenceintheweakinteractions. SM µ → µ ≃
The scattering of intense and polarized neutrino beam, 2G2FmeEνµ(gL2+gL2+gLgL), (2)
coming from the artificial neutrino source, on the po- 3π V A V A
larized electron target could detect the effects from the
but in the model-independent (MI) analysis we obtain:
ERWI. Presently the measurement of such observables
is only theoretically possible. We give an example how
σ (ν +e− ν +e−)
the polarized neutrino flux can be produced if the ex- MI µ µ
→ ≃
otic scalar coupling CR is present in the theory of muon 2G2m E
S F e νµ(gL 2+ gL 2+ gL gL cos(β )), (3)
capture interaction. 3π | V| | A| | V|| A| VA
The main goal of the first part of our paper (Section
I(Iν)eis−)toscsahtotweritnhgatofthleeftd-hiffaenrdenedtiaalncdrolsosngsietcutdioinnaflolyr pthoe- wcohueprleinggVLco=ns|tgaVLn|tse,iβRVLe,(ggALLg=L∗)|g=AL|egiLβALgaLrecoths(eβcom)palnedx
larµized muon neutrinos on the PET may be sensitive to β =βL βL is the relaVtivAephas|eVb|e|twAe|entheVgAL and
theCP-violatingeffects,ifoneassumesthecomplexstan- gLVAcouplVin−gs. A V
A
3
The effective vector and axial-vector neutral coupling reversal violation has been published by CPLEAR Col-
constants obtained from the absolute neutrino-electron laborationin 1998[12]. Many non-standardmodels take
scattering event rate are intoaccountnewCP-violatingphases,andcanbeprobed
in observableswhere the SM CP-violationis suppressed,
gVL ≃0 , gAL ≃±0.5 or whilealternativesourcescangenerateasizableeffect,e.g.
gL 0.5 , gL 0 . (4) theelectricdipolemomentoftheneutron,thetransverse
V ≃± A ≃ leptonpolarizationinthree-bodydecaysofchargedkaons
However, from our MI expression (3) one can see that K+[13,14],transversepolarizationoftheelectronsemit-
the solution (with CP-violating phase): ted in the decay of polarized 8Li nuclei [15]. There is no
directevidence of CPviolationinthe leptonic processes,
π
gL = gL 0.35 and β = (5) i.e. a neutrino-electron scattering. However, the future
| V| | A|≃ VA ±2 superbeamandneutrino factoryexperiments [16]will be
abletomeasuretheCPviolatingeffectsintheleptonsec-
provides to the same total cross section value as the SM
tor,wherebothneutrinoandantineutrinooscillationwill
fit (4). In the next subsection we present how the ex-
beobserved. Weindicatethatthescatteringofneutrinos
istence of non zero β phase is related to CP-odd in-
VA
on the PET has similar scientific possibilities.
terference contribution in the differential cross section.
The fermion-antifermion pair production cross-sections
have only T-even contributions, but their experimental
observations are essential to determine a single soultion
A. CP violation in standard νe scattering
from possible parameters (4). Even if β =0 the scat-
VA
tering of left-handed neutrinos on the PET provides a
new approachto decide whichofthe two couplingtypes, In this subsection, we consider the possibility of the
(mainly) pure gL or pure gL coupling, is realized in na- CP violation in the ν e− scattering, when the incoming
A V µ
ture. This approachis model independent in contrast to muon neutrino beam consists only of the L-handed and
e+e− experiments which make the assumption that the longitudinallypolarizedneutrinos. Weassumethatthese
neutral current is dominated by the exchange of a single neutrinos are detected in the standard V A NC weak
Z0. interactions with the PET and both the r−ecoil electron
Asiswell-known,CPviolationhasbeenobservedonly scattering angle θ′ and the azimuthal angle of outgoing
e
in the decays of neutral kaons and B-mesons. The Stan- electron momentum φ′ shown in Fig. 1 are measured
e
dardModel describes the existing data by a single phase with a good angular resolution. Because we allow for
of the CKM matrix, [10]. However, the baryonasymme- the non-conservationofthe combinedsymmetry CP, the
try of the Universe can not be explained by the CKM amplitude includes the complex coupling constants de-
phase only, and at least one new source of CP violation noted as gL,gL respectively to the initial neutrino of L-
V A
is required [11]. The first direct confirmation of a time chirality:
G
Mνµe = √F2(gVL(ue′γαue)(uνµ′γα(1−γ5)uνµ)+gAL(ue′γ5γαue)(uνµ′γ5γα(1−γ5)uνµ)), (6)
whereue andue′ (uνµ and uνµ′)arethe Diracbispinors andIm(gVLgAL∗)=|gVL||gAL|sin(βVA)),proportionaltothe
of the initial and final electron (neutrino) respectively. magnitudeofthetransverseelectrontargetspinpolariza-
GF =1.16639(1)×10−5GeV−2 [7]istheFermiconstant. tion, with ηˆe ⊥qˆ is of the form:
The formula for the differential cross section includ-
ing the CP-odd contribution (qˆ (ηˆe×pˆe′) is T-odd
·
d2σ E m G2 2m m
(cid:18)dydφ′e(cid:19)(VA) = 4νπ2e 2F(1−ηˆν ·qˆ)(|gAL|2(cid:20)−ηˆe·pˆe′r Eνe +y( y3−2√y)+ Eνey+(y−2)y+2(cid:21)
p
2m m
+|gVL|2 y2−ηˆe ·pˆe′ y3 E e +y−y(Ee +2)+2
(cid:20) r ν ν (cid:21)
p
4
FIG.1: Figureshowsthereactionplanefortheνµe− scattering,ηˆν -theunit3-vectoroftheinitialneutrinopolarization inits
rest frame, ηe⊥ -thetransverseelectron polarization vectorof target andtheproductionplaneof νµ-neutrinosfor thereaction
of µ−+p→n+νµ. In Section II, thescattering of left handed neutrinosis considered, which havenotransverse polarization
ην⊥ =0. FortheconsiderationsdescribedinSectionIII,themuoncapturereactionisusedasasourceoftransverselypolarized
neutrinos.
2m
+Im(gVLgAL∗)qˆ·(ηˆe ×pˆe′) y( E e +y) (7)
r ν
2m
+Re(gVLgAL∗)(cid:20)ηˆe·pˆe′(y−1)ry( Eνe +y)+(2−y)y(cid:21))
where ηˆν qˆ = 1 is the longitudinal polarization of the Itvariesfrom0to2/(2+me/Eν). θe′ -thepolaranglebe-
· −
incoming L-handed neutrino, q - the incoming neutrino tween the direction of the outgoing electron momentum
momentum, pe′ - the outgoingelectronmomentum, ηˆe - pˆe′ and the direction of the incoming neutrino momen-
the unit3-vectorofthe initialelectronpolarizationinits tumqˆ (recoilelectronscatteringangle),m -theelectron
e
restframe,seeFig.1. Themeasurementofthe azimuthal mass.
angleofoutgoingelectronmomentumφe′ isonlypossible
when the electron target polarization is known. The po-
larization vector for electrons is parallel to the magnetic
field vector. The variable y is the ratio of the kinetic
energy of the recoilelectron T to the incoming neutrino
e
energy E :
ν After the simplification of the vector products and us-
ing the complex number identities in the formula (7) for
y ≡ ETeν = mEνe(1+ mE2νeco)2s2−θec′os2θe′. (8) the cross section, we obtain with ηˆe ⊥qˆ the new form:
5
2,19
2,18
1 ]
-d 2,17
a
r
2 2,16
m
c
2,15
3
4
-0
1 2,14
[ (cid:73)'e 2,13
d
/(cid:86) 2,12
d
2,11
2,10
-1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0
Azimuthal angle (cid:73) [ (cid:83)rad ]
e'
FIG.2: Plotofthe ddφσ′e(VA) asafunctionoftheazimuthalangleφe′ forthe(νµe−)scattering,Eν =1 GeV,y=0.5,ηˆν·qˆ =−1
and|ηˆe⊥|=1: a)thecaseofthepureandrealaxial-vectorcouplingi.e. gAL =0.5andgVL =0(solidline),b)thecaseofthepure
and real vector coupling i.e. gL = 0 and gL = 0.5 (dotted line), c) CP violation, the case of the complex coupling constants
A V
|gVL|=|gAL|=0.354 with the relative phase βVA = π2 (dashed line).
d2σ E m G2 m m
(cid:18)dydφ′e(cid:19)(VA) = 4νπ2e 2F(1−ηˆν ·qˆ)(|ηe⊥|rEνey[2−y(2+ Eνe)] (9)
· cos(φe′) 2|gVL||gAL|cos(βVA)y+(2−y)|gAL|2−y|gVL|2 −2|gVL||gAL|cos(φe′ +βVA)
(cid:20) (cid:21)
(cid:0) (cid:1)
m
+ gL 2+ gL 2 y2 2y+2 +2gL gL cos(β )y(2 y) ey gL 2 gL 2 .
(cid:20) | V| | A| − | V|| A| VA − − Eν | V| −| A| (cid:21))
(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)
It can be noticed that the interference terms between III. SCATTERING OF TRANSVERSELY
the standard gL couplings depend on the value of the POLARIZED MUON NEUTRINO BEAM ON A
V,A
β phase. However, the angular asymmetry of recoil POLARIZED ELECTRON TARGET
VA
electrons is not vanishing even if β = 0. The CP-
VA
violating phase enters the cross section and changes the So far the scattering of left-handed and longitudinally
angleatwhichthenumberofrecoilelectronswillbemax- polarized neutrino beam on a polarized electron target
imal (φme′ax). (SLoPET) was proposed to probe the neutrino mag-
netic moments [17, 18] and the flavor composition of a
(anti)neutrino beam [19].
There were also the ideas of using the scattering of
transversely polarized neutrino beam on the unpolar-
ized electron target to probe the nonstandardproperties
6
of neutrinos. Barbieri et al. proposed to measure the new tests of the Lorentz structure of the charged- and
azimuthal asymmetry of recoil electrons caused by the neutral-current weak interactions. An admittance of all
non-vanishinginterferencebetweentheweakandelectro- the ERWI does not change qualitatively the conclusions
magnetic interaction amplitudes [20, 21]. Ciechanowicz from the investigations.
et al. [22] indicated that the azimuthal asymmetry of To show how the recoil electron spectrum may de-
recoil electron event rates could be generated by the in- pend on the angle between the transverse neutrino po-
terferencebetweenthe standard(V,A)L couplingsofthe larizationandtransversetargetelectronpolarization,we
L-handedDiracneutrinosandexotic(S,T,P)Rcouplings use the muon neutrino beam produced in the reaction
of the R-handed ones in the laboratory differential cross where the proton at rest captures the polarized muon
section. All the terms with interference, proportional to (µ−+p n+ν ),seeAppendixA.Theproductionplane
µ
→
the magnitude of the transverse neutrino polarization, is spanned by the direction of the initial muon polariza-
do not vanish in the massless neutrino limit and depend tion Pˆ (the muon is fully polarized, i.e. P = 1) and
µ µ
| |
on the azimuthal angle between the transverse neutrino of the outgoing neutrino momentum qˆ, Fig. 1. Pˆ and
µ
polarizationandtheoutgoingelectronmomentum. How- qˆ areassumedtobe perpendiculartoeachotherbecause
ever, in both cases, the neutrino detectors with a good this leads to the unique conclusions as to the possible
angular resolution would have to measure both the re- presenceoftheR-handedneutrinos. Whenadmittingad-
coil electron scattering angle and the azimuthal angle of ditionalexoticscalarcouplingCR inmuoncaptureinter-
S
outgoing electron momentum. action, Eq. (A.1), the outgoing muon-neutrino flux is a
There existthe non-standardmodels, in whichthe ex- mixtureofthe L-handedneutrinosproducedinthe stan-
oticcouplingsoftheRight-handedneutrinoscanappear. dard V A charged weak interaction and the R-handed
We mean here three classes of such models; left-right ones pro−duced in the exotic scalar S charged weak in-
symmetric models (LRSM), contacts interactions (CI) teraction (the transition amplitude and the neutrino ob-
and leptoquarks (LQ). For example, the CI can be in- servables are presented in Appendix A). This mixture is
troduced both for the vector coupling of the L-handed detected in the neutral current (NC) weak interaction.
neutrinos and scalar, tensor couplings of the R-handed We mean that the incoming L-handed neutrinos are de-
ones, [23]. Such interactions would allow to probe the tected in the standard V A neutral weak interaction,
scale for compositeness of quarks and leptons. As to the whiletheinitialR-handed−onesaredetectedintheexotic
LQ models, if the R-handed neutrinos are taken into ac- scalarS one. Theninthefinalstatealltheneutrinosare
count, there are possible couplings of these neutrinos to L-handed. Below we give the transition amplitude for
the scalar and vector LQ, [23]. An original discussion this type of neutral current:
concerning the LQ did not allow for the R-handed neu-
trinos, [24]. Left-right symmetric models were proposed G
to explain the origin of the parity violation, [25, 26]. Mνµe = √F2{(ue′γα(gVL −gALγ5)ue)(uνµ′γα(1−γ5)uνµ)
In such models the R-handed neutrino can couple to R-
1
handed gauge boson with a mass larger than for the ob- + 2gSR(ue′ue)(uνµ′(1+γ5)uνµ)}, (10)
servedstandardboson(m =80GeV). RecentlyTWIST
1
Collab. [27] has measured the Michel parameter ρ in
Thecouplingconstantsaredenotedwiththesuperscripts
the normal µ+ decay and has set new limit on the L and R as gL, gL and gR respectively to the incoming
WL WR mixing angle in the LRSM. Their result ρ = neutrino of LV- anAd R-chirSality. Standard couplings gL,
0.75−080 0.00044(stat.) 0.00093(syst.) 0.00023 is in gL are assumed to be real, i. e. βL =0,βL =0. V
goodagr±eementwiththe±SMpredictionρ=±3/4,andsets A V A
new upper limit on mixing angle χ <0.030(90%CL).
| |
In this part of our paper we show that the scattering
A. Laboratory differential cross section
of transversely polarized muon neutrino beam on a po-
larizedelectrontarget(SToPET)maybesensitivetothe
interference effects betweenthe L- andR-handed neutri- Because we consider the case when the outgoing elec-
nos in the differential cross section for the (ν e−) scat- tron direction is not observed, the formula for the lab-
µ
tering process. Our analysis is made for the case, when oratory differential cross section is presented after inte-
the outgoingelectrondirectionis notobserved. Itmeans gration over the azimuthal angle φe′ of the recoil elec-
thattheazimuthalangleoftherecoilelectronmomentum tronmomentum. Theresultofthecalculationperformed
would not be measured and nevertheless the new effects with the amplitude M , in Eq. (10), is divided into
νµe
could be observed. We consider the minimal extension three parts, standard (V,A), exotic (S) and interference
of the standard V A weak interaction to indicate the (VS+AS):
−
7
dσ dσ dσ dσ
= + + , (11)
dy dy dy dy
(cid:18) (cid:19)(V,A) (cid:18) (cid:19)(S) (cid:18) (cid:19)(VS+AS)
with,
(cid:18)ddσy(cid:19)(V,A) = E2νπmeG22F(1−ηˆν ·qˆ)( gVL +gAL 2(1+ηˆe·qˆ)+ gVL −gAL 2(cid:20)1−(ηˆe·qˆ)(cid:18)1− mEνe(1−y y)(cid:19)(cid:21)(1−y)2
(cid:0) (cid:1) (cid:0) (cid:1)
− gVL 2− gAL 2 (1+ηˆe·qˆ)mEνey), (12)
h(cid:0) (cid:1) (cid:0) (cid:1) i
where,ην qˆ isthelongitudinalpolarizationoftheincomingL-handedneutrino. Weshallpointoutthatthestandard
·
(SM) parthas beenalreadypublished inRef. [17]. The exoticpartofthe crosssectionmay contributemerely for the
R-handed neutrino scattering (ηˆν qˆ +1):
· ≃
dσ E m G2 1 m
dy = 2νπ e 2F(1+ηˆν ·qˆ)|gSR|28 2Ee +y y, (13)
(cid:18) (cid:19)(S) (cid:18) ν (cid:19)
In the interference part we have angular correlationswith the transversecomponent of the neutrino polarizationη⊥,
ν
both T-odd and T-even. The correlationcoefficients depend linearly on the exotic coupling constant gR. Hence, this
S
contribution could be a tool suitable to investigate the effects due to scalar interactions of the R-handed neutrinos:
dσ E m G2 m m
= ν e Fy qˆ (η⊥ η⊥) Im(gLgR∗) 1+ e y +Im(gLgR∗) 1 e (y 4)
(cid:18)dy(cid:19)(VS+AS) − 4π 2 ( · e × ν (cid:20) V S (cid:18) 2Eν (cid:19) A S (cid:18) − 2Eν − (cid:19)(cid:21)
m m
+(η⊥ η⊥) Re(gLgR∗) 1+ e y +Re(gLgR∗) 1 e (y 4) . (14)
e · ν (cid:20) V S (cid:18) 2Eν (cid:19) A S (cid:18) − 2Eν − (cid:19)(cid:21))
Itcanbenoticedthattheoccurrenceoftheinterference interactions, there is no interference between the gL
V,A
termsbetweenthestandardgVL,A andexoticgSRcouplings and gSL couplings in the differential cross section, when
doesnotdependontheneutrinomassandtheypertainin m 0. We do not consider this scenario.
ν
themasslessneutrinolimit. Theindependenceonthem →
ν
makes the measurement of the relative phases between
thesecouplingspossible. Thetermswiththeinterference
betweenthe standardgL andexoticgR couplings,Eqs.
V,A S
(14), include only the contributions from the transverse
B. CP conservation in νe scattering at low-energy
component of the initial neutrino polarization η⊥ and
ν
thetransversecomponentofthepolarizedelectrontarget
η⊥. Bothtransversecomponentsareperpendicularwith In this subsection, we will considerthe CP-symmetric
e
respect to the qˆ. scenariowiththe standard(V A) and S weak inter-
L R
−
If one assumes the production of only L-handed neu- actions. From the general formula for the cross section,
trinosinthe standard(V A)andnon-standardS weak we get with ηˆe qˆ for |ηe⊥|=1:
− ⊥
dσ dσ dσ dσ
= ( ) +( ) +( ) , (15)
dy dy (V,A) dy (S) dy (VS+AS)
dσ E m G2 m m
( ) = ν e Fy η⊥ cos(α)gR (1+ e y)gL +(1 e (y 4))gL , (16)
dy (VS+AS) − 4π 2 | ν | | S|{ 2E | V| − 2E − | A|}
ν ν
where(dσ/dy) ,(dσ/dy) aregivenbyEqs. (12,13) see that the CP-even interference terms enter the cross
(V,A) (S)
and α is the angle between the η⊥ and η⊥, Fig. 1. We
ν e
8

2.3 2.3
2.2 2.2
2.1 2.1
2.0 2.0
2 ] 1.9 1.9
m
1.8 1.8
c
3
4 1.7 1.7
-0
1
[ y 11..56 11..56
d
/
σ
d 1.4 1.4
1.3 1.3
1.2 1.2
1.1 1.1
1.0 1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
y
FIG. 3: Plot of the ddσy as a function of y for the (νµe−) scattering after integration over φe′, Eν = 100MeV: a) SM with
the L-handed neutrino (solid line), b) CP conservation, the case of the exotic S coupling of the R-handed neutrinos for α=0
(long-dashed line), α=π (short-dashed line) and α=π/2,3π/2 (dotted line), respectively, c) c) CP violation, the case of the
exoticScouplingoftheR-handedneutrinosforα+βVS =0andα+βAS =0(long-dashedline),α+βVS =π andα+βAS =π
(short-dashed line), respectively.
sectionandwillbelargeatthe α=0,π,andthey vanish verse and longitudinal components for the neutrino in
for α=π/2,3π/2. µ-capture: ην⊥ = 0.318, ηˆν qˆ = 0.948. Finally, with
AnalyzingtheassumptionthattheR-handedneutrinos these estima|tes|and the coup·lings g−′L, g′L and gR , the
arecreatedanddetectedintheexoticS weakinteraction, correlation coefficients, Eq. (16), anVd thAe cross|sSec|tion,
wehaveusedthesameupperlimitontheNCcouplinggSR Eq. (11),havebeencalculatedinordertoshowtheeffect
as for the µ-capture coupling CSR, i.e. |gSR|<0.974,if al- coming from the R-handed muon-neutrinos.
lowing for the qualitative argument of weak interactions C. CP violation in νe scattering at low-energy
universality. Moreover, with the presence of the exotic
scalar coupling gR =0.974, we have shifted the numer-
| S|
ical values of the V and A couplings to the new ones: Inthis subsection,we analyzethe caseofthe violation
g′L = 0.041,g′L = 0.517,whichstillliewithintheex- ofcombined symmetry CP. The formula for the differen-
pVerime−ntalbarsAonth−eSMresults: gL = 0.040 0.015, tial cross section including the interference contribution
dgALent=ly,−i0n.50A7p±pe0n.d0i1x4A[7,],wwehhenavηˆeνe·sVtqˆim=a−t−ed1.thIen±dterpaenns-- bteertawceteionntshweitshtaηnˆdeardqˆ(Vfor−|ηAe⊥)L|=an1disexooftticheSfRorwme:ak in-
⊥
dσ dσ dσ dσ
= ( ) +( ) +( ) , (17)
dy dy (V,A) dy (S) dy (VS+AS)
dσ E m G2 m m
( ) = ν e Fy η⊥ gR (1+ e y)gL cos(α+β )+(1 e (y 4))gL cos(α+β ) ,(18)
dy (VS+AS) − 4π 2 | ν || S|{ 2E | V| VS − 2E − | A| AS }
ν ν
where β βL βR, β βL βR are the rela- respectively.
tive phaVseSs ≡betwVee−n thSe gLA,Sg≡R anAd−gLS, gR couplings,
V S A S
9
It can be seen that the CP-odd interference contribu- torydifferentialcrosssectiondependontheangleα,Fig.
tion enters the cross section and will be substantial at 1,betweenthetransverseincomingneutrinopolarization
the α+β = 0,π and α+β = 0,π, and it vanishes andthetransverseelectronpolarizationofthetargetand
VS AS
for the α+β =π/2 and α+β =π/2, respectively. are present even in the massless neutrino limit. The ob-
VS AS
The situation is illustrated in the Fig. 3 for the same servation of the dependence on this angle α in the recoil
limits asforthe CP-symmetriccasewith E =100MeV electron energy spectrum would be a clear signal of the
ν
(long-dashed and short-dashed lines, respectively). The R-handed neutrinos in the νe scattering.
phasesβVS andβAS inEq. (18),whendifferentfrom0or It can be noticed that the disagreement with the SM
π, may result from CP-violationin NC weak interaction would be substantial for the small polar angle θe′, both
(νµe−). The angle α is defined in accordance with Fig. for the CP-even and CP-odd cases, Fig. 3.
1 and relates the direction of η⊥ to the direction of η⊥.
ν e TosearchfortheeffectsconnectedwiththeERWI,the
Sowiththe properchoicesofα,the phasesβ andβ
VS AS strong polarized neutrino beam and the polarized elec-
couldbe detected bymeasuringthe maximalasymmetry
tron target is required. The electron target should be
of the cross section dσ/dy.
polarized perpendicular to the direction of the incoming
On the other hand, if knowing these phases prior neutrino beam, ηˆe qˆ=0,because it leads to the unique
to (ν e−) scattering, it would be possible to test CP- ·
µ conclusions as to the R-handed neutrinos. If one has
symmetry in muon capture. In case of CP-violation,
the polarized artificial neutrino source, the direction of
the neutrino transverse polarization vector η⊥ would be
ν the transverse neutrino polarization with respect to the
turned aside from the production plane (qˆ,Pˆµ), hav- production plane will be fixed. So having the assigned
ingtheCP-breakingcomponent<S (Pˆ qˆ)> ,see direction of the polarization axis of electron target and
ν · µ× f
Fig. 1 and Eq. (A3). For illustration purposes let us turning the polarization axis of neutrino source, the de-
take β =β =0(i.e. CP-symmetry inν e−). Next, pendence of the event number on the angle α could be
VS AS µ
we measure the angular correlation η⊥ η⊥ cos(α), in tested.
order to see the direction of η⊥ aloneg·whνic∼h the maxi- It seems worthyof exploring high energy regionin the
e
mal asymmetry is oriented. If this direction were turned L- andR- handedneutrinos interference. Because of the
aside from Pˆ , it would be evidence for CP-breaking. angularcorrelationbetweenthetransversespinpolariza-
µ
tionsoftheneutrinoandelectronandatthelargekinetic
energy transfer to the recoil electron, we see strong an-
IV. CONCLUSIONS gularasymmetryinthecrosssectiondσ/dy forthe small
values of the polar recoil angle, see Eq. (16, 18). We
Inthefirstpartofthepaper,weshowthattheSLoPET expect for this fact some interest in the accelerator lab-
can be used to measure the CP violation in the pure oratories working with neutrino beams, accompanied by
leptonic process, Fig. 2. The azimuthal asymmetry of theprogressinthespinpolarizationengineering. Forthe
therecoilelectronsdoesnotdependontheneutrinomass futureoutline,weshallinspecttheotherexamples,which
and is not vanishing even if β = 0. The CP-breaking could be interesting from the point of observable effects
VA
phase β could be detected by measuring the maximal caused by the exotic neutrino states. We plan to work
VA
asymmetry of the cross section. The future superbeam mainlyonthe weakinteractionprocessesthatareknown
andneutrinofactoryexperimentswillprovidetheunique from experiment or have been already under considera-
opportunity for the leptonic CP violation studies, if the tion in the literature.
large magnetized sampling calorimeters with good event
reconstruction capabilities are build.
In the second part, we show that the SToPET may
Acknowledgments
be used to detect the effects caused by the interfering
L- and R-handed neutrinos. In spite of the integration
over the azimuthal angle φe′ of the recoil electron mo- This work was supported in part by the grant 2P03B
mentum, the terms with the interference between the 15522 of The Polish Committee for Scientific Research
standard (V,A) and exotic S couplings in the labora- and by The Foundation for Polish Science.
L R
APPENDIX A: MUON CAPTURE BY PROTON
Theamplitudeforthemuoncapturebyproton(µ−+p n+ν ),asaproductionprocessofmassivemuon-neutrinos,
µ
→
is commonly of the form:
q
Mµ− = (CVL +2MgM)(uνγλ(1−γ5)uµ)(unγλup)+(CAL+mµ2MgP)(uνiγ5γλ(1−γ5)uµ)(uniγ5γλup)
10
+CR(u (1 γ )u )(u u ), (A1)
S ν − 5 µ n p
where the fundamental coupling constants are denoted as CL, CL and CR respectively to the outgoing neutrino of
V A S
L- and R-chirality. Since we do not preclude the CP-asymmetry between SM and exotic sectors, we allow for these
coupling constantsto be the complex numbers. g ,g - the induced weakcouplings ofthe left-handed neutrinos,i.e.
M P
the weak magnetism and induced pseudoscalar, respectively; m , q, E , m , M - the muon mass, the absolute value
µ ν ν
of the neutrino momentum, its energy, its mass and the nucleon mass; u , u - the Dirac bispinors of initial proton
p n
and final neutron; u , u - the Dirac bispinors of initial muon and final neutrino.
µ ν
Following the results of Ref. [28], in the case of non-vanishing neutrino mass (m = 0), we take the transverse
ν
6
components of the neutrino spin polarization, T-even:
<S Pˆ >
ν · µ f
q q 1m q
= P (1+ )Re((CL +2Mg )CR∗)+ ν(CL+2Mg 2 CL+m g 2+ CR 2) , (A2)
| µ|{ E 2M V M S 2 E | V M| −| A µ2M P| | S | }
ν ν
and T-odd:
q q
<S (Pˆ qˆ)> = P ( + )Im((CL +2Mg )CR∗). (A3)
ν · µ× f −| µ| E 2M V M S
ν
Here, S is the neutrino spin operator,s=1/2is the neutrino spin; Pˆ is the unit vectorof the muonpolarizationin
ν µ
the muonic atom 1s state, and qˆ is the unit vector of the neutrino momentum. Pˆ and qˆ are perpendicular to each
µ
other, Pˆ qˆ=0.
µ
·
Itcanbenoticedthatinthelimitofvanishingneutrinomass,theseobservablesconsistonlyoftheinterferenceterm
between the standard CL coupling and exotic CR one. There is no contribution to these observables from the SM in
V S
which neutrinos are only L-handed and massless. The neutrino mass terms, m /E , in the above observables give a
ν ν
very small contribution in relation to the main one coming from the interference terms and they are neglected in the
considerations. As the induced weak couplings enter additively the fundamental CL couplings, they are omitted in
V,A
theconsiderations,fortheirpresencedoesnotchangequalitativelytheconclusionsconcerningthetransverseneutrino
polarization.
Using the current data [7], we calculate the lower limits on the SM couplings: CL > 0.850(4G /√2)cosθ and
| V| F c
CL >1.070(4G /√2)cosθ , and upper limit on the exotic scalar: CR <0.974(4G /√2)cosθ . Now, we may give
| A| F c | S| F c
the upper bound on the magnitude of the transverse neutrino polarization in the massless neutrino limit:
η⊥ = 1 <S (Pˆ qˆ)>2+<S Pˆ >2. (A4)
ν ν µ ν µ
| | s · × ·
q
The transverse components, Eqs. (A.2) and (A.3), are calculated with the amplitude Mµ− and normalized with the
µ-capture probability <1>f:
1 CR q q q CL 2 CR 2 q CL −1
η⊥ = S 1+ 1+ + 3+ A + + S 2 A cos(αL ) . (A5)
| ν | s (cid:12)CVL(cid:12) 2M " M M (cid:12)CVL(cid:12) (cid:12)CVL(cid:12) − M (cid:12)CVL(cid:12) AV #
(cid:12) (cid:12)(cid:16) (cid:17) (cid:16) (cid:17)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
After insertingfrom(cid:12)(cid:12)abov(cid:12)(cid:12)ethe limits oncouplingconstants(cid:12)(cid:12)and(cid:12)(cid:12)withth(cid:12)(cid:12)e rel(cid:12)(cid:12)ativepha(cid:12)(cid:12)seb(cid:12)(cid:12)etweenthe standardCAL and
CVL couplings αLAV ≡ αLA−αLV =π, under the condition that |ηˆν|= 1, one obtains; |ην⊥|≤ 0.318, which means that
the value of the longitudinal neutrino polarization is equal to ηˆν qˆ = 0.948.
· −
APPENDIX B: FOUR-VECTOR NEUTRINO POLARIZATION AND MICHEL-WIGHTMAN DENSITY
MATRIX
The formulas for the 4-vector of the massive neutrino polarization S in its rest frame and for the initial neutrino
moving with the momentum q, respectively, are as follows:
S = (0,ηˆν), (B1)
S′ = ηˆν ·q 1 Eν + 0 ηˆν ·q 0 , (B2)
Eν · mν q ! ηˆν !− Eν(Eν +mν) q!
S0′ = |q|(ηˆν qˆ), (B3)
m ·
ν
E
S′ = ν(ηˆν qˆ)qˆ+ηˆν (ηˆν qˆ)qˆ, (B4)
m · − ·
ν