- Email: [email protected]
Neutrino counting in e+e− collisions
Nuclear Physics B286 (1987) 293-305 North-Holland, Amsterdam
NEUTRINO COUNTING IN e + e - COLLISIONS* M. CAFFO 1"2, R. GATTO 3 and E. REMIDDI t'2"4 I INFN, Sezione di Bologna, 1-40126 Bologna, Italy 2 Dipartimento di Fisica, Universitgt di Bologna, 1-40126 Bologna, Italy 3D~partetnent de Pt~vsique Th~orique, Unwersit~ de Genk~,e, CH-1211 Genet.'a 4, Switzerland 4CERN, Ctt-1211 Genet,a 23, Switzerland Received 30 September 1986
We consider neutrino counting through e+e ~ , v 3 , both for unpolarized beams and longitudinally polarized electron beam. We fully discuss the main background, from radiativc Bhabha scattering, which we calculate for the relevant kinematical configurations. Possible measurements immediately beyond the Z ° peak or well above it are discussed and compared. The background would in principle be eliminated by taking cross section differences for opposite helicities leaving the statistical error only. Our formulae and program can be used to discuss the electromagnetic background to similar high energy reactions e" e ~ Xy, where X is the missing energy-momentum.
I. Introduction Experimental upper limits on the total number of light neutrino types have recently been given by UA1 [1] and UA2 [2]. The UA1 collaboration gives an upper limit of 7 additional neutrino types besides the three known types. The result has 90% confidence level and uses an assumption on the top mass (top mass taken at 40 GeV). The UA2 collaboration, under the same conditions, gives an upper limit of at most 2.6 +_ 1.7 light additional neutrino types. We also recall that there exists a cosmological limit from the He abundance suggesting a total of 4 light neutrino types [3]. According to the standard model any additional light neutrino coupled to the Z ° would add about 175 MeV to the Z ° width, so that a measurement of such a width to, say, 50 MeV, would also provide for crucial information. A more direct method for counting the number of neutrino types is the measurement of e+e - ~ ~uy * Partially supported by the Swiss National Science Foundation. 0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(1.1)
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M. Caffo et al. / Neutrino counting
which is dominated by the Z ° diagrams [4]. In this paper we shall deal with that reaction and with the calculation of its most dangerous background e ' e ~ e ~e - y with unseen final e+e -. We shall review the results of previous calculations and describe the relevant formulae and computations. We shall also discuss the possibility of an independent and virtually background free measurement with longitudinally polarized electron beam. In sect. 2 we illustrate the present situation for the process and for its background, then we comment on different experimental proposals. In sect. 3 we give the formulas, in case of longitudinal polarization of the incoming electron, for both the process and the background and we examine the possibility for independent measurement. We will use the following notation and numerical values: electron mass, m e ; W boson mass, Mw; Z ° boson mass, M z = 93.2 GeV; total decay width of Z°: F z = 2.8 GeV (currently proposed value); electroweak mixing angle, 0w, with sin20,, = 0.223; Fermi coupling constant (running), Gv; electromagnetic (running) coupling constant at Z ° mass: a = (137) i (see also the discussion in sects. 2 and 4); number of low mass neutrino types, N, = 3. Kinematics for the generic reaction, e ( P t ) e ' ( p 2 ) - - , f ( q l ) f ( q 2 ) 7 ( k ) , where f = p,e-; beam energy, E: s = - ( p l + p 2 ) 2 4E 2, p2 E 2 2. photon energy, ~o: x = ~o/E, s I = - ( q l + q2) 2 = s(1 - x); photon angle with respect to the direction of the incident electron, 07, y = cos 07; half opening angle of disappearance cone, 0cone; minimum angle of photon detection, 0vmin; final electron energy, E~, p) = E~ - m~; final electron angle with respect to the direction of the incident electron, 0~; azimuthal angle of final electron, q,~; angle between photon and final electron direction, 0e. 7; other invariant variables: t = - ( P 2 - q 2 ) 2, tl = - ( P l - ql) 2, u = - ( P 2 - q~)2, ux= - ( P l - q 2 ) 2, k = - p l . k , k = - p 2 . k , h = - q l . k , h + = - q 2 - k .
2. Unpolarized beams; discussion of radiative Bhabha scattering The standard model Feynman diagrams contributing to the radiative neutrino pair production are shown in fig. 1. Neglecting the last graph with double W boson propagator and also neglecting terms of order t / M ~ . , tl/M2w the cross section for
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M. Caffo et al. / Neutrino counting
e'~Te e
7e
e-/'N~,li\Ve e- /
\re
e* e- /
\vl
Fig. 1. Lowest order F e y n m a n diagrams for the process e - ( Pl )e' ( Pz ) "" v( ql ) b( q2 )'¢( k ).
the reaction is given by [5]
d2oo
dxdy
aGes ( 1 - x ) [ ( 1 - ~ x ) l 6~r 2
[
2+¼x2y2]
x(1 _ y 2 )
N~(g v + g~) + 2(g v + ga)[1 - s(1 - x ) / U } ]
+ 2],
(2.1)
with gv = - ½ + 2sin28w and gA = -- 1. The term proportional to N~ is dominant and comes from the square of the Z ° annihilation amplitudes, the next term proportional to (gv + gA) comes from the Z ° - W interference, the added constant "2" is all what is left of the square W exchange amplitude in the limit described above. Indeed neglecting t / M 2 and tl/M2w terms is justified only in view of the smallness of the W contribution itself as compared to the Z ° one. The main source of background to the reaction (1.1) is the radiative Bhabha scattering, e - e - ~ e + e - y , at the kinematical configuration in which the final positron and electron go undetected in forward and backward cones of half opening angle Ocon~ around the beam line, and the emitted photon is detected in an angular r a n g e 0yrain < 0~, < (180 ° - O~m), where 0rmm> 0co,e. The standard model Feynman diagrams contributing at lowest order to the process are shown in fig. 2. For the kinematical configuration relevant for the background the dominant contribution is given by the t-channel with virtual photon exchanged, so that the calculation for the background actually reduces to the evaluation of the QED part only (see below for the calculation of the other contributions and their numerical estimates). For hard (~ > me), non-collinear (0y >> m J E e ) photons the QED bremsstrahlung formula
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M. Caffo et al. /Neutrtno counting
e° e*i~e#
e÷'~ e°
,Z* e
~,Z° e" e"
-',Z* e~/
e,l.~e .I.
e,i.~
e~
e
e- ~t'~
~x~Z,
e-
e-
e-
e-
Fig. 2. Lowest order Feynman diagrams for the process e ( Pl )e ' ( p~ ) ---,e ( q l ) e ' (q2) Y(k).
simplifies into [6, 9] d 20(QED
O3~
dxdy
~s
foa'°"'sin 0. d 0. fO2 ~ d ~ 2
EE ex x(i~c-osO~,r)X"'
Xo = ¼A W m ,
[ss,(s 2 + s ? ) + ,,,(t 2 + ,?) + uu,(u 2 + u?)] A=
SSltt~ s
WIR=
sI
k.----~+h+h
t
k)h,
tI
u
k h +k~h
uI
+k h,
]
J"
(2.2)
To obtain the background in the mentioned kinematical configuration the integration of eq. (2.2) has to be done also for the above specified range of y = cos Oy and for some chosen range of x = ~/E. That operation requires a numerical treatment (we have chosen a Monte Carlo method and the program RIWIAD [10]) and much caution. While the angle of the emitted photon is large enough for avoiding collinear singularity problems, the proper treatment of the forward (or backward) electron (or positron) peak accompanying backward or forward emission of the photon is slightly more delicate. Indeed we find it convenient to integrate only on a reduced region of the phase space and then to exploit the symmetry of the integrand to recover the full value of the integral. In fact, as the differential cross section for photon emission is symmetric for 07 ~ (180 ° - Ov), the integral over Ormi" < 07 < 90 ° is equal to the integral over 90 ° < 07 < (180 ° - oTmin); our integration variables are the photon energy, the photon angles and the electron angles 0e
M. Caffo et al. / Neutrino counting
297
and q'e, so that the integration over 0e introduces an asymmetry in the procedure. As a consequence, we find that when integrating separately over the two 0r regions requiring the same Monte Carlo statistical precision, the hemisphere containing the electron gives a contribution systematically smaller than the other hemisphere, where convergence of the Monte Carlo integration is faster. The effect disappears for 0~m~n approaching 90 °, increases for decreasing 0~min. At a given 0vm~, the difference between the two results decreases when asking for an increased precision, the second of the two being very stable, the first slowly increasing towards the value of the other. After having verified this pattern in a number of typical configurations, we found it convenient for the evaluation of our tables to integrate only in the region 90 ° < 0r < (180 ° - 0 ~ ) an then to multiply by a factor of two the result. The other obvious symmetry requirement on the azimuthal angle q~e of the electron, on the other hand, comes out to be always satisfied at any requested precision and is of no help for speeding up the integration. The QED differential cross section becomes very large but finite at 0c = 0. At variance with Coulomb scattering where the square momentum transfer between initial and final electron vanishes as -4pZsin2(½0e), in the bremsstrahlung case one has ( E - Ee) 2 t 1 = - 2m~ ( EEe + PPe- m2e) - 4pPesin2( ½0e)
=-me
2(EEe)2 (EEl)
4EEesin2(½0c)
(2.3)
When expressed in terms of the independent variables x, O~, O¢, ff~, eq. (2.3) reads =
t1
z
x2E(1 + c°s 0~,r) 2
1 - 4EEesin2(7Oe),
-me 4Ee[1-½x(1-cosOe,v)]
(2.4)
2
where E(1 - x)
m~xcosO¢.v
Ee= [ 1 - ~ x ( 1 - c o s 0 e , y ) ] + 4 E ( 1 - x ) cos 0~.r = cos Oecos0r + sin O~sinOrcoS ,~,
' (2.5)
and a corresponding formula holds for t, by consistently replacing the electron variables by the positron ones. Therefore, when the photon is radiated in the positron hemisphere (cos0e, r <0), the 1 / t t peak strongly dominates on 1/t. It is clear that the numerical evaluation of the momentum transfers appearing as denominators of Feynman propagators must be carried out identically in m~; on the
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M. Caffo et al. / Neutrino counting
contrary m 2 can be disregarded in the numerators in first approximation (see below). We can conclude that with the warnings and prescriptions illustrated the integration variables we used are suitable for proper treatment of the dominant contribution from the region of very small values of 0e. We want to stress furthermore that eq. (2.2) is so compact that numerical contributions can be kept under control in every situation. In the expression for X0 the square bracket in A is composed of terms which are always positive so that no loss in precision is expected there. On the contrary the expression for W~R in the highly peaked region of small t and t 1 exhibits quite large cancellations, checked to be of 8 digits at worst. Since we use VAX double-precision arithmetic with 15 decimal digits, that poses no problems for a requested one percent precision. To show the result, a particular case is presented in table 1; other cases, in m c = 0 approximation, have already been reported [11]. We have mentioned that all rnc2 terms have been dropped from the numerator in eq. (2.2) and we note that no term with 1 / t 2, which has the leading behaviour in the chosen configuration, is present in X 0. An explicit calculation shows that, in the considered kinematical region (where the electron is almost aligned with the beam line and the photon in the position hemisphere) the leading rn~/t 2 I2 term, to be added to X 0 (eq. (2.2)) is given by 2 S2 + u 2 S 2 + U2 t 2 _ 2SSl _ 2UUl ] me + - + ] 2t? k2+ hl k+h+ '
Xm
(2.6)
an analogous formula holding or the leading m ~ / t 2 term in the symmetric situation with the photon in the electron hemisphere. The double poles in k+ and h~ agree with ref. [7], when crossing from s-channel to t-channel; we find no agreement however with the m~ terms as given in refs. [8,9], where probably only double poles in k+ and h÷ have been accounted for. The contribution of Xm to the cross section is always negative, usually very small or completely negligible against that of X0, to which it has to be summed; Xm is however very strongly peaked at 8e = 0, where it almost compensates the X0 peak, the sum being of course always positive. The contribution of the forward X,, peak is approximately given by QED d 2o,;, -
-
dxdy
Of3
-
7rs
= -
_
to,.,
r2,,
Jo" sin0~ dOe Jo dq,¢
m2 ft~"" C(tl) ~t~ dt 1 t~ TM
(14(1
EE~x 2-x(l~cosO~,v)
X''
m2C(O) t~ n
x(l_ -y)l
(1+
[1 -
'
},
(2.7)
where C(t~), not written explicitly for short, stands for the t~ slowly varying part of
299
M. Caffo et al. / Neutrino counting TABLE 1 Cross sections (in nanobarns) as functions of beam energy (in GeV) and of photon energy bins (in GeV) Beam rgy
0.10-0.25 0.25-0.50 0.50-1.0 1.0 - 2.0 2.0 -3.0 3.0 -4.0 4.0 - 5.0 5.0 -6.0 6.0 - 7.0 7.0 -8.0 8.0 -9.0 9.0 - 10.0
47
48
49
50
51
52
0.070 0.019 0.058 0.014 0.061 0.013 0.048 0.010 0.013 0.000 0.005 0.000 0.002 0.000 0.001 0.000 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.020 0.018 0.017 0.013 0.021 0.013 0.034 0.010 0.032 0.000 0.018 0.000 0.007 0.000 0.003 0.000 0.001 0.000 0.001 0.000 0.000 0.000 0.000 0.000
0.009 0.018 0.007 0.013 0.008 0.012 0.011 0.010 0.011 0.001 0.014 0.000 0.017 0.000 0.010 0.000 0.004 0.000 0.002 0.000 0.001 0.000 0.000 0.000
0.005 0.017 0.004 0.012 0.004 0.012 0.005 0.010 0.004 0.001 0.004 0.000 0.006 0.000 0.009 0.000 0.012 0.000 0.007 O.(XIO 0.(X)3 0000 0.001 0.000
0.003 0.017 0.002 0.012 0.(X)3 0.011 0.003 0.010 0.002 0.001 0.002 0.000 0.002 0.000 0.003 0.000 0.004 0.000 0.007 0.000 0.008 0.~.)0 0.004 0.000
0.002 0.016 0.002 0.011 0.002 0.011 0.002 0.009 0.001 0.001 0.001 0.000 0.001 0.000 0.001 0.000 0.002 0.0(X) 0.(X)2 0.000 0.003 0.000 0.(X)6 0.000
The first entry is for the process e ~e- ---,bvy, with M z = 93.2 GeV, I"z = 2.8 GeV, sin2(0,,.) = 0.22, a = ~ 7 and 3 neutrino types. The second entry is for the background process e ' e -~ e- e- y, with fin',d electron and positron escaping detection inside a cone, centered on the beam direction, of half angle 0co,~~ = 2 °. For the background process we take a = ~ as more appropriate to low-momentum transfer. Both entries are for a photon emission angle 0v between 0yrain and (180 ° with 0~ i"= 45 °. The value for the background includes also the electron mass correction.
-0groin),
the integrand. T h e actual value of t~ ~ is irrelevant, provided that Itt(8c = 0) I = It~'i"l << It~'~l . The integration over y and x can be achieved analytically giving Ot3 f omQED(Xmax) -- o Q E D ( X m i n ) = - - 2 S ~ [ 2 X ( X -- 4 ) + 8 1 n ( x ) ]
[ - x( x - 2)ln(1-
y m ~ ) + 2 Li2_
~ - Yma~
where Ym~x = COS 0Vmi" and Li2(x ) is the Euler dilogarithm.
Ymax (Ymax -- 1)
~ xm,.
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M. Caffo et al. / Neutrino counting
The limit of application of eq. (2.8) to the computation of the background is given by transverse momentum conservation, since for growing values of w more and more electrons and positrons go outside the disappearance cone, so that eq. (2.8) is to be used only if sin 0cone ¢0 < E sin 0~rain " (2.9) For larger values of ¢0 the value from eq. (2.8) is however an upper limit. Numerically the X,, contribution amounts to a few percent of the X 0 contribution at most, and is included in the values of table 1. If one wants to take into account also the contribution coming from the graphs with Z ° exchange the following part should be added to the square matrix element xo
[F-(s, s)(t2 + t?) + F+(s, s)(u2 + u2)] 4sh +h
g Z =
+ [F-(s, si)(t2+t?)+f"(s,s,)(u2+u21)] 4ssl [F-(t, t)(s :~+ s?) + F+(t, t)(u 2 + u?)]
4tk h_
[ F - ( s l , s l ) ( t l + t21) + F+(sl,Sl)(ul + u2)]
4slk ,k_ [
u uI k Sh_ + k_h+ --
t,.]
t
k~h+
k h
[ F- (tl,t,)(s2 + s?) + F+(tl,tt)(u 2 + u?)] 4txk+h
+ [F-(t, ll)(s2+s2)+4tq F+ (t, tl)(u2+u 2 )][ U__+k~h k_h+Ul + --+k+k_S
+¼(u2+u~
+
)(F'(s,t)[ u s - -st - - k_h÷ + - - h4h
F-+ (-s l , t ) [
sit
U- l k+h
-
+
-
- Sl k.k
t] k h _
_
h hSl ]
t ] F÷(s,q)[ u, s -k.h_ + - -st I h+h k-~_ +
+
F'(sx,tl)[kU_____~ sit I
+
s, k,k
-
-
t, k~h~
t, )} k+h~ ' -
-
(2.1o) where
L D(s)
+
+
• ,[(s-
+
2 2 Mzr ]
o(,)o(t) (2.11)
M. Caffo et al. / Neutrino counting
301
and D(s) = (s- M~)z+
2 2 MzFz,
cy - Cv+ 4- __
C A ~ '
(2.12) C~z=(C2 + C~)2+4C~C 2 ,
(1 - 4 sin20~,) Cv = - (4 sin Owcos Ow) '
CA=-
1 U4 1 % c OS,w,. v
(2.13)
(2.14)
Even if the Z ° annihilation contribution at the resonance is larger than the QED annihilation graph, the kinematical configuration under consideration is entirely dominated by the t-channel (Coulomb) exchange, so that the Z ° contributions are negative and still much lower than the pure QED part (in absolute value by a factor < 10-3). That is different from the eve .- -o ~ + ~ - 7 reaction, in which the t-channel is absent and the Z ° exchange does dominate. Eq. (2.10) is in agreement with ref. [8]. The contributions due to finite width effects, which correspond to an absorptive term, have also been investigated and found to be negligible (in absolute value by a factor < 10-6). Concerning the running value of the fine structure constant a, for e re- ---, ~uy 1 the obvious choice is a = 1_,-ff3, corresponding to a subtraction point at the square Z ° mass. On the contrary, the background amplitude is strongly peaked corresponding to extremely small values of the momentum transfers t, t t (exceptional momenta in the terminology of renormalization group). For those variables a value of a running at lower momenta seems more natural; for that reason, we use a = ~ for the background process. When using a running coupling corresponding to a high energy subtraction point, larger radiative corrections are expected to show up. In ref. [11] we decided to subtract, for simplicity, both the neutrino counting reaction and the electromagnetic background at the square Z ° mass (a = ~-~g) and to systematically include all deviations into the definition of the radiative corrections. Concerning the QED radiative corrections, the calculation of those relevant to the process e÷e -o ~v7 have recently been completed [12]; the QED radiative corrections to the background, on the contrary, are not known at present and their complete calculation may still take some time. From the experimental point of view there are two possible configurations of detection: (i) well above the Z ° peak with detection of high energy photons [13]; (ii) just above the Z ° peak with detection of relatively low energy photons [14]. The signal in case (i) has a cross section which is 10 times smaller than in case (ii). The background in case (i) is less than 5% of the signal, while in case (ii) it is comparable or larger. In case (i) therefore the knowledge of the leading part only of the background is enough to reliably evaluate the number of neutrino types. On the contrary, in case (ii) also the knowledge (now not available) of the QED radiative corrections to the background could perhaps be required. The signal corresponding to the existence of a 4th neutrino type is about 30% of the rates for 3 neutrinos only
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M. Caffo et al. / Neutrino counting
which we present in table 1. It is therefore essential, for the proper interpretation of the measurements, to know the QED background with a theoretical uncertainty much less than that signal. The configuration (i) is therefore in principle better suited for the neutrino counting experiments. In the configuration (ii) it is advisable to drastically cut the background by squeezing the e+e - disappearance cone and limiting the photon acceptance region to large angles. The ensuing reduction on the counting rate is in our opinion compensated by the increase in the purity of the signal.
3. Longitudinally polarized electron beam We consider now the case of longitudinally polarized electron and positron. The following remarks are of interest due to the planned longitudinal polarization at SLC [15] and may be also at LEP in a later stage [16]. The general structure of high energy behaviour of QED and of the standard model decouples incoming electron and positron pairs with equal helicity, so that polarization of the positron beam too would be of no use. In fact, if h e and hp are the helicities of the electron and the positron and have the value + 1, in the same limit of eq. (2.1) the cross section for the process e +e- ---, kvv with longitudinally polarized electron and positron is
dxdy
aGes he ~6¢r
d2oo
d2OLp
(1-hphe)
dxdy
(1- x)[(1-
~-x)2+ }x2y 2]
x(1 _ y 2 )
[N"2gvgA+2(gv+ga)[1--s(l--x)/M~] X[ ~:s--~Zx~ZM~z]~+--~2.7~z
]) +2
. (3.1)
When averaging on hp the coefficient (1 - hphe) gives 1, then averaging on h e the result of eq. (2.1) is obviously recovered. The difference between the distributions corresponding to opposite electron helicity and averaged over positron helicity is d26Lp(h¢ = - 1 ) APe
dxdy o@ = APe2 6~r 2
×
d26u,(he = + 1 ) ]
-
-dx-dy
1
(i- x)[O- x)2+ x(1 _ y 2 )
"N,2gvga + 2(gv+gA)[l - s ( 1 - x ) / M -2 2 [1 - s ( 1 -x)/M~]2+Iz/Mz
2]
]
+ 2]
(3.2)
where AP e = Pc(he = - 1) - P~'(he = - 1) is the difference of the fraction of electrons with helicity say h e = - 1 in two runs of opposite dominant helicity. The actual value of (3.2) depends critically on the obtainable polarization and on the actual value of sin2Ow, as gv vanishes at sin20,~. = 4.~ With 100% of polarization
M. Caffo et al. / Neutrinocounting
303
and a value of sin20w between 0.23 and 0.22 the ratio between (3.2) and (2.1) varies between 30% and 50% leaving us with reasonable counting rates. It is not necessary to present explicitly the actual numerical values: due to the smallness of the W contributions, they are immediately obtained by multiplying by the ratio [(4gvgAAPe)/(g2 + gA2)]= 0.5APe the values for the unpolarized case in table 1. The problems arising when the QED background is large compared to the rate of (2.1) have been extensively discussed in the previous section. We note here that the QED background is independent of the electron helicity, therefore the signal (3.2) is virtually background free. More precisely the largest contribution comes from the helicity-asymmetric background due to the interference between pure QED and Z ° exchange graphs, which turns out to be very small (by a factor < 10- 4). The cross section for this contribution with me = 0 in the numerator and averaged over positron helicity is d2°L~Ddxdy
{ ×
f0
IrsCt--~s °~"°sin0~d0~
F ° ( s , s) F°(st,s~) + -~-~-h-_ + sak+k_
F ° ( t , t) tk_h_
d ' ~ 2 - x(1 - cOSOe,v) [ X ° + X Z + h e X L p ] '
F°(s,s~)[ ss I
tlk+h ,
k_h,
+
u kS h
F°(ti, tl) +
o st,[u
+ - st
f0
__ tt 1
s
+
u1 k. h+
+ __
k.h_
k_h+
t I+
h,h_
k_h_
st I
k k_
k_h_
sxt t
t k+h.
+ __
+
k+k - - + k+h_
tt ] k_h _
h h_
h ~h
k+h+
k~k
k~h.
+ sit
k+h_
(3.3) where F°(s, t) is like in (2.11)-(2.14) with a suffix 0 in place of +_, and C°z = CvCA ,
C°z = ( C 2 + C2 ) C v CA.
(3.4)
As before, modifications arising from the finite width of the Z ° have been computed and turn out to be negligible (by a factor < 10-6). The disappearance of the background to the signal (3.2) still requires a careful consideration of the statistical error in the difference between the rates from the two polarization states, which practically remains the same as for the larger separate rates for h c = + 1, corresponding to the sum of the signal (3.1) plus the background (3.3). Therefore the possible preference to the method of the previous section has to be carefully considered in every kinematical configuration. In any case, the informa-
'
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M. Caffo et al. / Neutrino counting
tion offered by this second method is independent from the previous one and can provide an additional cross check to the neutrino counting experiment. 4. Conclusions
As it is well known, the main background to the neutrino counting reaction e+e ---o ~v3, comes from radiative Bhabha scattering e+e ---, e+e-'y with undetected final positron and electron in the forward and backward cones of small opening angle 0c around the beam line. The photon is detected by the apparatus for angles between some O~in and (180 ° -0ymin). The integration of the radiative Bhabha background in the relevant configuration is delicate, in particular because of the very small values that can take the square momentum transfer between the initial and final electron. Higher order corrections to the radiative Bhabha scattering in the considered kinematical configurations have not yet been calculated. The numbers we give in table 1 correspond to a = T2~T for the neutrino counting reaction e+e ----, ~vT and to ot = ~ for the electromagnetic background reaction, where low transferred momenta dominate. Measurements well above the Z ° peak with detection of high-energy photons are safer for neutrino counting than measurements just above the Z ° peak with low-energy photons. Such low photon energy measurements, however, will presumably be carried out first; in such a case to get safer configurations one should use smaller e~-e- disappearance cones and larger angle photons. An independent measurement is obtained by means of longitudinally polarized electron beam. The main theoretical advantage is the complete elimination of the QED background in the difference between opposite helicities data (the helicity asymmetric background from the interference between QED and Z ° exchange graphs is very small). The statistical error will however remain, playing a relatively larger role on the difference as compared to the separate data sets. We have performed a careful analysis of the radiative Bhabha QED background e+e ----, e+e-T to the neutrino counting reaction e+e --, ~v3'. We have presented the results of numerical calculations relevant to some of the proposed experiments. The choice of the experimental cuts 0co,e and 0vmin has only to be performed in view of reducing the overall statistical error originated from radiative Bhabha events. Our formulas and computer programs are available on request for a finer tuning of the predictions to the details of the experimental setups. They can of course be used for calculating the electromagnetic background to any similar reaction e4e ----, XT, where X is missing energy-momentum. Some of such conjectured missing energymomentum processes are expected to be depressed by large factors with respect to the neutrino counting reaction. In the most favorable case, where one may use longitudinally polarized beams, one has to take into account the above referred large statistical error in the difference of the rates for opposite polarizations, so that practical background control may be very difficult for such conjectured processes.
M. Caf/o et al. / Neutrino counting
305
We thank G. Barbiellini, M. Bourquin, D. Boutigny, F. Palmonari, D. Perret-Gallix for useful conversations. We also thank the members of the LEP 200 Electroweak Radiative Corrections Working Group (conveners F. Dydak and R. Kleiss) for the many useful discussions. Note added We have been informed that D. Karlen of SLAC has recently compared our results with histograms obtained by an event generator program and found that they interpolate the histograms accurately. We thank F.A. Berends for the information. We also thank G. Bonvicini of MARK II for the information that another confirmation of our numerical results has also been obtained by H. Vehman. Numerical discrepancies with our results (in particular in the energy photon dependence) have been claimed by Mana and Martinez in a recent preprint [17], but we are unable to understand the origin of the discrepancy. We have carried out several additional checks to investigate possible sources of loss of numerical precision and they all confirm our results. References [1] G. Arnison et al., Nucl. Phys. B276 (1986) 253 [2] J.A. Appel et al., CERN EP/85-166 [3] G. Steigman, K.A. Olive, D.N. Schramm and M.S. Turner, University of Chicago preprint (1986) and references therein [41 E. Ma and J. Okada, Phys. Rev. Lett. 41 (1978) 287 [5] K.J.F. Gaemers, R. Gastmans and R.M Renard, Phys. Rev. D19 (1979) 1605 [6] F.A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans and T.T. Wu, Phys. Lett. 103B (1981) 124 [7] F.A. Berends, R. Kleiss and S. Jadach, Nucl. Phys. B202 (1982) 63 [8] F.A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans, W. Troost and T.T. Wu, Nucl. Phys. B206 (1982) 61 [9] F.A. Berends and R. Kleiss, Nucl. Phys. B228 (1983) 537 [10] B. Lautrup, RIWIAD, CERN-DD Long Writeup Dl14 [11] M. Caffo, R. Gatto and E. Remiddi, Phys. Lett. 173 (1986) 91 [121 F.A. Berends, G.J.H. Burgers and W.L. van Ncerven, Leiden Univ. preprint (June 1986), Phys. Lett. B, to be published: and communication to LEP 200 Electroweak Radiative Corrections Working Group; M. Igarashi and N. Nakazawa, preprint TKU-HEP 86/01 (Februa~ 1986), Nucl. Phys. B, to be published [13] G. Barbiellini, B. Richter and J.L. Siegrist, Phys. Lett. 106B (1981) 414 [14] J. Bartels, A. Fridman, A. Schwarz and T.T. Wu, Z. Phys. C23 (1984) 295 [15] C.Y. Prescott, preprint SLAC-PUB-3728 (July 1985) [16] C. Bovet et al., in Physics at LEP, ed. J. Ellis and R. Peccei, CERN 86-02, vol. 1, p. 58 [17] C. Mana and M. Martinez, DESY preprint DESY 86-062, Nucl. Phys. B, to be publishcd
NEUTRINO COUNTING IN e + e - COLLISIONS* M. CAFFO 1"2, R. GATTO 3 and E. REMIDDI t'2"4 I INFN, Sezione di Bologna, 1-40126 Bologna, Italy 2 Dipartimento di Fisica, Universitgt di Bologna, 1-40126 Bologna, Italy 3D~partetnent de Pt~vsique Th~orique, Unwersit~ de Genk~,e, CH-1211 Genet.'a 4, Switzerland 4CERN, Ctt-1211 Genet,a 23, Switzerland Received 30 September 1986
We consider neutrino counting through e+e ~ , v 3 , both for unpolarized beams and longitudinally polarized electron beam. We fully discuss the main background, from radiativc Bhabha scattering, which we calculate for the relevant kinematical configurations. Possible measurements immediately beyond the Z ° peak or well above it are discussed and compared. The background would in principle be eliminated by taking cross section differences for opposite helicities leaving the statistical error only. Our formulae and program can be used to discuss the electromagnetic background to similar high energy reactions e" e ~ Xy, where X is the missing energy-momentum.
I. Introduction Experimental upper limits on the total number of light neutrino types have recently been given by UA1 [1] and UA2 [2]. The UA1 collaboration gives an upper limit of 7 additional neutrino types besides the three known types. The result has 90% confidence level and uses an assumption on the top mass (top mass taken at 40 GeV). The UA2 collaboration, under the same conditions, gives an upper limit of at most 2.6 +_ 1.7 light additional neutrino types. We also recall that there exists a cosmological limit from the He abundance suggesting a total of 4 light neutrino types [3]. According to the standard model any additional light neutrino coupled to the Z ° would add about 175 MeV to the Z ° width, so that a measurement of such a width to, say, 50 MeV, would also provide for crucial information. A more direct method for counting the number of neutrino types is the measurement of e+e - ~ ~uy * Partially supported by the Swiss National Science Foundation. 0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(1.1)
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M. Caffo et al. / Neutrino counting
which is dominated by the Z ° diagrams [4]. In this paper we shall deal with that reaction and with the calculation of its most dangerous background e ' e ~ e ~e - y with unseen final e+e -. We shall review the results of previous calculations and describe the relevant formulae and computations. We shall also discuss the possibility of an independent and virtually background free measurement with longitudinally polarized electron beam. In sect. 2 we illustrate the present situation for the process and for its background, then we comment on different experimental proposals. In sect. 3 we give the formulas, in case of longitudinal polarization of the incoming electron, for both the process and the background and we examine the possibility for independent measurement. We will use the following notation and numerical values: electron mass, m e ; W boson mass, Mw; Z ° boson mass, M z = 93.2 GeV; total decay width of Z°: F z = 2.8 GeV (currently proposed value); electroweak mixing angle, 0w, with sin20,, = 0.223; Fermi coupling constant (running), Gv; electromagnetic (running) coupling constant at Z ° mass: a = (137) i (see also the discussion in sects. 2 and 4); number of low mass neutrino types, N, = 3. Kinematics for the generic reaction, e ( P t ) e ' ( p 2 ) - - , f ( q l ) f ( q 2 ) 7 ( k ) , where f = p,e-; beam energy, E: s = - ( p l + p 2 ) 2 4E 2, p2 E 2 2. photon energy, ~o: x = ~o/E, s I = - ( q l + q2) 2 = s(1 - x); photon angle with respect to the direction of the incident electron, 07, y = cos 07; half opening angle of disappearance cone, 0cone; minimum angle of photon detection, 0vmin; final electron energy, E~, p) = E~ - m~; final electron angle with respect to the direction of the incident electron, 0~; azimuthal angle of final electron, q,~; angle between photon and final electron direction, 0e. 7; other invariant variables: t = - ( P 2 - q 2 ) 2, tl = - ( P l - ql) 2, u = - ( P 2 - q~)2, ux= - ( P l - q 2 ) 2, k = - p l . k , k = - p 2 . k , h = - q l . k , h + = - q 2 - k .
2. Unpolarized beams; discussion of radiative Bhabha scattering The standard model Feynman diagrams contributing to the radiative neutrino pair production are shown in fig. 1. Neglecting the last graph with double W boson propagator and also neglecting terms of order t / M ~ . , tl/M2w the cross section for
295
M. Caffo et al. / Neutrino counting
e'~Te e
7e
e-/'N~,li\Ve e- /
\re
e* e- /
\vl
Fig. 1. Lowest order F e y n m a n diagrams for the process e - ( Pl )e' ( Pz ) "" v( ql ) b( q2 )'¢( k ).
the reaction is given by [5]
d2oo
dxdy
aGes ( 1 - x ) [ ( 1 - ~ x ) l 6~r 2
[
2+¼x2y2]
x(1 _ y 2 )
N~(g v + g~) + 2(g v + ga)[1 - s(1 - x ) / U } ]
+ 2],
(2.1)
with gv = - ½ + 2sin28w and gA = -- 1. The term proportional to N~ is dominant and comes from the square of the Z ° annihilation amplitudes, the next term proportional to (gv + gA) comes from the Z ° - W interference, the added constant "2" is all what is left of the square W exchange amplitude in the limit described above. Indeed neglecting t / M 2 and tl/M2w terms is justified only in view of the smallness of the W contribution itself as compared to the Z ° one. The main source of background to the reaction (1.1) is the radiative Bhabha scattering, e - e - ~ e + e - y , at the kinematical configuration in which the final positron and electron go undetected in forward and backward cones of half opening angle Ocon~ around the beam line, and the emitted photon is detected in an angular r a n g e 0yrain < 0~, < (180 ° - O~m), where 0rmm> 0co,e. The standard model Feynman diagrams contributing at lowest order to the process are shown in fig. 2. For the kinematical configuration relevant for the background the dominant contribution is given by the t-channel with virtual photon exchanged, so that the calculation for the background actually reduces to the evaluation of the QED part only (see below for the calculation of the other contributions and their numerical estimates). For hard (~ > me), non-collinear (0y >> m J E e ) photons the QED bremsstrahlung formula
296
M. Caffo et al. /Neutrtno counting
e° e*i~e#
e÷'~ e°
,Z* e
~,Z° e" e"
-',Z* e~/
e,l.~e .I.
e,i.~
e~
e
e- ~t'~
~x~Z,
e-
e-
e-
e-
Fig. 2. Lowest order Feynman diagrams for the process e ( Pl )e ' ( p~ ) ---,e ( q l ) e ' (q2) Y(k).
simplifies into [6, 9] d 20(QED
O3~
dxdy
~s
foa'°"'sin 0. d 0. fO2 ~ d ~ 2
EE ex x(i~c-osO~,r)X"'
Xo = ¼A W m ,
[ss,(s 2 + s ? ) + ,,,(t 2 + ,?) + uu,(u 2 + u?)] A=
SSltt~ s
WIR=
sI
k.----~+h+h
t
k)h,
tI
u
k h +k~h
uI
+k h,
]
J"
(2.2)
To obtain the background in the mentioned kinematical configuration the integration of eq. (2.2) has to be done also for the above specified range of y = cos Oy and for some chosen range of x = ~/E. That operation requires a numerical treatment (we have chosen a Monte Carlo method and the program RIWIAD [10]) and much caution. While the angle of the emitted photon is large enough for avoiding collinear singularity problems, the proper treatment of the forward (or backward) electron (or positron) peak accompanying backward or forward emission of the photon is slightly more delicate. Indeed we find it convenient to integrate only on a reduced region of the phase space and then to exploit the symmetry of the integrand to recover the full value of the integral. In fact, as the differential cross section for photon emission is symmetric for 07 ~ (180 ° - Ov), the integral over Ormi" < 07 < 90 ° is equal to the integral over 90 ° < 07 < (180 ° - oTmin); our integration variables are the photon energy, the photon angles and the electron angles 0e
M. Caffo et al. / Neutrino counting
297
and q'e, so that the integration over 0e introduces an asymmetry in the procedure. As a consequence, we find that when integrating separately over the two 0r regions requiring the same Monte Carlo statistical precision, the hemisphere containing the electron gives a contribution systematically smaller than the other hemisphere, where convergence of the Monte Carlo integration is faster. The effect disappears for 0~m~n approaching 90 °, increases for decreasing 0~min. At a given 0vm~, the difference between the two results decreases when asking for an increased precision, the second of the two being very stable, the first slowly increasing towards the value of the other. After having verified this pattern in a number of typical configurations, we found it convenient for the evaluation of our tables to integrate only in the region 90 ° < 0r < (180 ° - 0 ~ ) an then to multiply by a factor of two the result. The other obvious symmetry requirement on the azimuthal angle q~e of the electron, on the other hand, comes out to be always satisfied at any requested precision and is of no help for speeding up the integration. The QED differential cross section becomes very large but finite at 0c = 0. At variance with Coulomb scattering where the square momentum transfer between initial and final electron vanishes as -4pZsin2(½0e), in the bremsstrahlung case one has ( E - Ee) 2 t 1 = - 2m~ ( EEe + PPe- m2e) - 4pPesin2( ½0e)
=-me
2(EEe)2 (EEl)
4EEesin2(½0c)
(2.3)
When expressed in terms of the independent variables x, O~, O¢, ff~, eq. (2.3) reads =
t1
z
x2E(1 + c°s 0~,r) 2
1 - 4EEesin2(7Oe),
-me 4Ee[1-½x(1-cosOe,v)]
(2.4)
2
where E(1 - x)
m~xcosO¢.v
Ee= [ 1 - ~ x ( 1 - c o s 0 e , y ) ] + 4 E ( 1 - x ) cos 0~.r = cos Oecos0r + sin O~sinOrcoS ,~,
' (2.5)
and a corresponding formula holds for t, by consistently replacing the electron variables by the positron ones. Therefore, when the photon is radiated in the positron hemisphere (cos0e, r <0), the 1 / t t peak strongly dominates on 1/t. It is clear that the numerical evaluation of the momentum transfers appearing as denominators of Feynman propagators must be carried out identically in m~; on the
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M. Caffo et al. / Neutrino counting
contrary m 2 can be disregarded in the numerators in first approximation (see below). We can conclude that with the warnings and prescriptions illustrated the integration variables we used are suitable for proper treatment of the dominant contribution from the region of very small values of 0e. We want to stress furthermore that eq. (2.2) is so compact that numerical contributions can be kept under control in every situation. In the expression for X0 the square bracket in A is composed of terms which are always positive so that no loss in precision is expected there. On the contrary the expression for W~R in the highly peaked region of small t and t 1 exhibits quite large cancellations, checked to be of 8 digits at worst. Since we use VAX double-precision arithmetic with 15 decimal digits, that poses no problems for a requested one percent precision. To show the result, a particular case is presented in table 1; other cases, in m c = 0 approximation, have already been reported [11]. We have mentioned that all rnc2 terms have been dropped from the numerator in eq. (2.2) and we note that no term with 1 / t 2, which has the leading behaviour in the chosen configuration, is present in X 0. An explicit calculation shows that, in the considered kinematical region (where the electron is almost aligned with the beam line and the photon in the position hemisphere) the leading rn~/t 2 I2 term, to be added to X 0 (eq. (2.2)) is given by 2 S2 + u 2 S 2 + U2 t 2 _ 2SSl _ 2UUl ] me + - + ] 2t? k2+ hl k+h+ '
Xm
(2.6)
an analogous formula holding or the leading m ~ / t 2 term in the symmetric situation with the photon in the electron hemisphere. The double poles in k+ and h~ agree with ref. [7], when crossing from s-channel to t-channel; we find no agreement however with the m~ terms as given in refs. [8,9], where probably only double poles in k+ and h÷ have been accounted for. The contribution of Xm to the cross section is always negative, usually very small or completely negligible against that of X0, to which it has to be summed; Xm is however very strongly peaked at 8e = 0, where it almost compensates the X0 peak, the sum being of course always positive. The contribution of the forward X,, peak is approximately given by QED d 2o,;, -
-
dxdy
Of3
-
7rs
= -
_
to,.,
r2,,
Jo" sin0~ dOe Jo dq,¢
m2 ft~"" C(tl) ~t~ dt 1 t~ TM
(14(1
EE~x 2-x(l~cosO~,v)
X''
m2C(O) t~ n
x(l_ -y)l
(1+
[1 -
'
},
(2.7)
where C(t~), not written explicitly for short, stands for the t~ slowly varying part of
299
M. Caffo et al. / Neutrino counting TABLE 1 Cross sections (in nanobarns) as functions of beam energy (in GeV) and of photon energy bins (in GeV) Beam rgy
0.10-0.25 0.25-0.50 0.50-1.0 1.0 - 2.0 2.0 -3.0 3.0 -4.0 4.0 - 5.0 5.0 -6.0 6.0 - 7.0 7.0 -8.0 8.0 -9.0 9.0 - 10.0
47
48
49
50
51
52
0.070 0.019 0.058 0.014 0.061 0.013 0.048 0.010 0.013 0.000 0.005 0.000 0.002 0.000 0.001 0.000 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.020 0.018 0.017 0.013 0.021 0.013 0.034 0.010 0.032 0.000 0.018 0.000 0.007 0.000 0.003 0.000 0.001 0.000 0.001 0.000 0.000 0.000 0.000 0.000
0.009 0.018 0.007 0.013 0.008 0.012 0.011 0.010 0.011 0.001 0.014 0.000 0.017 0.000 0.010 0.000 0.004 0.000 0.002 0.000 0.001 0.000 0.000 0.000
0.005 0.017 0.004 0.012 0.004 0.012 0.005 0.010 0.004 0.001 0.004 0.000 0.006 0.000 0.009 0.000 0.012 0.000 0.007 O.(XIO 0.(X)3 0000 0.001 0.000
0.003 0.017 0.002 0.012 0.(X)3 0.011 0.003 0.010 0.002 0.001 0.002 0.000 0.002 0.000 0.003 0.000 0.004 0.000 0.007 0.000 0.008 0.~.)0 0.004 0.000
0.002 0.016 0.002 0.011 0.002 0.011 0.002 0.009 0.001 0.001 0.001 0.000 0.001 0.000 0.001 0.000 0.002 0.0(X) 0.(X)2 0.000 0.003 0.000 0.(X)6 0.000
The first entry is for the process e ~e- ---,bvy, with M z = 93.2 GeV, I"z = 2.8 GeV, sin2(0,,.) = 0.22, a = ~ 7 and 3 neutrino types. The second entry is for the background process e ' e -~ e- e- y, with fin',d electron and positron escaping detection inside a cone, centered on the beam direction, of half angle 0co,~~ = 2 °. For the background process we take a = ~ as more appropriate to low-momentum transfer. Both entries are for a photon emission angle 0v between 0yrain and (180 ° with 0~ i"= 45 °. The value for the background includes also the electron mass correction.
-0groin),
the integrand. T h e actual value of t~ ~ is irrelevant, provided that Itt(8c = 0) I = It~'i"l << It~'~l . The integration over y and x can be achieved analytically giving Ot3 f omQED(Xmax) -- o Q E D ( X m i n ) = - - 2 S ~ [ 2 X ( X -- 4 ) + 8 1 n ( x ) ]
[ - x( x - 2)ln(1-
y m ~ ) + 2 Li2_
~ - Yma~
where Ym~x = COS 0Vmi" and Li2(x ) is the Euler dilogarithm.
Ymax (Ymax -- 1)
~ xm,.
300
M. Caffo et al. / Neutrino counting
The limit of application of eq. (2.8) to the computation of the background is given by transverse momentum conservation, since for growing values of w more and more electrons and positrons go outside the disappearance cone, so that eq. (2.8) is to be used only if sin 0cone ¢0 < E sin 0~rain " (2.9) For larger values of ¢0 the value from eq. (2.8) is however an upper limit. Numerically the X,, contribution amounts to a few percent of the X 0 contribution at most, and is included in the values of table 1. If one wants to take into account also the contribution coming from the graphs with Z ° exchange the following part should be added to the square matrix element xo
[F-(s, s)(t2 + t?) + F+(s, s)(u2 + u2)] 4sh +h
g Z =
+ [F-(s, si)(t2+t?)+f"(s,s,)(u2+u21)] 4ssl [F-(t, t)(s :~+ s?) + F+(t, t)(u 2 + u?)]
4tk h_
[ F - ( s l , s l ) ( t l + t21) + F+(sl,Sl)(ul + u2)]
4slk ,k_ [
u uI k Sh_ + k_h+ --
t,.]
t
k~h+
k h
[ F- (tl,t,)(s2 + s?) + F+(tl,tt)(u 2 + u?)] 4txk+h
+ [F-(t, ll)(s2+s2)+4tq F+ (t, tl)(u2+u 2 )][ U__+k~h k_h+Ul + --+k+k_S
+¼(u2+u~
+
)(F'(s,t)[ u s - -st - - k_h÷ + - - h4h
F-+ (-s l , t ) [
sit
U- l k+h
-
+
-
- Sl k.k
t] k h _
_
h hSl ]
t ] F÷(s,q)[ u, s -k.h_ + - -st I h+h k-~_ +
+
F'(sx,tl)[kU_____~ sit I
+
s, k,k
-
-
t, k~h~
t, )} k+h~ ' -
-
(2.1o) where
L D(s)
+
+
• ,[(s-
+
2 2 Mzr ]
o(,)o(t) (2.11)
M. Caffo et al. / Neutrino counting
301
and D(s) = (s- M~)z+
2 2 MzFz,
cy - Cv+ 4- __
C A ~ '
(2.12) C~z=(C2 + C~)2+4C~C 2 ,
(1 - 4 sin20~,) Cv = - (4 sin Owcos Ow) '
CA=-
1 U4 1 % c OS,w,. v
(2.13)
(2.14)
Even if the Z ° annihilation contribution at the resonance is larger than the QED annihilation graph, the kinematical configuration under consideration is entirely dominated by the t-channel (Coulomb) exchange, so that the Z ° contributions are negative and still much lower than the pure QED part (in absolute value by a factor < 10-3). That is different from the eve .- -o ~ + ~ - 7 reaction, in which the t-channel is absent and the Z ° exchange does dominate. Eq. (2.10) is in agreement with ref. [8]. The contributions due to finite width effects, which correspond to an absorptive term, have also been investigated and found to be negligible (in absolute value by a factor < 10-6). Concerning the running value of the fine structure constant a, for e re- ---, ~uy 1 the obvious choice is a = 1_,-ff3, corresponding to a subtraction point at the square Z ° mass. On the contrary, the background amplitude is strongly peaked corresponding to extremely small values of the momentum transfers t, t t (exceptional momenta in the terminology of renormalization group). For those variables a value of a running at lower momenta seems more natural; for that reason, we use a = ~ for the background process. When using a running coupling corresponding to a high energy subtraction point, larger radiative corrections are expected to show up. In ref. [11] we decided to subtract, for simplicity, both the neutrino counting reaction and the electromagnetic background at the square Z ° mass (a = ~-~g) and to systematically include all deviations into the definition of the radiative corrections. Concerning the QED radiative corrections, the calculation of those relevant to the process e÷e -o ~v7 have recently been completed [12]; the QED radiative corrections to the background, on the contrary, are not known at present and their complete calculation may still take some time. From the experimental point of view there are two possible configurations of detection: (i) well above the Z ° peak with detection of high energy photons [13]; (ii) just above the Z ° peak with detection of relatively low energy photons [14]. The signal in case (i) has a cross section which is 10 times smaller than in case (ii). The background in case (i) is less than 5% of the signal, while in case (ii) it is comparable or larger. In case (i) therefore the knowledge of the leading part only of the background is enough to reliably evaluate the number of neutrino types. On the contrary, in case (ii) also the knowledge (now not available) of the QED radiative corrections to the background could perhaps be required. The signal corresponding to the existence of a 4th neutrino type is about 30% of the rates for 3 neutrinos only
302
M. Caffo et al. / Neutrino counting
which we present in table 1. It is therefore essential, for the proper interpretation of the measurements, to know the QED background with a theoretical uncertainty much less than that signal. The configuration (i) is therefore in principle better suited for the neutrino counting experiments. In the configuration (ii) it is advisable to drastically cut the background by squeezing the e+e - disappearance cone and limiting the photon acceptance region to large angles. The ensuing reduction on the counting rate is in our opinion compensated by the increase in the purity of the signal.
3. Longitudinally polarized electron beam We consider now the case of longitudinally polarized electron and positron. The following remarks are of interest due to the planned longitudinal polarization at SLC [15] and may be also at LEP in a later stage [16]. The general structure of high energy behaviour of QED and of the standard model decouples incoming electron and positron pairs with equal helicity, so that polarization of the positron beam too would be of no use. In fact, if h e and hp are the helicities of the electron and the positron and have the value + 1, in the same limit of eq. (2.1) the cross section for the process e +e- ---, kvv with longitudinally polarized electron and positron is
dxdy
aGes he ~6¢r
d2oo
d2OLp
(1-hphe)
dxdy
(1- x)[(1-
~-x)2+ }x2y 2]
x(1 _ y 2 )
[N"2gvgA+2(gv+ga)[1--s(l--x)/M~] X[ ~:s--~Zx~ZM~z]~+--~2.7~z
]) +2
. (3.1)
When averaging on hp the coefficient (1 - hphe) gives 1, then averaging on h e the result of eq. (2.1) is obviously recovered. The difference between the distributions corresponding to opposite electron helicity and averaged over positron helicity is d26Lp(h¢ = - 1 ) APe
dxdy o@ = APe2 6~r 2
×
d26u,(he = + 1 ) ]
-
-dx-dy
1
(i- x)[O- x)2+ x(1 _ y 2 )
"N,2gvga + 2(gv+gA)[l - s ( 1 - x ) / M -2 2 [1 - s ( 1 -x)/M~]2+Iz/Mz
2]
]
+ 2]
(3.2)
where AP e = Pc(he = - 1) - P~'(he = - 1) is the difference of the fraction of electrons with helicity say h e = - 1 in two runs of opposite dominant helicity. The actual value of (3.2) depends critically on the obtainable polarization and on the actual value of sin2Ow, as gv vanishes at sin20,~. = 4.~ With 100% of polarization
M. Caffo et al. / Neutrinocounting
303
and a value of sin20w between 0.23 and 0.22 the ratio between (3.2) and (2.1) varies between 30% and 50% leaving us with reasonable counting rates. It is not necessary to present explicitly the actual numerical values: due to the smallness of the W contributions, they are immediately obtained by multiplying by the ratio [(4gvgAAPe)/(g2 + gA2)]= 0.5APe the values for the unpolarized case in table 1. The problems arising when the QED background is large compared to the rate of (2.1) have been extensively discussed in the previous section. We note here that the QED background is independent of the electron helicity, therefore the signal (3.2) is virtually background free. More precisely the largest contribution comes from the helicity-asymmetric background due to the interference between pure QED and Z ° exchange graphs, which turns out to be very small (by a factor < 10- 4). The cross section for this contribution with me = 0 in the numerator and averaged over positron helicity is d2°L~Ddxdy
{ ×
f0
IrsCt--~s °~"°sin0~d0~
F ° ( s , s) F°(st,s~) + -~-~-h-_ + sak+k_
F ° ( t , t) tk_h_
d ' ~ 2 - x(1 - cOSOe,v) [ X ° + X Z + h e X L p ] '
F°(s,s~)[ ss I
tlk+h ,
k_h,
+
u kS h
F°(ti, tl) +
o st,[u
+ - st
f0
__ tt 1
s
+
u1 k. h+
+ __
k.h_
k_h+
t I+
h,h_
k_h_
st I
k k_
k_h_
sxt t
t k+h.
+ __
+
k+k - - + k+h_
tt ] k_h _
h h_
h ~h
k+h+
k~k
k~h.
+ sit
k+h_
(3.3) where F°(s, t) is like in (2.11)-(2.14) with a suffix 0 in place of +_, and C°z = CvCA ,
C°z = ( C 2 + C2 ) C v CA.
(3.4)
As before, modifications arising from the finite width of the Z ° have been computed and turn out to be negligible (by a factor < 10-6). The disappearance of the background to the signal (3.2) still requires a careful consideration of the statistical error in the difference between the rates from the two polarization states, which practically remains the same as for the larger separate rates for h c = + 1, corresponding to the sum of the signal (3.1) plus the background (3.3). Therefore the possible preference to the method of the previous section has to be carefully considered in every kinematical configuration. In any case, the informa-
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tion offered by this second method is independent from the previous one and can provide an additional cross check to the neutrino counting experiment. 4. Conclusions
As it is well known, the main background to the neutrino counting reaction e+e ---o ~v3, comes from radiative Bhabha scattering e+e ---, e+e-'y with undetected final positron and electron in the forward and backward cones of small opening angle 0c around the beam line. The photon is detected by the apparatus for angles between some O~in and (180 ° -0ymin). The integration of the radiative Bhabha background in the relevant configuration is delicate, in particular because of the very small values that can take the square momentum transfer between the initial and final electron. Higher order corrections to the radiative Bhabha scattering in the considered kinematical configurations have not yet been calculated. The numbers we give in table 1 correspond to a = T2~T for the neutrino counting reaction e+e ----, ~vT and to ot = ~ for the electromagnetic background reaction, where low transferred momenta dominate. Measurements well above the Z ° peak with detection of high-energy photons are safer for neutrino counting than measurements just above the Z ° peak with low-energy photons. Such low photon energy measurements, however, will presumably be carried out first; in such a case to get safer configurations one should use smaller e~-e- disappearance cones and larger angle photons. An independent measurement is obtained by means of longitudinally polarized electron beam. The main theoretical advantage is the complete elimination of the QED background in the difference between opposite helicities data (the helicity asymmetric background from the interference between QED and Z ° exchange graphs is very small). The statistical error will however remain, playing a relatively larger role on the difference as compared to the separate data sets. We have performed a careful analysis of the radiative Bhabha QED background e+e ----, e+e-T to the neutrino counting reaction e+e --, ~v3'. We have presented the results of numerical calculations relevant to some of the proposed experiments. The choice of the experimental cuts 0co,e and 0vmin has only to be performed in view of reducing the overall statistical error originated from radiative Bhabha events. Our formulas and computer programs are available on request for a finer tuning of the predictions to the details of the experimental setups. They can of course be used for calculating the electromagnetic background to any similar reaction e4e ----, XT, where X is missing energy-momentum. Some of such conjectured missing energymomentum processes are expected to be depressed by large factors with respect to the neutrino counting reaction. In the most favorable case, where one may use longitudinally polarized beams, one has to take into account the above referred large statistical error in the difference of the rates for opposite polarizations, so that practical background control may be very difficult for such conjectured processes.
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We thank G. Barbiellini, M. Bourquin, D. Boutigny, F. Palmonari, D. Perret-Gallix for useful conversations. We also thank the members of the LEP 200 Electroweak Radiative Corrections Working Group (conveners F. Dydak and R. Kleiss) for the many useful discussions. Note added We have been informed that D. Karlen of SLAC has recently compared our results with histograms obtained by an event generator program and found that they interpolate the histograms accurately. We thank F.A. Berends for the information. We also thank G. Bonvicini of MARK II for the information that another confirmation of our numerical results has also been obtained by H. Vehman. Numerical discrepancies with our results (in particular in the energy photon dependence) have been claimed by Mana and Martinez in a recent preprint [17], but we are unable to understand the origin of the discrepancy. We have carried out several additional checks to investigate possible sources of loss of numerical precision and they all confirm our results. References [1] G. Arnison et al., Nucl. Phys. B276 (1986) 253 [2] J.A. Appel et al., CERN EP/85-166 [3] G. Steigman, K.A. Olive, D.N. Schramm and M.S. Turner, University of Chicago preprint (1986) and references therein [41 E. Ma and J. Okada, Phys. Rev. Lett. 41 (1978) 287 [5] K.J.F. Gaemers, R. Gastmans and R.M Renard, Phys. Rev. D19 (1979) 1605 [6] F.A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans and T.T. Wu, Phys. Lett. 103B (1981) 124 [7] F.A. Berends, R. Kleiss and S. Jadach, Nucl. Phys. B202 (1982) 63 [8] F.A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans, W. Troost and T.T. Wu, Nucl. Phys. B206 (1982) 61 [9] F.A. Berends and R. Kleiss, Nucl. Phys. B228 (1983) 537 [10] B. Lautrup, RIWIAD, CERN-DD Long Writeup Dl14 [11] M. Caffo, R. Gatto and E. Remiddi, Phys. Lett. 173 (1986) 91 [121 F.A. Berends, G.J.H. Burgers and W.L. van Ncerven, Leiden Univ. preprint (June 1986), Phys. Lett. B, to be published: and communication to LEP 200 Electroweak Radiative Corrections Working Group; M. Igarashi and N. Nakazawa, preprint TKU-HEP 86/01 (Februa~ 1986), Nucl. Phys. B, to be published [13] G. Barbiellini, B. Richter and J.L. Siegrist, Phys. Lett. 106B (1981) 414 [14] J. Bartels, A. Fridman, A. Schwarz and T.T. Wu, Z. Phys. C23 (1984) 295 [15] C.Y. Prescott, preprint SLAC-PUB-3728 (July 1985) [16] C. Bovet et al., in Physics at LEP, ed. J. Ellis and R. Peccei, CERN 86-02, vol. 1, p. 58 [17] C. Mana and M. Martinez, DESY preprint DESY 86-062, Nucl. Phys. B, to be publishcd