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Testing for the presence of new neutrinos and additional bosons
ANNALS OF PHYSICS210, 388-4Ol (1991)
Testing for the Presence of New Neutrinos and Additional Bosons TARANJEET
Devi
Ahilya
CHHABRA
University,
AND P. RAM BABU
School of Physics, Khandwa Road,
Indore
452001,
India
Received December 26. 1990 We relate the observables of G,,,e scatterings, e+e- +p’+pL and e+e- +vi$ to look for the existence of new neutrinos coupling to additional and/or standard Z-bosons. The observables in ve scattering and e+e- +vVy are sensitive to the presence of new neutrinos and the ones in e+e- +pc+pc are sensitive to additional Z-bosons. Hence the relations between low energy observables of these reactions can be used to test the existence of new neutrinos coupling to standard Z-bosons and/or additional Z-bosons. A way to use the observables determined from experiments conducted at high energies in testing these hypothesis is indicated. 0 1991 Academic press, IIIC.
INTRODUCTION
The interest in the question whether a single Z-boson mediates neutral weak interactions or many Z-bosons mediate the neutral weak interaction has received more attention mainly due to superstring models which can lead to a low energy gauge group larger than SU(3), x SU(2), x U(l),. As new hadronic facility and highly precise experiments in e+e- are reached, we expect to verify whether these models are correct or not [ 11. The additional neutral gauge bosons could also have a different origin such as compositeness model, e.g., see L. Epele et al. [2]. Whatever may be the origin of the additional gauge boson, phenomenologically the question is an important one. In the past many tests [3] have been suggested which are relations between various observables of reactions such as ve, VP scattering and e+e- + p+p-, hadrons, etc. Both in superstring and composite models [2], additional neutrinos beyond the known three families are allowed. In composite models they could be excited states and could be massive; in other models they could have additional families. These additional neutrinos, as experimental data indicates (pointed out below) may not couple to the standard Z-boson with the same strength as the other neutrinos, but could couple to additional Z-bosons as in Ref. [2] or they could be right-handed and could couple to additional Z-bosons as in the Seesaw model [4]. The additional neutrinos could be massive (of the order of 50 GeV) and/or mixing with the standard neutrinos, with mixing parameters as low as 10pi3. The experimental study of Z”-decays by aleph collaboration [S] rules out the stable 388 OOO3-4916/91 $7.50 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
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389
BOSONS
neutrinos up to masses of 42.7 GeV with mixing parameters down to lo- 13. There are severe experimental constraints on the number of types of neutrinos. In fact the recent experiment by L3 [6] collaboration, reports N, = 3.29 f 0.17. They rule out the possibility of four or more neutrino flavours at the 4a confidence level. The restrictive limits on the number of neutrinos essentially come from the measurements of width of Z”-resonance apart from cosmological constraints. These limits do not apply to the case when the additional neutrinos couple to the additional Z-bosons. In fact, the determination of (dt + d:) from ve scatterings [7] and Z-boson decay [S] are consistent within the error limits for the existence of additional Z-bosons. It is interesting to note that within the error bars the data are consistent with the existence of a heavier Z-boson with mass limits (2 132 GeV and d207 Gev), in addition to the standard Z-boson. Hence one can say that the data at present do not rule out the existence of additional Z-bosons. Such a possibility does exist in some compositeness and superstring inspired models [2,4]. The question of the existence of additional neutrinos to additional Z-bosons is of importance. Such questions were posed earlier in the literature and modification in neutrino counting due to additional Z-bosons are investigated in a large class of two Z”-boson models which are motivated by superstring theory. The investigations revealed that the experimental number N, will not shift considerably due to additional bosons (~0.4) [9]. There are many similar investigations in the literature. Withholding a Z-boson resonance would essentially mean suppressing the contributions of other Z-bosons, hence, insensitivity to the additional neutrinos coupling to these Z-bosons. In this paper we consider the possibility that the new neutrinos couple to additional Z-bosons and we suggest a way of testing this hypothesis when the additional neutrinos are massless, or of a very low mass.’ The tests are relations between observables of purely leptonic reactions e+e- -+ v~y, ve scatterings and e +e ~ + ,U+p ~ The tests involve the observables of the low energy experiments in which the propagator effects could be neglected. The inclusion of propagator effects will modify our results. In fact at high energies, where propagator effects of Z-bosons could become important, such tests are not possible. We get different relationships between the observables for the cases: (a) The standard case, i.e., single Z-boson and no additional neutrinos. (b) A single Z-boson coupling to standard and additional neutrinos. (c) Additional neutrinos and standard neutrinos couple to many Z-bosons. (d) Standard neutrinos and the additional neutrinos couple to different sets of Z-bosons. The different relationships observed for these different hypotheses (a-d) are the tests we propose. We also consider the effects of new W-bosons coupling to electronic type neutrinos in all the cases above. ’ The case that these neutrinos is the subject of another paper.
are massive
(stable
or unstable,
within
the detector)
is interesting
and
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AND
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In Section 1, we write down the general amplitude that is useful for calculating the observables of the reaction e+e- -+ vVy, ve scatterings. We also give the relationships between the coupling constants for the hypotheses (ad). In Section 2 we present the observables of the reactions and in Section 3 we give the relation between the observables. Section 4 is for some comments and discussion. In the Appendix we present the modification of the observables due to propagator effects. 1
In writing down the amplitude for reactions e+e- + vVy, and ve scatterings, we assume many Z-bosons mediating the electron neutrino sector and the existence of new neutrinos. We also allow for possibilities of nondiagonal neutrino interactions in the neutrino sector. However, we do not allow violation of ,LJ- e - r universality which has been experimentally established with reasonably good precission. The Feynman diagrams that contribute to the reaction e+e- --) vVy are [lo] as in Fig. (1). Using these diagrams one can write down the amplitudes for the above reactions. In writing down amplitudes, v, - v, - v, and p-e, universality is assumed. We write here the amplitude that contributes to neutrinoelectron scatterings and e+e- + iVN, vV, vN, A%. The amplitude for the photon tagged reaction has the obvious inclusion of electromagnatic Bremsstrahlung term:
+Cey”~(d:+d:.l,)ey,Nl.‘f(l-y,)N k
(1.1) The symbols e, v, N stands for electron, neutrino, and new types of neutrino’s, respectively. The first term in Eq. (1.1) is the coupling of electrons to standard neutrinos mediated by i, Z-bosons with relative strengths cli, the second term is non-diagonal neutrino neutral coupling mediated by j, Z-bosons with relative strengths /Ij, the third is the coupling of new neutrinos mediated by k, Z-bosons with relative strengths yk. The sets of Z-bosons i, j, k may not be mutually
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Y. N
(a)
(b)
69
(d
FIG. 1. The Feynman diagrams tions due to (1.d) are not considered “w” could be neglected.
contributing to e+e- --t vVy, N&J, V&J, NC?. In the text because in the low-energy approximation the propagator
contribueffects of
exclusive. Similarly the last three terms allow the possibility of many IV-bosons mediating .Fv, &V couplings. The relative strength of last two terms could be unrelated because the set of m, IV-bosons that couple to Fv currents may not have some elements in common with the set of n, IV-bosons that couple to FiV currents. In case the m and n sets are identical, and I= 1 or 6,= 1 for all 1 the strength of the sixth term is the square of the strength of the fifth term. Now we consider the four hypotheses regarding the Z-bosons mediating the electron-neutrino sector, including the new types of neutrinos. In the charged current sector many IV-bosons are allowed, but the IV-bosons coupling to old and new neutrinos are common.
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The relation between the coupling constants d’;(kk and cx,,Bj, yk are given below in four different cases when hypotheses (ad) hold. The relations between the fermicoupling constants and the coupling constants in Eq. (1.1) are: Case a.
(i) (ii)
a,d2,,=dv.A
(iii)
and
ai
db,,=O
Vif
1;
pi=yi=o.
(1.2a)
Case b.
GF
(i)
g2
-----y 3-W
(ii)
Cs,+vlm+t”l
ald>.A=dv,A;
bld\.,=Dv,A;
G:,,
= Gv,,.
(1.2b)
Case c.
where i is the number of IV-bosons.
(0
(ii) 1 aidL,A = dy.a; i
CB;dL%.=D,.; i
CYid’,,,=Gv,A.
(1.2c)
i
Case d.
(ii)
1 aidL,A = dv,A; i
CBjdL,,=Dv..; .i
Cy/cdk,A=Gv,A; k
’
(1.2d)
dv,,ZDv,,ZGv,A.
It can be seen from the above relations, that if the IV-boson sector is kept unchanged then many of the above relations are simplified. However, the resulting simplification may not give rise to new observable relations among the observables (see Section 3). 2 The neutrino electron scatterings are very well studied when both diagonal and non-diagonal neutrino neutral currents are present [ 111. In the Ref. [ 11 J the initial
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NEUTRINOS
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ADDITIONAL
393
BOSONS
neutrino arising out of charged current decays is always taken to be the standard neutrino because it is assumed that the new neutrinos couple only to Z-bosons. Here we assume a finite coupling strength of new neutrinos to W-bosons also. However, the non-standard neutrino coupling in both neutral and charged sectors is assumed to be an order of magnitude smaller than standard neutrino couplings. Hence, the flux of new neutrinos arising out of the charged current decays will be an order of magnitude smaller. Due to this reason our observables of neutrinoelectron scatterings are considered to be the sum of the observables of the reactions ve + ve; ve - Ne, reducing to the case considered in Ref. [ 121. The final recoil electron energy distribution, neglecting the mass of the neutrinos in comparison to the energy of the beam and recoil energy, and using the amplitude given in Eq. (1.1 ), can be written as (2.1)
where i may be either e, ,u, or T. And the A, B, and C’s in the different cases are: Case a. A.=(d:,+d2,)+2(dv+d,+1) B,=2d,,d,+2(d,+dA+
1)
C,=(d$-d2,)+2(dv-d,J (2.2)
A,=(d$+dZ,) B, = 2d,d, C, = (d;,-
d;).
Cases b, c, and d. A,= (d$+d2,)+26,(dc,+dA B,=2d,,d,
+26,(d,+d,
+6,)+N’(DZ,.+D:)+2N’q,(Dv+ +6,)+2N’D,,D,
+2N’q,(D,+
D, +q,) D, +ij,)
C,=(d;-d:,)+26,(d,r-d,)+N’(D;-DZ,)+2N’q,(D,r-D,) A,, = (df, + d;) + N’(D;. + 0;) B, = 2d,d,
(2.3)
+ 2(D,D,)N’
C,,=(d;,-d:,)+N’(D;.-D:,). Where d,,, D,, ‘s for the Cases (b, c, d) are as defined in Eqs. (1.2bk( 1.2d), respectively, and N’ is both the standard and the additional neutrino couplings. The reaction e+e- -+ vi”y, is suggested mainly to count the number of neutrino generations. Even though this reaction could be studied at low energies (E,,, is 40 to 50 GeV), the separation of the events from the background could be a hard task.
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AND
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Hence it was suggested that this reaction can be experimentally studied siting on the Z-boson resonance. But by doing so, one is counting or measuring the coupling strengths of only those neutrinos that couple to standard Z-bosons. As pointed out earlier, the possibility that there are new neutrinos coupling to new Z-bosons does exist in many superstring and compositeness inspired models, and the experiments of e’e- + vVy on Z-boson resonance are insensitive to such a possibility. Here we write down an angle and energy distribution of the tagged photon when the center of mass energy is small compared to the mass of any Z-boson and it is given by da dxdY
-2
$
O,(l -h+k)+O,(h+
x (1 -x)(1 [
-x/2)2+;x2(1
x (x(1 -x)[S+
X(1-
S x(1 - YZ)
-IL) -x)
03s 1 1
Y2 +
4( 1 - Y2)
Y*)+2@~~&3+]}
,
(2.4)
where x is E,IE, Y is Cos 8, Q is three momenta of photons, and Or, 02, O3 in the different cases are Case a. 0,=3(d:+d:,)+2(d,+dA+1) 02=6d,d,+2(d,+d,+l)
(2.5)
03=3(d;-d;)+2(dv-da). Cases b, c, d. 0, =3(d;+d;)+26Adv+dA+6,)+2N’(D2y+D;)+4N’q,JDv+ + NW:+ 0,=6d,d,
G:) + 2N<,,(G,+
+26,(d,+d,
+ 2NG,G,
G, + 5,)
+6,)+4N’D,D,
+ 2Nt,(G,+
D,,,+q,)
+4N’qm(Dv+
G, + t,)
D, +q,) (2.6)
O,=3(d;-d;)+26,(d,-d,)+2N’(D2,-D2,)+4N’q,,(Dv-DA) + N(G;-
GZ,) +2N(,(G,-
GA).
Where d,,, hAT GA ‘s for the Cases b, c, and d are as defined in Eqs. (1.2b)-( 1.2d), respectively. N is the number of additional neutrino couplings and N’ is the standard and additional neutrino couplings. E.g., in models with the three heavy neutrinos respecting flavour conservation (per), N is three and N’ is also three. But if additional neutrinos have different flavours with flavour conservation then N’ is zero.
NEW
NEUTRINOS
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BOSONS
395
In writing down Eqs. (2.5) to (2.6) we neglected the masses of the electron and the neutrino. We kept the electron-positron polarization and summed over the polarizations of the photon. The observables 0,) 02, 0, contain the contributions of all the channels, i.e., e+e- -+ vVy, N&, vi@, NGy. Hence ve scattering observables are, in general, unrelated to the above observables because of the additional channel Ni@, which was neglected in the ve scatterings. However, some inequalities could be there and these are our tests for the various hypotheses. The reaction e+e- +,u+P- has been very well studied experimentally and the effects of many Z-bosons mediating neutral weak interaction have also been pointed out. The observables arising out of measurements of deviation of the total cross sections from electromagnatic cross sections, angular asymmetry, and parity violation are K, L, and M, respectively. If many Z-bosons are coupling to electron current and pe universality is valid then Cuses a, b. K= dt. L = d,d,
(2.7)
M=d’,. Case c. K=cd’cz, L = C d;,d; M=C
(2.8)
da’.
Case d.
(2.9)
In general, L2 < KM and the equality sign holds good in the case of a single Z-boson hypothesis. Relating the e+e- -+ pfpP reaction to ve scatterings and e+e- + XT7 is very useful, since observables of this reaction are affected by many Z-bosons but are not affected by the existence of new neutrinos.
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3 The observables 0,) O,, 0, obtained from e+e- + vVy with unpolarized, longitudinally polarized, and transversely polarized electron-positron beam and the observables A,,u, B,,p, C,,, obtained from ve scattering the recoiled electron energy distribution are expressed in terms of (see Eqs. (2.2), (2.3), (2.5), (2.6)) the coupling constants appearing in Eq. (1.1). These expressions for the observables are different for the different hypothesis (a-d), (see Eq. (2.4)). The observables K, L, M obtained from e+e- + p+p ~ also contained the same coupling constants appearing in the electron sector. These observables are not affected by the likely presence or absence of new neutrinos. The different relations (see Eqs. (3.11) to (3.35)) between the observables in the hypotheses (a-d) lead to different relations between observables for the cases when the hypothesis (a) or (b) or (c) or (d) are true and are as given below: (a)
Case a. L2=KM
(3.11)
(A,+2A,)-O,=O
(3.12)
@,+2B,)-02=o
(3.13)
(C,+2C,)-o,=o
(3.14)
(A, + B,)(A, - B,) - C: = 0
(3.15)
(A,+B,)(A,-B&C:,=0
(3.16)
(0, +
-
O,)(O,
02)
I(O,+O,)-(A,+C,)I
I(0,-0,)-(A,-C,)I-(02-B,)2=0
01 --A, =- 0,-B, A,
(3.17)
> 0:
=-
03-C<
BP
CP
.
(3.18) (3.19)
Case b. L2=KM
(3.20)
(A,+2A,)-0,
(3.21)
(B,+2B,)-02#0
(3.22)
(C,+2C,)-03fO
(3.23)
(A,+B,M--B,)fC:
(3.24)
(A, + B,M,
- BP) = C:
(3.25)
(0, + 02)(0,
- 02) > 0:
(3.26)
I(O, + 0,) - (A, + C,)l I(O, - 0,) - (A, - C,)l > (0, - B,J2.
(3.27)
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Cases c and d. L2
(3.28)
(A.+2A,,)-O,
(3.29)
(B,+2B,)-O,#O
(3.30)
(C, + 2C,,) - 0, # 0
(3.31)
(A.+B<,)(A<,-B&4’;
(3.32)
(A, + B&A,, - B,,) > Cf
(3.33)
(0, +
(3.34)
O,)(O,
-
02)
I(O,+O,)-(A,+C<,)I
>
0:
ItO,-O,)-(A-C,)I
>(0z-BeJ2
(3.35)
The relation between the observables of e+e- + pL+pP are same in Cases b, c, and d; i.e., these relations are mainly effected by whether a single Z-boson is mediating the electronic neutral current or not. This has been pointed out already in the literature [13]. The relation within ve electron scatterings are same in the cases whether single Z-boson mediating neutral weak interaction or multi-Z-bosons are mediating neutral weak interaction. The relation between these observables are sensitive to the existence of new neutrinos. The same comment hold good for the observables within e +e ~ + vVy. If p - e - t universality is assumed (see Eqs. (3.11) to (3.35)) our relations (3.15) to (3.18) hold good in Case a, as well as in a theory with multi-Z-bosons. The existence of new neutrinos coupling to the standard Z-bosons is ruled out by the experiments measuring the decay width of Z-boson [6]. Hence the cases that are interesting in neutral weak interaction mediated by a single Z-boson with no new neutrinos, and the multi-Z-bosons theory with new neutrinos; i.e., experimentally our hypothesis b is already ruled out. However, the relation in this case is presented for completeness. In addition the relations (3.21), (3.22) (3.23) in Case b can be use to test the hypothesis that the new neutrinos couple to standard Z-bosons with very low coupling strength; i.e., the new neutrinos need not be a member of an additional neutrino family with universality. The interesting multi-Z-bosons theory with new neutrinos could have two special cases, i.e., our hypotheses c and d. Within hypotheses c and d, also, there could be different cases depending on whether there are additional W-bosons coupling to new neutrinos or not. 3.1. Distinction between Hypotheses a, c, and d. Here we discuss the different observable relations in Cases a, c, and d, when the W-bosons do not couple to new neutrinos. We find that the relations between the observables of these three reactions cannot distinguish between hypotheses c and d. It is clear in the sense that the observables of e+e- --+p+p are only sensitive to the cases with single Z-bosons versus those
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with many Z-bosons. Whereas the observables of the other two reactions are sensitive only to the fact of whether new neutrinos are present or not. Hence no relation between these observables can distinguish between the different types of coupling of multi-Z-bosons to old and new neutrinos. Henceforth we discuss the ways to distinguish between hypotheses a and c. The relations (3.11) and (3.28) clearly differ in the two cases but are not sensitive to the presence of new neutrinos. The relations (3.12)-(3.14) and (3.29)-(3.31) are also different in the two cases and these are sensitive to the fact that neutrinoelectron scattering, the initial neutrino which arises in charged current decays, cannot be a new neutrino. The relations (3.15), (3.16), (3.18) and (3.32), (3.33), (3.35) are known in the literature [12, 141 and are sensitive to the existence of new neutrinos and their coupling to new Z-bosons. The experimental verification of the relations (3.28), (3.32), (3.33), (3.35) would indicate the presence of additional neutrinos and additional Z-bosons. However, if (3.28) and (3.15), (3.16), (3.18) are experimentally obeyed, then the multi-Z-boson theory without new neutrinos is indicated. If experimentally (3.28), (3.32), (3.33), (3.35) are valid then the assumption that the new neutrino does not couple to a IV-boson is wrong. Here we have only tried to indicate how the relations between observables could be used to test the various sets of theories. If the relations of Case a are verified, then it may not rule out the derivations from the standard model; i.e., the deviations from the standard model can satisfy some “conspiratorial conditions” to mimic the standard theory. However, such a thing is very unlikely. In Cases c and d the coupling of new neutrinos to IV-bosons will not alter our relations. However, as pointed out earlier in that case, if the IV-boson coupling strengths to new neutrinos are not very small and new IV-boson masses are not very large, then the contribution of the reaction Ne -+ ve, Ne could have been neglected in writing Eqs. (2.2) and (2.3).
4.
COMMENTS
AND
DISCUSSION
In summary we have tried to suggest a way to distinguish between hypotheses a, b, c, and d, i.e., (a) The standard case, i.e., single Z-boson and no additional neutrinos. (b) A single Z-boson coupling to standard and additional neutrinos. (c) Additional neutrinos and standard neutrinos couple to many Z-bosons. (d) Standard neutrinos and the additional neutrinos couple to different sets of Z-bosons. By relating the observables in e+e- + vVy, electron-neutrino e+e- + p+p-. The following comments are due:
scattering,
and
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BOSONS
399
1. The observables in our test (Eqs. (2.2), (2.3), (2.5), (2.6)) are obtained from low energy experiments. The precision at present may not be sufficient to decide unambiguously any of these hypotheses. Our tests point out the possibility of deciding between these hypotheses if the measurement of these observables is possible with sufficient precision. The tests are interesting in a sense that they bring about the importance of low energy precision measurements in testing the theories beyond the standard model. 2. The possibility that the new neutrinos could also be right-handed is not considered. However, is very easy to make this extension and this will not make any difference in the relations for hypothesis a obviously. For hypotheses c and d, also, allowing new neutrinos to be right-handed will not alter the relations (eqs. (3.28), (3.29)). In Case b, which is experimentally uninteresting, there are some changes but if the new neutrinos are purely right-handed [4] then the relations in Case b also do not change. 3. The possibility that the new neutrinos are massive is not considered in the paper. However, if the masses there are not very large, i.e. (of the order of few million electron volts) then, also, our conclusions hold good. In that case extraction of the observables A,,P, B,+,, C,,, from the experimental data will be different. 4. The conspiracy conditions which hypothesis c could satisfy to mimic hypothesis a do exist but these are very unlikely. In any case to rule out hypotheses c and d may not be strictly valid if the relations (3.11) to (3.19) are experimentally verified. 5. The fact that our relations involve observables from different reactions could be a disadvantage. However, with observables obtained from precision experiments, a careful analysis using our relation could distinguish between the phenomenologically important questions. 6. In the literature, consequences of a specific model in different reactions are reported. Here we do not assume any specific model and suggest tests by relating the observables of different reactions. In a specific reaction such tests could become possible at very high energies; e.g., if a model predicts that the second Z-boson mass is 300 GeV then the experiments at energies beyond Z-boson resonance could see some effects of the second Z-boson. 7. If one withholds a Z-boson resonance and observes the decay products, e.g., p +pP etc., the determined coupling constant combinations will not be sensitive to the existence of additional Z-bosons. But the reaction e + e - -+ vVy studied at the center-of-mass energies, i.e., away from Z-boson resonance, could reveal a different photon energy spectrum in the case when additional Z-bosons are present2. Thomas G. Rizzo [9] has conducted a similar study within some superstring inspired multi-Z-boson models. However, his study concentrated on changes in the ‘A detailed study of photon Z-bosom model, independently,
595/210/2-l I
energy is being
angular distribution carried out at present
to test the existence and can be reported
of an additional on later.
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determination of the number of neutrinos as revealed by a shift in the total cross section. His conclusion was that the additional Z-bosons do not considerably affect determination of the number of neutrino types with the present day experimental precision.
APPENDIX
For low center-of-mass energies of the beam effects just reduce to M in the denominator. (S N MS,), one has to include propagator effects magnitude of the cross section. Inclusion of the the following replacements:
454+
particles (S+ MS) the propagator But in the high energy region of - Zi bosons which enhances the propagator effects can be done by
-d’,,.&, [S(l-x)-M;,]+iTZ,M,,
In ev + ev
where M,., r,, are the mass and width of the Zi boson coupling to the diagonaltype neutrino neutral current. Similarly, M,,, rzB are the mass and width of the Z, boson coupling to the non-diagonal neutrino neutral current. After the above replacements, since d iv,A are, in general, complex, (dL,,)* and (d’,dL) can be Because of these propagator effects the replaced by (dig, d’,,) and Re(d’s’,). correlations in e+e- + vVy, like contributions to “time reversal violating” (% x 3, ) .o, P + . (3, x s_ ), s, . (P, x $), become non-zero. These kinematical correlations multiply the coupling constant combinations Im(d’,*d’,), Im(D&*Ds), and Im(G$*G$). In the case of non-diagonal neutrino neutral currents without and propagator effects, D$,A and G$,A are, in general, complex, but Cij Im(Df*D$) CijIm(G$*G5) vanish because D$,A = D$,: and G$*A = G$,%.
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BOSONS
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Testing for the Presence of New Neutrinos and Additional Bosons TARANJEET
Devi
Ahilya
CHHABRA
University,
AND P. RAM BABU
School of Physics, Khandwa Road,
Indore
452001,
India
Received December 26. 1990 We relate the observables of G,,,e scatterings, e+e- +p’+pL and e+e- +vi$ to look for the existence of new neutrinos coupling to additional and/or standard Z-bosons. The observables in ve scattering and e+e- +vVy are sensitive to the presence of new neutrinos and the ones in e+e- +pc+pc are sensitive to additional Z-bosons. Hence the relations between low energy observables of these reactions can be used to test the existence of new neutrinos coupling to standard Z-bosons and/or additional Z-bosons. A way to use the observables determined from experiments conducted at high energies in testing these hypothesis is indicated. 0 1991 Academic press, IIIC.
INTRODUCTION
The interest in the question whether a single Z-boson mediates neutral weak interactions or many Z-bosons mediate the neutral weak interaction has received more attention mainly due to superstring models which can lead to a low energy gauge group larger than SU(3), x SU(2), x U(l),. As new hadronic facility and highly precise experiments in e+e- are reached, we expect to verify whether these models are correct or not [ 11. The additional neutral gauge bosons could also have a different origin such as compositeness model, e.g., see L. Epele et al. [2]. Whatever may be the origin of the additional gauge boson, phenomenologically the question is an important one. In the past many tests [3] have been suggested which are relations between various observables of reactions such as ve, VP scattering and e+e- + p+p-, hadrons, etc. Both in superstring and composite models [2], additional neutrinos beyond the known three families are allowed. In composite models they could be excited states and could be massive; in other models they could have additional families. These additional neutrinos, as experimental data indicates (pointed out below) may not couple to the standard Z-boson with the same strength as the other neutrinos, but could couple to additional Z-bosons as in Ref. [2] or they could be right-handed and could couple to additional Z-bosons as in the Seesaw model [4]. The additional neutrinos could be massive (of the order of 50 GeV) and/or mixing with the standard neutrinos, with mixing parameters as low as 10pi3. The experimental study of Z”-decays by aleph collaboration [S] rules out the stable 388 OOO3-4916/91 $7.50 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
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389
BOSONS
neutrinos up to masses of 42.7 GeV with mixing parameters down to lo- 13. There are severe experimental constraints on the number of types of neutrinos. In fact the recent experiment by L3 [6] collaboration, reports N, = 3.29 f 0.17. They rule out the possibility of four or more neutrino flavours at the 4a confidence level. The restrictive limits on the number of neutrinos essentially come from the measurements of width of Z”-resonance apart from cosmological constraints. These limits do not apply to the case when the additional neutrinos couple to the additional Z-bosons. In fact, the determination of (dt + d:) from ve scatterings [7] and Z-boson decay [S] are consistent within the error limits for the existence of additional Z-bosons. It is interesting to note that within the error bars the data are consistent with the existence of a heavier Z-boson with mass limits (2 132 GeV and d207 Gev), in addition to the standard Z-boson. Hence one can say that the data at present do not rule out the existence of additional Z-bosons. Such a possibility does exist in some compositeness and superstring inspired models [2,4]. The question of the existence of additional neutrinos to additional Z-bosons is of importance. Such questions were posed earlier in the literature and modification in neutrino counting due to additional Z-bosons are investigated in a large class of two Z”-boson models which are motivated by superstring theory. The investigations revealed that the experimental number N, will not shift considerably due to additional bosons (~0.4) [9]. There are many similar investigations in the literature. Withholding a Z-boson resonance would essentially mean suppressing the contributions of other Z-bosons, hence, insensitivity to the additional neutrinos coupling to these Z-bosons. In this paper we consider the possibility that the new neutrinos couple to additional Z-bosons and we suggest a way of testing this hypothesis when the additional neutrinos are massless, or of a very low mass.’ The tests are relations between observables of purely leptonic reactions e+e- -+ v~y, ve scatterings and e +e ~ + ,U+p ~ The tests involve the observables of the low energy experiments in which the propagator effects could be neglected. The inclusion of propagator effects will modify our results. In fact at high energies, where propagator effects of Z-bosons could become important, such tests are not possible. We get different relationships between the observables for the cases: (a) The standard case, i.e., single Z-boson and no additional neutrinos. (b) A single Z-boson coupling to standard and additional neutrinos. (c) Additional neutrinos and standard neutrinos couple to many Z-bosons. (d) Standard neutrinos and the additional neutrinos couple to different sets of Z-bosons. The different relationships observed for these different hypotheses (a-d) are the tests we propose. We also consider the effects of new W-bosons coupling to electronic type neutrinos in all the cases above. ’ The case that these neutrinos is the subject of another paper.
are massive
(stable
or unstable,
within
the detector)
is interesting
and
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In Section 1, we write down the general amplitude that is useful for calculating the observables of the reaction e+e- -+ vVy, ve scatterings. We also give the relationships between the coupling constants for the hypotheses (ad). In Section 2 we present the observables of the reactions and in Section 3 we give the relation between the observables. Section 4 is for some comments and discussion. In the Appendix we present the modification of the observables due to propagator effects. 1
In writing down the amplitude for reactions e+e- + vVy, and ve scatterings, we assume many Z-bosons mediating the electron neutrino sector and the existence of new neutrinos. We also allow for possibilities of nondiagonal neutrino interactions in the neutrino sector. However, we do not allow violation of ,LJ- e - r universality which has been experimentally established with reasonably good precission. The Feynman diagrams that contribute to the reaction e+e- --) vVy are [lo] as in Fig. (1). Using these diagrams one can write down the amplitudes for the above reactions. In writing down amplitudes, v, - v, - v, and p-e, universality is assumed. We write here the amplitude that contributes to neutrinoelectron scatterings and e+e- + iVN, vV, vN, A%. The amplitude for the photon tagged reaction has the obvious inclusion of electromagnatic Bremsstrahlung term:
+Cey”~(d:+d:.l,)ey,Nl.‘f(l-y,)N k
(1.1) The symbols e, v, N stands for electron, neutrino, and new types of neutrino’s, respectively. The first term in Eq. (1.1) is the coupling of electrons to standard neutrinos mediated by i, Z-bosons with relative strengths cli, the second term is non-diagonal neutrino neutral coupling mediated by j, Z-bosons with relative strengths /Ij, the third is the coupling of new neutrinos mediated by k, Z-bosons with relative strengths yk. The sets of Z-bosons i, j, k may not be mutually
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Y. N
(a)
(b)
69
(d
FIG. 1. The Feynman diagrams tions due to (1.d) are not considered “w” could be neglected.
contributing to e+e- --t vVy, N&J, V&J, NC?. In the text because in the low-energy approximation the propagator
contribueffects of
exclusive. Similarly the last three terms allow the possibility of many IV-bosons mediating .Fv, &V couplings. The relative strength of last two terms could be unrelated because the set of m, IV-bosons that couple to Fv currents may not have some elements in common with the set of n, IV-bosons that couple to FiV currents. In case the m and n sets are identical, and I= 1 or 6,= 1 for all 1 the strength of the sixth term is the square of the strength of the fifth term. Now we consider the four hypotheses regarding the Z-bosons mediating the electron-neutrino sector, including the new types of neutrinos. In the charged current sector many IV-bosons are allowed, but the IV-bosons coupling to old and new neutrinos are common.
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The relation between the coupling constants d’;(kk and cx,,Bj, yk are given below in four different cases when hypotheses (ad) hold. The relations between the fermicoupling constants and the coupling constants in Eq. (1.1) are: Case a.
(i) (ii)
a,d2,,=dv.A
(iii)
and
ai
db,,=O
Vif
1;
pi=yi=o.
(1.2a)
Case b.
GF
(i)
g2
-----y 3-W
(ii)
Cs,+vlm+t”l
ald>.A=dv,A;
bld\.,=Dv,A;
G:,,
= Gv,,.
(1.2b)
Case c.
where i is the number of IV-bosons.
(0
(ii) 1 aidL,A = dy.a; i
CB;dL%.=D,.; i
CYid’,,,=Gv,A.
(1.2c)
i
Case d.
(ii)
1 aidL,A = dv,A; i
CBjdL,,=Dv..; .i
Cy/cdk,A=Gv,A; k
’
(1.2d)
dv,,ZDv,,ZGv,A.
It can be seen from the above relations, that if the IV-boson sector is kept unchanged then many of the above relations are simplified. However, the resulting simplification may not give rise to new observable relations among the observables (see Section 3). 2 The neutrino electron scatterings are very well studied when both diagonal and non-diagonal neutrino neutral currents are present [ 111. In the Ref. [ 11 J the initial
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NEUTRINOS
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393
BOSONS
neutrino arising out of charged current decays is always taken to be the standard neutrino because it is assumed that the new neutrinos couple only to Z-bosons. Here we assume a finite coupling strength of new neutrinos to W-bosons also. However, the non-standard neutrino coupling in both neutral and charged sectors is assumed to be an order of magnitude smaller than standard neutrino couplings. Hence, the flux of new neutrinos arising out of the charged current decays will be an order of magnitude smaller. Due to this reason our observables of neutrinoelectron scatterings are considered to be the sum of the observables of the reactions ve + ve; ve - Ne, reducing to the case considered in Ref. [ 121. The final recoil electron energy distribution, neglecting the mass of the neutrinos in comparison to the energy of the beam and recoil energy, and using the amplitude given in Eq. (1.1 ), can be written as (2.1)
where i may be either e, ,u, or T. And the A, B, and C’s in the different cases are: Case a. A.=(d:,+d2,)+2(dv+d,+1) B,=2d,,d,+2(d,+dA+
1)
C,=(d$-d2,)+2(dv-d,J (2.2)
A,=(d$+dZ,) B, = 2d,d, C, = (d;,-
d;).
Cases b, c, and d. A,= (d$+d2,)+26,(dc,+dA B,=2d,,d,
+26,(d,+d,
+6,)+N’(DZ,.+D:)+2N’q,(Dv+ +6,)+2N’D,,D,
+2N’q,(D,+
D, +q,) D, +ij,)
C,=(d;-d:,)+26,(d,r-d,)+N’(D;-DZ,)+2N’q,(D,r-D,) A,, = (df, + d;) + N’(D;. + 0;) B, = 2d,d,
(2.3)
+ 2(D,D,)N’
C,,=(d;,-d:,)+N’(D;.-D:,). Where d,,, D,, ‘s for the Cases (b, c, d) are as defined in Eqs. (1.2bk( 1.2d), respectively, and N’ is both the standard and the additional neutrino couplings. The reaction e+e- -+ vi”y, is suggested mainly to count the number of neutrino generations. Even though this reaction could be studied at low energies (E,,, is 40 to 50 GeV), the separation of the events from the background could be a hard task.
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Hence it was suggested that this reaction can be experimentally studied siting on the Z-boson resonance. But by doing so, one is counting or measuring the coupling strengths of only those neutrinos that couple to standard Z-bosons. As pointed out earlier, the possibility that there are new neutrinos coupling to new Z-bosons does exist in many superstring and compositeness inspired models, and the experiments of e’e- + vVy on Z-boson resonance are insensitive to such a possibility. Here we write down an angle and energy distribution of the tagged photon when the center of mass energy is small compared to the mass of any Z-boson and it is given by da dxdY
-2
$
O,(l -h+k)+O,(h+
x (1 -x)(1 [
-x/2)2+;x2(1
x (x(1 -x)[S+
X(1-
S x(1 - YZ)
-IL) -x)
03s 1 1
Y2 +
4( 1 - Y2)
Y*)+2@~~&3+]}
,
(2.4)
where x is E,IE, Y is Cos 8, Q is three momenta of photons, and Or, 02, O3 in the different cases are Case a. 0,=3(d:+d:,)+2(d,+dA+1) 02=6d,d,+2(d,+d,+l)
(2.5)
03=3(d;-d;)+2(dv-da). Cases b, c, d. 0, =3(d;+d;)+26Adv+dA+6,)+2N’(D2y+D;)+4N’q,JDv+ + NW:+ 0,=6d,d,
G:) + 2N<,,(G,+
+26,(d,+d,
+ 2NG,G,
G, + 5,)
+6,)+4N’D,D,
+ 2Nt,(G,+
D,,,+q,)
+4N’qm(Dv+
G, + t,)
D, +q,) (2.6)
O,=3(d;-d;)+26,(d,-d,)+2N’(D2,-D2,)+4N’q,,(Dv-DA) + N(G;-
GZ,) +2N(,(G,-
GA).
Where d,,, hAT GA ‘s for the Cases b, c, and d are as defined in Eqs. (1.2b)-( 1.2d), respectively. N is the number of additional neutrino couplings and N’ is the standard and additional neutrino couplings. E.g., in models with the three heavy neutrinos respecting flavour conservation (per), N is three and N’ is also three. But if additional neutrinos have different flavours with flavour conservation then N’ is zero.
NEW
NEUTRINOS
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BOSONS
395
In writing down Eqs. (2.5) to (2.6) we neglected the masses of the electron and the neutrino. We kept the electron-positron polarization and summed over the polarizations of the photon. The observables 0,) 02, 0, contain the contributions of all the channels, i.e., e+e- -+ vVy, N&, vi@, NGy. Hence ve scattering observables are, in general, unrelated to the above observables because of the additional channel Ni@, which was neglected in the ve scatterings. However, some inequalities could be there and these are our tests for the various hypotheses. The reaction e+e- +,u+P- has been very well studied experimentally and the effects of many Z-bosons mediating neutral weak interaction have also been pointed out. The observables arising out of measurements of deviation of the total cross sections from electromagnatic cross sections, angular asymmetry, and parity violation are K, L, and M, respectively. If many Z-bosons are coupling to electron current and pe universality is valid then Cuses a, b. K= dt. L = d,d,
(2.7)
M=d’,. Case c. K=cd’cz, L = C d;,d; M=C
(2.8)
da’.
Case d.
(2.9)
In general, L2 < KM and the equality sign holds good in the case of a single Z-boson hypothesis. Relating the e+e- -+ pfpP reaction to ve scatterings and e+e- + XT7 is very useful, since observables of this reaction are affected by many Z-bosons but are not affected by the existence of new neutrinos.
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3 The observables 0,) O,, 0, obtained from e+e- + vVy with unpolarized, longitudinally polarized, and transversely polarized electron-positron beam and the observables A,,u, B,,p, C,,, obtained from ve scattering the recoiled electron energy distribution are expressed in terms of (see Eqs. (2.2), (2.3), (2.5), (2.6)) the coupling constants appearing in Eq. (1.1). These expressions for the observables are different for the different hypothesis (a-d), (see Eq. (2.4)). The observables K, L, M obtained from e+e- + p+p ~ also contained the same coupling constants appearing in the electron sector. These observables are not affected by the likely presence or absence of new neutrinos. The different relations (see Eqs. (3.11) to (3.35)) between the observables in the hypotheses (a-d) lead to different relations between observables for the cases when the hypothesis (a) or (b) or (c) or (d) are true and are as given below: (a)
Case a. L2=KM
(3.11)
(A,+2A,)-O,=O
(3.12)
@,+2B,)-02=o
(3.13)
(C,+2C,)-o,=o
(3.14)
(A, + B,)(A, - B,) - C: = 0
(3.15)
(A,+B,)(A,-B&C:,=0
(3.16)
(0, +
-
O,)(O,
02)
I(O,+O,)-(A,+C,)I
I(0,-0,)-(A,-C,)I-(02-B,)2=0
01 --A, =- 0,-B, A,
(3.17)
> 0:
=-
03-C<
BP
CP
.
(3.18) (3.19)
Case b. L2=KM
(3.20)
(A,+2A,)-0,
(3.21)
(B,+2B,)-02#0
(3.22)
(C,+2C,)-03fO
(3.23)
(A,+B,M--B,)fC:
(3.24)
(A, + B,M,
- BP) = C:
(3.25)
(0, + 02)(0,
- 02) > 0:
(3.26)
I(O, + 0,) - (A, + C,)l I(O, - 0,) - (A, - C,)l > (0, - B,J2.
(3.27)
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397
Cases c and d. L2
(3.28)
(A.+2A,,)-O,
(3.29)
(B,+2B,)-O,#O
(3.30)
(C, + 2C,,) - 0, # 0
(3.31)
(A.+B<,)(A<,-B&4’;
(3.32)
(A, + B&A,, - B,,) > Cf
(3.33)
(0, +
(3.34)
O,)(O,
-
02)
I(O,+O,)-(A,+C<,)I
>
0:
ItO,-O,)-(A-C,)I
>(0z-BeJ2
(3.35)
The relation between the observables of e+e- + pL+pP are same in Cases b, c, and d; i.e., these relations are mainly effected by whether a single Z-boson is mediating the electronic neutral current or not. This has been pointed out already in the literature [13]. The relation within ve electron scatterings are same in the cases whether single Z-boson mediating neutral weak interaction or multi-Z-bosons are mediating neutral weak interaction. The relation between these observables are sensitive to the existence of new neutrinos. The same comment hold good for the observables within e +e ~ + vVy. If p - e - t universality is assumed (see Eqs. (3.11) to (3.35)) our relations (3.15) to (3.18) hold good in Case a, as well as in a theory with multi-Z-bosons. The existence of new neutrinos coupling to the standard Z-bosons is ruled out by the experiments measuring the decay width of Z-boson [6]. Hence the cases that are interesting in neutral weak interaction mediated by a single Z-boson with no new neutrinos, and the multi-Z-bosons theory with new neutrinos; i.e., experimentally our hypothesis b is already ruled out. However, the relation in this case is presented for completeness. In addition the relations (3.21), (3.22) (3.23) in Case b can be use to test the hypothesis that the new neutrinos couple to standard Z-bosons with very low coupling strength; i.e., the new neutrinos need not be a member of an additional neutrino family with universality. The interesting multi-Z-bosons theory with new neutrinos could have two special cases, i.e., our hypotheses c and d. Within hypotheses c and d, also, there could be different cases depending on whether there are additional W-bosons coupling to new neutrinos or not. 3.1. Distinction between Hypotheses a, c, and d. Here we discuss the different observable relations in Cases a, c, and d, when the W-bosons do not couple to new neutrinos. We find that the relations between the observables of these three reactions cannot distinguish between hypotheses c and d. It is clear in the sense that the observables of e+e- --+p+p are only sensitive to the cases with single Z-bosons versus those
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with many Z-bosons. Whereas the observables of the other two reactions are sensitive only to the fact of whether new neutrinos are present or not. Hence no relation between these observables can distinguish between the different types of coupling of multi-Z-bosons to old and new neutrinos. Henceforth we discuss the ways to distinguish between hypotheses a and c. The relations (3.11) and (3.28) clearly differ in the two cases but are not sensitive to the presence of new neutrinos. The relations (3.12)-(3.14) and (3.29)-(3.31) are also different in the two cases and these are sensitive to the fact that neutrinoelectron scattering, the initial neutrino which arises in charged current decays, cannot be a new neutrino. The relations (3.15), (3.16), (3.18) and (3.32), (3.33), (3.35) are known in the literature [12, 141 and are sensitive to the existence of new neutrinos and their coupling to new Z-bosons. The experimental verification of the relations (3.28), (3.32), (3.33), (3.35) would indicate the presence of additional neutrinos and additional Z-bosons. However, if (3.28) and (3.15), (3.16), (3.18) are experimentally obeyed, then the multi-Z-boson theory without new neutrinos is indicated. If experimentally (3.28), (3.32), (3.33), (3.35) are valid then the assumption that the new neutrino does not couple to a IV-boson is wrong. Here we have only tried to indicate how the relations between observables could be used to test the various sets of theories. If the relations of Case a are verified, then it may not rule out the derivations from the standard model; i.e., the deviations from the standard model can satisfy some “conspiratorial conditions” to mimic the standard theory. However, such a thing is very unlikely. In Cases c and d the coupling of new neutrinos to IV-bosons will not alter our relations. However, as pointed out earlier in that case, if the IV-boson coupling strengths to new neutrinos are not very small and new IV-boson masses are not very large, then the contribution of the reaction Ne -+ ve, Ne could have been neglected in writing Eqs. (2.2) and (2.3).
4.
COMMENTS
AND
DISCUSSION
In summary we have tried to suggest a way to distinguish between hypotheses a, b, c, and d, i.e., (a) The standard case, i.e., single Z-boson and no additional neutrinos. (b) A single Z-boson coupling to standard and additional neutrinos. (c) Additional neutrinos and standard neutrinos couple to many Z-bosons. (d) Standard neutrinos and the additional neutrinos couple to different sets of Z-bosons. By relating the observables in e+e- + vVy, electron-neutrino e+e- + p+p-. The following comments are due:
scattering,
and
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NEUTRINOS
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BOSONS
399
1. The observables in our test (Eqs. (2.2), (2.3), (2.5), (2.6)) are obtained from low energy experiments. The precision at present may not be sufficient to decide unambiguously any of these hypotheses. Our tests point out the possibility of deciding between these hypotheses if the measurement of these observables is possible with sufficient precision. The tests are interesting in a sense that they bring about the importance of low energy precision measurements in testing the theories beyond the standard model. 2. The possibility that the new neutrinos could also be right-handed is not considered. However, is very easy to make this extension and this will not make any difference in the relations for hypothesis a obviously. For hypotheses c and d, also, allowing new neutrinos to be right-handed will not alter the relations (eqs. (3.28), (3.29)). In Case b, which is experimentally uninteresting, there are some changes but if the new neutrinos are purely right-handed [4] then the relations in Case b also do not change. 3. The possibility that the new neutrinos are massive is not considered in the paper. However, if the masses there are not very large, i.e. (of the order of few million electron volts) then, also, our conclusions hold good. In that case extraction of the observables A,,P, B,+,, C,,, from the experimental data will be different. 4. The conspiracy conditions which hypothesis c could satisfy to mimic hypothesis a do exist but these are very unlikely. In any case to rule out hypotheses c and d may not be strictly valid if the relations (3.11) to (3.19) are experimentally verified. 5. The fact that our relations involve observables from different reactions could be a disadvantage. However, with observables obtained from precision experiments, a careful analysis using our relation could distinguish between the phenomenologically important questions. 6. In the literature, consequences of a specific model in different reactions are reported. Here we do not assume any specific model and suggest tests by relating the observables of different reactions. In a specific reaction such tests could become possible at very high energies; e.g., if a model predicts that the second Z-boson mass is 300 GeV then the experiments at energies beyond Z-boson resonance could see some effects of the second Z-boson. 7. If one withholds a Z-boson resonance and observes the decay products, e.g., p +pP etc., the determined coupling constant combinations will not be sensitive to the existence of additional Z-bosons. But the reaction e + e - -+ vVy studied at the center-of-mass energies, i.e., away from Z-boson resonance, could reveal a different photon energy spectrum in the case when additional Z-bosons are present2. Thomas G. Rizzo [9] has conducted a similar study within some superstring inspired multi-Z-boson models. However, his study concentrated on changes in the ‘A detailed study of photon Z-bosom model, independently,
595/210/2-l I
energy is being
angular distribution carried out at present
to test the existence and can be reported
of an additional on later.
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determination of the number of neutrinos as revealed by a shift in the total cross section. His conclusion was that the additional Z-bosons do not considerably affect determination of the number of neutrino types with the present day experimental precision.
APPENDIX
For low center-of-mass energies of the beam effects just reduce to M in the denominator. (S N MS,), one has to include propagator effects magnitude of the cross section. Inclusion of the the following replacements:
454+
particles (S+ MS) the propagator But in the high energy region of - Zi bosons which enhances the propagator effects can be done by
-d’,,.&, [S(l-x)-M;,]+iTZ,M,,
In ev + ev
where M,., r,, are the mass and width of the Zi boson coupling to the diagonaltype neutrino neutral current. Similarly, M,,, rzB are the mass and width of the Z, boson coupling to the non-diagonal neutrino neutral current. After the above replacements, since d iv,A are, in general, complex, (dL,,)* and (d’,dL) can be Because of these propagator effects the replaced by (dig, d’,,) and Re(d’s’,). correlations in e+e- + vVy, like contributions to “time reversal violating” (% x 3, ) .o, P + . (3, x s_ ), s, . (P, x $), become non-zero. These kinematical correlations multiply the coupling constant combinations Im(d’,*d’,), Im(D&*Ds), and Im(G$*G$). In the case of non-diagonal neutrino neutral currents without and propagator effects, D$,A and G$,A are, in general, complex, but Cij Im(Df*D$) CijIm(G$*G5) vanish because D$,A = D$,: and G$*A = G$,%.
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BOSONS
401
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