Physics of mirror fermions

PHYSICS OF MIRROR FERMIONS

J. MAALAMPI Research Institute for Theoretical Physics, University of Helsinki, Siltavuorenpenger 20C, SF-OO1 70, Helsinki, Finland and M. ROOS Department of High Energy Physics, University of Helsinki, Finland

I

NORTH-HOLLAND

PHYSICS REPORTS (Review Section of Physics Letters) 186, No. 2 (1990) 53—96. North-Holland

PHYSICS OF MIRROR FERMIONS J. MAALAMPI Research Institute for Theoretical Physics, University of Helsinki, Siltavuorenpenger 20C, SF-00170, Helsinki, Finland

and M. ROOS Department of High Energy Physics, University of Helsinki, Finland Received March 1989

Contents: 1. Introduction 2. Family unification in orthogonal groups 2.1. Spinor representations of SO(n) 2.2. SO(18) models 3. Other theories involving mirror fermions 3.1. Family unification in SU(N) groups 3.2. Extended supersymmetry 3.3. Kaluza—Klein theories 3.4. E 8 x E8 superstring models

55 57 58 60 64 64 66 66

4. Low-energy effects of mirror mixing 4.1. Low-energy lagrangian 4.2. Experimental tests of the V—A structure 5. Constraints for mirror neutrinos 5.1. Constraints at low mass 5.2. Constraints at high mass 6. Production and decay signatures of mirror leptons 7. Summary

69 70 72 78 81 84 88 91

67

References

93

Abstract: We review the theoretical motivation and phenomenological implications of mirror fermions. Mirror fermions are a new class of particles having V + A weak interactions. They are predicted by many recent particle theories, including orthogonal and unitary symmetries for family unification, Kaluza—Klein theories, extended supersymmetry and composite models. The mass of mirror particles breaks the electroweak symmetry and hence they must be lighter than about 300 GeV. Charged mirror leptons must be heavier than about 27 GeV. Mixing with ordinary quarks and leptons, the mirror fermions would cause violation of the V—A rule in weak interactions. We present an overall fit to experimental data and derive upper bounds for such mixing. We also make a survey of various laboratory experiments, as well as astrophysical and cosmological observations to find constraints for mirror neutrino masses, lifetimes and mixing angles. We conclude that mirror neutrinos N with masses in the ranges mN ~ I eV and mN ~ 18 GeV could mix appreciably with ordinary neutrinos. In the ranges 50MeV ~ m~~ 18 GeV and I eV ~ m,~ ~ 25 eV the mixing is 10~—10~ or less. Elsewhere mirror neutrinos are forbidden by astrophysical or cosmological arguments. Mirror neutrinos with masses m~~ 25 eV or mN 5 GeV are WIMP candidates; to make them responsible also for the solar neutrino deficiency problem would be rather contrived, however. No limits can be inferred for mirror quarks which must have very small mixing with ordinary quarks. We describe the experimental signatures of and prospects for production of mirror fermions in the new generation of accelerators.

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J. Maalampi and M. Roos, Physics of mirror fermions

55

1. Introduction One of the greatest surprises in the history of particle physics was the discovery [1] that parity is violated in weak interactions. Before this observation it was believed, according to Fermi’s original hypothesis [2], that weak interactions have purely vectorial (or axial vectorial) parity conserving Lorentz structure. In 1956 Lee and Yang [3] presented their revolutionary theory Where the coupling between a charged lepton and a neutrino contains both a vector (V) part and an axial vector (A) part, so that the fermion current has a V— A structure. Thus leptons feel the weak force only through their left-handed chirality component and parity is maximally violated. The V A hypothesis was quickly generalized [4] to the sector of nucleons and turned out to describe the existing experimental data successfully. Since Lee and Yang presented their theory, many new particles have been discovered, and all of them were found to follow the V A rule within experimental accuracy. Hence the chiral character of the weak force is a central ingredient in any successful particle theory. In the Glashow—Weinberg—Salam model [5] (GWS model or standard model), the famous SU(2)L X U(1) gauge field theory unifying electromagnetism and weak interactions, the V— A structure is built in by assuming that only the left-handed fermions transform nontrivially under the gauge group SU(2), and only they couple to the weak gauge bosons W~,mediators of the weak force. The right-handed fermions transform as singlets and therefore do not feel the W field. Also the so-called grand unified theories (GUTs), such as the SU(5) model [6], which attempt to unify the electroweak and strong forces within a common framework, have a similar structure. Actually the requirement of chirality is very restrictive for gauge theories, because the complex representation pattern it demands can easily lead to axial anomalies [7] and make the theory nonrenormalizable. Nevertheless, one may ask whether there possibly exists a second class of fermions coupling to the W boson with V + A currents in a mass range inaccessible to present experiments. Such particles have been called mirror fermions. In fact, Lee and Yang themselves were the very first to pose this question in their classical paper [3]. Their argument was purely aesthetical: the existence of mirror fermions would make the world left—right symmetric in a broad sense. The present interest in mirror fermions has a more technical origin. We already mentioned the question of anomalies. Ordinary and mirror fermions are conjugate to each other with respect to the gauge symmetry group, and consequently a fermion representation including both of them is real, in which case the anomaly cancellation is automatic. This mechanism for anomaly cancellation is almost as old as the subject of anomalies itself [8]. One of the first unified models where mirror fermions were introduced for this purpose, i.e. to save an otherwise anomalous theory, was due to Pati and Salam [9], who considered a “maximal” lepton—quark unification within an SU(32) gauge group. In many recent theoretical schemes mirror fermions arise more naturally. Such a case is the so-called family unified models where all the fermion families are incorporated within one spinor representation of a large orthogonal group [10—17],or within high-dimension representations of large unitary groups [18—23]or exceptional groups [24].Actually, the only models considered so far explaining the repetition of fermion families without the existence of mirror fermions are some superstring models [25]. Mirror fermions necessarily appear in the particle spectrum also when an extended supersymmetry (N 2) [26] is imposed on a gauge theory, as well as in the Kaluza—Klein theories [27].They appear also in some composite models [28] and play an essential role in nonperturbative regularization in lattice theories as they allow for a formulation with an exact local chiral symmetry [29]. There are, on the other hand, some theoretical and phenomenological arguments speaking against the existence of mirror fermions. It is odd that no mirror fermions have been detected, all fermions —

—

56

J. Maalampi and M. Roos, Physics of mirror fermions

observed so far having V— A weak interactions. It is difficult to understand why mirror fermions would have systematically higher masses than the observed fermions. Also odd is the fact that mirror fermions and ordinary fermions do not seem to mix at any observable level. We also know that in the standard model the structure of a fermion family is such that the anomalies are cancelled in a nontrivial manner. In the presence of mirror fermions there would be no need for any such complicated pattern of states. Of course, none of these arguments proves that mirror fermions do not exist. In any case, mirror fermions do exist in quite a variety of theoretical models, as we have just noted. A study of their properties and phenomenological signatures is thus well motivated and even necessary in order to test how well those theories match the facts of nature. In our definition, mirror fermions have exactly the same quantum numbers with respect to the gauge group of the standard model as their ordinary counterparts, the known quarks and leptons, except that they have the opposite handedness. Explicitly, under the SU(3)~x SU(2)L x U(1)~gauge symmetry of the strong and electroweak interactions, an ordinary V— A family (the electron family for example) transforms according to (~)~(1,2,_~),

(~)

~(3,2,

eH(1,1,1),

~(1,1,0), (1.1)

~),

u~L~(3, 1,

-i),

d~L~(3*, 1,

and the corresponding V + A mirror family transforms according to (~)~(1,2,

~),

EL~(l,l,—l),

NL~(l,l,O),

DC (U~)~(3*,2,_~),

(1.2) ~L~(3,1,3),

D~L~(3,1,—~).

Here all fields are two-component Weyl spinors and, for example, u1 stands for the up-quark and u~for the up-antiquark, i = 1, 2, 3 being a colour index.2)L Therepresentations three numbersand in the (1.1) and the brackets value of in ~Y,eqs. respectively. (1.2) electric give thecharge multiplicity of the The is defined as SU(3)~ Q = T and SU( 2)L. 3L + ~Y where T3L =is 2°3 is the diagonal of (1.2) SU( by The transformation law of the right-handed Weyl spinors obtained from eqs.generator (1.1) and conjugation. Hence we see that the right(left)-handed components of the mirror fermions transform similarly as the left(right)-handed components of the corresponding ordinary fermions. *) Mirror fermions are very interesting objects from a phenomenological point of view. The reason is that they cannot be heavier than about 300 GeV, since their masses are generated during the spontaneous symmetry breaking of the SU(2)L x U(1)~.gauge symmetry. One thus expects to produce *) In the literature also another definition of mirror fermions exists. Kobzarev, Okun and Pomeranchuk [301 introduced, following the suggestion of Lee and Yang [3], a new fundamental symmetry, the symmetry under “mirror transformations” A, and hypothesized that the world should obey CPT x A invariance. As a consequence of this requirement, a particle and its mirror counterpart should be degenerate in mass, and they would mix maximally through the ordinary interactions, thus making the theory vector-like. A solution to this dilemma would be that the mirror fermions of Kobzarev et al. must not interact in the normal way but must have “photons”, “weak bosons” and ‘gluons” of their own. The only manner in which the “mirror world” and the known world would communicate is through gravity. There is a certain resemblance between this “mirror world” and the “shadow world” arising in the E 8 x E8 superstring models [31]. Of course, mirror fermions of this kind would in spite of their possible cosmological implications (32] be quite uninteresting from the experimental point of view and they are not considered further in this review.

J. Maalampi and M. Roos, Physics of mirror fermions

57

them copiously in the new generation of colliders TRISTAN, LEP, SLC, Tevatron, HERA and SSC. On the other hand, mirror fermions may manifest themselves already at lower energies. It is quite possible, and even probable, that mirror neutrinos are much lighter than charged mirror fermions, which in turn must be heavier than about 40 GeV according to the UA1 experiment [33] (this limit assumes that mirror neutrinos are very light), or heavier than about 26 GeV according to the first results from the TRISTAN experiments [34]. Light mirror neutrinos could be produced e.g. in p.., ‘n~and K decays. Charged mirror fermions, if they mix with their ordinary partners, would manifest themselves as deviations from the purely left-handed structure of the known fermions [35—38]. The purpose of this paper is to review the theoretical motivations of the mirror fermion hypothesis and to describe their possible phenomenological implications and experimental signatures. In section 2 we consider family unification theories, concentrating on models based on the large orthogonal groups such as SO(18) where the appearance of mirror fermions is particularly natural and suggestive. In section 3 other theoretical schemes where mirror fermions may appear are briefly discussed. We start our phenomenological considerations in section 4 where we study effects of mixing between ordinary and mirror fermions at low energies. We derive best-fit values for the appropriate mixing angles by combining all relevant experimental information on the Lorentz structure of the weak currents. In section 5 we consider properties and possible observational consequences of mirror neutrinos in laboratory experiments, astrophysics and cosmology. Prospects for production and detection of mirror fermions in future colliders are discussed in section 6. We conclude in section 7 with a summary.

2. Family unification in orthogonal groups The matter we see around ourselves in everyday life consists mainly of the first generation of fermions, the e-family (u, d, e, Ve), but we know that there exist at least two heavier fermion generations: the p..-family (c, s, p, v~)and the ‘r-family (t, b, ‘r, ~T) manifesting themselves through high-energy particle collisions or in cosmic rays. It has turned out to be a nontrivial problem to understand this repetition of families. In the standard SU(3)~x SU(2)L x U(1) model [5] one does not address this question at all, and the same is true for the standard grand unified schemes within SU(5) [6] and SO(10) [10, 39]. In the latter models each family occupies its own representation, 1 + 5* + 10 in SU(5) and 16 in SO(10), as would any further generation to be discovered. The number of generations which is an important parameter, e.g., in cosmology and for asymptotic freedom of the color force, is totally free in these theories. Family unified models were introduced in attempts to solve this problem. In such models not only the forces, but also the fermions are unified in the sense that they all are incorporated into a single irreducible representation (or into a small set of such representations) of a simple gauge group G. The dimension of the representation determines the number of fermion families. Furthermore, since all fermions are within one representation the different families are intimately connected and one could hope to get some understanding in this way of, for example, the mass relations between the generations. Several Lie groups have been suggested as the family unified symmetry G. These include orthogonal groups such as SO(14) [40], SO(15) [14], SO(16) [41] and SO(18) [13, 42—44], unitary groups such as SU(7) [19, 20], SU(8) [21, 22] and SU(11) [18] and exceptional groups such as E 8 [24]. The most appealing models are those based on the orthogonal groups, especially on SO(18). Our treatise will devote most time to the family unification in orthogonal groups.

58

J. Maalampi and M. Roos, Physics of mirror fermions

The existence of mirror fermions is a common feature to all family unified models. It can be shown [45]that SU(5) and SO(10) are the only grand unified models giving a correct description of the known fermion families given in eq. (1.1), but not incorporating mirror fermions. In the orthogonal groups the situation is economical in the sense that the only nonstandard fermions appearing are the mirror fermions; in models based on other symmetry groups, the particle spectrum is in general less appealing. As was already mentioned in the introduction, the recently invented E8 x E8 heterotic superstring theory [31] may provide low-energy models [25,46] predicting three families of ordinary fermions as a result of the fundamental mathematical structure of the theory. A “family” in these models is, however, not the standard one but contains eleven additional so far unobserved new particles. Furthermore, some remnants of mirror families usually remain light even in these models, leptons most often. In this section we will consider the orthogonal models. We start by giving a short account of the relevant group theory of spinors*) in SO(n) and thereafter we describe some concrete models based on SO(18). 2.1. Spinor representations of SO(n)

—

The attractiveness of the orthogonal groups to family unification stems from a unique property of their spinor representations [12, 14, 47]. The spinor representations decompose into several copies of SO(10) spinors, thus offering a natural explanation for the repetition of families. A spinor representation is connected to the Clifford algebra associated with the group SO(n) in the 2 = (x~)2, following Define n hermitian calledthat generalized Dirac~matrices, that xalgebra where x isway. an n-dimensional vector.matrices It then~,follows the matrices obey thesuch Clifford {yj,y}2~J,

i,j=1,...,n.

(2,1)

If a matrix 0 is the representation matrix of transformations on the vector, x—~Ox, one can define a similarity transformation R(O) on the y-matrices as follows: y,—* 0 11y’

=

R’(O)y,R(O).

(2.2)

For infinitesimal transformations 1 , (2.3) R(O) = 1 + ~ic~y~y where is a parameter. The R(O) is called a spinor representation of SO(n), and an object ~1i transforming as

(2.4) is called a spinor. In GUT applications the group SO(n) is taken as a symmetry group of gauge transformations, and the elements of the spinor ~(i are identified with fermions; each element of the ~1s is associated with a two-component Weyl spinor, a left-handed component of a fermion or of an antifermion. *)

A more complete presentation can be found, e.g., in refs. [12, 14].

I. Maalampi and M. Roos, Physics of mirror fermions

59

It follows from eq. (2.1) that the Dirac matrices of SO(21) and SO(2l + 1) are 21 x 2’ matrices and the spinor ~i has 2’ components. The Dirac matrices ~ (i = 1, , 2!) of SO(21) can be constructed from the Dirac matrices ~, (i = 1, , 21 — 2) of SO(21 — 2) according to the prescription . .

.

)y—0

.

i1,...,2l2,

3®7,,

.

.

721_r0t®l,

(2.5)

72,02®l.

One can start to iterate the y~from the Pauli matrices o~and cr2 which are the generalized Dirac matrices in the case 1 = 1. The Dirac matrices of SO(21 + 1) consist of the matrices 7~, , y~,of SO(21) and the matrix .

721+1

=

(—i)’(7172 x

x y~,),

. .

(2.6)

corresponding to the y~matrix of the conventional Dirac algebra. As can be seen from eq. (2.3), the antisymmetric tensors = (112i)[y,, y~] , i, j = 1,..., n

(2.7)

can be taken as the generators of the group SO(n) in the spinor representation. In the construction (2.5) the generators p12, F34~. 1’21~12l are diagonal, indicating that the rank of SO(21) and SO(21 + 1) is I. One can readily see that the “chirality” matrix 721+ 1 commutes with generators of SO(2l). Since (721+1)2 = 1, the spinor i4’ can be split into two irreducible spinor representations, =

~(1 + y~)~i,

~

=

~(1 —

(2.8)

+~‘~

which are sometimes called semispinors. Both of them have dimension 21_i, but they are inequivalent representations. However, for an odd 1, ~i = (l/ç)*. The spinor representation of SO(21 + 1) consists of ~ and ~/i_, but it is irreducible because the group contains transformations which take ~/‘~to ~i and vice versa. For example, in SO(10) and SO(11) the dimension of the spinor representation is 32, but since the rank 1 = 5 is odd the spinor of SO(10) reduces to two sixteen-dimensional semispinors, ~ In the SO(1O) grand unified model [10, 39] the semispinor representation 16 is used for fermion representation of the theory, since only it provides a correct assignment for the known fermions in one family. It decomposes under the SU(5) subgroup as follows, 16=1+5*+10,

(2.9)

and one can assign (i colour index) 1=

5*

(deL

~‘L’

e~),

10 = (u~L,U~L,d,1, e~)

(2.10)

.

It will be useful to identify the fermions according to their SO(10) quantum numbers, i.e. in terms of the eigenvalues e, = ±1of the diagonal generators ‘i-12i~ By noting from eqs. (2.6) and (2.7) that 721+1 = I’12 x x ‘~1121, we see that the states in 16 have e1 x X e~= +1. Thus .•.

60

J. Maalampi and M. Roos, Physics of mirrorfermions

1

=

++++

+)

5* =

,

+

—

—

— —)

+

4 permutations,

10

=

I++ +

—

—

~+ 9 permutations,

(2.11) where ± stand for ±1. The negative chiral projection spinor 16*, however, is not suitable for the known left-handed fermions or antifermions, but it has the quantum numbers of a mirror family 16*=l+5+10*,

(2.12)

with the assignment 1

=

NL

~

(D1~,N~,E~),

10* = (U~L, UIL, D,CL, EL). (2.13) 2)L X U(1) quantum numbers of mirror fermions were given in eq. (1.2). The weight The SU(3)~ x SU( vectors of 16* are obtained from eq. (2.11) by interchanging + and — signs. We have seen that the SO(10) grand unified model avoids mirror fermions only because its spinor representation is reducible. For any larger orthogonal group, the model will necessarily incorporate mirror fermions. Indeed, it is easy to see from the iterative structure of the Clifford algebra that the SO(10 + 2k + 1) spinor ~~1o+2k+i decomposes under SO(10) as follows: ~‘iO+2k+1

=

2”(16

+

16*)

and the semispinors of SO(10

(2.14) +

2k) as follows

(~c)lo+2k=2~i(16+l6*),

(~1_)1o+2k2~l(16+16*).

(2.15)

This structure is very suggestive for family unification. Nothing similar happens in the other Lie groups, mainly because the cancellation of triangle anomalies then requires a conspiracy of many different representations and a plethora of exotic new particles [see the discussion on the SU(N) models in section 3.1]. The smallest spinor representations which can accommodate the known three families are 64 + 64* of SO(14) and 128 of SO(15). These representations are real, however, and there is no natural mechanism (except supersymmetry) which would prevent mirror fermions and ordinary fermions from pairing up through 16~16* terms and from acquiring a large mass. The lowest-dimensional group which can circumvent this difficulty in a satisfactory manner is SO(18). The uniqueness of SO(18) stems from the decomposition SO(18) DS0(10) x SO(8) and the special property of SO(8) that its spinor and vector representations are of the same dimension and connected by an outer automorphism [13]. Due to this property, one can arrange at least some of the fermion and mirror fermion families to remain massless until the electroweak symmetry is broken. In the next subsection we will show how this is achieved. 2.2. S0(18) models The 256-dimensional (semi)spinor of SO(18) decomposes under the SO(10) x SO(8) C SO(18) subgroup as follows: 256__*(16,8+)+(16*,8_),

(2.16)

J. Maalampi and M. Roos, Physics of mirror fermions

61

where 8+ are two inequivalent spinor representations of SO(8). Since the spinor 256 is complex in this decomposition, the bare mass term 256 x 256 is not invariant under the gauge symmetry and thus forbidden. This enables at least part of the families to avoid superheavy masses of the order of the unification scale 1015 GeV or more. Not all of the families can remain light, however, since otherwise the great number of light particles would cause the coupling constants to blow up at typically a few hundred TeV, thereby preventing a perturbative unification [42, 43]. In a realistic theory one therefore needs a mechanism making some of the families very heavy and decoupling them from low-energy physics, while keeping the others light. We will now describe three models proposed in the literature providing such mechanisms. Mode! 1. Gell-Mann, Ramond and Slansky [13] have proposed a model, where some part of the horizontal symmetry SO(8) in the decomposition SO(18) D SO(10) x SO(8) acts as a symmetry group of a hypercolor (HC) force. This new strong force is supposed to confine those particles which are nonsinglets under the HC symmetry, and to decouple them from the world seen at low energies. The special character of the SO(8) group mentioned above plays a crucial role in this scheme. Consider for example the subgroup SO(6) SU(4) of SO(8). The “standard” embedding of SO(6) into SO(8) would correspond to the decomposition 8+ —*4 + 4* of the semispinors. However, thanks to the automorphism mentioned, also a peculiar embedding such that 8~—*6 + 1 + 1 (which is the decomposition rule for the vector representation 8~in the standard embedding) is possible as well. Equation (2.16) then implies that an SO(6) HC force would leave two ordinary families unconfined. Clearly, a viable HC symmetry leaving three (or more) families light must be some subgroup of SO(6). Several symmetry patterns are possible: (i) SO(8)—*SU(4)----*SU(2) x SU(2), (ii) SO(8)—*SU(4)----*Sp(4), and (iii) SO(8)—* SU(4)—* SU(2). The corresponding decompositions of the semispinors are ‘~

(i) 8~—*6+1 + 1—*(2,2)

+

(1,1)

+

(1,1)

+

(1,1) + (1,1),

8_~~~*4+4*_~*(0,2)+(0,2)+(2,0)+(2,0); (ii) 8~—*6+1+1--~5+1+1+1,

8_*4+4*~~~~*4+4*;

(iii) 8~—*6+1+1----*3+1+1+1+1+1,

(2.17)

8__*4+4*~~~*2+2+2*+2*.

These alternatives hence predict four, three and five light V— A families, respectively. The remaining V— A families and all the V + A families are hypercolour nonsinglets and do decouple. The HC picture has a serious flaw, however. Renormalization group analysis shows [47] that the hypercolour force becomes strong typically at the scale 1010 GeV, which is unacceptably high because for example the 16. 16_ condensates would cause breaking of the electroweak symmetry at this same scale. The reason is that these condensates, while singlet under the HC group, do not contain a singlet piece under SU(2)L x U(1). No consistent model avoiding this difficulty has been presented so far. Model 2. The HC assumption is not the only possibility to split the spinor 256. In ref. [42] Bagger and Dimopoulos have made a search for such continuous family symmetries H C SO(8), under which the spinor representation 256 is complex. Such a symmetry would allow some of the V — A and V + A families to pair off and acquire a large mass through 16~ 16 mass terms, while banning these terms from the rest of the families. There are altogether six such symmetries H in SO(8): SU(3) x U(1), SU(2) X U(1) and four different U(1) groups. It was found that families appear in all these cases pairwise, i.e. for each light normal family there is a light mirror family. For the first two symmetries six V— A and V + A family pairs remain massless down to the electroweak scale, but this option must be rejected since it turns out [42] that the strong coupling constant a5 blows up at 10” GeV because of too .

62

J. Maalampi and M. Roos, Physics of mirror fermions

many light particles. The same is true for two of the four U(1) symmetries. The two remaining U(1), specified in ref. [42], give an acceptable theory predicting four light V A and V + A families which constitute a complex representation under the family symmetry group. The breaking of the horizontal symmetry SO(8) to one or the other of these family symmetries U(l)F can be achieved by a Higgs multiplet transforming as a totally antisymmetric tensor in the symmetric part of the product 256 256, —

(256•256)~=F+I+I,

(2.18)

preserving SO(10) x U(l)F. Here .f~denotes a totally antisymmetric tensor representation of the order n with dimensions (~), (~15~)and ~( ~) for 1~,17~and 1, respectively. The .f contains a piece (1,56) under SO(10) x SO(8) which leaves SO(10) unbroken, and where the 56 has a component which breaks SO(8) to U(1)F. The U(l)F symmetry cannot, however, remain unbroken because the corresponding gauge boson obviously mediates flavour changing neutral currents. Experimental limits require [42] that the U(l)F symmetry must be broken by a Higgs particle, x~at a scale (x) ~ i05 GeV. Nevertheless, this does not necessarily generate mass terms connecting ordinary and mirror fermions which would spoil the whole scenario, since it is possible to choose the U(l)F charge n of x such that the coupling 16~ 16_x is forbidden. In other words, there remains a discrete symmetry Z,, U(l)F which continues to forbid the undesired mass terms. The Z,~symmetry does not necessarily forbid the usual 16~ 16~and 16 16 mass terms below the weak scale, and it may even happen that Zn is broken at this scale. When the discrete symmetry is broken also the terms 16~ 16. become allowed and the ordinary and mirror families will mix. If the Z,~symmetry remains unbroken, however, the mirror mixing can occur only through unrenormalizable interactions, implying typically a mixing

c

.

.

(2.19)

~F—MW/MXlO,

where M~ iO’5 GeV is the grand unification mass scale. Why then are the mirror families systematically heavier than the ordinary families? A possible answer is evident from the multiplication rules of the SO(8) spinors 8~•8~=1+28+35,

8•8=1+28+35’.

(2.20)

If we choose the representations H 8 + and H’ 8 to take care of the fermion masses, and assume (H) ~ (H’), the desired splitting is obtained. Instead of introducing two widely separated scales one could also assume (H) = 0 and argue that the masses of the ordinary fermions are induced by the large scale (H’) through radiative corrections. A typical diagram contributing to the V—A fermion masses Mf is then given in fig. 2.1. From it one can estimate -~

Mf

(aGUT/21r)MF.

(2.21)

With aGUT 0.1 this predicts mirror masses MF in reasonable agreement with experimental limits. One should also note that as soon as the discrete family symmetry Zn is broken, radiative corrections would also generate mixing between ordinary and mirror fermions at some level. Model 3. Senjanovié, Wilczek and Zee [43] have proposed still another method to split the spinor 256. In their model the splitting is done by attaching a Peccei—Quinn symmetry [48] U(1)~ 0to the

J. Maalampi and M. Roos, Physics of mirror fermions

(~)

63

(w~

~

I?

~uJJ~(

0

(H> Fig. 2.1. A typical radiative mass term for an ordinary fermion f induced by mirror mixing. Here F denotes a mirror fermion, G a gauge boson, H a Higgs particle responsible of electroweak symmetry breaking, and ‘I’ a Higgs particle giving mass to the gauge boson.

gauge symmetry SO(18). The Peccei—Quinn symmetry may be necessary to solve the strong CPproblem. It is assumed that a linear combination of U(1)~0and a certain U(1) subgroup of the horizontal symmetry SO(8), the one generated by the charge QH = ~‘11,12 + ‘~314 +

‘516

+

p718,

(2.22)

remains unbroken at the grand unification scale. This symmetry protects some of the families against large masses. More explicitly, if the Peccei—Quinn charge Q~,0 is taken to be unity for all fermions, the combined charge QF3,= QH +H-++—)+3 Q~,0 takes the following values: permutations, QF=

QF =

(2.23) —1,

+ — — —)

+ 3 permutations,

for the V— A families,

++++),

QF=5’ QF =

1,

++——) + 5 pennutations,

QF=—~,

(2.24)

H———)~

for the V + A families. One can see that conservation of the QF charge allows four V — A families with QF = —1 and four V + A families with QF = 1 to pair off and acquire a large mass, as well as two V— A families with QF = 3 and two V + A families with QF = —3. The rest of the fermions, three ordinary families with QF = 3 and three mirror families, two with QF = 1 and one with QF = 5, are protected by the symmetry and are light. The family charge QF will be broken by the Higgs fields which are responsible for the masses of the SU(2)L singlet neutrinos VR and NL. The VR obtains its mass inside the SO(10) through the Yukawa coupling =

+Cyyjykylym~+(~l 26)hJktm,

(2.25)

64

J. Maalampi and M. Roos, Physics of mirror fermions

where 4’126 is a Higgs multiplet in a totally antisymmetric 126-dimensional tensor representation, and C is the generalized charge conjugation matrix [12, 14]. In the normalisation used, Q~,0= —2 for both 4126 and

The left-handed mirror neutrino NL obtains its mass similarly through the coupling ~ Both representations 128 and 128*, in fact many of them, can be found in the i; tensor representation of SO(18), and it turns out that there exists a pair for which QH(~i26) = QH(4i26~) = —4. By letting i; acquire a vacuum expectation value in these directions, one has QF(( 1)) = —6(1;), 3). and the family symmetry brokencross downterms to abetween discrete regular symmetry Z6 generated byfamilies. D exp(iITQF/ This discrete symmetry stillis forbids families and mirror It must be broken ultimately, however, since it also forbids the pure mirror mass terms for which D(qi, 4126*.

6..ifr16~)=

exp(±i2ir/3).One should also notice that D(~i16~i16) = 0, which shows that the ordinary and the mirror families will obtain their masses from different Higgs representations. This may give a possible explanation for their mass difference. One novel feature of the model is the number of light neutrinos. From eqs. (2.23) and (2.24) we can see that all three right-handed neutrinos VR obtain a mass proportional to (1;), but of the mirror neutrinos NL only two do. The remaining NL will couple to its right-handed partner NR to form a Dirac particle with a mass of the same order of magnitude as the mirror up-quark. Due to the well-known see-saw mechanism [13, 49] the model hence predicts five light neutrino states, the three predominantly left-handed neutrinos ~L and the two predominantly right-handed mirror neutrinos NR. These examples show that a realistic model incorporating mirror families may be conceived within an SO(18) gauge theory. We should emphazise, however, that none of these models has been analysed in full detail so far. Two conclusions are nevertheless evident: that some of the mirror families should have masses at the O(M~)scale and that mixing between ordinary and mirror families should occur at some level.

3. Other theories involving mirror fermions It seems to be a general “rule” that the more sophisticated the symmetry of a theory is, the more left—right symmetric is the particle spectrum. We shall now list and briefly describe a quite diverse set of recently studied theories which all predict mirror quarks and leptons. Apart from the theories to be considered below also some composite models predict mirror fermions [281. 3.1. Family unification in SU(N) groups As mentioned in the previous chapter, also large unitary groups SU(N) have been considered as possible gauge groups for family unification. The idea of using SU(N) groups looks natural when one recalls the fact that all the low energy gauge symmetries are unitary and that grand unification of gauge interactions can be constructed within a unitary group, SU(5). One difficulty is, however, obvious from the beginning: going beyond SU(5), the pattern of families does not any more enter in any simple way in the representations of the gauge group, in contrast to what happens in the orthogonal groups. In order to accommodate three or more ordinary families, one is compelled to employ a set of several irreducible representations of the group, but as a result one also has in hand a large and colourful zoo of unwanted particles. There is no unique and automatic rule for arriving at a particular choice of fermion representations in an SU(N) model. Instead, a set of guiding principles have been formulated, some of which may

J. Maalampi and M. Roos, Physics of mirror fermions

65

sound quite arbitrary. The usual assumptions are: (i) Only totally antisymmetric representations are allowed. This guarantees that fermions are only singlets, triplets or antitriplets with respect to the colour subgroup SU(3)~. (ii) The representations should contain an equal number of colour triplets and antitriplets to ensure a nonchiral colour. (iii) Triangle anomalies must cancel among fermions. (iv) The representation of left-handed fermions should be complex with respect to SU(N) to guarantee the V— A structure of weak couplings. (v) Each irreducible representation may appear only once so as to prohibit a trivial replication of families. In some cases it has turned out to be useful to relax some of these principles, usually the last two. The extra fermions predicted by the SU(N) family unified models are made “harmless” by applying the survival hypothesis [18], according to which any subset of the left-handed fermion representation, real with respect to the SU(3)~x SU(2)L x U(1) symmetry, gets a mass of the order of the unification scale 1015 GeV. The aim is to find a theory where the non-real part of the fermion representation consists just of three V — A families, while all other states have their masses in an inaccessible range. In ref. [181Georgi studied family unification within the symmetry SU(11). A three-family model was achieved by choosing the fermion representation to be Rf = [4] + [8]+ [9] + [10]= 330

+

165 + 55

+

11,

(3.1)

where [m] denotes the m-fold totally antisymmetric tensor representation of SU(11) with dimension 11!Im!(11 — m)!. In general, the anomaly of a representation [m] of SU(N) is given by [50] AN[m]

=

(N

—

2m)(N —3)! /(N — m — 1)!(m

—

1)!,

(3.2)

and thus R~in eq. (3.1) is anomaly-free. The model is the minimal three-family model, which obeys the rules (i) to (v). If one relaxes the condition (v) smaller gauge symmetries become possible. In ref. [19]it was shown that already the SU(7) symmetry allows any number of V— A generations. In particular, three families result if one chooses the following anomaly free combination: Rf=2[2]+[3]+8[6]=2•21+35+8.7.

(3.3)

The total dimension of this representation is D = 133 which might be more acceptable than D = 561 of the SU(11) model. Had we forbidden duplication of representations, we would have ended up with a model [20] where R~=[0]+[2]+[4]+[6]=1+21+35+7.

(3.4)

The combination (3.4) includes two ordinary families and two mirror-like families. [Note that (3.4) is exactly the decomposition of a spinor representation 64* of SO(14) with respect to the subgroup SU(7).] If the survival hypothesis were valid, there would be no light families in this model. It was shown in ref. [20], however, that the survival hypothesis can be evaded by adopting a particular nonstandard assignment for the electric charge operator due to which mirror fermions have nonconventional charges. Charge conservation then forbids the pairing of V — A and V + A fermions.

66

J. Maalampi and M. Roos, Physics of mirror fermions

The most economical SU(8) model [21] is the one where R1=[3]+[6]+[7]=56+28*+8*,

(3.5)

which predicts three V— A families plus additional SU(2)L doublets, which have either V + A or mixed V, A weak interactions. Other SU(8) models include the following two, studied by Chaichian, Kolmakov and Nelipa [22]: R;[1]+[31+[5]+[7]8+56+56*+8*,

(3.6)

R~=[2]+[4]+[6]=28+70+28*,

(3.7)

both of which are trivially anomaly free. These models predict four V— A families and four V + A families, all light. A further discussion about family unification in SU(N) groups can be found in ref. [23]. 3.2. Extended supersymmetry Making a grand unified theory supersymmetric [51] provides a solution to the hierarchy problem, because it protects scalar masses against radiative corrections [52]. This is the main motivation for introducing supersymmetry in the context of gauge theories. The simplest supersymmetric theory contains just one supersymmetry charge (N = 1), but also the extended supersymmetric theories [53] with 2 ~ N < 8 have attracted interest because they provide a possibility to construct a finite field theory besides solving the hierarchy problem. The structure of such theories is more restricted than that of the N = 1 theory [54, 55]. In particular they are necessarily vector-like, implying the existence of mirror fermions. Let us consider as an example the N = 2 theory [54, 55]. The particle spectrum is given by an N = 2 supermultiplet, which consists of an N = 2 vector multiplet and N = 2 hypermultiplets. The N = 2 vector multiplet in turn contains an N = 1 vector multiplet V= (v, A) and an N = 1 chiral multiplet ~ = (~, ~ both transforming as an adjoint representation under the gauge symmetry. Here v is a spin-i particle, A and ç1i~ are spin- ~ particles and ~‘ a spin-0 particle. The hypermultiplets consist of two left-handed N = 1 chiral multiplets X = (x, ‘I’+) and Y = (y, One of the two supersymmetric charges generates transitions within each N= 1 multiplet, e.g. the transitions ~—* ~ x~—*~ and y~—*çL,~,, whereas the other one generates transitions V~—*4 between the N = 1 components of the vector multiplet and transitions X*-* Y between the N = 1 components of the hypermultiplet. For consistency the latter require that X and Y transform according to representations which are conjugates of each other. This property of hypermultiplets, characteristic of all N 2 theories, implies that each fermion has its mirror counterpart. A solution to the mass splitting between ordinary and mirror fermions was suggested by del Aguila et al. [55].The electroweak symmetry was enlarged to SU(4) x U(1), and it can be argued [56]that such a low energy symmetry follows from a finite SO(12) grand unified theory. ~

3.3. Kaluza—Klein theories The idea of the so-called Kaluza—Klein theories [27]is to establish a connection between gravity and other particle forces by postulating an extra dimension for space—time. The original inventor of this idea was a Finnish physicist G. Nordstrom [57],who introduced an extra space—time dimension into his

J. Maalampi and M. Roos, Physics of mirrorfermions

67

scalar theory of gravity, so as to unify gravity and electromagnetism. The first attempt in the context of Einstein’s general relativity was due to Kaluza [58], who assumed that space—time is five-dimensional, with the line element d~2= gM~(z)dzM dzN,

(3.8)

where z = (xiL, x5) and M, N = 1,... ,5. Kaluza assumed that the components ~ = g,~(x)of the metric field ~ should be interpreted as a spin-2 graviton, = A~(x)as a spin-i photon and ~ = as a spin-0 dilaton. The fifth dimension then manifests itself as electromagnetism in the fourdimensional world. Klein [59] developed further Kaluza’s theory by giving an explanation as to why ~ should be a function of x~only, as Kaluza had conjectured, with no dependence on the extra dimension x5. Klein suggested that the fifth dimension is “compactified” to a circle S’ at each point of the four-dimensional space—time. The radius of the circle would naturally be of order M~flCk. The tensor ~ can then be represented by the following Fourier expansion: ,~

++

5

(n)

Inmx5

g~~(x,x)=g~~(x)e

The massless particles — graviton, photon and dilaton — are just the n = 0 modes of the expansion and hence independent of x5. In the modern versions of the Kaluza—Klein theory [60] one tries to unify all the SU(3)~x SU(2)L X U(i) gauge forces with gravity, and hence more than one extra dimension is obviously needed. The most promising models also involve supersymmetry, which places an upper limit for the space—time dimensionality, d 11. This is because an N = 1 supersymmetry in eleven dimensions corresponds to an N = 8 maximal supersymmetry in four dimensions. The seven extra dimensions in the d = 11 Kaluza—Klein theory are thought to be compactified “spontaneously” so that the splitting of the eleven-dimensional space M 11 to an M4 x M7 ground state can appear naturally. The physics in four dimensions (e.g. gauge interactions) are determined by the geometry of the manifold M7, usually taken as a seven-sphere. What is essential to our present study is that in the d = 11 theory the fermion spectra tend to be nonchiral. The basic reason is that in odd dimensions one cannot define a chiral operator which would appear as the (1 7~)projection operator in four dimensions. This traces back to the fact that the spinor representation of SO(11) is irreducible (see section 2). It has been shown [60] that Kaluza—Klein theories can produce chiral fermions only if the fermions are chiral already in the original high dimension and the chirality is protected by a topological twist through the compactification. In the d = 10 theory one can define a chirality operator, ~(1 ± yii), but no chirality preserving compactification has been found. On the other hand, chiral fermions in higher dimensions may result in chiral anomalies in hexagon diagrams which may destroy not only the Yang—Mills gauge invariance but also, and what is perhaps more serious, Einstein’s general covariance. The appearance of chiral V— A families in four dimensions in the Kaluza—Klein theories thus necessarily implies the existence of mirror fermions. The mechanism, which would split the nonchiral fermion spectra into two chiral sets of states is, however, unknown. —

3.4.

E8

x E8 superstring models

A candidate for a consistent quantum theory of all forces including gravity is the E8 X E8 heterotic string theory [31] defined in a ten-dimensional manifold M10. This theory has vacuum states with

68

J. Maalampi and M. Roos, Physics of mirror fermions

Kaluza—Klein structure, that is, M,0 = M4 x K, where M4 is a flat four-dimensional Minkowski space and K is a compact six-dimensional space of SU(3) holonomy, a Calabi—Yau manifold. The effective four-dimensional theory is an N = 1 supergravity and has E6 x E8 gauge symmetry, where the E8 is associated to a “shadow world” felt by the observable world only through gravitational interactions. Nonsinglet scalar superfields, which should include the scalars and fermions of the standard model, form 27 and 27* representations of the E6 [61]. These decompose under the SO(10) subgroup of E6 as follows: 27=1+10+16,

27*=1+10+16*.

(3.10)

Thus they contain a fermion and a mirror fermion family, respectively, and a set of “exotic” new fields. What is interesting is that the number of families, n(27), and mirror families, n(27*) determined by the topological properties of the compact manifold K, are not generally equal. In fact, the difference is determined by the Euler characteristic ~(K) of the manifold [31, 46] n(27)

—

n(27*)

=

~(K)I2.

(3.11)

For a typical Calabi—Yau manifold the Euler characteristic is often unreasonably large (x 100), but if we consider quotient manifolds KIG, where G is a discrete symmetry of K with no fixed points, a more acceptable number of families may emerge n(27) — n(27*)

=

~(K)I2N(G),

(3.12)

where N(G) is the number of elements in G. The fact that quotient manifolds KIG are multiply connected offers a simple mechanism to break the E6 down to a realistic low-energy gauge symmetry G0, as pointed out in ref. [62]. The usual Higgs mechanism does not work since the 27 and 27* do not contain any suitable scalar fields for that purpose. The remnant symmetry G0, not broken by uncontractable Wilson loops, has rank five or six. In the simplest case G0 = SU(3)~x SU(2)L x U(1) x U(1). Not all of the n(27) families and n(27*) mirror families necessarily remain massless in the breaking E6—* G0, but at least the lx(K)/N(G) 27-plets do so. If we denote the decompositions of 27 and 27* under G0 by E, r. and ~, r’~’,respectively, the massless matter fields are [~(K) 12N(G)] •27 +

~

n.(r~+ r~),

(3.13)

where the multiplicities n1 depend on the Calabi—Yau manifold at hand. In most cases some of the multiplicities n, vanish. Besides a (perhaps suitable) number of V— A generations in the 27, one may thus have some incomplete families, always accompanied by the corresponding mirror fields. The number of broken families as well as the set of fields remaining light varies from one Calabi—Yau manifold to another, as can be seen from concrete examples studied in the literature [46]. It is important to note that r~and r~cannot pair up and form vector-like states since the 27 27* couplings are absent from the superpotential of the theory. Some mixing may however be generated through higher-order operators, such as 2 (27*)2 , (3.14) .

(i/MPlaflck)27

J. Maalampi and M. Roos, Physics of mirror fermions

69

when 27 and 27* develop vacuum expectation values, but the mixing would necessarily be very small (—M,~/ MPlanCk). To summarize, in superstring theories one may have a net number of chiral V— A families with no accompanying V + A families as a solution to the family replication problem. In general, however, some remnants of mirror families survive to the low mass scales as well. It still has to be noted that not all of the fermions in the left-handed 27 have the standard V — A couplings to the W boson. Indeed, the SO(i0) representations 10 in the decomposition (3.10) are real under the low-energy symmetry G0 and contain, for example, the SU(2)L doublets (NE, E)L and (E’~,N~)L.The E’ has V + A weak couplings; it pairs up with E to form a vector-like Dirac field. It is possible in principle that E’ also does mix with the ordinary V—A leptons in the 16-plet part of the 27, giving rise to a small right-handed component in their weak couplings. This kind of mixing would lead [63] to a slightly different parametrization of the weak lagrangian than considered below in this paper.

4. Low-energy effects of mirror mixing We have seen in the previous sections that a great variety of models predict mirror fermions at a low mass scale. To maintain their chiral character mirror fermions, just like the ordinary V — A fermions, obtain their mass through mass terms breaking the electroweak gauge symmetry SU(2)L x U(1). The upper limit for the mass would then be mF ~ 0(300 GeV), if we want to keep the validity of the perturbation theory by not allowing too large Yukawa couplings. As mentioned in the introduction, charged mirror quarks and leptons should be heavier than 26 GeV to account for the fact that they have not been detected so far in e~e collisions [34]. The limits for the mass of mirror neutrinos depend, as we will see, crucially on the value of the mirror mixing angle. In this section and in section 5 we will consider various effects of mirror fermions detectable in the laboratory already at present energies, or observable in astrophysics or in cosmology. It was described above how one can circumvent the survival hypothesis by prohibiting the mass terms which would mix ordinary and mirror fermions. This could be done, e.g. with the help of a discrete remnant symmetry inherited from the original gauge invariance or due to supersymmetry. It usually happens that this protection is evaded during the electroweak symmetry breaking so that Yukawa couplings mixing fermions and mirror fermions start appearing. Alternatively, the mixing is induced radiatively or through higher-order couplings. An immediate consequence with great phenomenological importance would then be that the Lorentz structure of the weak interactions would no longer be purely chiral, but there would be a finite (probably small) V + A piece in addition to the standard V— A term [35—38]. One should recall at this stage that also in the so-called left—right symmetric models (LR-models) [64] one has right-handed interactions. The LR-models are based on the gauge symmetry SU(2)L X SU(2)R x U(i), where SU(2)R corresponds to a new electroweak force. Right-handed components of leptons and quarks are arranged into doublets under the SU(2)R group, and thus couple to the gauge bosons WR, ZR of that group in V + A currents. These new interactions become operative at some higher energy scale (MWR, MzR ~ 1 TeV), but a possible mixing of the “right-handed” vector bosons and the known weak bosons W and Z would cause a deviation from the pure V— A structure of the ordinary weak interactions. In general, the mirror model and the LR-model will, however, lead to

70

J. Maalampi and M. Roos, Physics of mirrorfermions

different parametrizations, except in the case where M~,M~—*~ with a finite (W, WR) and (Z, ZR) mixing (this corresponds to the model c to be defined below). In the following we will present a general analysis to find out how large deviations from pure left-handed weak couplings the existing laboratory data allow. As a result we obtain upper limits for the mixing angles between fermions and their mirror counterparts for different assumptions about the mirror neutrino masses. This analysis updates our earlier work presented in refs. [35, 37, 38], but this time we will also take into account the electroweak radiative corrections [65],which are known to play a crucial role in the standard model in achieving agreement between theory and experiment [66, 67]. 4.1. Low-energy lagrangian The transformation properties of the ordinary fermions and mirror fermions under the electroweak symmetry group SU(2)L x U(1) were given in eqs. (1.1) and (1.2), respectively. The electroweak lagrangian is accordingly given for the charged currents by ~,CC

=

(g/2\ )Wfa{i~,y(1— y~)+ üy~(1— 75)d

+

Nya(1 + 75)L + Uy~(1+ 75)D} + h.c., (4.1)

and for the neutral currents by ~NC=(g/4cos~)Z~[~.y(4x_1_7s)?+ +

iyn(17~)V

t~ya(-~~x+ 1—y5)u+ dya(~x—1+ 75)d

+ Lya(4x

—

1 + 75)L

+ Uyn(—~x+ 1

+

+

Neya(l + y5)N~

y5)U + Dya(IX —1— 75)D].

(4.2)

The weak eigenstates appearing in the lagrangians are not necessarily equal to the mass eigenstates because of mixing. In the following we will only take into account mirror mixing, ignoring possible mixing between flavours. In terms of the SO(10) representations, mirror mixing arises from the 16 16* terms, as discussed above. The most general mass lagrangian for an “extended” family 16 + 16* has the generic form 6 16* + ~ 16* + h.c. (4.3) m11l6~16 + mFFl The diagonal masses mff and mFF of fermions and mirror fermions, respectively, have the upper bound mff, mFF ~ 0(300 GeV). The value of the mixing mass m~ is notorrestricted by gauge 2)L x1,,instead, U(i) symmetry, SO(10). However, invariance, since that term does not break either the SU( the existence of light fermions with approximately V — A couplings requires mfF ~ mFF. Consider for example the mirror mixing of the electron. The mass lagrangian is given by =

=

~

+

h.c.,

(4.4)

where we now use primes to indicate weak eigenstates [those appearing in the original weak lagrangians

I. Maalampi and M. Roos, Physics of mirror fermions

71

(4.1) and (4.2)]. We will assume that the mass matrix appearing in (4.4) is hermitian. Then the diagonalization is characterized by one rotation angle and one finds e=cosOee’+sinOeE’

E=~sinOee’+cosOeE’,

,

where e and E are the mass eigenstates and the mixing angle tan 20~= 2meEI(mEE

—

(4.5) °e is

given by

mee)~ 2~meE/mEE~.

(4.6)

The eigenvalues of the mass are mEe

=

~(mEE+ mee)±~\I(mEE— mee)2

+4meE

(4.7)

.

Similarly one can define mirror mixing for the other fermions. The mixings appropriately modify the gauge interaction lagrangians (4.1) and (4.2). For the charged currents we have [36, 37] ~~CC=

(g/2\r2)Wta{~~y~[co5(4,~0~) cos(4.~+ —

—

+

v~y[sin(4~ — 0~)+ sin(4~+

+

I ~7~,[sin(4~,, 0~)+ sin(4~+ Oe)ys] —

+ Ney[cos(q5~ — 0~)+ cos(4 1 + 0~)y5]L + uya[cos(4~u — Od)

—

+ UYa [—sin(4~ — Od)

cos(4~+ Od)ys]d +

+

i~’ya[sin(4u

sin(4~+ Od)7S]d +

tb/a

—

[cos(4~

Od) + —

Od)

sin(4~+ Od)7S]D + cos(4~+ Od)ys]D} + h.c. (4.8)

and for the neutral currents [38] ~NC

=

(g/4cos ~)Za[1ya(4X —1

+

cos 201 75)~+ Eya(sin 20~y5)L

+ Ly(sin2Oeys)é+Lya(4x—1—cos2Oeys)L +

(l

COS 24~~ y~)ii~ +

(sin 24L1 75)N

+Ny(sin2~eys)pe+Ney(i+cos24eys)N +

u7a(x+1cos24~u7s)u+u7a(sin24~uys)U

+ Uy(sin2~y5)u+Uy(—~x+1+cos2~y5)U +d7(~x—1+cos20dyS)d+d7(sin2Od7S)D 20d ‘y + Dya(sin20d 75)d + Dya(~x—1— cos 5)D].

(4.9)

0d denote the vN, L, uU and dD mixing angles respectively. We have assumed ~, 0~, 4’~and thatHere neutrinos are Dirac particles.

72

J. Maalampi and M. Roos, Physics of mirrorfermions

Besides the modifications in the Lorentz structure of the currents, another consequence of mirror mixing is the existence of nondiagonal neutral currents (U’, Nv etc.) which are forbidden in the standard model as a result of the GIM mechanism [68]. In mirror mixing the GIM mechanism does not work, because the mixed objects have opposite axial vector couplings, leaving a net nondiagonal axial vector current. All the neutral vector couplings including the electromagnetic couplings are in contrast untouched by the mixing. The frustration of the GIM mechanism is a novel feature of mirror mixing and can be used to distinguish mirror !eptons and quarksfrom the heavy fermions of ordinary type. To give an example, the decay channel L —~ I’~I[~ris possible (though suppressed by a mixing factor) for a mirror lepton, but not for a sequential lepton. 4.2. Experimental tests of the V

—

A structure

A straightforward manner to study effects of mirror fermions is to look for possible deviations from the V— A structure of weak couplings of the ordinary fermions [35,36]. (As we have pointed out already, such deviations can also be signatures of a right-handed gauge boson WR in a left—right symmetric gauge model, but we stress again that these two models lead in general to different parametrizations of the currents and can be distinguished from each other.) Extensive analyses carried out in refs. [35, 37, 38], updated in this section, confirm that as a whole the low-energy data are consistent with a purely left-handed structure of currents, but the experimental uncertainties still do allow for quite considerable right-handed pieces in the lepton currents. As far as mirror mixing of quarks is concerned, there is one laboratory constraint surpassing all the others, namely the nonobservation of flavour changing neutral currents (FCNC) [38, 69]. We know that the ordinary quarks d and s mix, as specified by the Cabibbo—Kobayashi—Maskawa matrix, but the direct FCNC couplings ~dZ are absent due to the GIM cancellation. However, as we saw above, the GIM mechanism is not valid for mirror mixing, so if s and d quarks contain a mirror impurity, the ~dZ coupling may be induced by mirror mixing. As a result a KL — K~mass difference is generated by a Z exchange diagram, see fig. (4.1). Comparison with experimental results implies typically a mixing angle

[38]

0q~10~~~

(4.10)

Even if the FCNC currents were absent at the tree level, they would be induced by radiative corrections, as we know from the standard model. In the standard model the KL—Ks mass difference is generated by weak interactions of second order in GF due to the famous box diagrams [70]with up-type quarks u, c and t as internal states. In the presence of mirror mixing, the internal states may be replaced by heavier mirror quarks U, C and T (see fig. 4.2) with appropriate mixing factors in the vertices. A comparison of the ensuing mass difference to the measured value leads to the mixing angle bound [69] 0q~10~4~

(4.11)

for mirror quark masses 0(100 GeV). If one takes this bound as a generic one for all the quark—mirror— quark mixing angles, there is no chance to see deviations from the V A structure of quark currents with the experimental sensitivity available at present. We will therefore concentrate in the following only on the leptonic currents. The effects of mirror mixing of quarks in the deep inelastic neutrino nucleon scattering have been studied in ref. [71]. —

J. Maalampi and M. Roos, Physics of mirror fermions

Fig. 4.1. A flavour changing neutral current interaction inducing K°—K°mixing. This GIM rule violating diagram is possible if the quarks d and s mix with their mirror counterparts.

73

Fig. 4.2. Box diagram for K°—K°mixing in the presence of mirror quarks and mirror mixing.

Mirror mixing in neutrino beams will lead to oscillations between ordinary neutrinos and mirror neutrinos of the same family. Traditionally neutrino oscillation has been divided into two categories: “first class” oscillations or flavour oscillations between lepton families, and “second class” oscillations within a single family. The former conserve the total lepton number, whereas the latter represent oscillations between a neutrino and its antineutrino, possible in the presence of right-handed neutrino fields in the theory; thus it breaks total lepton number conservation by two units. When mirror neutrinos are present a “third class” of oscillations can take place: neutrinos oscillate into mirror neutrinos conserving the total leptonic number (and family-specific leptonic numbers as well) [72].For this to occur, the mass lagrangian need contain only Dirac mass terms. If there are both Dirac and Majorana mass terms, also second class oscillations between neutrinos and mirror antineutrinos of the same family can occur, again breaking total leptonic number by two units [72].The pattern of oscillations could be quite complicated since nothing forbids several classes of oscillations from being present simultaneously. It is clear that the constraints one obtains for mirror mixing from experimental data depend on the unknown masses of mirror neutrinos. For example, if a mirror neutrino is light, neutrino beams obtained from pseudoscalar meson decay may contain a mirror component. Rather than taking the mirror neutrino masses as free parameters into our analysis, we follow refs. [37, 38] and consider separately the following limiting mass configurations, or models for the e and ~i families: model(a) mN,mN mK, model (c) m~, mN > m~, model(d) mN >mK, mN
= ~

=

1

= ~

=

~ sin2 ~

—

—

[1 cos(2~rx/L)]. —

(4.12)

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J. Maalampi and M. Roos, Physics of mirror fermions

Here x is the distance between production and interaction vertices and L = 4rrE5Iz~m2is the oscillation length. The T family will not be considered here because the precision of the experimental information is much inferior to that of the e and p families. Constraints on mirror mixing in the ‘r family have been studied in refs. [38, 74]. 4.2.1. Charged current processes The leptonic and semileptonic processes with information about the V, A structure of the charged weak currents of the e and p. families are the following: (1) pseudoscalar t2-decays (‘rr, K—* ev and IT, K—* pv), (2) muon ~3-decay(p—+evi.i), (3) nuclear ~3-decay(superallowed 0~—*0~ transitions such as ‘~O—~ 14N*ev), (4) inverse muon decay (v~+ e p + The relevant quantities in the first process are the universality ratios R~= F(rr—* ev)IF(i’r—* p..v) and RK = F(K—t. ev)IF(K—* pv), and the longitudinal polarization of the muon in ii, K—* pv. In the muon decay the relevant parameters are p, ~ ô, P~and GF, where p, ~ and ~ are the famous Michel parameters, P~-is the longitudinal polarization of the decaying muon, ~e is that of the final state electron and GF is the Fermi constant. The quantity sensitive to the V, A structure of the interaction in the nuclear 3-decay is the longitudinal polarization Pe of the decay electron. Let us note that the longitudinal lepton polarizations measured in different processes have the same theoretical values since they are determined by the same basic currents. —~

Table 1 Theoretical expressions of parameters using eq. (4.13) for

e

=

e, ~i

Pseudoscalar and nuclear beta decays Model~

R

P_

—

(clv)P 5_

a b c d

I

2I(V~+ A~)

a~ a f3~

(V~+ A~)I(V~ + A~) l(V~+A~)

f3~ a~

Muon decay

P

Model

p

a cb d

~(1+/3,/35)2~3,—f3~

~(1+f35a5) ~(1 + a~a) l(1 + a..13 5u) Inverse muon decay Model a b c d

2f3,—a~ 2a~ — a 2a5

—

$

~(~,+f35)I(2$,—$5) i(13,+a5)/(2135—a5) ~(a~ — a 5)/(2a, — a5) i(a, + $,jI(2a5 — $5u)

Pc —j3~ —a5 _a~

~ 2(V~+A~) ]G2(V,~+ A~)(V~ + A~) ~G2(V,~+ A~) ]G

S

{[(V~+ A)12l2[(1 + a2)(1 + c) + 2a~$,(1- c)] ~[(1 + a~3(1+ c) + 2a~p..(1—c)] l[(1+a2)(1+c)+2a 2[(1 5a5(1—c)I + a~)(1+ c) + 2a~a(1— c)] ] {[(V~ + A)12] ‘~Definition of the models a—d is given in the text. ~

+

[(~ + A)/2]2[(1

+

[(V~,+

+

~)(1

+

c)

+

2~~$~(1 - c)]}

A ~)I2]~[(1+ &~j(1+ c) + 2a~c~~(1c)]} —

J. Maalampi and M. Roos, Physics of mirror fermions

75

The interesting quantity in the inverse muon decay is the differential cross section ratio S = (do’idy)I(do’idy)5~~ mod’ where y = E,L/EV. It should be emphazised that the possible deviation from V — A enters here not only through the vertices responsible for the reaction itself, but it is also inherent to the initial neutrino polarization, reflecting the production process of the neutrino beam from pion and kaon decays. The theoretical expressions for the parameters entering above are given in table 1, where the following notation is used: V1

=

cos(01

—

c~),

V1

=

sin(01

—

A1

~

=

cos(01

+

cb1),

A1

=

sin(01

+

(4.13) a1

=

2V1A 1I(V~+ A~),

&,~s =

1~= V1A1

2V1A 1I(V~+ A~),

—

The experimental data used in our fit are given in table 2. 4.2.2. Neutral current processes The processes which test the neutral current lagrangian (4.9) are the following: (1) elastic (anti)neutrino scattering (~v~e ~-*‘v~e,~v~e—* (2) eke-annihilation (e~e—*p.~p.,cë, bb),

Table 2 Experimental data used for the fit Quantity

Value

Muon decay and $ decay p 0.7517 ± 0.0026 6 0.7496±0.0034 0.998±0.042 1.0027±0.0084 0.99860±0.00082 = P5~ 1.004 ± 0.006 Inverse muon decay S

0.967 ± 0.085

Pseudoscalar meson decay R~,”/R’’°° 0.996±0.018 0.978 ± 0.044 P5c(ir-decay) 1.04 ± 0.09 P 5c(K-decay) 0.974 ± 0.040

Ref.

Quantity

115 116 115 117 116, 118 119

Neutrino—quark neutral current process c,~ —0.181 ± 0.056 cld 0.327±0.050 c25—lc2d —0.101±0.242 u~ 0.1269±0.0136 d~ 0.1732±0.143 u~ 0.0261 ± 0.0087 d~ 0.0041 ± 0.0100

120

value

Ref.

Electron—quark neutral current processes 0.605±0.152 h~A 0.513±0.105

66 66 66 66 66 66

66

66 66

115, 121 37, 115 122 115

Neutrino oscillation 2 çb,) R(1 — l sin

123

B A

124, 125 124

Weak bosom

M~

(80.9±1.4)GeV

67

126

M5

(92.1

67

1.000

±

0.047

127

e -scattering 6cm2 Leptonic neutral current process (rr5e—si~5e)) (1.5—3.0 GeV) (7.6±2.2)x I0~ u(v 42E, cm2/GeV 5e —s v5e) (1.50 ± 0.25) x 10 (1.75±0.79)x1042E~cm2/GeV o-(v 5e —s ye) 1.44 ± 0.22 u(v5e

—‘ye

)

1.34±0.96 0.56 ± 0.13

±

1.8) GeV

66 66

76

J. Maalampi and M. Roos, Physics of mirror fermions

(3) (anti)neutrino—hadron scattering (~v~X—* ~v~X), (4) elastic scattering of polarized leptons on nuclei (e1D—* eD, p1C—* pC). In the first process, experiments constrain the total cross sections o’(iee), u(vee), o-(i~e)and o(v~.e) (averaged over the beam energy distribution); in the second process they constrain20 the partforward— of the backward asymmetry and the parameter the standard 1 + cos model predicdifferential cross sectionAFB do~(e~e —* p..4~p..)/d cos ~0. Thethe onlycoefficient alteration for to the tions for the latter two quantities is due to the modification of the Fermi constant GF. We remind the reader that, as extracted from the p decay rate, GF may also be affected by mirror mixing. In the (anti)neutrino—hadron scattering the relevant quantities are the ratios R~= uNc(vP.X)Iucc(VP.X), the corresponding ratio R,~for the i~Xscattering and R’ = (oNc(v~X)± UNC(V~X))I(O~CC(V.~X) + ucc(i~X)),where X = p or n and X denotes an isoscalar target. In the e 1D scattering one measures the assymmetry AD = [dcr(e~) dff(e~)]/[du(e) + do-(e1~)], and in the p1N scattering the assymmetry B(PI, P2) = [do’(p~,P,) — da’(p, P2)]I[do-(p~, P1) + da’(p , P2)], where o-(p. , P~~) is the cross section wlth polarization P1(2) for p.. X—* p.. X. In our analysis of neutral currents we will take advantage of the recent comprehensive studies of Costa et al. [66] and Amaldi et a!. [67], where the standard model is confronted with up-to-date combined experimental neutral current data. The authors have taken properly into account the electroweak radiative corrections, the existence of which they establish at the 3u level. Although the radiative corrections in the standard model and in the model with mirror fermions are not exactly the same, we will apply the results of ref. [66] as a first approximation. To see under which assumptions this approximation is justified, let us look briefly into the standard model2 defined radiativebycorrections. [65] A measure of radiative corrections are the parameters Ar and As —

M~= A

2= A 0/sin ~ (1

—

Ar)~

0/sin 0°,,,,, (1 0w 0 COS 0,,,,, 0 (l

M~/cosO~= A0!sin 2= 37.28 GeV and 0 = (ITaI’s/~GF)”

— 6)1/2

(4.14) 1/2

=

where A

2

6w5(~~T),’5m

.20

0,,,,,, (4.15)

=

Ar

—

[A2(1

—

Ar)/sin2 O~,,](1— tan2 O°~,, — As2Icos2O°~,,).

The weak mixing angle sin2 O°~,,,,is obtained from deep inelastic scattering when all radiative corrections are ignored and As2 measures the radiative corrections, thus sin2Ow

=

sin2O~,,,+ As2

In the standard model [65] Ar

(4.16) 7% and As2

—1% for the generic values m

1 = 45 GeV and MH = 100 GeV of the top quark and the physical Higgs scalar, respectively. The main contribution to Ar comes from renormalization of the electromagnetic coupling constant a due to ioop diagrams involving light fermions. The Higgs mass dependence of the correction Ar is very small as long as M~< 1 TeV [67]. For heavy fermions like the t quark and mirror fermions, the sensitivity is more important and depends on the mass splitting between weak isospin doublet states. As long as the splitting obeys

J. Maalampi and M. Roos, Physics of mirror fermions

m,,

+1/2

—

m,,

~1/2I

77

~ 100 GeV, the effect is small. For larger splittings the contribution is given by [65]

2 ~(Ar) —(3a/6ir)(cos2 0~/sin40~)jV~1I(m 1

~+1/2

—

The effect increases rapidly so that for mq — mq.l> 180 GeV and mL — mMI >310GeV the data are inconsistent with the theory (for MH = 100 GeV). It is then justified to use the standard model radiative corrections for the mirror model as long as the mirror fermions are lighter than 100 GeV, or the mass splittings within weak doublets are not too large. In the following we will assume that this approximation is valid. A conventional model-independent parametrization of the effective weak lagrangian is given by ~ff

=

—(GFI”/~)[iy,~(1 yS)v][uLuy~(1 — y5)u + —

+ dLdy~(1 75)d —

+

dRdy~(1+ y5)d

+

gLeY

u~uy

(1

+

y5)u

(1 — y5)e + g~~~(1 + y5)e], (4.17)

=

(G,iV~)[ëy,~ y5e(C1~üy~u + Ciddy~d)

+ ey~e(C25uy~y~u + C2ddy~LySd)+ ëy~e(h~AI~y~LySu + h~Ady~ySd)]. The radiatively corrected values of the coupling parameters uL,. . . , h~ have been extracted from experimental data in ref. [66]. Other relevant parameters measured in experiments are the forward— backward asymmetry parameters A and B defined by [66] 4 GeV2)s + (B/b8 GeV4)s2. (4.18) A~~(s)(A/b For the process e + e ÷p.~p. these parameters are in the mirror model given by -

—

A

— —

3CO52OeCOS2O 2GeV\2 1j. (1OM~ j 32 sin20~(1— sin20~)~

—

B—

—

3COS2OeCOS2O,,. [102GeV~4 512 sin4Ow (1— sin2Ow)2 ~ M~ ~ (4.19)

In the case of light mirror neutrinos new channels open for elastic neutrino—electron scattering, yielding quite different predictions from the standard model. We cannot therefore use the results of the standard model analysis of ref. [66]for these processes. The radiative corrections are however known to be negligibly small for these specific reactions compared to the experimental uncertainties, and can safely be ignored. We therefore just update our earlier tree-level analysis [38],including also the result of the recently measured vee—* vee scattering cross section. The results of our analysis of the V, A structure of the weak currents are summarized in table 3. The table gives the 1 o upper limits of the leptonic mixing angles for the different mirror-neutrino mass alternatives (a)—(d). We can see that in all cases the results are consistent with no mirror mixing within simultaneous 90% confidence limits. It should be noted that considering the individual processes separately, or including for example only the CC data, would allow much larger mirror mixings than those quoted in table 3. To give an example, if both the mirror neutrinos Ne and N~are heavy, the CC data leave the V — A limit arbitrary in the sense that many nontrivial values of mixing angles yield the V— A prediction [36]. As is seen from the table, even including all the data, the muon neutrino mixing

78

J. Maalampi and M. Roos, Physics of mirror fermion~ Table 3 Mirror mixing angles (in degrees) and simultaneous 90% confidence limits (change in x’ of 6.25 in model a, 7.78 in models b and d, 9.24 in model c), except for the last line where we give the “conventional” errors in model c, 2 of 1.0 corresponding to a change in x Mirror mixing angles (in degrees) Model~ sin2O,~

X

~

a~ 05

d,5,55

0.229 0.229 0.229 0.229 0.226 0.229 0.229

c

0.228

afl..Oh

b,05 b,COh c d~05

0(<2.8)

i.4(<2.3)

—

—

—

—

0 (<25.3)

4.1 (<20.3) 0 (<11.5) 0 (<11.5) 7.3 (<18.4)

0 (<2.7) —

12.9 (<24.9) 4.8 (<13.1) 4.8 (<13.1) ±

0.006

13.3 ±

0 (<11.6) 0 (<11.4) 10.2 (<24.8) i.4(<2.4) —

11.9 ±

—

0 (<27.2) 0 (<19.1) 0(<12.8) 0 (<12.8) 0.0

±

9.0

—

3.3 (<21.4) 5.4

±

2.8

6.6 9.0 9.3 9.3 8.0 6.6 9.0 10.2

Definition of the models a—d is given in the text.

angle ~ still remains undetermined. This happens in the cases of coherent neutrino beams, when the neutrino and the mirror neutrino are nearly degenerate. The weakest bounds occur in the model c, where the mirror neutrinos are heavy, and where the larger number of free parameters dilutes the experimental information. The upper limits for the different mirror mixing angles range from 14.3°to 25.5°. Note, however, the last line in table 3: if one defines errors “conventionally”, three of the four mixing angles are nonvanishing at 2o- significance. These limits are the tightest bounds one obtains when the mirror neutrinos Ne and N,~are very heavy, mN ~ 18 GeV, or very light, m~~ 1 eV. In the range between, 1 eV ~ mN ~ 18 GeV, these bounds are surpassed by the results of searching for production and subsequent decay of new neutral leptons and by the astrophysical and cosmological constraints. These bounds are the subject of the following section.

5. Constraints for mirror neutrinos Since mirror neutrinos may be very light, there is evidently a much larger variety of experimental constraints for them than for the other mirror fermions. Information on their masses and on the mirror mixing angles comes from terrestrial laboratory experiments, from astrophysics and from cosmology. In this section we collect all the experimental and observational constraints and derive bounds in the planes of mass versus mixing-angle and mass versus lifetime for different types of mirror neutrinos. Obviously one obtains constraints not only for the neutrino mixing angle 4~but also for the charged lepton mixing angle 0. As before, we neglect all mixing between different flavours, and assume that the mirror mixing is the only mixing present. In the literature there exist many studies on constraints for a new “heavy” neutrino [75]. Usually the new neutrino is, 2)L however, be a sequential neutrino (v4L) transforming atransforming member of aasnew doublettaken or atosinglet neutrino (or right-handed neutrino) as ~R an left-handed SU( The essential differences between such neutrinos and a mirror neutrino N are the SU(2)L singlet. following: *) *) Differences

between ordinary neutrinos and mirror neutrinos with respect to CP-violation have been considered in ref. 176].

J. Maalampi and M. Roos, Physics of mirror fermions

79

(i) In the sequential neutrino case the GIM mechanism is operative forbidding nondiagonal Z couplings Zv4v1, = 1, 2, 3. In the mirror neutrino case nondiagonal couplings ZvN may appear. 2O, where 0 is the (ii) For a singlet neutrino VR the Z-coupling ZVRVR is suppressed by a factor sin mixing angle between VR and a doublet neutrino VL~ For the mirror neutrino the coupling ZNN is of full weak interaction strength. These differences make our study different in details from previous analyses, but the quantitative changes turn out to be quite minor. The major techniques applied in the laboratory to obtain evidence for a new neutrino and its mixing to the known neutrinos are the following [75]. If the neutrino is light enough to be produced in nuclear beta decay, it will show up as a second threshold in the Curie plot or cause oscillation phenomena in reactor neutrino experiments. If the neutrino is heavier, say MN ~ 10 MeV, one may search for a monochromatic peak in electron or muon momentum spectra from pion and kaon decay at rest. One can also study the decay-in-flight in an intensive beam of these same hadrons and search for a secondary vertex downstream to identify a heavy-neutrino decay. Still larger neutrino masses can be studied by using beams of charmed mesons. Finally, one can produce heavy neutrinos in pairs or together with a light neutrino in e + e - collisions, extending the range of investigation still further. The astrophysical constraints for neutrino masses and mixings come from the following sources. Neutrino production in the interior of white dwarfs will contribute to the cosmic diffuse X- and y-ray background from neutrino decay; comparing the expected contribution with the observational bounds gives a constraint. The measured neutrino flux from the recently collapsed supernova SN1987A gives information on the Ve oscillation into a neutrino which behaves inertly in terrestrial detectors. In big bang cosmology mirror neutrinos would have profound effects on the expansion of the universe, on the primordial nucleosynthesis and on the formation of large scale structures in the universe [77].The constraints from cosmology often depend on the lifetime of the neutrinos, in our case on the mirror mixing angles. Important bounds on the mass versus mixing angle follow from the effects on the cosmic photon flux of the radiative decay of a heavy neutrino v 2, v2—*vly.

(5.1)

This decay will occur in the standard model if neutrino flavours mix [78], and in the mirror model if leptons and mirror leptons mix [38, 79]. The process in the mirror case differs, however, essentially from that in the standard model and merits a closer look. The effective matrix element for the decay (5.1) compatible with gauge invariance is of the general form M = r~q~(p1)r~5(a+ by5)v2(p2), (5.2) 5 = (p 5 are the polarization vector and the momentum of the photon, respectivewhere r~and 2 — p1)the radiative width [78] ly. From (5.2)q one derives T1

=

F(v2~v1~) = (1/8IT)[(m~ — m~)/m

3(~a~2 + b~2)= (E~IIT)(~a~2 + b~2),

(5.3)

2] where E 0 is the energy of the photon. The process proceeds (in the unitary gauge) through the two one-loop diagrams shown in fig. 5.ia. If we write the charged current lagrangian of leptons in the general form

80

J. Maalampi and M. Roos, Physics of mirrorfermions

a

‘~tt\

b

VL

Fig. 5.1. (a) Diagrams for the radiative neutrino decay v

2—s v1’~in the standard model. (b) A diagram for the radiative mirror neutrino decay N—s vy

as a result of L mirror mixing.

t”

—~=r

$1’°° W

2

~

~

i~y~(V 11 — A~y5)e,

i=1,2

(5.4)

then the parameters a and b appearing in (5.2) are in leading order given by a=

—

eGF

2

8V~

~

m1f(r1)(V11V~ A11A2),

b

—

= —

eGF

2 ~

8V~1T

m1f(r1)(V11A2

—

A1V12),

where f(r1) is a smooth function of the order of unity given in ref. [78].These leading terms obviously vanish in the standard model where V1~IA~ = 1. The couplings of the mirror model (identifying v~ = v and v2 = N) can be read off from eq. (4.8): =

sin(41

—

0~),

A11

=

—sin(41

+

0~),

V12

=

cos(t~1— 0)

,

A12

=

—cos(Ø1 + O~) (5.6)

From (5.3), (5.5) and (5.6) one can see that the rate vanishes if the charged lepton mixing angle 01 For the rate of the radiative decay NR—*VLY (or vL—*NR-y) (~‘=e) we obtain [79, 80] ~i(N~)K(lOXbO16s)~1(mNrnv)(mE)f(r)2sin22O(l+cos22~)

=

0.

(5.7)

where 4r f(r)=

2 2+ 3r 2lnr 1—r (1—r )

Here the constant K = 1 for Dirac neutrinos and K this decay is presented in fig. 5.lb.

(5.8) =

2 for Majorana neutrinos. A typical diagram for

J. Maalampi and M. Roos, Physics of mirror fermions

81

Let us note that if N and v are Dirac particles there will be, at least in some simple models, a connection between the masses m~and m5 on the one hand, and the size of the mixing angle 4~on the other hand, such that a small mass difference m~— mj implies a small mixing angle [81]. In the Majorana case, where both N and v may get their masses separately (e.g. through the see-saw mechanism) one would not expect any such relation. There are two reasons why the radiative decay rate (apart from the coupling constants) is many orders of magnitude larger in the mirror model than in the standard model [79]. The first reason is that there is no need for “chirality matching” between the two lepton—W vertices in the diagrams in fig. 5.1, thus one picks up the mass part of the internal lepton propagator. The natural scale of the width is hence F—~~ In the standard model one has F— ~ The second reason is the further suppression of the standard model rate due to the GIM cancellation, as a result of which the rate 2. actually is proportional to F— G~m~(m1IM~) Let us remark that not only the radiative decay rate but also the magnetic moment of a neutrino could as a result of mirror mixing be orders of magnitude larger than in the standard model [80, 82]. A magnetic moment of the order of ~ 10b0 ~ possible in the presence of mirror leptons, would solve the solar neutrino problem [83]. Let us now go in closer detail through the experimental, astrophysical and cosmological constraints on the masses m~and mN , the mixing angles °e’ 0~of charged mirror leptons, and the mixing angles of mirror neutrinos. We will consider two ranges of mirror neutrino masses separately: a low mass range from 0.1 eV to 100 eV and a high mass range from 5 MeV upwards (in practice to about 20 GeV). Constraints for neutrinos in the low mass range come mainly from astrophysics and cosmology. Laboratory experiments in turn give important bounds when the neutrinos lie in the high mass range. The intermediate range which falls outside the figures is excluded mainly for cosmological reasons. Precise of beta decay spectra in the interval 100 eV ~ mN ~s4 MeV yield a 2 4~emeasurements ~ 10_2_103 [84]. constraint sin 5.1. Constraints at low mass In the low mass range most of the relevant processes do not differentiate between neutrino flavours. We will therefore assume e—p. universality. Consequently fig. 5.2, which presents boundaries for the neutrino mixing angle ~1 as a function of mN, refers to both electron and muon mirror neutrinos. Log sin2~ S\~4

~

-2

10

100

mN

levi

Fig. 5.2. Bounds on the neutrino mixing angle f as a function of the mirror neutrino mass mN. A: laboratory experiments measuring the V. A structure of weak couplings; B: the non-observation of radiative mirror neutrino decays in the cosmic photon flux [85];C: cosmological limit for the sum of the masses of stable neutrinos [86];D: upper limit for the oscillation of the supernova SN1987A neutrinos v, to mirror neutrinos [87];E: measurement of ultraviolet light of extragalactic origin 188].

82

J. Maalampi and M. Roos, Physics of mirror fermions

The boundary A is obtained from terrestrial experiments measuring the V, A structure of weak interactions, the processes already considered in section 4. These processes do differentiate between flavours, but in this plot we still assume e—p.. universality. The astrophysical boundary B is obtained from the nonobservation of radiative mirror neutrino decays (5.1) in the cosmic photon flux [85]. According to eq. (5.7) this information gives primarily a constraint on the charged lepton mixing angle 0~.However, a charged lepton mixing always induces radiatively (cf. fig. 5.3) a contribution to the neutrino mixing angle 4~, of the order of ç~~ (aU1)2. (Similarly one has O~~(a~ 2 when the data constrain ~ Therefore the data give an indirect 1) the neutrino mixing angle q boundary, curve B, also for 1. The curve C is the well known cosmological upper limit for the sum of the masses of stable neutrinos [86], ~m5~25eV,

(5.9)

which is obtained from the observed total energy of the Universe, and from the requirement that the observed large-scale structures of the Universe be permitted to develop within the lifetime of the Universe. Other estimates of this neutrino mass sum up to 100 eV exist. As a consequence of mirror mixing, ordinary neutrinos would oscillate into mirror neutrinos. In detectors where one expects to see the ordinary neutrinos convert into visible charged leptons by their charged current interactions, the mirror neutrinos would be inert and go undetected. This effect has been explored both for laboratory neutrinos and extraterrestrial neutrinos. Thus the curve D in fig. 5.2 is obtained [87]from the recent supernova SN1987A collapse by estimating the maximum amount of Ve neutrinos that could have oscillated into inert mirror neutrinos N5. The observed neutrino flux would have been smaller than expected on the grounds of the dynamics of a collapsing star. Curve E in fig. 5.2 follows from the measurement [88] of ultraviolet light of extragalactic origin aboard the Space Shuttle 1985. This result improves considerably the earlier constraint B, although in a quite narrow mass range 13.5eV~mN~17.5eV. Note that curves B, D, and E are strictly speaking not constraints on the mass2.m~of the decaying neutrino: B and Eofconstrain (m~ — m~)/mN, hereasmirror D constrains The oscillation laboratory neutrinos intowinert neutrinos(m~ gives m~)” bounds in the plot of Am2 versus sin2 24~shown in fig. 5.4. The disappearance of accelerator as measured in several experiments [89]is summarized by curve A. Curve B summarizes data on reactor i~ 5disappearance [90]. The information from fig. 5.4 has not been redrawn in fig. 5.2 because the v~oscillation bound is included in the low energy bound A of fig. 5.2, and the ~e oscillation bound is too weak to be of interest. 0e and 0~as a function of mirror neutrino mass are Constraints on 5.5, the where chargedwelepton angles c—p universality. The curves A to E refer to the presented in fig. againmixing have assumed —

v~ LLR l~ ~NR LR Fig. 5.3. Radiative corrections inducing mixing of an ordinary lepton and mirror lepton in the case when the corresponding SU(2)L doublet partners mix.

J. Maalampi and M. Roos, Physics of mirror fermions

83

10~

C:Cosmology

102

10

-

Log sin?B

-

0

5?~—.

-10

1~1 —

0.2

2 2ø~ 0.6

04 sin

-

-

-12-

—

0.8

1

1.0

Fig. 5.4. Limits for the mass difference ~m’ of oscillating laboratory and reactor neutrinos as a function of mixing angle 4,. A: the disappearance of accelerator v5 [89];B: the disappearance of reactor i~[80]; C: cosmological upper limit [86].

-11.

1

10 I

100 m .

-

5[eV] Fig. 5.5. Bounds on the charged lepton mixing angle 0 as a function of the mirror neutrino mass m,~.A to E: as in fig. 5.2; F: limit from superallowed 0 * —s 0 beta decays [91].

same processes as in fig. 5.2. The constraint F is a limit for 0e~based on a detailed study of superallowed —* 0~beta decays [91]. The constraint B from the cosmic photon flux [85]is orders of magnitude stronger than2 the corresponding the bound neutrino angle since thecollapse experimental 0e directly (see eq.limit 5.7).forThe D mixing from the supernova [87] is limit now constrains sin suppressed by a radiative factor, as compared to the corresponding limit on 4~e’and so is the constraint E. The constraints A—F in fig. 5.5 can equally well be plotted as mN versus radiative lifetime, T(N—* v-y). From eq. (5.7) it is obvious that a reduction in Oe at constant mN implies an increase in T. In fig. 5.6 we show a straight line corresponding to eq. (5.7) with maximal mirror mixing, sin2 20e = 1, as well as constraints B, C and E from fig. 5.2. Constraints A, D and F are weaker and are therefore left out of fig. 5.6. One conclusion from figs. 5.2, 5.4 and 5.6 is that there is room for a light mirror neutrino with radiative lifetimes of 1018 s or more. In the mass range m~~ 1 eV it could mix appreciably with the ordinary neutrinos, as is evident from table 3. Between 1 eV and 25 eV mixing angles in the range sin2 0 ~ i04_108 are allowed. This region is interesting for three reasons: the missing dark matter in

84

J. Maalampi and M. Roos, Physics of mirror fermions mN I

I

I

I

I

I

1 TeV

do,P

p

W

2

Y

1

LogB—P ~N”~ as in figs. 5.2 and 5.7, 5.9 (below); V: Lee—Weinberg limit for Dirac Fig. 5.6. Constraints for mirror neutrino mass versus radiative lifetime: neutrinos [110];W,: the limit of total energy density stored in mirror neutrinos at time later than 100 s [108];W 2: limit for the photofission of deuterium [109];X,: limit for ionizing extragalactic photons [85];X2: limit for ionizing photons from the direction of the Virgo cluster [85]; Y: the absence of distortions in the cosmic background radiation; Z: constraint from supernova energetics [107].

the Universe, the missing neutrino radiation from the sun, and the prospects of cosmic photon flux detection in the near ultraviolet, the visible and the infrared parts of the spectrum. To be WIMP candidates (weakly interacting missing particles) the mirror neutrinos must be in the 10—25eV mass range where cosmology sets the upper limit (5.9). This is clearly still possible. To explain the solar neutrino deficiency problem by the Wolfenstein—Mikheyev—Smirnov 2 = (mN m mechanism 2 must be [92], and yet not exceed the SN1987A oscillation limit, the mass difference Am 5) of the order of i0~eV2 [87, 93]. If m 5 mN, this implies lifetimes T> 1022 s and cosmic radiation in the infrared region far beyond detection. Such mirror neutrinos would clearly not be WIMP candidates. If one requires the mirror neutrinos to be responsible for both dark matter and the solar deficiency problem, mN2 and m,, are both in the 10—25 eV mass range, but they are accidentally degenerate = b0~eV2 or less. Noting that the SN1987A observations are already in some difficulty to Am to within accommodate the solar neutrinos, we tend to conclude that mirror neutrinos may be light WIMPs but —

‘~

probably not the solution to the solar problem. 5.2. Constraints at high mass Consider now upper bounds for the mirror mixing angles in the case of mirror neutrino masses in the range m~~ 5 MeV. Figure 5.7 presents upper limits on the neutrino mixing angle 4~ie.Practically the same limits are valid also for the charged lepton mixing angle 0e The reason is that mirror mixing enters in all the processes under study through an eNeW vertex, originating either via VeNe mixing from the original eveW vertex, or via an eE mixing from the original ENeW vertex. Consequently all rates are proportional, to a good approximation, to 0~+ 4~. The same is true for the rates in muon neutrino processes, to be considered later on; they are proportional to O~+

J. Maalampi and M. Roos, Physics of mirror fermions

tog2e,

sin

III,,,,

II~II’~I

-1

A

85

—

~

-2

__

~

10MeV

100MeV

1GeV

10GeV

mN

Fig. 5.7. Bounds on the mixing angle 0, of electron and mirror electron as a function of mirror neutrino mass m 5. A: laboratory experiments measuring the V A structure of weak couplings; G: leptonic (2-decays of pions at rest [94];H: leptonic (2-decays of kaons at rest [96];I: leptonic (2-decays of pions in flight [97, 98]; J: leptonic (2-decays of kaons in flight followed by leptonic decays of the final state mirror neutrino [97. 98]; K: leptonic (2-decays of kaons in flight followed by semileptonic decays of the final state mirror neutrino [97,98]; L: D-meson decays in a beam dump followed by leptonic decays of the final state mirror neutrino [99]; M: D-meson decays in a beam dump followed by semileptonic decays of the final state mirror neutrino [100];N: limit for mirror neutrino production in ee —~i,N, due to monojet data [102];0: search for secondary vertices due to a final state N, decay in e e - —s N,N, 1103]; P: limit for mirror neutrino production in ee —s i,N, and e *e - ~ N,N, in a searchfor leptonic and semileptonic decays of the final state mirror neutrino [1041.The region below the curves is allowed.

The boundary A in fig. 5.7 is the mass-independent constraint obtained from studies of the V, A structure of weak currents. The curve G is derived from an experimental study of leptonic 2-decays of the pion at rest [94].The method is to search for a secondary monochromatic peak in the final-state electron or positron momentum spectrum, which would indicate a decay to the mirror neutrino channel, i~—~ eNe. The rate of this decay is given by F(’rr—s’eN5)= ~

(5.10)

where Ve and A5 are defined in eq. (4.13), Pe = Pe(m~5,m,,) is a kinematical factor given in ref. [95] and F(ir—~eve) is a factor (1 — O(O~,~~)) times the rate in the standard model. The curve H is obtained by studying mirror lepton production in the decay K—~.eNe of kaons at rest [96]. One searches for a secondary peak in the momentum spectrum of the electron. Also the pion and kaon decays in flight have been studied. Intensive IT and K beams may, depending on the amount of mixing, produce a considerable amount of mirror neutrinos, whose subsequent decays could be detected downstream. The PS191 experiment [97, 98] at CERN had a setup for detecting such decays; no events observed translate into the boundaries I (IT-decay) and J, K (K-decay) in fig. 5.7. Curves L and M follow from two experiments searching for charmed D-meson decays to an electron and a heavy neutrino with subsequent decay of the neutrino. The proportion of heavy neutrinos in the

86

J. Maalampi and M. Roos, Physics of mirrorfermions

neutrino flux is enhanced in these experiments by using a beam dump to absorb longer-lived hadrons before they decay. The CHARM experiment [99] (curve L) considers the decay channels Ne~e~e~ve

or ep~v~,

(5.11)

whereas the BEBC experiment [100] (curve M) detects the same channels, and in addition the semileptonic final states Ne e ~‘ ~ Still higher mass regions are reached by considering heavy lepton production in electron—positron collisions. A mirror neutrino Ne can_be produced either together with a light ordinary antineutrino, e~e—+i~eNe, or pairwise in e~e—~N5N5. The first process proceeds via both Wand Z exchange (see —~

- ‘~

fig. 5.8). Note that in the sequential and singlet neutrino cases only the W-exchange graph is allowed. In pair production, the dominant contribution comes from Z-exchange just as in the sequential model (in the singlet model the pair production is strongly suppressed). The experimental signal of Ne production via e~e ~eNe would be a monojet event [101] with missing energy due to a i~on one side, and if the production does not occur too close to threshold, a jet due to Ne decay on the other side. The curve N is based on a search of such events at PEP [102]. The pair production is not suppressed by mirror mixing. The decay rate of Ne, however, is suppressed and therefore one can get information on the mixing angles also in this process. The method is to set limits for the Ne lifetime by measuring the distance between the e~e collision point and a possible secondary vertex where Ne decays. Such secondary vertices have been searched for in the MARK II detector at PEP [103], but none has been found. Thus if mirror neutrinos (or other heavy neutrinos) are produced, they decay either too close to the primary vertex or outside the fiducial volume of the vertex detector. The result rules out a region in the mixing angle versus mass plane, shown as curve 0 in fig. 5.7. The curve P is obtained by the CELLO experiment [104] from search of leptonic and semileptonic decays of heavy neutrinos produced in e~e collisions through e~e —~ N1N1 or VeNe. Let us now consider constraints on the mixing angles in the muonic sector, when the mirror neutrino is heavy. As in the electron case, both of the mixing angles, çfr~and O~,are restricted by practically the same bounds, presented in fig. 5.9. The curves A to P refer to the same experiments as before. The following remarks are in order. The single N~production in electron—positron collision (curve M), e~e ~ is now possible only via Z-exchange. (We note that in the sequential neutrino model the corresponding process is strictly forbidden due to the GIM cancellation.) Similarly the Z-exchange is the only contribution to the pair production, e~e—~N~N~ (curve 0). Constraint 0 is a new result from the PS191 experiment [105] for K decay in flight, K~—÷ based on search for semileptonic decays of a mirror neutrino N~.The previous less stringent limits (curves J and R) were obtained by looking for leptonic decay channels [97, 98]. —~

WI:: Fig. 5.8. Diagrams for the reaction ee

J. Maalampi and M. Roos, Physics of mirror fermions 2$~

Logsin

I

‘‘I’’’’I

I

‘‘‘III’

87

‘‘‘I’’’~

I

A P

U

-2 I

10MeV

100MeV

1GeV

10GeV

mN

Fig. 5.9. Bounds on the mixing angle 0, of muon and mirror muon as a function of mirror neutrino mass m 5. A to P: as in fig. 5.7; Q: limit for mirror neutrino production in K~—s ~‘N,, followed by semileptonic decays of the final state mirror neutrino [105]:R: limit for mirror neutrino production in K* —~ ~i. N, in a search for decays N,—’ ~ e * v of the final state mirror neutrino [981;S: limit for mirror neutrino production in v,X—’N,,X in a search for secondary decays N, —~ ~X’[99];T: limit for mirror neutrino production in ~ N,X in a search for secondary decays N,~ti*l~_v,[106]; U: limit for mirror neutrino production in v,X—~N,Xin a search for a secondary vertex [106].

Since the nondiagonal Z-couplings are possible in the mirror model, one could produce mirror neutrinos also in deep inelastic neutrino—nucleon neutral current scattering (5.12)

v~X—~N~X,

where X stands for a nucleon and X for undetected particles. One signature of this process would be two showers, one for the debris of X and one for the debris of X’ in the decay N~ pX’, and a muon pointing to the vertex of the second shower. The CHARM experiment [99] has searched for heavy neutrinos using this signature. Their result is given by the curve S in fig. 5.9. The CCFRR experiment [106]at Fermilab has used two different signatures: the curve T is obtained by searching for dimuon decays ~ p..~p..v1,.and the curve U by searching for a secondary vertex downstream of the primary production vertex. All the constraints referred to in this subsection of high mass neutrinos can be translated into constraints in the plot of m~versus radiative lifetime, fig. 5.6. To simplify, in fig. 5.6 we only draw two summary curves: one for curves 0 and P and one for curves G—M. Figure 5.6 also contains some further constraints of astrophysical or cosmological origin. Astrophysics provides the constraint Z imposed by supernova energetics [77],the constraint Y imposed by the absence of distortions in the cosmic background radiation [107], and two constraints imposed by nucleosynthesis: the limit W1 that the total energy density stored in mirror neutrinos should not exceed the energy density stored in radiation at time later than lOOs [108], and the limit W2 on the photofission of deuterium [109]. The limit X1 refers to ionizing extragalactic photons [85] and the limit X, refers to ionizing photons from the direction of the Virgo cluster [85]. Finally the curve V corresponds to the Lee—Weinberg limit for massive Dirac neutrinos [110]. —~

88

J. Maalampi and M. Roos, Physics of mirror fermions

Above the Lee—Weinberg limit, mirror neutrinos with lifetimes exceeding the age of the Universe (denoted t0 in fig. 5.6) could still be WIMP candidates. We note that the mirror mixing angle would 2 20e ~ i0~°. then be exceedingly sin constraints [111], but they are all weaker in the sense that they would Actually there aresmall, still more fall inside the region already excluded in fig. 5.6. We therefore do not draw them here.

6. Production and decay signatures of mirror leptons If mirror fermions exist and are not too heavy they would be copiously produced in eke, ep and p~ collisions in the next generation of accelerators. In principle the most straightforward method to distinguish them from ordinary fermions would be to search for a right-handed interaction in the angular distributions of the decay products. Checking the validity of the V— A rule in this manner belongs, in fact, to the normal procedure of experimental analysis when a new fermion candidate is discovered. The angular distributions in heavy lepton decays with V— A and V + A interactions have been compared in ref. [112]. One clear signal distinguishing mirror and ordinary fermions would be visible in the forward— backward asymmetry AFB in e~e collisions. In mirror fermion production e*e FF, AFB has the same absolute value as the asymmetry in the production e + e ff of ordinary fermions of the same mass, however with the opposite sign [113] —*

-

AFB(e~e—~FF) = —AFB(e~e—-~ff).

—~

(6.1)

As pointed out in ref. [113] this effect would be enhanced if the electron beam were polarized, because then the asymmetry would be substantial also on the Z-peak where the cross section is high. The signal would be cleanest for the fermions which are effectively stable, that is, for the lightest mirror lepton and quark when the mirror mixing is very small or vanishes. Neutrino oscillations do not distinguish mirror neutrinos from neutrinos very well [72]. In all classes of oscillations the parent neutrinos disappear. The signature distinguishing first class oscillations is the appearance of wrong-family charged leptons. The appearance of wrong-sign charged leptons of the parent family may be a second-class signal of mirror neutrinos; however, the left—right symmetric model has the same second-class signature. Third-class oscillations involving mirror neutrinos do not distinguish themselves by any charged lepton appearance [72]. In the following we shall describe the main signatures of mirror leptons based on their characteristic decay patterns. Mirror leptons exhibit a pattern of allowed decays and suppressed or forbidden decays which is different from that of sequential V— A leptons. Let us first consider the case where the mirror leptons L and N are sufficiently light so that they could appear in the final states of weak gauge boson decays. The dominant decay channels for a charged mirror lepton (assuming mL> mN) are L—*N~ 1 (6.2)

J. Maalampi and M. Roos, Physics of mirror fermions

and for a mirror neutrino (assuming mN>

89

mL)

(6.3) where q stands for ordinary quarks. In case these processes are kinematically suppressed, the dominant channels would be Ee~

N—~e~v1

(6.4,5)

~~4velve

~~+VeVlVl

—+eqq

—~v~qq.

These channels have larger phase space than the processes (6.2) and (6.3), but the decays are slowed down by a mixing angle. The decay signature of a charged mirror lepton would thus be an isolated light lepton plus missing energy or two quark jets plus an isolated lepton or missing energy. The signature of a mirror neutrino would be a lepton pair plus missing energy. In a real experimental situation the hadronic decays of L and N may often appear as containing just one jet, since the jet separation efficiency is limited. The main reactions for mirror lepton production in high energy accelerators would be the weak boson decays

W—~LN,

Z—~LL~—~NN,

(6.6)

which proceed with full weak interaction strength. The decay rates as functions of mirror lepton masses m~and mN are given by -

GFMW1

--

F(W —s~L N)=

12\/~

(,,1—

m~+m~ 2

2 A12) 3

-

~

+

A~2)(1 mi~ -

-

(m~~)

mLmN]

(6.7)

+ (~V,~

F(Z—+LL~)= (GFM~/24V2ir)(1—4m~/M~,j”2 x [(Iv,..,V+ ae~2)(1 m~/M~) + (~vel2 —

where OF

—

a~2)

3m2jM~],

(6.8)

g2I4\/~M~,,,,, V 1 and A1 are the charged current couplings given in eq. (4.13) and v1 and a1

are neutral current couplings given by u1=4sin Ow—i,

a,=cos2O1.

(6.9)

90

J. Maalampi and M. Roos, Physics of mirror fermions

If the mirror neutrino N is a Dirac particle the decay rate for Z—~NN is obtained from (6.8) by replacing m~by m~and v1 by unity. If N instead is a Majorana particle, the rate is given by 312~a 2. (6.10) F(Z~4NN)Maj = (GFM~/6V~1r)(i 4m~/M~) 1~ If the decays (6.6) are kinematically forbidden or highly suppressed, the mixed channels —

W—~N

Z—*v 1N+i1N

—~L~ + L~

—*Li~1,

(6.11)

would dominate. These channels are slowed down by a mixing angle. For example, F(W-~rI~) =

(1-

~

+

~)[~!2

A~2)(1

-

-

2M~~)]’ (6.12)

-

F(Z —IL

+

-

GFM~ ( )=24~

m~\[

[k11~ —

2/

m~

m~\1

2 201. A striking signature of mrror neutrino production in Z-decay would be a monojet where a1 = sin plus missing energy due to the process Z—s. vN, N—s. where the two quark jets are not separated. This process is not possible for a sequential heavy neutrino since it requires nondiagonal Z-couplings. Haber and Reno [114] have estimated the monojet cross section due to the presence of a mirror neutrino to be iq~j

o(Z—s. monojet + missing energy) ~ 0.220.12, u(Z—s. veve)

(6.13)

for mN = 20—40 GeV, and with çb~,0,,, ~ i0_2 for the mirror mixing angles. Another signal relatively free from background is the purely leptonic Z decay, for which the cross section is estimated to be [114] r(Z—s. NN—s. ~~v o(Z—s. 1)~1)~) -

il) ~ (5.5—2.8) x i0~3,

(6.14)

under the same assumptions as above. The same process is possible also for a sequential neutrino, the upper limit for the cross section (for a v 1N mixing angle ~i0_2_10_1) being [114] one order of magnitude larger than in (6.14). Another purely leptonic channel is Z—*NN—s.

~

+

neutrinos,

(6.15)

with a signature of two isolated charged leptons plus missing transverse energy. The same final state would arise also from a pair production of charged mirror (or sequential) leptons, but the angular distributions of the final-state charged leptons would differ in these two cases. Finally, if the mirror leptons were heavier than the weak bosons, decays to a light fermion plus a

J. Maalampi and M. Roos, Physics of mirror fermions

91

weak boson, L—s.v1W —s.Z,

N-s.W~ —~‘v1Z,

(6.16)

are important. Note that the modes with a Z-boson in the final state are only possible for mirror leptons. The mirror lepton signals are still below the sensitivity of the present experiments, such as the SPS collider experiments at CERN. They should, however, be easily seen in the new accelerators where weak gauge bosons would be copiously produced and where the collision energies are much higher than those achieved today.

7. Summary In this work we have considered the hypothesis of mirror fermions, a class of particles having V + A weak interactions. All leptons and quarks known today have V— A couplings, but the existence of mirror fermions at a higher mass scale is not ruled out. The mirror fermions which were first suggested merely for aesthetical reasons to balance the asymmetric nature of weak interactions, now enter many recent theories which go beyond the standard model. The family unified models which are constructed to explain the repetitive appearance of fermion generations, necessarily contain mirror families. Most suggestive is the situation in models based on an orthogonal gauge symmetry, such as SO(18), where the spinor representations always incorporate an equal number of V— A and V + A families (but no other fermions). Also Kaluza—Klein theories and the theories with an extended N 2 supersymmetry, as well as some composite models involve mirror fermions. It may be possible that in the framework of the E8 X E8 superstring theory one can construct models without mirror fermions, though realistic candidates for such models are still lacking. A nontrivial problem in models with mirror fermions is how to circumvent the survival hypothesis, that is, how to prohibit a V + A particle from pairing up with a V— A particle and forming a vector-like extra heavy state. We showed that for this the theory should have an additional symmetry unbroken down to the electroweak scale. Just like the ordinary V—A fermions, mirror leptons and quarks obtain their masses as a result of the electroweak symmetry breaking. Consequently, mirror fermions are light, m~~ 300 GeV. It is therefore a task for the new generation of accelerators, TRISTAN, LEP, HERA, SLC, Tevatron and SSC, to prove or disprove the existence of mirror fermions. If mirror fermions exist, they must mix with the ordinary fermions. As a result of such “mirror mixing” the Lorentz structure of the weak interactions of the ordinary particles would deviate from the conventional V— A form. Depending on the theory, mirror mixing may follow from Yukawa couplings directly, or it may be induced by loop corrections or through nonrenormalizable terms. The size of the mixing is thus very model dependent. Another consequence of the mixing is that neutrinos in neutrino beams may disappear by oscillation into inert mirror-neutrino states. We have analyzed charged and neutral current data and found that the possible V + A impurity in the weak amplitudes is typically less than 10—15%. The result of our overall fit in terms of mirror mixing angles is presented in table 3. However, these results are only interesting in the mass ranges ~ 1 eV or mN ~ 18 GeV where no stronger constraints exist.

92

J. Maalampi and M. Roos, Physics of mirror fermions

Since no charged mirror fermions have so far been observed in e~e collisions one knows that their mass must exceed 26 GeV. This limit does not apply, however, to mirror neutrinos which could be much lighter and which could be observable already now in various processes. The limits one can derive for mirror neutrino masses mN depend on the size of mirror mixing. In the low mass range m~~ 100 eV the boundaries in the m~versus mixing angle plane (see figs. 5.2, 5.5, 5.6) follow mainly from astrophysics and cosmology: from the nonobservation of radiative neutrino decays in the cosmic photon flux, from the cosmological upper limit for the sum of the masses of stable neutrinos, from the maximum amount of SN1987A supernova Ve neutrinos that could have oscillated into mirror neutrinos N5, and from the measured ultraviolet light of extragalactic origin. In the high mass range m~ 5 MeV—18 GeV the mirror mixing angles are bounded within ~10_4_b0_8 (see figs. 5.7, 5.9) by laboratory experiments: searches for mirror neutrinos in the leptonic decays of pseudoscalar mesons and searches for mirror neutrino production in e~e collisions. Mirror neutrinos in the range m~~ 5 GeV are cold WIMP candidates, in the range 10 eV ~ mN ~ 25 eV hot WIMP candidates. Measurements of the extragalactic uv photon flux on board the last space shuttle in 1985 [88] now excludes the range 13.5 eV ~ m~~ 17.5 eV and radiative lifetimes (4.5—5.8) X 23 10 5. To explain the solar neutrino deficiency problem by mirror neutrinos and2 the Wolfenstein— ~ i0~eV2, of no Mikheyev—Smirnov mechanism [92] one is forced either to very small masses, Am cosmological interest, or neutrinos and mirror neutrinos of mass 10—25 eV must be accidentally mass-degenerate to Am2. Since also the neutrino oscillation limit from supernova SN1987A has to be pushed to its extreme, we think that mirror neutrinos are unlikely candidates for the solar problem and the missing dark matter problem simultaneously. We note that the current limit on the number of neutrino species can tolerate at least one mirror neutrino in addition to the known three ordinary neutrinos. Production and decay of mirror fermions would exhibit a few clear signatures by which they could be distinguished from ordinary fermions. The signatures are consequences of two facts of fundamental importance. Firstly, the weak interactions of ordinary fermions and mirror fermions have opposite chiral structure yielding distinct angular distributions. Secondly, the mirror mixing frustrates the GIM mechanism, that is, the nondiagonal neutral couplings like vNZ and LZ are allowed. A good experimental signature based on the first property would be the forward—backward asymmetry in e~e collisions, which would have opposite sign for production of ordinary fermions and mirror fermions. A striking signature based on the second property would be Z-boson decay into a final state consisting of a monojet plus missing energy. This process requires a nondiagonal Z-coupling of neutrinos and would therefore be a unique signal of mirror neutrino production. In conclusion we stress that the possible existence of mirror particles deserves serious consideration in future high energy experiments. If none of the new generation accelerators produces mirror fermions, it is very probable that they are ruled out for good. Such a negative result would be valuable for the further development of particle theory.

Acknowledgements It is a pleasure to express our gratitude to Kari Enqvist and Kalevi Mursula who have participated in most of our previous work on mirror fermions.

J. Maalampi and M. Roos, Physics of mirror fermions

93

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