Nuclear Physics B209 (1982) 427-432 © North-Holland Publishing Company
G R A N D UNIFICATION A N D MIRROR PARTICLES A.S. SCHWARZ and Yu.S. TYUPKIN Worm Data Center, Molodezhnaya, 3, Moscow, 117296, USeR
Received 23 July 1982 The grand unificationmodel containingstandard and mirror particlesis constructed. In models of this kind there exist such strings that a standard particle transforms into a mirror particle in going around the string.
Lee and Yang [1] have conjectured that the spatial reflection can transform standard particles into so-called mirror particles. Of course, the coupling constant of the interaction between mirror and standard particles must be very small. Then the existence of mirror particles does not contradict experimental data [2, 3]. It will be shown in this paper that one can construct a grand unification theory containing both standard and mirror particles. There exist topologically non-trivial strings in such theories. Some of these strings admit the following remarkable property. The standard particle going around this string transforms into the mirror particle. It is natural to call such strings Alice strings [4] because Alice might go through the looking-glass by means of this string. As example of grand unification theory including standard and mirror particles we consider a theory based on the gauge group SO(20). Therefore, we begin with some remarks on strings in theories based on the group SO(N). As mentioned in [5], topologically non-trivial strings in theories based on simple connected gauge groups exist in the case when the group of unbroken symmetries is disconnected (i.e. there exist unbroken discrete symmetries). The group SO(N) is not simple connected; however, it can be replaced by locally isomorphic simple connected group Spin(N). The subgroup of SO(N) consisting of unbroken symmetries will be denoted by H, the corresponding subgroup of Spin(N) will be denoted by I~I. Let us suppose that every closed path in H is contractible in SO(N) [for example H is contained in a simple connected subgroup of SO(N)]. Then one can assert that the connected subgroup I7/con of 1-71is isomorphic to the connected subgroup of H. However, 17t contains an additional discrete symmetry tr because the non-trivial element of Spin(N) corresponding to unity in SO(N) does not lie in I-'/.... We see that in the case under consideration there exist topologically non-trivial strings corresponding to the discrete symmetry o- (in the case of the SO(10) model such strings were considered in [6]). If the group H is connected there exist only 427
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topologically non-trivial strings of this kind. However, we are interested in strings of another kind arising in the case when the group H is disconnected. The gauge group SO(20) contains two-independent subgroups SO(10) acting on the first ten and last ten coordinates in 20-dimensional space, respectively. Let us denote by a the element of SO(20) interchanging the first and last ten coordinates. For every subgroup G c SO(10) we consider the subgroup L(G) c SO(20) generated by the subgroup G x G c SO(10) x SO(10) c SO(20) and the discrete symmetry a. We will break the group SO(20) step by step to the groups H0 = L(G0), H~ = L(G~), H2 L(Gz), where Go = SU(5), G1 = SU(3) × SU(2) × U(1) and G2 SU(3) × U(1). The generators of two subgroups U(1) lying in HE can be identified with standard electric charge and mirror electric charge, respectively. The discrete symmetry a remains unbroken. The strings corresponding to this discrete symmetry are Alice strings. In the grand unification of standard and mirror particles we will assume that fermions are transformed as chiral spinors by the transformations of SO(20) and that scalar fields are transformed as antisymmetric tensors. It is convenient to realize the spinors as elements of the space E of the Fock representation of canonical anticommutation rela~lc, ls [7, 8]: =
=
+
[a+,,a, ]+=[a,,a,]+=O,
[a+,,a,]+=6,,,
1~
(The group SO(20) can be considered as the group of linear canonical transformations. Every canonical transformation generates a unitary transformation of Fock space which is defined up to a sign.) We have assumed that fermions are chiral spinors of SO(20); therefore, we can identify the fermion fields with fields taking values in the space E . . . . c E spanned by the vectors 1 10), ~-~. Ta,+~a,~[0)= Ill,
1
i2) . . . . . 10! a'+~"" a'+l°lO)= li~. . . . . i~o),
where the Fock vacuum 10) satisfies a,[0)= 0 and l<~ik <<-10. In such a way the fermion field ~ can be represented in the form ItS) = ~o[0)+ t~,1,21il, i 2 ) + " ' " +tp,
,o[il . . . . . ilo) = ~A[A),
where A = (il . . . . . i~) is multi-index, tPA = t0, ,k and IA) = lil . . . . . ik). Of course, the fermion fields have both isotopic and Lorentz indices. We assume that the fermion fields are two-component (Weyl) spinors with respect to the Lorentz group. The spinor representation of SO(20) in Fock space E generates the representation T of SO(20) in the space of operators acting in E. This representation T is reducible, it splits into a direct sum of antisymmetric tensor representations. (Every operator can be represented as a sum of operators having the form
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where s, = 1 . . . . . 2 0 ,
F2, = i(a + - a , ) .)
F:,-1 = a + + a , ,
We will assume that the scalar field in our model transforms with respect to the representation T. The interaction between fermion and scalar fields will be taken in the form 1 ,1 ,k11 12o_k+ h.c. Y"k t ( 2 0 - k ) ! 0,1 ,~(&0)~l ,~o-~e,~oe Here a, /3 = 1, 2 are Lorentz indices. This interaction can also be represented in the following form:
(O *lBOlO ) + h.c. , where ( I ) denotes the scalar product in Fock space,
B=I-IF~. s=odd
One can consider a more general interaction by assuming that irreducible components of the field ~b interact with fermion fields with different coupling constants. In other words the interaction can be taken in the form E X,(0*IB4,,I0) + h . c . , n
where &, are irreducible components of ~b. It is evident that the field & containing an odd number of Fs do not interact with fermion fields. The covariant derivative of the scalar field can be written as
D~4~=O~O+ig[A~,c~],
~ =1 ..... 4,
where A~ = (A~,)sI~2F~lF~2 and (A~,)~ls2 is the gauge field taking values in the Lie algebra of SO(20). The self-interaction of the scalar field must satisfy the following conditions. For the first step the group SO(20) must be broken to the group H0 = L(SU(5)) by means of a scalar field with vacuum expectation value (v.e.v) having the form
O .(1) . . . = M l ( a ~ " " a 5 + + a 6+ " " a l o+ + h.c.)
(1)
In the following step we break the symmetry by means of a field with v.e.v. ~(2)
+
+
+
. . . . = M 2 { ( a l a 2 a 3 + h . c . ) ( n 4 + n 5 ) + (a6aTas+ + + +h.c.)(n9+nl0)} .
(2)
The group H1 = L(SU(3) x SU(2) x U(1)) is unbroken at this step. It is obvious that (1) and (2) do not contribute to the masses of fermions. To generate fermion masses we must use the components of the scalar field containing even number of Fs. Let
43D
A.S. Schwarz, Yu.S. Tyupkin / Grand umficatton and rmrror particles
ds define E1 as the subspace of E . . . . spanned by vectors [kl, k2), [kl . . . . . k4),
k, = 1 . . . . . 5.
The space E2 will be defined as the space obtained from E1 by means of the transformation a. The charged fermions corresponding to the elements of E~ will be identified with standard particles. The charged fermions corresponding to the elements of E2 will be identified with mirror particles. The interaction between standard and mirror particles is small because it is transferred by very heavy gauge bosons. [The masses of these bosons are generated by (1) where M1 must be chosen very large.] For the last step we b r e a k the gauge group to H2 = L ( S U ( 3 ) x U(1)). The masses of fermions are generated simultaneously at this step. It is important to note that we can obtain arbitrary mass terms by taking appropriate v.e.v, because all possible interactions of scalar fields with fermions are included in our lagrangian. This follows from the assertion that irreducible components of the tensor square of the chiral spinor representation are antisymmetric tensors. Therefore we can describe the mass term instead of the v.e.v. We will take the mass term in the form ~. MAt~.4~bA -t- h . c . ,
(3)
A
where A = (il, • • •, ik) is an arbitrary multi-index and A is a multi-index corresponding to A. (The construction of A will be described below.) We assume for definiteness that the indices in the multi-index are ordered. Then the multi-index can be identified with the subset of the set {1, 2 , . . . , 10}. Expression (3) can be invariant with respect to the group SU(3) x U(1) acting on the first four indices in the case when v,(A)+ v,(fi,)= 1 and also in the cases when v,(A) = u,(.4) = 0 or v,(A)= u,(A) = 1. (Here i = 1 . . . . . 4 and u1(A) is equal to 1 if A contains index ] and u~(A)= 0 if A does not contain ] for/' = 1 . . . . . 10.) If one of these conditions is fulfilled one can assert that (3) is SU(3)× U(1) invariant if MA has an appropriate symmetry with respect to permutation of the indices 1, 2, 3. The description of expressions (3) which are invariant with respect to the group SU(3) x U(1) acting on the indices 6 , . . . , 9 is similar. The mass term (3) must be invariant with respect to two groups SU(3) × U(1) and with respect to the transformation a. Let us consider first the case when the multi-index A is a-invariant. To ensure the a-invariance of ~A~t~Awe must require that P~ is a-invariant too and that the integers ½[AIand ½1AI have the same parity. (We denote by Ial the number of indices in A.) To construct A~ in such a way in this case we can take u~(/~) = 1 - u~(A) for / ' = 1 . . . . . 4 , 6 . . . . . 9 and us(A)=v5(A)=vio(A)=ulo(A). If A is not ainvariant there exists a freedom in a choice of /k. One must impose only the condition a(A) = a (A) and MA = :t:M,~(A). [The choice of the sign in the last equation must guarantee the a-invariance of (3).] To ensure that the mass term on the subspace E , coincides with the mass term in the standard SU(5) model we must require that v, (A) = 1 - u , (A) for i = 1 . . . . . 4, us(Ak) = us(A) and v/ (A) = 0 for j >t 6
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431
in the case when vl(A)= 0 for ]/>6. In the case when the multi-index A contains both indices ~<5 and indices I>6 we assume that MA is large enough to explain that particles corresponding to the field 0A are not observed. Let us note that there exist two massless two-component neutrinos for our choice of the mass term; they correspond to the fields 01234 (standard neutrino) and I//6789 (mirror neutrino). One can replace these two neutrinos by one massive neutrino by including in (3) the expression ml//1234~6789+h.c. This massive neutrino interacts with both standard and mirror particles. To give appropriate masses to W- and Z-bosons we must use the freedom in the choice of Yukawa interaction coupling constants. T o study the strings in the model under consideration we can apply the general results of [5, 9]. Using these results we find that there exist strings corresponding to the discrete symmetry a. These strings can be written in the form
ao(r, O)= U(O),X,o(r)U-I(o) , A, =az =0,
(4)
O(r, O)= T(U(O))O(r) . Here U(O) = e °M c SO(20) and M can be chosen as
o'/ where
I =
-E5 E5
E5 ) -E5 '
the matrix E5 being the unit matrix of rank five, I the matrix of rank ten and M the matrix of rank 20. In other words U(O) can be considered as linear canonical transformation
a,(O) = K,l(O)a,, where ((cos 210 + i sin ¼0 cos ¼O)E5 (sin 2 ¼O- i sin ¼0 cos ¼0)E5'~ g (0) = \ (sin 2 ¼0 - i sin 10 cos ¼0)E5 (cos 210 + i sin 41-0cos ¼0)E5]" By an appropriate choice of the functions .4o(r), ,~(r), fields (4) satisfy the equations of motion. It is shown in [9] that the particle going around the string corresponding to the discrete symmetry a transforms into the particle obtained from the initial particle by means of the transformation a. Therefore the standard particle going around the string (4) transforms into the mirror particle. We see that (4) can be considered as an Alice string. The possibility of observation of mirror particles and Alice strings is discussed in [3, 9].
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We would like to express our gratitude to M.Yu. Khlopov and L.B. Okun' for helpful discussions. References [1] T.D. Lee and C N. Yang, Phys. Rev. 104 (1956) 254 [2] Yu. Kobzarev, L. Okun' and I. Pomeranchuk, Yad. Ftz. 3 (1966) 1154 [3] S.I. Bhnnlkov and M.Yu. Khlopov, Preprint ITEP-11 (1982), Yad. Fiz., to be pubhshed [4] L. Carrol, Through the looking-glass and what Alice found there (Macmillan, December, 1871) [5] A.S. Schwarz and Yu.S. Tyupkin, Phys. Lett. 90B (1980) 135 [6] T.W.B. Kibble, G. Lazarides and Q. Shaft, Imperial College preprint ICTP/81/82-18 (1981) [7] F.A. Berezin, Method of second quantization (Moscow, 1965) [8]_R.N. Mohapatra and B. Sakita, Phys. Rev. D21 (1980) 1062 [9] A S. Schwarz, Preprint ITEP-19 (1982), Nucl Phys. B208 (1982) 141