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Neutral currents and grand unification
Volume 92B, number 3,4
PHYSICS LETTERS
19 May 1980
NEUTRAL CURRENTS AND GRAND UNIFICATION S.M. BARR and A. ZEE Department o f Physics, University o f Pennsylvania, Philadelphia, PA 191 74, USA Received 24 December 1979
We show that in certain interesting classes of gauge models useful information about grand unification may be obtained from careful neutral current measurements at low energies.
1. We are now in the unprecedented situation of having a physical theory, the SU(3) X SU(2) X U(1) gauge theory, that is consistent with all non-gravitational physical phenomena. Recent neutral current experiments have dramatically corroborated the validity of this theory at low energy. Moreover, there are cogent theoretical grounds for believing that these gauge interactions may be unified within an SU(5) gauge theory [1 ] at superheavy energy scales [2] (near 1015 GeV). The (perhaps disheartening) suggestion has even been made that nothing interesting may remain undiscovered below .1 this superheavy unification scale of energies. This might well be true if the SU(5) theory really incorporated all (non-gravitational) gauge interactions. At the same time, on the experimental front, neutrino scattering experiments soon to be performed should substantially improve the precision to which neutral current couplings are known (for a review, see ref. [4] ). The question presses itself upon us whether such measurements are likely to afford us decipherable clues to what lies beyond or whether we must be resigned to the difficult task of probing the structure of physics at 1015 GeV without the aid of accelerators. In this note we wish to make two simple but encouraging remarks. First, there exist plausible reasons for anticipating that significant deviations from the simplest Weinberg-Salam neutral current predictions may be seen in more precise experiments. And second, such deviations, if of certain simple forms, may provide information about physics at superheavy energies. Z In the SU(5) model the superheavy breaking down to SU(3) X SU(2) X U(1) is most economically achieved through a higgs field in the adjoint (24) representation acquiring a vacuum expectation value of the form diag (1, 1, - 2 / 3 , - 2 / 3 , - 2 / 3 ) ~ . Notice that SU(5) is not broken into SU(3) X SU(2) but into SU(3) × SU(2) X U(1). In this note we wish to argue that additional U(1) factors may be present and may provide a clue to physics beyond SU(5). Let us suppose that the grand unified group is something larger than SU(5), say SU(N), N > 5. Then it seems not improbable that the superheavy breaking of this larger group is also accomplished by a higgs in the adjoint representation of which the (24) higgs in the SU(5) model is merely a part. In such a case what would remain of the SU(N) group after superheavy breaking would be of the general form .2
*1 For a contrary view see ref. [3]. ,2 The breaking of SU(N) by adjoint higgs has been analyzed by Li and Kim [5]. If one restricts oneself to quartic polynomial higgs potentials, then in the notation of eq. (1), q = 0 and p = 3, as was pointed out by Li. If however we adopt the view that the adjoint higgs (which may not be an elementary field in any case) can have more general couplings then more complicated breakings may occur. 297
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p
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p-1
-~[-1
×
where ]~Pl=Ikl + q = N . Notice that the number of unbroken diagonal generators is ]~Pl(kl_ - 1) + q + (p - 1) = N - 1. In other words all N - 1 diagonal generators of SU(N) commute with the diagonal vacuum expectation value of the adjoint higgs field. It is easy to understand eq. (1). The VEV of the adjoint higgs field (call it gZ~t~) has N diagonal entries. Some of these will be degenerate. If some set of k l > 1 of these entries are degenerate then an SU(kl) subgroup is left unbroken. In eq. (1) we suppose there are p such non-abelian groups. If an entry, say (~2m), is unique (not degenerate with another entry) on the other hand, then a U(1) subgroup is left unbroken corresponding to the generator Xa~ = 6aO - N6C~m3m ~. These unbroken subgroups are denoted by Ilqm=lUm(1 ) in eq. (1). Since the total number of diagonal entries of ~2~t~ is N we have 2;/=1 k I + q = N. However, as already remarked, all N - 1 diagonal generators of SU(N) are.unbroken. This accounts for the last factor in eq. (I) of (p - 1) unbroken U(1) subgroups. For example the SU(5) model breaks down to SU(3) X SU(2) X U(1). This is clearly seen to be a special case of eq. (1) with q = 0 , p = 2 and k l = {2,3}. As another example suppose an SU(5) is broken by (f2a 3) -- diag ( - 4 , 1, 1, 1,1), t h e n q = 1,p = 1 and k 1 = 4. So that SU(5) --* SU(4) X U(I)'. Now just as in the SU(5) model the further breaking of SU(3) X SU(2) X U(1) happens at low (that is, not superheavy) energies, so we might expect that the residual group of SU(N) is also broken at low energies (perhaps in the TeV range). The unbroken non-abelian factors, one assumes, confine so that the presently known quarks and leptons are singlets under these groups, except for color SU(3). This is in accordance with the technocolor philosophy [6] : all fermions which are nonsinglet under one of these new (non-color) unbroken non-abelian groups will be assumed to be confined in high-mass bound states. Thus one expects in this scheme that the lowenergy phenomenology of quarks and leptons would be describable by the usual SU(3)c X [SU(2) X U( 1)] ws model with some number of additional broken U(1) factor groups. These additional U(1) gauge interactions would modify, at some level, the SU(2) X U(1) neutral current predictions. In this way we might hope to learn something about grand unification from neutrino scattering experiments. The above is meant to be a motivation for contemplating deviations from the standard SU(2) X U(1) neutral current predictions. In fact, much of what follows will be independent of the specific picture described above. We will show that rather non-trivial constraints on experiments can follow from some simple theoretical remarks. The basic idea is extremely simple. According to a general analysis [7], the neutral current interaction hamiltonian at zero momentum transfer is given by 1
HNC = 2 ~
(~Tuni ~ ) ( U - 2 ) i / ( ~ ' U n / i f ) "
(2)
The notation is as follows. Let the electricially neutral gauge bosons A '~ of the theory couple to g~ T~. T~ are electrically neutral generators and the electric charge operator is Q = L'uaCa Tc~. Pick out one generator contributing to the charge operator Q out of the set Ta ; call it TO, and refer to the rest as Ti's. (/~-2)i / is some symmetric inverse mass matrix. Then the operators n i are given by (3)
ni = gi [T. - (eZ/g2)C i Q] .
This structure is in fact already quite restrictive. For instance, assuming the existence of one extra neutral gauge boson beyond the photon and the Z boson and that Q = T3 + Y/2, we have HNC up to a normalization, given schematically by (1 + e l ) [ Q z ] [Qz] + e2([Qz] [X] + IX] [ Q z l ) + e 3 [ X ]
[x].
(4)
Here Qz denotes T 3 - (e2/g2)Q and X the generator corresponding to the extra gauge boson. Thus, the accessible 298
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neutral current couplings may be parametrized in terms of a handful of unknowns. We now analyze the situation in more detail.
3. Case (i). First let us see what follows from assuming that the SU(5) model is extended to SU(5) × IIiUi(I) [which breaks down to SU(3)c X U(1)e m ]. We only assume that these interactions are family independent*3,4. We do not assume for now that they are unified in any larger group. Because the Ui(1) generators commute with all of the SU(5) generators it is clear that each of the Ui(1) gauge bosons (call them Bi) couples with equal strength to every fermion which belongs to a certain representation of SU(5). Let us denote the strength with which Bi couples to a fermion belonging to a representation, r, of SU(5) by gi Xi(r), where gi is the coupling constant of Ui(1). We can absorb gi into a redefined X i = Xi(r) - g i X i ( r ) / g so that Bi couples togXi(r) where g is the coupling constant of the SU(2) weak-isospin group. The fact that these couplings depend only on the SU(5) representation to which the fermions belong considerably restricts the form of the deviations from the simplest Weinberg-Salam neutral current predictions. Let us use the following notation. B and A/are the weak-hypercharge and weak-isospin gauge bosons;g' and g are the corresponding gauge couplings, A = the photon = (g'A 3 +gB)/(g 2 +g,2)1/2, DZZ - (OI T {ZZ}lO)q= O,
DBiBj -~ (0[ T {B i B.}lO)q=0'
Z--- (gA 3 - g'B)/(g 2 +g,2)1/2,
DZB i =--DB i Z =--(01T(ZBi}IO)q= O, Qz -- T3 - sin20w Q"
Here T 3 is the SU(5) generator corresponding to the third component of weak isospin and sin20w ~ e2/g 2 where e is the electromagnetic coupling constant and g is the coupling constant of the weak-isospin group as already mentioned. Notice that sin20w can be determined in principle from charged-current experiments (after the W+mass is measured). Now using eqs. (2) and (3) we can write, for two fermions 41 and 42 belonging, respectively, to the SU(5) representations r 1 and r2, the coefficient of (~ l')'u 41) (~2 ')'u 42) in the low-energy, neutral-current effective hamiltonian g 2 ( Q z ( ~ 1)
X l ( r 1)
X2(r1)
...)/
DZZ
/ coS20w I l
DZB1 c°S0w
DZB1
DZBz
c°s0 w cos0 w
DB1BI
X l ( r 2)
DB1Bz /
DZBz cos0 w
DBIB 2
DBzB 2
!
!
:
".
//l
~
/
Let us define Qx(r) and DZB by Qx(r)DzB -= ~iXi(r)DzB ' and Qx(5) - 1, and further define Q2(r, r') and 2 , _ ]~] , " - - _72 DBB by Q (r, r )DBB = Gi Xi(r)DBiB j Xj(r ) and Q}(5,5) = 1 We can rewrite the above more simply as ,3 We shall say nothing about new U(1) gauge interactions that couple differently to different families. Such family-dependent interactions can lead to complicated deviations from the simplest Weinberg-Salam neutral current predictions. This is because the observed mass eigenstates u, d, c, and s might be complicated linear combinations of the states of definite charge under the new U(1) group(s). Therefore we restrict the discussion to family-independent U(1) groups. For some examples of new familydependent U(1) groups proposed in the literature see ref. [8]. ,4 In this note, we are also ignoring possible effects from horizontal gauge bosons (see for example ref. [9] ). Their effects are probably negligible in view of the existing constraints on flavor-changing neutral currents. 299
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g2 {Qz(~l)Qz(~2)(Dzz/COS20w ) + [ Q z ( f f l ) Q x ( r 2 ) + Qx(rl)Qz(~#2)] (DzB/COSOw) + Q2(rl, r2)DBB}.
(6)
The factors of cos 0 w come from the fact that the Z field couples to gQz/cos 0 w. In conventional notation M2w × Dzz/COS20w = P. We are only interested in the SU(5) representations 5 (and 5) and 10 (and ]-0) so that we can simplify notation by defining A = Qx(10) = - Q x ( ] ' 0 ) , B - 92(3, 10) = _Q2(~, T-0), etc., and C =- 02(10,10) = -O2(10,]-0), etc. Also we define e =M2wDBz/COsOw and e =M2wDBB. Then the usual neutrino scattering parameters *s are given by eL(u) = (-~ - ~sin20w)P + (1 - ~sin20w + eR(u)
A)e'+ (2B)e",
= (---~sin20w)P +(--~sin20w - A ) e ' + ( - 2 B ) e ,, ,
eL(d)=(- ½ +~sin20w)P+(-1
+~sin2Ow+A)e r+ (2B)e I t ,
• 2 eR(d) =(-~sin20w)0 +(--1 + 2~sln 0w)e , + ( - 2 ) e
-
,
g~ = (--½ + 2 sin20w)P + (4 sin20w - A)e' + 2(1 - B)e", gAe = ( _ l ) p + (A)e'+ 2(1 + B ' ) e ' , Clu = (-½ + 4sin20w)P + (1 - ~ s i n 2 0 w ) ( 1 C2 u =(_!2 +2sin20w)P + [1 + ( - 3
-A)e',
+8sin2Ow)A]e'+4(B- C)e",
C1 d = (1 _ _~sin20w)P + [(-1 + ~sin20w ) + ( - 2 + 4sin20w)A] e ' + 2(C - 1)e", C2 d = (1 _ 2 sin20w)P + [(-1 + 4 sin20w ) + (4 sin20w)A] e ' + 2(1 - C)e".
(7)
e We see that these ten quantities .6 , eL(u), eR(u), eL(d), eR(d), g ve ' gA' Clu' C2u' Cld, C2d' can be fit to six parameters, p, e', e", A, B and C. Thus there exist four relations among them. (Of course, i f M w is not measured independently the ten neutral current couplings will have to be fit to seven parameters, including sin20w.) Given that the present data favor the one-parameter SU(2) X U(1) predictions one can suppose that e', e", and p - 1 are small. In practice, to have the ten couplings measured accurately enough to allow the sort of simultaneous fitting we have in mind would hardly be a simple task .7 . The parameters e' and e" probably will have to be of order 10% for the present analysis to be useful at all. However, see the appendix where a more practical relation is derived. Case (ii). If there is only one "extra" U(1) group coupling to quarks and leptons then we have a sort of factorization: B = A and C = A 2, so that the ten measured quantities can be fit with only the four parameters p, e', e" and A (again not counting sin20w). Thus in this case there are six relations among them. Case (riO. Now let us assume that the extra U(1) groups are unified with SU(5) in some larger grand unified group SU(N) as discussed in the introduction. Let us suppose that the known fermions are neutral under the q groups, which were denoted by llqm=1 Urn(1 ) in eq. (1). (This can happen i f q = 0 or if the known fermions have only SU(5) indices.) Then there are further constraints on the neutral current couplings. There are two points to be noticed. First, only one linear combination of the p - 1 gauge bosons of the groups denoted Un(1) in eq. (1) ,5 This is the notation of Langacker et al. [4]. ,6 We have not included interference effects in e+e- ~ ~÷~- on the grounds that any such effects are probably only visible near resonance. . 7 This is of course assuming that all other ( " k n o w n " ) corrections, due to strong and higher order electro-weak can be disentangled and subtracted out.
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couples to ordinary quarks and leptons. Thus we have all the constraints of case (ii) above. And second, this boson couples to the quinticity [10] of the ordinary fermions. By quinticity [in analogy with triality for SU(3)] is meant the operator that counts the number of SU(5) indices. Therefore we have one further constraint, namely 4:8 A = - 2 . In this case then the ten quantities in eq. (4) can be fit with only three parameters p, e' and e". (It should be emphasized that this result is unaffected even if the electric charge operator is of the form Q = 13 + Y/2 + X , where 13 and Y/2 are the usual SU(5) generators corresponding to weak isospin and weak hypercharge.) Now if the known quarks or leptons are charged under one or more of the groups denoted by U m (1) in eq. (1) (but family independence still assumed) then we are back to case (i) or case (ii) depending on whether there are several or only one "extra" U(1) groups coupling to the known fermions. We should mention that even if SU(5) is totally irrelevant any detected deviations in the neutral current would still have to exhibit a definite pattern. For instance, suppose the correct low energy theory .9 is SU(3) X SU(2) X U(1) X U(1). Referring to eq. (4) we see that certain constraints follow simply because X(UL) = )((dE) and X(UL) = X(e t ) . In this case, the ten neutral current couplings could be fit with seven parameters. Thus we may summarize in the following way. (I) If the deviations of the neutral current couplings from the simplest Weinberg-Salam predictions can be fit with six parameters in the manner shown in eq. (7) [case (i)] then it suggests both that SU(3)c × SU(2)L X U(1) are indeed unified in SU(5)and that there is m o r e than one "extra" U(1) group coupling to the known fermions. (2) If the more restrictive four parameter fit [case (ii)] is good then i.t suggests both that SU(5) is good and that there is only one extra U(1) coupling to the known fermions. (3) And finally if the three-parameter fit [case (iii)] is good it further suggests that SU(5) and the extra U(1) groups are embedded in a simple group larger than SU(5). In other words, Weinberg-Salam gave us a zero-parameter (not counting sin20w !) fit to neutral current couplings. I f experimentalists should see any deviations from this fit, they should attempt a three-parameter fit [eq. (3) above.] If that fails, they should then go to a four-parameter, and finally to a six-parameter fit [eqs. (2) and (1) above]. For a more practical test of the data see the appendix. 4. It may be useful to discuss a specific example to illustrate the foregoing statements. Our numerological instinct suggests that the sequence U(1) × SU(2) X SU(3) should be extended to U(1) × SU(2) X SU(3) × SU(4) X SU(5) × .... Let us succumb to this temptation and consider SU(9) ~ SU(4) × SU(3) X SU(2) × U(1) X U(1). This is a special case of eq. (1) with N = 9, k 1 = 2, k 2 = 3, k 3 = 4, q = N - Z k i = 0 and p = 3. In a notation in which the weak hypercharge is given by II/2 = diag ( - I / 2 , - 1 / 2 , 1/3, 1/3, 1/3, O, O, O, 0), the generator of the additional U(1) is given by X = diag (4,4, 4, 4 , 4 , - 5 , - 5 , - 5 , - 5 ) / 6 x / - 6 . Notice that this couples to the quinticity of the quarks and leptons. (Note also that ordinary quarks and leptons are singlets under SU(4) as SU(4) is assumed to confine at some large energy scale.) To further break the symmetry we introduce a higgs in the fundamental representation 6/1, A = 1 ..... 9 (the Weinberg-Salam higgs field) with (¢1> = o 4= 0 and a higgs transforming as a totally antisymmetric five index tensor c~[ABCDE] with (~b12345> = w >~ o. The reason ~b12345 is chosen is that it is at once an SU(4) singlet and an SU(5) singlet. Define a = ~ ( 1 + 25 w2/02). Then after some arithmetic the neutral current interaction hamiltonian for fermions ~1 and ~2 in SU(5) representations r 1 and r 2 is found to be HNC = [ ( ~ l Tut~l)(~27t~t~2)/2v2(~ a - s~)] X { a Q z ( ~ l ) O z ( f f 2 2 ) - [X(rl)Qz(ff2) + Q z ( O l ) X ( r 2 ) ] / 3 x / ~
+ ¼X(rl)X(r2) }.
In the limit a ~ oo we recover the standard SU(2) X U(1) result where p ~ 1 and e', e" ~ 0. We see also that this conforms to the pattern expected for case (iii) with the three unknown parameters p, e', e" having definite values 4:8 There are other interesting ways of determining A. For instance, if SO(10) is broken down by the 24 adjoint of SU(5) into SU(3) x SU(2) X U(1) X U(1) one getsA = -1/3 [11]. 4:9 There have been a large number of models based on SU(2) × U(1) x U(1) in the literature. However, our analysis is general. 30t
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since we are working with a specific model. Note that we do not choose to break SU(4) X SU(3) X SU(2) X U(1) X U(1) down to SU(4) X SU(3) c X U(1)e m by letting (q526789) = w ~ v as then in the limit ~ ~ ~ we do not recover the usual predictions of the Weinberg-Salam model. The above is meant merely as an illustration of our general remarks and is not put forward with any expectation that it is the correct model. Obviously, our analysis would go through for other groups as well. 5. We have tried to show, first, that in certain plausible extensions of the SU(5) unified gauge model deviations from the shnplest Weinberg-Salam neutral current predictions may be expected. And, second, we have shown that if these deviations have certain definite forms they can provide information on whether the SU(5) model is correct, whether there are further gauge interactions beyond SU(5), the number of such interactions and whether such interactions are unified with SU(5) within some larger simple group than SU(5). We end with a speculation which suggests itself. We have seen that when SU(N) is broken down by an adjoint ttiggs, there naturally arise several U(1) factors. The traditional view is that the only exact abelian gauge symmetry breaking is the electromagnetic U(1). We would like to suggest that this may well not be the case! There could be additional species of "photons" in the world. For instance, an SU(N) gauge theory may be broken into SU(n 1) X SU(n2) X SU(3) X SU(2) X U(1) X U(1) X U(1). One of the three U(1)'s only couples to technifermions and not to ordinary fermions, and thus could be left as an exact gauge theory. However, technifermions are assumed to be bound in oppositely "charged" pairs. Ordinary fermions can couple to this massless gauge boson via a loop involving supermassive gauge bosons [which are not in SU(5)] and thus couple via magnetic moment or charge radius. We thank John Kim for a careful reading of the manuscript. The work of S.M.B. was supported by the National Science Foundation and that of A.Z. in part under contract no. EY-76-C-02-3071 of the US Department of Energy. Appendix. We may use the relations in eq. (7) to derive a formula of more practical usefulness. The parameters Clu , C2u , Cld and C2d are not as well known and are harder to measure than the neutrino scattering parameters. Also note that g~¢ and g~ are given exactly as linear combinations of the other parameters: g ev = - 2 e X [eL(u ) + eR(u)] -- [eL(d ) + eR(d)], gA = eL(d) -- eR(d)" Therefore any useful relation must involve only the epsilons. Fortunately there are such relations:
NUMERATOR
/CASErIT
1.0
1.0 .,
~
I
I > DENOMINATOR
-- ~L( u ) + 2 ~R ( u ) + ,~L(d) +5fl.~R (d)
kWEINBERG - SALAM
302
Fig. 1. A plot of neutrino scattering parameters. The straight
line represents the relation derived in the appendix for case (iii) [see eq. (A2)]. The origin is the exact Weinberg-Salam prediction. The shaded area is experiment [4] and represents one standard deviation.
Volume 92B, number 3,4 2eL(U ) -- eR(u ) + 2eL(d ) = 5Ae'+ lOBe",
PHYSICS LETTERS eL(u ) + 2eR(u ) + eL(d ) + 5eR(d ) = - - 5 e ' - - 10e".
19 May 1980 (A1)
Taking the ratio of these equations we find 2eL(u ) -- eR(U ) + 2eL(d ) ratio =
eL(U ) + 2eR(u ) + eL(d ) + 5eR(d)
Ae' + 2Be" - e' + 2e"
= - , 4 [case (ii)] = +2 [case (iii)].
(A2)
If case (ii) obtains and there is one additional U(1) gauge boson coupling to ordinary fermions then A = B and the r.h.s, is equal to - A . If the more restrictive case (iii) obtains [additional U(1) groups are unified with SU(5) in some larger group] then the r.h.s, o f eq. (A2) is equal to +2. Using the values o f eL(u), eR(u), eL(d ) and eR(d ) from Langacker et al. [4] we find that ratio = (0.052 + 0 . 1 4 ) / ( - 0 . 4 7 9 -L-_0.25).
(A3)
Clearly the error on these measurements are too large to draw any conclusion. Both numerator and denominator are consistent with zero (the exact Weinberg-Salam result). However, if the precision with which the neutrino scattering parameters are known is increased, then some definite conclusions can be drawn. See fig. 1.
References [1] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [2] H. Georgi, H. Quinn and S. Weinberg, Phys. Rev. Lett. 33 (1974) 451. [3] S. Glashow, Old and new directions, HUTP-79/A029; F. Wilczek and A. Zee, to be published. [4 ] J.1. Sakurai, UCLA/79/TEP/18 ; P. Langacker, J.E. Kim, M. Levine, H.H. Williams and D.P. Sidhu, Univ. of Pennsylvania preprint COO-3071-243, to be published in Neutrino-79. [5] L.F. Li, Phys. Rev. D9 (1974) 1723; 1. Kim, Univ. of Pennsylvania preprint UPR-0141T. [6] S. Weinberg, Phys. Rev. D13 (1976) 974; L. Susskind, SLAC-PUB-2142-1978. [7] H. Georgi and S. Weinberg, Phys. Rev. D17 (1978) 275. [8] M.A.B. Beg and H.-S. Tsao, Phys. Rev. Lett. 41 (1978) 278; S. Barr and P. Langacker, Phys. Rev. Lett. 42 (1979)1654. [91 F. Wilczek and A. Zee, Rev. Lett. 42 (1979) 421. [10] J. Kim and A. Zee, Univ. of Pennsylvania preprint UPR-0132T. [11] F. Wilczek and A. Zee, Phys. Rev. Lett. 43 (1979) 1571.
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NEUTRAL CURRENTS AND GRAND UNIFICATION S.M. BARR and A. ZEE Department o f Physics, University o f Pennsylvania, Philadelphia, PA 191 74, USA Received 24 December 1979
We show that in certain interesting classes of gauge models useful information about grand unification may be obtained from careful neutral current measurements at low energies.
1. We are now in the unprecedented situation of having a physical theory, the SU(3) X SU(2) X U(1) gauge theory, that is consistent with all non-gravitational physical phenomena. Recent neutral current experiments have dramatically corroborated the validity of this theory at low energy. Moreover, there are cogent theoretical grounds for believing that these gauge interactions may be unified within an SU(5) gauge theory [1 ] at superheavy energy scales [2] (near 1015 GeV). The (perhaps disheartening) suggestion has even been made that nothing interesting may remain undiscovered below .1 this superheavy unification scale of energies. This might well be true if the SU(5) theory really incorporated all (non-gravitational) gauge interactions. At the same time, on the experimental front, neutrino scattering experiments soon to be performed should substantially improve the precision to which neutral current couplings are known (for a review, see ref. [4] ). The question presses itself upon us whether such measurements are likely to afford us decipherable clues to what lies beyond or whether we must be resigned to the difficult task of probing the structure of physics at 1015 GeV without the aid of accelerators. In this note we wish to make two simple but encouraging remarks. First, there exist plausible reasons for anticipating that significant deviations from the simplest Weinberg-Salam neutral current predictions may be seen in more precise experiments. And second, such deviations, if of certain simple forms, may provide information about physics at superheavy energies. Z In the SU(5) model the superheavy breaking down to SU(3) X SU(2) X U(1) is most economically achieved through a higgs field in the adjoint (24) representation acquiring a vacuum expectation value of the form diag (1, 1, - 2 / 3 , - 2 / 3 , - 2 / 3 ) ~ . Notice that SU(5) is not broken into SU(3) X SU(2) but into SU(3) × SU(2) X U(1). In this note we wish to argue that additional U(1) factors may be present and may provide a clue to physics beyond SU(5). Let us suppose that the grand unified group is something larger than SU(5), say SU(N), N > 5. Then it seems not improbable that the superheavy breaking of this larger group is also accomplished by a higgs in the adjoint representation of which the (24) higgs in the SU(5) model is merely a part. In such a case what would remain of the SU(N) group after superheavy breaking would be of the general form .2
*1 For a contrary view see ref. [3]. ,2 The breaking of SU(N) by adjoint higgs has been analyzed by Li and Kim [5]. If one restricts oneself to quartic polynomial higgs potentials, then in the notation of eq. (1), q = 0 and p = 3, as was pointed out by Li. If however we adopt the view that the adjoint higgs (which may not be an elementary field in any case) can have more general couplings then more complicated breakings may occur. 297
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p-1
-~[-1
×
where ]~Pl=Ikl + q = N . Notice that the number of unbroken diagonal generators is ]~Pl(kl_ - 1) + q + (p - 1) = N - 1. In other words all N - 1 diagonal generators of SU(N) commute with the diagonal vacuum expectation value of the adjoint higgs field. It is easy to understand eq. (1). The VEV of the adjoint higgs field (call it gZ~t~) has N diagonal entries. Some of these will be degenerate. If some set of k l > 1 of these entries are degenerate then an SU(kl) subgroup is left unbroken. In eq. (1) we suppose there are p such non-abelian groups. If an entry, say (~2m), is unique (not degenerate with another entry) on the other hand, then a U(1) subgroup is left unbroken corresponding to the generator Xa~ = 6aO - N6C~m3m ~. These unbroken subgroups are denoted by Ilqm=lUm(1 ) in eq. (1). Since the total number of diagonal entries of ~2~t~ is N we have 2;/=1 k I + q = N. However, as already remarked, all N - 1 diagonal generators of SU(N) are.unbroken. This accounts for the last factor in eq. (I) of (p - 1) unbroken U(1) subgroups. For example the SU(5) model breaks down to SU(3) X SU(2) X U(1). This is clearly seen to be a special case of eq. (1) with q = 0 , p = 2 and k l = {2,3}. As another example suppose an SU(5) is broken by (f2a 3) -- diag ( - 4 , 1, 1, 1,1), t h e n q = 1,p = 1 and k 1 = 4. So that SU(5) --* SU(4) X U(I)'. Now just as in the SU(5) model the further breaking of SU(3) X SU(2) X U(1) happens at low (that is, not superheavy) energies, so we might expect that the residual group of SU(N) is also broken at low energies (perhaps in the TeV range). The unbroken non-abelian factors, one assumes, confine so that the presently known quarks and leptons are singlets under these groups, except for color SU(3). This is in accordance with the technocolor philosophy [6] : all fermions which are nonsinglet under one of these new (non-color) unbroken non-abelian groups will be assumed to be confined in high-mass bound states. Thus one expects in this scheme that the lowenergy phenomenology of quarks and leptons would be describable by the usual SU(3)c X [SU(2) X U( 1)] ws model with some number of additional broken U(1) factor groups. These additional U(1) gauge interactions would modify, at some level, the SU(2) X U(1) neutral current predictions. In this way we might hope to learn something about grand unification from neutrino scattering experiments. The above is meant to be a motivation for contemplating deviations from the standard SU(2) X U(1) neutral current predictions. In fact, much of what follows will be independent of the specific picture described above. We will show that rather non-trivial constraints on experiments can follow from some simple theoretical remarks. The basic idea is extremely simple. According to a general analysis [7], the neutral current interaction hamiltonian at zero momentum transfer is given by 1
HNC = 2 ~
(~Tuni ~ ) ( U - 2 ) i / ( ~ ' U n / i f ) "
(2)
The notation is as follows. Let the electricially neutral gauge bosons A '~ of the theory couple to g~ T~. T~ are electrically neutral generators and the electric charge operator is Q = L'uaCa Tc~. Pick out one generator contributing to the charge operator Q out of the set Ta ; call it TO, and refer to the rest as Ti's. (/~-2)i / is some symmetric inverse mass matrix. Then the operators n i are given by (3)
ni = gi [T. - (eZ/g2)C i Q] .
This structure is in fact already quite restrictive. For instance, assuming the existence of one extra neutral gauge boson beyond the photon and the Z boson and that Q = T3 + Y/2, we have HNC up to a normalization, given schematically by (1 + e l ) [ Q z ] [Qz] + e2([Qz] [X] + IX] [ Q z l ) + e 3 [ X ]
[x].
(4)
Here Qz denotes T 3 - (e2/g2)Q and X the generator corresponding to the extra gauge boson. Thus, the accessible 298
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neutral current couplings may be parametrized in terms of a handful of unknowns. We now analyze the situation in more detail.
3. Case (i). First let us see what follows from assuming that the SU(5) model is extended to SU(5) × IIiUi(I) [which breaks down to SU(3)c X U(1)e m ]. We only assume that these interactions are family independent*3,4. We do not assume for now that they are unified in any larger group. Because the Ui(1) generators commute with all of the SU(5) generators it is clear that each of the Ui(1) gauge bosons (call them Bi) couples with equal strength to every fermion which belongs to a certain representation of SU(5). Let us denote the strength with which Bi couples to a fermion belonging to a representation, r, of SU(5) by gi Xi(r), where gi is the coupling constant of Ui(1). We can absorb gi into a redefined X i = Xi(r) - g i X i ( r ) / g so that Bi couples togXi(r) where g is the coupling constant of the SU(2) weak-isospin group. The fact that these couplings depend only on the SU(5) representation to which the fermions belong considerably restricts the form of the deviations from the simplest Weinberg-Salam neutral current predictions. Let us use the following notation. B and A/are the weak-hypercharge and weak-isospin gauge bosons;g' and g are the corresponding gauge couplings, A = the photon = (g'A 3 +gB)/(g 2 +g,2)1/2, DZZ - (OI T {ZZ}lO)q= O,
DBiBj -~ (0[ T {B i B.}lO)q=0'
Z--- (gA 3 - g'B)/(g 2 +g,2)1/2,
DZB i =--DB i Z =--(01T(ZBi}IO)q= O, Qz -- T3 - sin20w Q"
Here T 3 is the SU(5) generator corresponding to the third component of weak isospin and sin20w ~ e2/g 2 where e is the electromagnetic coupling constant and g is the coupling constant of the weak-isospin group as already mentioned. Notice that sin20w can be determined in principle from charged-current experiments (after the W+mass is measured). Now using eqs. (2) and (3) we can write, for two fermions 41 and 42 belonging, respectively, to the SU(5) representations r 1 and r2, the coefficient of (~ l')'u 41) (~2 ')'u 42) in the low-energy, neutral-current effective hamiltonian g 2 ( Q z ( ~ 1)
X l ( r 1)
X2(r1)
...)/
DZZ
/ coS20w I l
DZB1 c°S0w
DZB1
DZBz
c°s0 w cos0 w
DB1BI
X l ( r 2)
DB1Bz /
DZBz cos0 w
DBIB 2
DBzB 2
!
!
:
".
//l
~
/
Let us define Qx(r) and DZB by Qx(r)DzB -= ~iXi(r)DzB ' and Qx(5) - 1, and further define Q2(r, r') and 2 , _ ]~] , " - - _72 DBB by Q (r, r )DBB = Gi Xi(r)DBiB j Xj(r ) and Q}(5,5) = 1 We can rewrite the above more simply as ,3 We shall say nothing about new U(1) gauge interactions that couple differently to different families. Such family-dependent interactions can lead to complicated deviations from the simplest Weinberg-Salam neutral current predictions. This is because the observed mass eigenstates u, d, c, and s might be complicated linear combinations of the states of definite charge under the new U(1) group(s). Therefore we restrict the discussion to family-independent U(1) groups. For some examples of new familydependent U(1) groups proposed in the literature see ref. [8]. ,4 In this note, we are also ignoring possible effects from horizontal gauge bosons (see for example ref. [9] ). Their effects are probably negligible in view of the existing constraints on flavor-changing neutral currents. 299
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g2 {Qz(~l)Qz(~2)(Dzz/COS20w ) + [ Q z ( f f l ) Q x ( r 2 ) + Qx(rl)Qz(~#2)] (DzB/COSOw) + Q2(rl, r2)DBB}.
(6)
The factors of cos 0 w come from the fact that the Z field couples to gQz/cos 0 w. In conventional notation M2w × Dzz/COS20w = P. We are only interested in the SU(5) representations 5 (and 5) and 10 (and ]-0) so that we can simplify notation by defining A = Qx(10) = - Q x ( ] ' 0 ) , B - 92(3, 10) = _Q2(~, T-0), etc., and C =- 02(10,10) = -O2(10,]-0), etc. Also we define e =M2wDBz/COsOw and e =M2wDBB. Then the usual neutrino scattering parameters *s are given by eL(u) = (-~ - ~sin20w)P + (1 - ~sin20w + eR(u)
A)e'+ (2B)e",
= (---~sin20w)P +(--~sin20w - A ) e ' + ( - 2 B ) e ,, ,
eL(d)=(- ½ +~sin20w)P+(-1
+~sin2Ow+A)e r+ (2B)e I t ,
• 2 eR(d) =(-~sin20w)0 +(--1 + 2~sln 0w)e , + ( - 2 ) e
-
,
g~ = (--½ + 2 sin20w)P + (4 sin20w - A)e' + 2(1 - B)e", gAe = ( _ l ) p + (A)e'+ 2(1 + B ' ) e ' , Clu = (-½ + 4sin20w)P + (1 - ~ s i n 2 0 w ) ( 1 C2 u =(_!2 +2sin20w)P + [1 + ( - 3
-A)e',
+8sin2Ow)A]e'+4(B- C)e",
C1 d = (1 _ _~sin20w)P + [(-1 + ~sin20w ) + ( - 2 + 4sin20w)A] e ' + 2(C - 1)e", C2 d = (1 _ 2 sin20w)P + [(-1 + 4 sin20w ) + (4 sin20w)A] e ' + 2(1 - C)e".
(7)
e We see that these ten quantities .6 , eL(u), eR(u), eL(d), eR(d), g ve ' gA' Clu' C2u' Cld, C2d' can be fit to six parameters, p, e', e", A, B and C. Thus there exist four relations among them. (Of course, i f M w is not measured independently the ten neutral current couplings will have to be fit to seven parameters, including sin20w.) Given that the present data favor the one-parameter SU(2) X U(1) predictions one can suppose that e', e", and p - 1 are small. In practice, to have the ten couplings measured accurately enough to allow the sort of simultaneous fitting we have in mind would hardly be a simple task .7 . The parameters e' and e" probably will have to be of order 10% for the present analysis to be useful at all. However, see the appendix where a more practical relation is derived. Case (ii). If there is only one "extra" U(1) group coupling to quarks and leptons then we have a sort of factorization: B = A and C = A 2, so that the ten measured quantities can be fit with only the four parameters p, e', e" and A (again not counting sin20w). Thus in this case there are six relations among them. Case (riO. Now let us assume that the extra U(1) groups are unified with SU(5) in some larger grand unified group SU(N) as discussed in the introduction. Let us suppose that the known fermions are neutral under the q groups, which were denoted by llqm=1 Urn(1 ) in eq. (1). (This can happen i f q = 0 or if the known fermions have only SU(5) indices.) Then there are further constraints on the neutral current couplings. There are two points to be noticed. First, only one linear combination of the p - 1 gauge bosons of the groups denoted Un(1) in eq. (1) ,5 This is the notation of Langacker et al. [4]. ,6 We have not included interference effects in e+e- ~ ~÷~- on the grounds that any such effects are probably only visible near resonance. . 7 This is of course assuming that all other ( " k n o w n " ) corrections, due to strong and higher order electro-weak can be disentangled and subtracted out.
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couples to ordinary quarks and leptons. Thus we have all the constraints of case (ii) above. And second, this boson couples to the quinticity [10] of the ordinary fermions. By quinticity [in analogy with triality for SU(3)] is meant the operator that counts the number of SU(5) indices. Therefore we have one further constraint, namely 4:8 A = - 2 . In this case then the ten quantities in eq. (4) can be fit with only three parameters p, e' and e". (It should be emphasized that this result is unaffected even if the electric charge operator is of the form Q = 13 + Y/2 + X , where 13 and Y/2 are the usual SU(5) generators corresponding to weak isospin and weak hypercharge.) Now if the known quarks or leptons are charged under one or more of the groups denoted by U m (1) in eq. (1) (but family independence still assumed) then we are back to case (i) or case (ii) depending on whether there are several or only one "extra" U(1) groups coupling to the known fermions. We should mention that even if SU(5) is totally irrelevant any detected deviations in the neutral current would still have to exhibit a definite pattern. For instance, suppose the correct low energy theory .9 is SU(3) X SU(2) X U(1) X U(1). Referring to eq. (4) we see that certain constraints follow simply because X(UL) = )((dE) and X(UL) = X(e t ) . In this case, the ten neutral current couplings could be fit with seven parameters. Thus we may summarize in the following way. (I) If the deviations of the neutral current couplings from the simplest Weinberg-Salam predictions can be fit with six parameters in the manner shown in eq. (7) [case (i)] then it suggests both that SU(3)c × SU(2)L X U(1) are indeed unified in SU(5)and that there is m o r e than one "extra" U(1) group coupling to the known fermions. (2) If the more restrictive four parameter fit [case (ii)] is good then i.t suggests both that SU(5) is good and that there is only one extra U(1) coupling to the known fermions. (3) And finally if the three-parameter fit [case (iii)] is good it further suggests that SU(5) and the extra U(1) groups are embedded in a simple group larger than SU(5). In other words, Weinberg-Salam gave us a zero-parameter (not counting sin20w !) fit to neutral current couplings. I f experimentalists should see any deviations from this fit, they should attempt a three-parameter fit [eq. (3) above.] If that fails, they should then go to a four-parameter, and finally to a six-parameter fit [eqs. (2) and (1) above]. For a more practical test of the data see the appendix. 4. It may be useful to discuss a specific example to illustrate the foregoing statements. Our numerological instinct suggests that the sequence U(1) × SU(2) X SU(3) should be extended to U(1) × SU(2) X SU(3) × SU(4) X SU(5) × .... Let us succumb to this temptation and consider SU(9) ~ SU(4) × SU(3) X SU(2) × U(1) X U(1). This is a special case of eq. (1) with N = 9, k 1 = 2, k 2 = 3, k 3 = 4, q = N - Z k i = 0 and p = 3. In a notation in which the weak hypercharge is given by II/2 = diag ( - I / 2 , - 1 / 2 , 1/3, 1/3, 1/3, O, O, O, 0), the generator of the additional U(1) is given by X = diag (4,4, 4, 4 , 4 , - 5 , - 5 , - 5 , - 5 ) / 6 x / - 6 . Notice that this couples to the quinticity of the quarks and leptons. (Note also that ordinary quarks and leptons are singlets under SU(4) as SU(4) is assumed to confine at some large energy scale.) To further break the symmetry we introduce a higgs in the fundamental representation 6/1, A = 1 ..... 9 (the Weinberg-Salam higgs field) with (¢1> = o 4= 0 and a higgs transforming as a totally antisymmetric five index tensor c~[ABCDE] with (~b12345> = w >~ o. The reason ~b12345 is chosen is that it is at once an SU(4) singlet and an SU(5) singlet. Define a = ~ ( 1 + 25 w2/02). Then after some arithmetic the neutral current interaction hamiltonian for fermions ~1 and ~2 in SU(5) representations r 1 and r 2 is found to be HNC = [ ( ~ l Tut~l)(~27t~t~2)/2v2(~ a - s~)] X { a Q z ( ~ l ) O z ( f f 2 2 ) - [X(rl)Qz(ff2) + Q z ( O l ) X ( r 2 ) ] / 3 x / ~
+ ¼X(rl)X(r2) }.
In the limit a ~ oo we recover the standard SU(2) X U(1) result where p ~ 1 and e', e" ~ 0. We see also that this conforms to the pattern expected for case (iii) with the three unknown parameters p, e', e" having definite values 4:8 There are other interesting ways of determining A. For instance, if SO(10) is broken down by the 24 adjoint of SU(5) into SU(3) x SU(2) X U(1) X U(1) one getsA = -1/3 [11]. 4:9 There have been a large number of models based on SU(2) × U(1) x U(1) in the literature. However, our analysis is general. 30t
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since we are working with a specific model. Note that we do not choose to break SU(4) X SU(3) X SU(2) X U(1) X U(1) down to SU(4) X SU(3) c X U(1)e m by letting (q526789) = w ~ v as then in the limit ~ ~ ~ we do not recover the usual predictions of the Weinberg-Salam model. The above is meant merely as an illustration of our general remarks and is not put forward with any expectation that it is the correct model. Obviously, our analysis would go through for other groups as well. 5. We have tried to show, first, that in certain plausible extensions of the SU(5) unified gauge model deviations from the shnplest Weinberg-Salam neutral current predictions may be expected. And, second, we have shown that if these deviations have certain definite forms they can provide information on whether the SU(5) model is correct, whether there are further gauge interactions beyond SU(5), the number of such interactions and whether such interactions are unified with SU(5) within some larger simple group than SU(5). We end with a speculation which suggests itself. We have seen that when SU(N) is broken down by an adjoint ttiggs, there naturally arise several U(1) factors. The traditional view is that the only exact abelian gauge symmetry breaking is the electromagnetic U(1). We would like to suggest that this may well not be the case! There could be additional species of "photons" in the world. For instance, an SU(N) gauge theory may be broken into SU(n 1) X SU(n2) X SU(3) X SU(2) X U(1) X U(1) X U(1). One of the three U(1)'s only couples to technifermions and not to ordinary fermions, and thus could be left as an exact gauge theory. However, technifermions are assumed to be bound in oppositely "charged" pairs. Ordinary fermions can couple to this massless gauge boson via a loop involving supermassive gauge bosons [which are not in SU(5)] and thus couple via magnetic moment or charge radius. We thank John Kim for a careful reading of the manuscript. The work of S.M.B. was supported by the National Science Foundation and that of A.Z. in part under contract no. EY-76-C-02-3071 of the US Department of Energy. Appendix. We may use the relations in eq. (7) to derive a formula of more practical usefulness. The parameters Clu , C2u , Cld and C2d are not as well known and are harder to measure than the neutrino scattering parameters. Also note that g~¢ and g~ are given exactly as linear combinations of the other parameters: g ev = - 2 e X [eL(u ) + eR(u)] -- [eL(d ) + eR(d)], gA = eL(d) -- eR(d)" Therefore any useful relation must involve only the epsilons. Fortunately there are such relations:
NUMERATOR
/CASErIT
1.0
1.0 .,
~
I
I > DENOMINATOR
-- ~L( u ) + 2 ~R ( u ) + ,~L(d) +5fl.~R (d)
kWEINBERG - SALAM
302
Fig. 1. A plot of neutrino scattering parameters. The straight
line represents the relation derived in the appendix for case (iii) [see eq. (A2)]. The origin is the exact Weinberg-Salam prediction. The shaded area is experiment [4] and represents one standard deviation.
Volume 92B, number 3,4 2eL(U ) -- eR(u ) + 2eL(d ) = 5Ae'+ lOBe",
PHYSICS LETTERS eL(u ) + 2eR(u ) + eL(d ) + 5eR(d ) = - - 5 e ' - - 10e".
19 May 1980 (A1)
Taking the ratio of these equations we find 2eL(u ) -- eR(U ) + 2eL(d ) ratio =
eL(U ) + 2eR(u ) + eL(d ) + 5eR(d)
Ae' + 2Be" - e' + 2e"
= - , 4 [case (ii)] = +2 [case (iii)].
(A2)
If case (ii) obtains and there is one additional U(1) gauge boson coupling to ordinary fermions then A = B and the r.h.s, is equal to - A . If the more restrictive case (iii) obtains [additional U(1) groups are unified with SU(5) in some larger group] then the r.h.s, o f eq. (A2) is equal to +2. Using the values o f eL(u), eR(u), eL(d ) and eR(d ) from Langacker et al. [4] we find that ratio = (0.052 + 0 . 1 4 ) / ( - 0 . 4 7 9 -L-_0.25).
(A3)
Clearly the error on these measurements are too large to draw any conclusion. Both numerator and denominator are consistent with zero (the exact Weinberg-Salam result). However, if the precision with which the neutrino scattering parameters are known is increased, then some definite conclusions can be drawn. See fig. 1.
References [1] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [2] H. Georgi, H. Quinn and S. Weinberg, Phys. Rev. Lett. 33 (1974) 451. [3] S. Glashow, Old and new directions, HUTP-79/A029; F. Wilczek and A. Zee, to be published. [4 ] J.1. Sakurai, UCLA/79/TEP/18 ; P. Langacker, J.E. Kim, M. Levine, H.H. Williams and D.P. Sidhu, Univ. of Pennsylvania preprint COO-3071-243, to be published in Neutrino-79. [5] L.F. Li, Phys. Rev. D9 (1974) 1723; 1. Kim, Univ. of Pennsylvania preprint UPR-0141T. [6] S. Weinberg, Phys. Rev. D13 (1976) 974; L. Susskind, SLAC-PUB-2142-1978. [7] H. Georgi and S. Weinberg, Phys. Rev. D17 (1978) 275. [8] M.A.B. Beg and H.-S. Tsao, Phys. Rev. Lett. 41 (1978) 278; S. Barr and P. Langacker, Phys. Rev. Lett. 42 (1979)1654. [91 F. Wilczek and A. Zee, Rev. Lett. 42 (1979) 421. [10] J. Kim and A. Zee, Univ. of Pennsylvania preprint UPR-0132T. [11] F. Wilczek and A. Zee, Phys. Rev. Lett. 43 (1979) 1571.
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