Families as radial excitations — Meeting with experimental bounds

Volume 155B, number 5,6

PHYSICS LETTERS

6 June 1985

F A M I L I E S AS RADIAL E X C I T A T I O N S MEETING WITH EXPERIMENTAL BOUNDS -

Christoph K O P P E R Max- Planck- lnstitut fi~r Physik und Astrophysik - Werner- Heisenberg- lnstitut fur Physik Fohringer Ring 6, D - 8000 Munich 40, Fed. Rep. Germany

Received 3 January 1985; revised manuscript received 11 March 1985

Interpreting higher families (or generations) as radial excitations of composite leptons and quarks we have to take into account experimental constraints, especially on family number changing processes. This letter comments on g - 2, ~t --, e'r, flavour changing neutral currents, Kobayashi-Maskawa mixing and related topics. A compositeness scale in the 1 TeV region seems sufficient to achieve consistency with experimental bounds.

In a previous paper [1] it was shown to be possible to produce eigenstates of relativistic wave equations which fulfill the experimental constraints on the mass spectrum and spatial extension of the charged leptons and quarks, namely: (A) There should not be low-lying orbital excitations (corresponding to higher spin quarks and leptons). (B) The extension of the wavefunctions is much below the Compton wavelength of the respective particle. (C) The level spacing increases considerably from one state to the next (for the low-lying tightly bound states). A typical spectrum is given in fig. 1. Here the existence of weakly bound states and their positions depend on the (unspecified) long-range part of the interaction. The number of tightly bound states is controlled by a criterion given in ref. [1]. The underlying physical picture was that of composite leptons and quarks being bound states o f a fermion and a boson as is the case in some models [2,3]. The strong binding forces were considered as mediated by gauge particle exchange. U-channel fermion exchange via Yukawa-vertices was also shown to provide strong binding forces but found to display some drawbacks in comparison with the first option to which we will restrict here. The conditions under which (A), (B),

log m ] TeV I00 GeV

m/. M

m/.

10 OeV -

-

rlq~

-

-

tlq v

-

-

m e

1 GeV 100 MeV

10 MeV 1 MeV

Fig. 1. Typical form o f the charged lepton spectrum according to ref. [ 1 ] for a compositeness scale of about 1 TeV. We expect two (rn4, m s ) or one (m4,) additional tightly bound state(s) and then weakly bound states also comprising higher spin composites. The details o f this part depend on the longrange forces whereas the tightly bound states are largely independent of those.

(C) hold are to have renormalizable strongly attractive interactions (a > 1 even for r ,~ l/A, A being the compositeness scale), asymptotic freedom (or at least an "undercritical" [1 ] ultraviolet fixed poin 0 and the requirement that each tightly bound state fixes its own scale through its mass at short distances. Especially the third issue requires investigation. 409

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Here we want to show that the picture is not in contradiction with some constraints on composite models, provided by the data on (g - 2)e,u, on/a -+ e't, on flavour changing neutral currents (FCNC) and on Kobayashi-Maskawa (KM) mixing. In closing we also shortly comment on some other constraints. Since our approximations are not really under good control the intention is just to show that a reasonably simplified appraoch already indicates qualitative agreement with experiment. g - 2. To evaluate the magnetic moments of the electron or the muon, one decomposes the matrix element of the electromagnetic currentju(x) making use of Lorentz invariance, current conservation, hermiticity and parity: O

v

=eiqxfi~'(?uFl(q2)+ i :ruuq- F2(q2))

(1)

q=p'-p.

6 J u n e 1985

suppressed by (approximate) chiral symmetry then m, m F are due to marginal breaking of chiral symmetry only and the leading contribution to F~ is of order (m/A) 2 instead of m/A. We want to argue here that a quadratic suppression factor does not imply chiral symmetry (and consequently parity doubling). Instead we invoke a theorem by Chiu which says that F~(q 2) vanishes if the bound state wavefunctions display an 0(4) symmetry in euclidean space [7]. In ref. [1] this was seen to be the case for massless BS bound states (independently of the ladder approximation). Now we use that (i) F~ is a scalar. (ii) 0(4) symmetry is broken in the BSE by linear (and higher) terms in the bound state four-momentum vector Pu I l l . (iii) These terms are less singular than the leading ones at short distances by at least one power and thus can be taken into account perturbatively. (ii) and (iii) imply that the BS wavefunction can be expanded in a Taylor series

Here ~ , , £~, denote the leptons in the spin-energymomentum states (/3, P'), (a, P), F 1 and F 2 are form factors, which in principle can be calculated from integrals over the Bethe-Salpeter (BS) wavefunction of the lepton, and m is the lepton mass. (g - 2) is given by [4]

with coefficients A u u (x). Thus F~ has also ~o"b~ analytic in P, and since it vanishes for P = 0 (i) implies

g - 2 = 2F2(0 ).

F~(q2 = 0, P #: 0) = O(p2/A 2) -- O(m2/A2),

(2)

Xp(X) = X0(X) + ~

n~ 1

A

un(X)P t ...P

/'tl "'"

(4)

~an

(5)

Apart from radiative terms F 2 will also receive a contribution F 2e from the composite structure. But it seems to be clear on physical grounds that this contribution to the anomalous magnetic moment vanishes if the bound state extension shrinks to zero because a pointlike composite cannot be distinguished from an elementary Dirac particle at any finite energy. So the question is only by which power of the compositeness scale F~ will be suppressed. Arguments by Shaw, Silverman and Slansky based on dispersion relations [5], and by Brodsky and Drell [6], indicated a suppression factor

thus necessitating a compositeness scale not too much below 1 TeV.

F~ ~ mmF[m2a

8(9 ~ eT) < 1.3 X 10- 10,

(8)

B0~ ~ e'yT) < 8.4 X 10-9,

(9)

(3)

(m F and m B are the fermion and boson masses, respectively), for a fermion-boson bound state of extension 0(1/mB), m B ~, m F. It was also argued [6] that if fermion masses are 410

where A again is the bound state extension. The experimental bounds [8] and QED calculations [9] imply IF~(0,Pe-)I ~ 10 -1°,

[ F ~ ( 0 , P - ) I ~< 10 -8,

(6,7)

/i ~ e?. The experimental data on the suppression of/a-number violating processes provide one of the most crucial constraints to our interpretation of families. We have [10]

B 0 a ~ e+e-e - ) < 2.4 X 10 -12.

(10)

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The two-photon decay could be mediated by an intermediate p-state lepton via a double E1 transition. (9) then gives a lower bound on the p-state mass of about 100 GeV as calculated by Vi~nji6 [1 1]. Thus (9) does not render difficulties since low-mass pstates do not appear in our model. As for (10) we argue for a strong suppression of the process by the same mechanism as the one suppressing (8) and through the creation of an additional lepton pair. To estimate the matrix element for/a --* e7 we make use of the LSZ-procedure and Mandelstam's prescription for evaluating electromagnetic current matrix elements between bound states [12]. With an outgoing photon of four-momentum k and polarization e we can write (eou t, (k, e)outl/ain>

=fd4x

6 June 1985

xe a(KaP- V) x ~ = 0 ,

P=PeorP=P

(in the sense of continuous matrix multiplication), where K, V are the kinetic and interaction terms of the respective BSE, in the case at hand [1 ] K = (r/~ + i ~x - mF) { [(1 -- r/) ( - i P ) - a x ] 2 + m 2 } , V = --ig27 v

+D

([ax~D,~(x) ]

[-2i(1 - , 0 e " + 2a~] }.

= -iefd4x

(14)

in which case (12) gives

f d4x ie(x)'r"(D + m 2 ) x , ( x )

e ikx d4x ' Xe(X - x ' )

X e i(Pe-et~)x (.(I--1x,

+m2B)X~(X -

X')

=

_i(ag2/ap,)fd4x ie(x)'r"

X {[axeD (x)]

= -ie(2n)4 6 ( P - Pe - k)

X f d 4 x ~(x) l(tq + m 2 ) x a ( x ) .

(11)

Xe, Xu denote the BS wavefunctions o f e and/a. Here we assumed the constituent charge to be carried by the fermion. Only the lowest order contribution has been taken into account (fig. 2). This is consistent with calculating the BS wavefunctions in the ladder approximation, and we live on the hope -substantiated to some degree in ref. [1 ] - that this already gives a good qualitative approximation to the real problem. On the other hand we have the following orthogonality relation for the wavefunctions ×e, Xu

+ 2Dv~ x)

P=P e ore=P,

P ~///////

////////r

g-p ~(k,c)

Fig. 2. Lowest order contr~ution to u ~ e3, in the radial excitation picture. The boson = is electrically neutral.

×~(x), (15)

since V depends on P only through the running coupling because of the scaling argument [mind (14)]. Making use of the fact that

ag/ae v ~ e v

(16)

for the running coupling, and of the transversality of the emitted photon = o

(17)

[e.g. Pu = (m, 0, 0, 0), Pe = (Ee, 0, 0, k), e 1 = (0, 1, 0 , 0 ) , e 2 = ( 0 , 0 , 1,0)] we have from (I 1),(15)--(17)
P.

(13)

Here we introduce x = x I - x 2, with Xl(X2) the fermion (boson) coordinate; 77 can be chosen between 0 and 1, (12) can be derived similarly as the normalization condition for X [13] , l . We choose n = I

eikX(eout le'j(x)llain)

(12)

v

(18)

So in the ladder approximation ta -+ e7 is completely forbidden independent of the ratio m~[A. There is some indication that (18) can be maintained to higher orders for the emission of a real photon. But we still have to face virtual photon emission leading to the decay (10). ,t See also chapter 10 of ref. [41. 411

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In any case making use of 0(4) invariance as for (g - 2) we get at least a form factor

F2(m2/A2);

(19)

in this case (10) implies A ~ 100 TeV. Further suppression will come from the fact that Xu(X) has one node whereby the overlap integral [e.g. (15)] is diminished. So without being able to make a strong quantitative statement we resume by stating that there is good indication for the decay modes ts -* 3e,/a -+ e7 to be strongly suppressed.

FCNC. In the same way as before 0(4) symmetry and possibly orthogonality will suppress virtual processes

U -+ eZ.

(20)

For real decays we have an additional suppression

(mu]Mz) 4. The same should hold true in the quark sector. The 75-term in the/aeZ-vertex does not cause any harm since the analysis for the 7u- and for the 7uTS-parts may be performed separately. Here we again assume the electroweak properties of the bound state to be carried by the fermion.

where lu°m) etc. denote the unperturbed quark states. The strong increase in size of the denominators for increasing generation index makes one expect strongly decreasing mixing angles for higher generations in • agreement with the facts [ 15 ]. Even in simple calculations using potential models [ 14,16] which do not display the correct spectrum and where isospin breaking is simulated by using different parameters in the potential for the two isospin sectors show this decreasing mixing pattern just because of a wavefunction effect: a sufficient difference in the number of nodes of the respective wavefunctions diminishes the value of the overlap integrals. However, as expected, the effect in this case is not sufficient. Unitarity of the mixing matrix can be inferred from the isospin algebra on neglect of continuum contributions [ 14, 17,3 ]. This approximation seems to be a good one for the tightly bound states, and to this degree we can also exclude FCNC (see above). In our picture isospin breaking could be due to different masses of the subconstituent fermion doublet and/or different effective fermion-boson coupling. Both might be related to electric charge. That this may produce a sufficient effect in spite of the smallness of (~elm could be related to the shortdistance behaviour of the wavefunctions [ 1]

X ~ ri~/h(r)-hcrit,

X(r) > ~crit'

X ~ r~/Xcrit-Mr),

X(r) < Xcrit ,

KM mixing. KM mixing in the generally accepted interpretation is due to a mismatch between mass eigenstates and eigenstates of the weak charged current. Following Terazawa [14] the elements Urnn of the KM matrix can be written as (um IFTu(I - 75 )FId n) = UmnfimTta( 1 - 75)dn, (Urn II+ldn) = Umn,

(21)

where F 7u(l - 75)F is the weak charged current at the subconstituent level, I+ the isospin raising operator, urn, d n denote the mth up-like or the nth downlike quark, respectively. In a hamiltonian formulation the deviation of U from unity is due to an isospin violating part HI, sufficient weakness of which allows for perturbative evaluation

-

Umn mum_ mun 412

dO z

0

( m I lid n)

mdrn _ m d n

,

(22)

(23)

where X(r) is related to the logarithmically running coupling so that a small change in X can considerably modify the wavefunctions at short distances. Finally the momentum-transfer part in the overlap integral exp (iqr) [ 14,17,3 ] can lead to small CP-violating imaginary parts of the entries in the KM matrix and to a momentum-dependent mixing pattern at high energies [ 14]. Similar considerations hold in the lepton sector in case of massive neutrinos where the largest mixing angle has to be much smaller however which might be related to the much larger mass ratios involved. For a review of the data see ref. [ 18].

Further comments. An unconventional process, peculiar to our model is the transition

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(qT::l)= (FB) (FB) -~ (FF)virt(BB).

L,

(24)

This is possible for sufficiently low-lying (Bg) states which could be produced b y the same gauge forces as the (FB) b o u n d states. A detailed inspection has not been performed so far, however. In any case (24) is suppressed by at least ( m q / A ) 2 [19] in the rate. This list of constraints is by n o means complete. As far as I can see, however, it contains in principle the mechanisms which, for example, can also be invoked to interpret various results and b o u n d s in the vast range o f k a o n physics, e.g. K L 7~/z+e - ,

K L ~/a+/a - ,

AmKL,K S.

All these seem n o t to imply more stringent b o u n d s on A than/~ -~ e3'. The conclusion ther. of this qualitative discussion is that we can make ends meet w i t h o u t stretching and b e n d i n g too much provided the compositeness scale is O(TeV). More precise theoretical statements have to be left for the future. On the other hand, experiment soon can provide for new i n f o r m a t i o n if A is sufficiently small. For example, we expect family n u m b e r violating processes in e+e - - or e - p-scattering. For a phenomenological analysis in view o f HERA cf. ref. [ 2 0 ] . Without being able to offer an explanation we would also like to draw a t t e n t i o n to recent CELLO measurements on e+e - -~ 4 leptons and e+e /z+/~-~, which show an excess of events in comparison to QED calculations in the region o f large masses for b o t h l e p t o n pairs or for ~3' at x/s > 30 GeV [21 ]. It would be natural in our model (and the more likely the larger A) to expect a fourth generation with masses again about a factor o f

O(mr/m ~ , mt/m c, mb/m s) above the third generation. Here more stringent statements are not only prohibited b y theoretical considerations, b u t one should also note that the n e u t r i n o masses are stilll u n k n o w n , m t is k n o w n up to large errors only [22] and n o t fully established, and the light quark masses render conceptual difficulties since we do n o t really k n o w n which mass to compare to the l e p t o n masses. Thus progress in the understandhag o f family structure can come from various branches o f theory and e x p e r i m e n t as well.

6 June 1985

This paper was outlined in discussions with H.P. Dtirr. The possibility o f low-lying bosonic states and the decay (24) was pointed out by L. Stodolsky. {1 ] C. Kopper, Families as radial excitations, Max-PlanckInstitut preprint MPI-PAE/PTh 69/84. [21 J.C. Pati, A. Salam and J. Strathdee, Phys. Lett. 58B (1975) 265; M. Veltman, in: Proc. Intern. Symp. on Lepton and photon interactions (Fermilab, 1979); E. Derman, Phys. Rev. D23 (1981) 1623; L. Abbott and E. Farhi, Phys. Lett. 101B (1981) 69; H. Fritsch and G. Mandelbaum, Phys. Lett. 102B (1981) 369. [3] O.W. Greenberg and J. Sucher, Phys. Lett. 99B (1981) 339. [4] C. ltzykson and J.B. Zuber, Quantum field theory (McGraw-Hill, New York, 1980) p. 160. [5] G.L. Shaw, D. Silverman and R. Slansky, Phys. Lett. 94B (1980) 57. [6] S.J. Brodsky and S.D. Drell, Phys. Rev. D22 (1980) 2236. [7] T.W. Chiu, A theorem on the Bethe-Salpeter equation in ladder approximation, University of California, lrvine, Technical Report 82-4. [8] P.B. Schwinberg, R.S. van Dyck and H.G. Dehmelt, Phys. Rev. Lett. 47 (1981) 1679; J° Bailey et al., Phys. Lett. 68B (1977) 191. [9IT. Kinoshita and W.B. Lindqvist, Phys. Rev. Lett. 47 (1981) 1573; T. Kinoshita, B. Ni~i~ and Y. Okamoto, Phys. Rev. Lett. 52 (1984) 717. [1Ol J.D. Bowman et al., Phys. Rev. Lett. 42 (1978) 556; G. Azuelos et al., Phys. Rev. Lett. 51 (1983) 164; SINDRUM Collab., W. Bertl et al., Phys. Lett. 140B (1984) 299. [111 V. Vi~nji~-Triantafillou,Phys. Lett. 95B (1980) 47. 1121 S. Mandelstam, Proc. R. Soc. A233 (1955) 248. 113] N. Nakanishi, Supp. Prog. Theor. Phys. 43 (1969) 1. [14l H, Terazawa and K. Akama, Phys. Lett. 101B (1981) 190. 1151 A.J. Buras, W. Slominsky and H. Steger, Nucl. Phys. B238 (1984) 529. [161 H. Katsumata and Y. Tomozawa, Ann. Phys. (NY) 149 (1983) 457. [17] V. Vi~nji~-Triantafillou,Phys. Rev. D25 (1982) 248. 118] J. Tran Thanh Van, ed., Proc. Moriond Workshop on Massive neutrinos (1984). [191 C. Kopper and L. Stodolsky, unpublished. 1201 R. Rfickl, Phys. Lett. 129B (1983) 363. 121 ] CELLO Collab., H. Behrend et al., A study of final states with four charged leptons in e÷e- interactions, CEN Saclay DPhPE 84-08, DESY 84-103 (1984); An investigation of the process e÷e- ~ tz~-, DESY 84-101 (1984). 1221 UA1 Collab., G. Arnison et al., Phys. Lett. 147B (1984) 493. 413