Charged scalar field and quantum number violations

Volume 161B, number 1,2,3

PHYSICS LETTERS

24 October 1985

C H A R G E D SCALAR FIELD AND Q U A N T U M N U M B E R V I O L A T I O N S A. Z E E Department of Physics, FM-15, University of Washington, Seattle, IYA 98195, USA Received 17 June 1985 We show that the introduction of a charged scalar particle into the standard theory leads to numerous phenomenological consequences. In particular, muon-neutrino scattering on electron can resonate in the s-channel, a fact which is potentially important in high-energy neutrino experiments and conceivably relevant in explaining the recently reported underground m u o n events from Cygnus X-3. We focus on the violation of various q u a n t u m numbers, including electron, muon, and tauon numbers.

The SU(3) X SU(2) x U(1) theory, having been confirmed resoundingly by experiment, is now enshrined as the standard theory. But we must keep in mind that the nature of the scalar sector remains shrouded in mystery and future experiments may well reveal a surprise or two in this sector. Over the years, we, along with other investigators, have tinkered :[:1 with the scalar sector. Several years ago, we considered a rather minimal modification by introducing [2] a charged + 1 scalar field h. In the standard theory, a single Higgs doublet changes the lepton doublets ~ to the lepton singlets cb. The coupling +L~gR -a is automatically diagonal in the family index a and the theory conserves electron number Le, muon number L,, and tauon number L, separately. Given the quantum numbers of the fermions in the theory, we can easily list all possible Yukawa terms. In the lepton sector, the doublet ~ transforms as (2, - ½) under SU(2) x U(1). Thus, we can couple two lepton doublets to a triplet scalar field (3,1) or to a singiet scalar field (1,1). In the name of simplicity, whatever that means, we opted for an SU(2) singlet, which we denote by h, with the coupling ij

a b (';,LC%L)h

[here 0' denote SU(2) indices]. The h field has ,1 For an example of a recent attempt, see ref. [1].

O)

charge + 1 and changes a charged lepton to an anti-neutrino. The only other terms allowed by SU(2) x U(1) are potential terms for the h field: h+h, (h+h) 2, and h+h¢+~. In particular, h cannot couple to quarks. (Charge conservation forbids h from having a vacuum expectation value.) Our model thus represents a "natural" and minimal extension of the standard theory. We have added one charged particle. Our model has the characteristic and striking feature that the h interaction necessarily flips family. Fermi statistics requires that the coupling f,b must be antisymmetric in a and b. The h field maximally defies the "kinship hypothesis" [3]. The h coupling reads in detail: 2[ fe~(ueCIx-

t,~Ce) + f~,(~,~C~- u¢CIx)

+ f , e ( u ~ C e - v~C~)]h.

(2)

(We have suppressed the ubiquitous subscript L reminding us that only left-handed leptons are involved.) As a first guess, we might imagine the three coupling strengths to be roughly equal. But considering how vastly the Yukawa couplings of the Higgs differ we may want to keep an open mind on this question.

Lepton number conservatives. Since we can simply define h to have lepton number L = L e + L~, + L~ = - 2, lepton number is not violated, thus guaranteeing that the neutrinos will remain

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z_+<

Table 1

Q.~

Q u a n t u m n u m b e r of h interaction terms. Term

Le

L#

L.

(et*h) (erh) (grh)

0 0 -1

0 1 0

0 +1 +1

(a)

Violation of electron, muon, and tauon numbers. Thus, h exchange could lead to numerous processes forbidden .2 by the standard theory such as v ~ e - ~ v,g-, v~e-~ v~r-, ~ e - - + ~ g , and v~e --, v~r (fig. 1). We mention in passing that the rather futuristic experiment searching for processes such as v~e --, v~r- has a neutrino energy threshold of m 2 / 2 m e = 4 x 10 3 GeV. (Unfortunately, energetically more accessible processes such as v ~ e - ~ vd~ cannot be distinguished from the standard process v,e + v d , - since the lepton fields involved are all left-handed. But, as we will mention presently, these nonstandard processes may be important in astrophysical processes.) The exchange of h also contributes to allowed processes such as v,e v~e . Here the energy and angular distributions m a y allow us to distinguish, in principle of course, the h contribution from the standard Z contribution.

Resonant v~e- scattering. A striking feature of our model is that v, scattering on electron occurs with h mediation in the s-channel. Thus, the scattering cross section of v~ in matter can be resonant for a neutrino energy equal to 2.3 × l0 s GeV ( m h / 1 5 GeV) 2. (3)

,2 This was not stated correctly in ref. [2].

142

(b)

Fig. 1. (a) h coupling and (b) resonant production of m u o n s in rue collision.

massless. To generate neutrino Majorana masses, we have to introduce additional structures [2] into the model. We will not do that here. By convention, we can say that, of the three terms contained in eq. (2), the (egh) term forces h to carry electron number Le = - 1, and L, = - 1, and L~ = 0. But then the other two terms will violate L~, L,, and L~ indicated in table 1.

E* = m 2 / 2 m e -

h-

(We will show below that m h may be of order 15 GeV.) We compute the cross section of v ~ e - ~ v ' f to be

o = ( iT
]

× [(s- mh)2+ m~,F~]-1

(4)

(Here f ' denotes the coupling of h to the u'g channel and F h the width of the h.) As expected, for m 2 >> s >> m 2, 0 ocs and for s >> m 2, 0 o: 1/s. M o r e interestingly, at resonance s = m 2, the cross section becomes enormous, ff....

=(8~r/m~)r(h--'v.e-)r(h~v'¢)/F2.

(5)

Since there are six outgoing channels, Ores.- 7 × 10-3°cm2 (15 G e V / m h ) 2, which may be compared to the deep inelastic cross section for muon production by a p, of energy E, - 7 X 10- 36 c m 2 ( E / 1 0 3 GeV). [The formula in eq. (4) does not hold if v'E= p,e- due to interference with the standard processes, but at resonance the difference is slight.] N o t e that a muon is ultimately produced in four of the six channels. The mean free path of a m u o n neutrino with energy E * in rock is as small as e * = ( m h / 1 5 GeV)2(1 km).

(6)

Possible relevance to Cygnus X-3. These features of our model m a y be relevant in explaining the recently reported observation *3 of high-energy m u o n s in the Soudan underground detector, apparently initiated by neutral primaries coming from Cygnus X-3. The X-ray binary Cygnus X-3 was discovered to a source of ultra high energy g a m a rays up to 106-107 GeV [5,6]. It is supposed [7] that the pulsar produces ultra high energy *3 We understand that there is also some data from the Irvine- M i c h i g a n - Brookhaven collaboration.

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protons which collide with the companion star, producing *r° 's, whose decays generate g a m m a ray, and rr + 's, whose decays generate v~ and F~. Thus, a flux of ultra high energy p, and ~, is expected [8]. Of course, we know of no reason why the neutrino energy spectrum should happen to peak around E * and we should actually integrate the spectrum ~ ' ( E ) against the cross section in eq. (4) in order to estimate the muon production rate:

f dE~(Elo(E) = (ffcf'2/32~r)mE(rr/2memhFh)o~(E*) =(ffef'Z/64)(mh/Fh)(1/me)~(E*).

(7)

N o t e that in the standard model ~,~ and ~ cannot scatter on e - via an s-channel resonance. One ,4 must count on a flux of ~e from the decay of the muons coming from ~r + decay which can scatter off the e resonating on the W - . The ve and ~¢ will have somewhat lower energies than the v, and ~,. Alternatively, one can assume oscillation of the ~,~ or ~, into ~¢. The resonance cross section in our model is equal to the resonance cross section in the standard theory multiplied by ( 3 " / w / / m h ) 2.

We now turn to the task of estimating m h and the three f 's. Charged scalars are produced in e + e - annihilation with a cross section o (e + e----, h + h - ) = ¼o(e+e ----, ~t+/~- ) and so will not be obvious in o ( e + e ---, all)/o(e+e ----,/~+bt ) given the present experimental errors. Recently, some precision measurements have been reported and we understand that [10] m h probably would have to be more than about 15 GeV. (The process e + e----, h +h could also occur via neutrino exchange in the t-channel.) Muon (g-2) provides another bound: the graph in which a virtual p h o t o n goes into a h+h - loop gives 6a -- l(a/,lr)Z(m~/mh) 2 -- l ( a / , / r ) 4 ( 1 5 GeV/mh) 2 well within the accepted limit [11]. The h also contributes to m u o n ( g - 2) via its coupling to m u o n and neutrino (fig. 2), by an amount of order :~4

The possible relevance of s-channel resonant particle production by high energy neutrinos has been suggested by Wilczek [9].

24 October 1985

tz -

vr

e-

Fig. 2. ~t ~ eT decay; a similar graph contributes to the magnetic moment of the muon.

(fez +f~)(m~/mh) 2, to be compared with the . weak correction - g2(m~/Mw) 2 which has not yet been seen. A more stringent bound comes from/~-decay. The exchange of h causes the muon to decay into four possible channels: /~---* e-~¢u,, e-~,p~, etc. But since the leptons involved are all left-handed, we recover the standard (V - A) form after Fierz re-arrangement; h exchange does not distort the angular and energy distributions in ~t decay. The rate of/~ decay is corrected approximately by the 2 2 2 factor [1 + 2(f~,/mh)/(g / 2 M w2) ] . Nominally, the data on universality allows a deviation of one percent or so, implying ffe/4~r _< 3 X 10-6(row/15 GeV) 2. (In fact, the strangeness changing element Uus suffers from large uncertainties. Note also that h exchange adds to the/L-decay amplitude and so we cannot simply increase Uub.) The h is weakly coupled but not excessively so. For comparison, the corresponding quantity for the Yukawa coupling of the Higgs doublet ranges from 10 a3 to 1 0 - 5 /~ --->eT decay. Since the electron and muon numbers are no longer separately conserved, we expect that infamous decay/~ --> eT. The decay amplitude (fig. 2) is proportional to f~,.J.~¢,in accordance with the general observation that q u a n t u m numbers are violated due to the "clash" between different terms in the lagrangian. In order for the branching ratio to be below the experimental bound of 1.7 x 10-1°, we must have f,,f~e < 2 X 1 0 - 8 ( m h / 1 5 GeV):. In this model, /~--, 3e occurs only by pair production from/~ --, eT. One might have expected neutrino oscillation in this model, but since the relevant one-loop 143

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graph does not break the gauge symmetry its effect is described by a common wavefunction renormalization on the lepton doublet.

h- I / e+ / e

"l"

/

Production and decay. We next turn to the production and decay of the h. Once produced, the h particle decays rapidly into channels such as e-~%, /~-pc, r - u , etc. with a partial width

r(h --, e'~,) = =

(f2/8~r)(1

(f2/8~r)mh

-

-

for m h

m ¢2/ m h )

2 2m h

p or5

(b)

(o) /

>>m,,. "

(There are six channels altogether.) Thus, h has a rather narrow width. In eq. (7), the factor (m h / F n ) compensates for one power of f 2. The narrow width of h, or equivalently the smallness of f 2, implies that the integrated cross section will be small, of course. The h m a y be produced pair-wise via a virtual photon or a virtual Z. Thus, we expect to see processes such as pp o r ~ p ~ h ÷ h - + X ~ (/~+/~or e r e - o r / t r e - or e+/~ - ) + X'. The Z coupling to h is determined by SU(2) × U(1) and we find F(Z

h+h -)=¼atan28Mz(1

- "Z 2] ' 3 / 2 , -- " t 4 m h2/ M

corresponding to F(Z ~ h + h - ) / F ( Z ~ tt+# - ) = 0.15 for M z >> m h. In electron-positron annihilation, the processes e + e - ~ h + h - ~ (/~+/~- or e+e - o r / t + e or e+/~ - ) + neutrinos would have to be separated from Tr ~.- production and decay. The cross section for e r e - is smaller by a factor of 4 but the ~"decays into a lepton plus neutrinos only 35% of the time. In the rest frame of the h the decay lepton emerges isotropically. Thus, the angular distribution will be different. Just above the h threshold, the two charged leptons emerge in a roughly uniform and uncorrelated angular distribution, while the two charged leptons from T decay will be more or less back to back. The h m a y also be produced singly off a lepton line in electron, muon, or neutrino deep inelastic collisions: ( e - o r / x - ) N ~ h - ~ X , v~N ~ h e+X, and so on (see fig. 3). The decay of the h leads to anomalous events such a s / ~ - N ~ e - X ' , p,N /~ e + X ' and so forth. The amplitudes for these anomalous processes are suppressed by a factor of f and by the fact that the lepton propagator is 144

e+/

~-. /Yh-.~e S

(c)

~'~

Fig. 3. Production of the h.

far off-shell. These anomalous processes may be c o m p a r e d to the structurally similar processes in which a W -+ is shaken off a lepton line in deep inelastic collisions.

Variations on a theme. Referring to table 1, we see that if we so choose, we have the option of conserving L e, L~, or L, by omitting one or more terms in eq. (2). We can go on to consider the other Yukawa terms allowed by SU(2) x U(1) and renormalizability. For instance, we can couple a doubly-charged scalar to lepton singlets, viz. Lb~C~bk. The coupling )~b has to be symmetric. We note immediately that k causes the decay /~ ~ e + e - e -, implying a stringent bound on f 2 / m 2 . N o t e that if both k and h are introduced, gauge symmetry allows the coupling M k + h h which violates L by two units. Bizarre but (probably) inaccessible processes such as "r+~ e #+/t+~,~ m a y proceed. (But who knows, the coupling M may be unexpectedly large.) Similar games may be played with quarks. We can introduce Yukawa couplings such as d RCd Rn, U RCd RP, U RC U Rr, and q LCq LS" The scalar fields n, p and r have charges 2/3, - 1/3, and - 4 / 3 respectively. Each of these fields transforms as color 6 if the corresponding couplings are symmetric in family, as color 3 if anti-symmetric in family. Similar remarks may be made for the field s. Since the couplings are parity violating, the effective coupling = (Yukawa coupling/m n,p,r,s) 2 must be less than the weak coupling. We must insure that

Volume 161B, number 1,2,3

PHYSICS LETTERS

the A S = 2 i n t e r a c t i o n (dRCdR)(~RCSR) is b e l o w the w e l l - k n o w n b o u n d d e d u c e d from K s - K L p h e n o m e n o l o g y . A l o n g this line, we m a y c o n s t r u c t m o d e l s of s u p e r w e a k C P violations. F o r the s a m e reason that the h c o u p l i n g b y itself does n o t violate L, these couplings d o not v i o l a t e b a r y o n n u m b e r B. Again, if we so choose, we c a n violate B b y two units with c o u p l i n g such as ( n p p ) o r ( r n n ) a n d generate n e u t r o n - a n t i n e u t r o n oscillation. W e have r e m a r k e d [13] earlier that if these exotic scalars t r a n s f o r m as color 6, then since n e i t h e r 6 × 3 n o r 6 × 3 c o n t a i n s a color singlet, the p r o t o n would not d e c a y a n d these s c a l a r p a r t i c l e s m a y be relatively light. In this way, we c a n c o n s t r u c t m o d e l s in which B a n d L can e a c h b e v i o l a t e d b u t in which the p r o t o n r e m a i n s stable. F o r e x a m p l e , b y i n t r o d u c i n g the c o u p l i n g ( p p p h ) we c a n have the d e u t e r o n d e c a y i n g into /~+ve. W e t h a n k G. Gabrielse, W. H a x t o n , J. R u t h e r f o o r d , a n d F. Wilczek for helpful discussions, M. R u d e r m a n a n d J. R u t h e r f o o r d for d i s c u s s i o n of the Cygnus X-3 data, a n d T.J. Lee for c h e c k i n g the sign of h exchange in/~ decay. N o t e added. The h resonance is extremely narrow. F o r muon production in rocks, the motion o f the electron broadens out the peak by a factor o f ~1,05, giving a width to E* o f order 10 4 GeV(mh/15 GeV) 2 . The possible relevance o f the h resonance to detection o f ultra-high-energy neutrinos from astrophysical sources depends cruciaUy on the neutrino energy spectrum 9r(E), o f course. In principle, the energy spectrum o f the produced muon would reveal the resonance effect, if any. Below resonance, the cross section o in equation (5) rises linearly but is much less than the cross section Oconv for the conventional process vN ~ / a X . Since Oconv rises logarithmically rather than linearly for neutrino energy E exceeding ~ 1 0 5

24 October 1985

GeV, one might imagine a regime of extreme energies in which o dominates Oconv. However, it turns out that E and m h both have to be enormous. One may also mention the possibility that f r e -= 0 which can be guaranteed by the conservation o f L e + L r - L u. In this case,fu r may be large without running afoul o f t h e / l -* e7 bound. With the f couplings small, the process r u e - ~ h7 may provide a more efficient source for muons. Finally, we mention that the azimuthal angle distribution o f the Cygnus X-3 events suggest that the primary cannot be a neutrino. However, the data in this regard is not firmly established. Some aspects o f the model o f ref. [2] have been analyzed by Petcov [14]. We have benefited from discussions with A. Dar, S. Barr, E. Kolb, M. Turner and L. Wolfenstein. References

[1] [2] [3] [4] [5] [6] [7] [8]

V. Silveria and A, Zee, UW preprint (1985). A. Zee, Phys. Lett. 93B (1980) 389. F. Wilczek and A, Zee, Phys. Rev. Lett. 42 (1979) 421. M.L. Marshak et al., Phys. Rev. Lett. 54 (1985) 2079. M. Samorski and W. Stamm, Astrophys. J. 268 (1983) 117. J. Lloyd-Evans et al., Nature 305 (1983) 784. D. Eichler and W.T. Vestrand, Nature 307 (1984) 613. T.K. Gaisser and T. Stanev, Phys. Rev. Lett, 54 (1985) 2265; E.W. Kolb, M.S. Turner, T.P. Walker, FNAL-Publ 85f40A; A. Dar, Technion-PH-85-25 (1985). [9] F. Wilczek, private communication and to be published. [10] J. Rutherfoord, private communication. [11] T. Kinoshita and J. Sapirstein, in: Atomic physic, eds. R.S. Van Dyck Jr. and E.N. Fortson (World Scientific, Singapore, 1984). [12] W.W.K.innison et al., Phys. Rev. D25 (1982) 2846; A.L. Hallin et al., in: Intersections between particle and nuclear physics 1984, p. 477. [13] D. Seckel and A. Zee, in: Proc. Neutron-antineutron oscillation 1982, eds. M.S. Goodman, M. Machacek, and P.D. Miller, p. 51, and references therein. [14] S.T. Petcov, Phys. Lett. 115B (1982) 401.

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