P H Y S I C A L R E V I E W D V O L U M E 11 , N U M B E R 9 1 N A Y 1 9 7 5
Cancellation of neutral A S # 0 hadronic processes without charmed particles*
Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11 794
(Received 14 January 1975; revised manuscript received 6 May 1975)
The SU, @ U, gauge model of weak and electromagnetic interactions introduced by Weinberg and Salam is extended to the group SU2@ UI @ U,. A natural cancellation of the effective Lagrangian for neutral strangeness-changing semileptonic processes is achieved without using charmed quarks. The additional neutral vector boson may have an arbitrarily small mass.
In the renormalizable unified theory of weak and electromagnetic interactions developed by Wein- berg and Salam,' the incorporation of hadrons r u n s into the problem of suppressing the AS = 1 neutral hadronic cur ren ts . A popular way of circumventing this difficulty i s by the addition of a t least one more "charmed" quark, B' , to the conventional @ , I, h tr ipleta2 This postulate h a s received much theoretical i n t e r e ~ t , ~ but it i s s t i l l necessary to explore al ternat ive mechanisms.
The possibility we investigate i s the extension of the model based on the group SU28 Ul to one based on the group SU2@ U,@ U,, where the additional vector boson i s responsible fo r the cancellation of A S = l neutral cur ren ts . This scheme was inves- tigated previously by Schechter and Ueda4 but their analysis w a s not sufficiently general . F o r example, they did not explore al l possible Higgs- Kibble systems. It will be shown that we can elim- inate the AS = l semileptonic effective Lagrangian in a natural manner.' The leptonic s e c t o r in this model i s compatible with present data on v, and v, ~ c a t t e r i n g , ~ whereas that of Schechter and Ueda i s not. The leptonic sec tor , fu r thermore , i s a lmost identical to the SU,B U,@ U, model of Dolgov e t al.7 However, their motivation for studying this model i s different f rom o u r s and their Higgs sys tem i s not a s general . In particu- l a r , they made no attempt to include hadrons. We d o not consider higher-order induced effects in this paper .
Since this work was commenced, the discovery of J and $ resonances with m a s s e s between 3 GeV and 4 GeV,' and the suspicion that they may have something to d o with weak interactions makes it interesting to investigate the possibility of a unified gauge theory of weak and electromagnetic interactions with a neutral vector boson much
lighter than the Z of the Weinberg-Salam model. The group SU,@ U,@ Ul i s a possible group f o r a model which exhibits this feature. It i s to be emphasized that the l a t t e r reason for the extension of the SU, 8 U, model to SU, 3 U, 8 U, i s logically independent of the f o r m e r , and that the possibility of a neutral vector boson lighter than the Z of the Weinberg-Salam model i s interestingg independent- ly of any identification with the new resonances.
The group SU,B U,@ U, i s a possible subgroup, for the weak and electromagnetic interact ions, of the gauge group of the world,'' r a t h e r than the group SU,@ U,. A model based upon the group SU,@ U18 U, has been discussed elsewhere within a different context."
11. THE SU28 Ul YJU, REPRESENTATIONS
In complete correspondence with the Weinberg- Salam model1 f o r the leptons we consider
a s a left-handed doublet, and e , = $(I - y, )e a s a right-handed singlet, under SU,. The muon i s t reated analogously. We group the conventional quark s ta tes , d',X,X, into a left-handed doublet,
a left-handed singlet, hCL, and th ree right-handed singlets, B,, X,,h,, where
a r e the Cabibbo rotated s ta tes . (0, i s the Cabibbo angle.) We a s s u m e an approximate SU,-symmetric
11 - C A N C E L L A T I O N O F N E U T R A L A S # 0 H A D R O N I C P R O C E S S E S . . .
strong-interaction Lagrangian. The choice of Higgs-Kibble s c a l a r fields,'' whose
vacuum expectation values break the symmetry down to the electromagnetic U, group and s imul- taneously give m a s s t e r m s to the leptons, quarks, and vector bosons in the Lagrangian, i s not un- ambiguous. There must be several s c a l a r s to pro- vide the four vector bosons with masses , and t h e r e must be a t l eas t one s c a l a r SU, doublet s o that the electron (and muon) and the 6' quark can obtain m a s s e s through the Higgs-Kibble mech- an i sm.
The following s imple possibilities for the s c a l a r fields exist:
(a) one complex doublet coupled to the leptons and the quarks , together with one singlet coupled to the quarks;
(b) two complex doublets, one coupled to the leptons and the other coupled to the quarks ;
(c) two complex doublets, one coupled to the leptons and the other coupled to the quarks, to- gether with a singlet coupled to the quarks.
We shal l d i scuss the situation in t e r m s of pos- sibility (b) s ince only in this c a s e does the natural cancellation of the effective Lagrangian occur . Other features discussed occur with a l l t h r e e pos- s ibi l i t ies , and c a s e (b) i s therefore representa- tive; only t r ivial modifications a r e required to t rans fe r to c a s e s (a) o r (c) .
111. THE TWO-DOUBLET REPRESENTATION
Denoting the gauge coupling constant of SU, by g , and absorbing the coupling constants of the U, groups into the hypercharges (qi , p i ) , the general renormalizable Lagrangian, f o r weak and electro- magnetic interactions, invariant under the group S U 2 ~ U , @ U , , i s C = 6 : ,,,, e + 6 : , , , ~ , , , + 6 : ~ e p t , , , + ~ ~ , d , , n , + C,,, + c,,, , where
1- - Cgauer = -4FpU. FPlJ - +B P LJ Bp"- icPu c'", (3.1) gscalars =-$[a,$,- ~is@,~~P,+i(q,~,+~,~,)P~l~
- %[a,q2 - iig@,.;~,+i(q~~,+~~~~)@~l~ , (3 -2)
ghadions = -$L;LY,[ap- +igi?,.? - i (q ,B ,+P,C, ) l i~
- XCLyu[a,- i(q,B,+ P,C, ) I b L - ~ ~ y ~ [ a , - i ( q , ~ , + ~ , c , ) p ~
- ~ ~ y ~ [ a , - i ( q , ~ , + P , ~ , ) 1 ~ R - XRyp [a, - i ( q , ~ , + P,C, ) I ~ R , (3.4)
-. The definitions of the field t ensors F,,, , B,, , C,,, in (3.1) a r e
We note that the gauge fields B,, C, may be r e - defined without l o s s of generality s o that
Spontaneous symmetry breaking occurs when the sca la r fields
a r e allowed to develop the vacuum expectation values
( i = 1, 2 ) . (3.7)
In the t r e e approximation, the charged vector mesons W i = ( l / f i ) ( W ; r zW;) now, by the Higgs- Kibble mechanism," acquire the m a s s
The neutral vector-meson m a s s matr ix i s given, f rom (3.2), by the expression ($.gwt + ~ B , + P , C , ) ~ F , ~ + ( i g ~ ~ + q B , , + p ~ C , ) ~ F , 2 , and w e define the photon field
where e2 = 4q2g2/ (g2 + 4q2) i s the square of the electr ic charge. We let the m a s s mat r ix of the vector mesons be diagonal on the basis A, , X u , Z,, where
where A+ and X - a r e r e a l f o r C P conservation, and
g 2 + 4 q 2 + h + h - = 0 (3.9)
for orthogonality. The eigenvalues 12- =Mx2 and A+=MZ2 of the m a s s matr ix a r e given by
D E S M O N D D A R B Y
A,=$[p:F? + p 2 2 ~ 2 2 + a ( g 2 +4q2)(F12 + F Z 2 ) ] which a r e necessary and sufficient to yield a di- agonal lepton and quark m a s s mat r ix upon symme- * 3[p l2F ,2 + PZ2F2' + a(g2 + 4q2)(F12 + F ~ ' ) ] ~ t ry breaking," implies that -
- (p, - p2)'(g + 4 q 2 ) ~ 1 2 ~ 2 2 ) 1 1 2 . (3.10) Pq=P3-Pl , A useful identity i s
P s = c - P z ,
where and it i s obvious by inspection of the m a s s mat r ix that p 6 = p B = p 9 = c .
s o that there i s no lower bound on MX2 when MZ2 i s sufficiently large. This may be seen by con- s idering (p, - 0,)' sufficiently smal l in (3.10). This i s in contrast to the model discussed in Ref. (9).
We now investigate the constraints on the hyper- charges. The c o r r e c t coupling of the leptons and the quarks to the photon implies that
Hence, only 6, viz., q ,p1,p2,p, , b, c, of the 18 hypercharges introduced a r e a rb i t ra ry .
C,,, = V(+,, Q2) i s the gauge-invariant potential which i s a polynomial in Q, and 9, of maximum dimension 4 for renormalizabi l l ty . The general polynomial satisfying these requirements can indeed have a minimum, in the t r e e approxima- tion, a t the vacuum expectation values ( @ i ) , given in (3.7). With this potential and the t e r m s in C,,,, , (3.5), a l l par t ic les in the model, except the photon and the neutrino, have a m a s s upon spontaneous symmetry breaking. The couplings G,, G,, Gi ( i = l , 5 ) a r e determined by the fermion m a s s e s and the Cabibbo angle, which a r e f r e e in the model.
The F e r m l coupling constant f o r p decay i s r e - covered in the local l imit by the identification
q 6 = q 8 = q 9 = b . G, - = - g2
Also, gauge invariance of the t e r m s in C,,,, , (3.5), fi 8Mw2'
IV. THE NEUTRAL SEMILEPTONIC LAGRANGIAN
The coupling of the leptons and the quarks to the bosons X and Z i s given by the following Lagrangian, using (3.3), (3.4), (3.8), (3.12), and (3.13):
A necessary, but of course not sufficient, condition for a conspiracy between X , and 2, to cancel semileptonic A S = 1 neutral p rocesses i s that the leptonic cur ren ts coupled to X , and 2, should be pro- portional. This follows if we d o not allow the leptons t o t ransform under the additional U, group, and implies that
p4 = p 3 = p l = 0 . (4.2)
We now evaluate the local limit of Eq. (4.1) t o find the effective Lagrangian for A S = 1 neutral semileptonic (s l ) p rocesses :
1g= ?z[(g2 + 4q2)D,5~g VL - (g2 - 4 q 2 ) z ~ Y l i e ~ + 8 q 2 ~ ~ ~ g e ~ ] , j t S = is in8, C O S ~ , ( % , ~ , ~ , +XLy,TiL).
It i s trivial to show that $fis"a 0 using Eqs. (3.9), (3.10) and (3.11). This i s surpris ing s ince one would expect, a s in Ref. 4, that this could only be achieved by s o m e prudent choice of the otherwise a r b i t r a r y p a r a m e t e r s q and p2. The can- cellation i s therefore in some sense natural.= All we need i s the condition that the leptons d o not t ransform under the additional U , group.
The problem of suppressing AS = 1 effects when the local limit i s not valid remains (e.g. in K decay where q2 =mK2) , especially if we wish M, to be smal l enough to associate X , with one of the new resonancess below 4 GeV. An investigation of higher-order p rocesses i s necessary to s e e if the induced A S = 1 semileptonic decays can be sufficiently suppressed.
V . THE ELIMINATION OF THE AS# 0 CURRENT
Even with the remarkable cancellation discussed in the l a s t section, A S = 2 neutral cur ren t process- e s s t i l l exis t , a s can be seen by inspection of Eq. (4.1). However, the question a r i s e s of eliminat- ing the strangeness-changing neutral cur ren t ent i rely by choosing pz and q such that the follow- ing relat ions a r e valid:
This implies that h + = h - s o X, and Z , become de- generate and the orthogonality condition (3.9) i s violated. Nor is i t possible to make the equal- i t ies of (5.1) only approximate and obtain suppres- sion of A S = 2 processes . In fact such processes in this model will be of o rder G,. This indicates that fu r ther s tudies a r e required to construct models where vector bosons, ra ther than charmed quarks, a r e responsible for suppression of strangeness-changing processes .
VI. BOUNDS FROM LEPTONIC SCATTERING
One of the problems encountered in the ea r ly attempt a t utilizing the S U , U,% U, model4 was compatibility with neutrino-electron scat ter ing experimental data.= The theory presented here i s
quite compatible with data on elast ic neutrino interactions. T o see this we examine the effective Lagrangian for the f our-fermion electron neutrino interaction which may be written a s [using Eqs. (3.141, (4.1)1,
We have used the following definitions:
In the c a s e discussed in Sec. IV, Eqs. (4.2) and (6.3) imply that p, =p2 = 0 , s o that
The condition P2 > 1 may be used to calculate the allowed region for the p a r a m e t e r s C, and C, which, on a (CA,C,) plot, i s the inter ior of the trapezoid bounded by
2556 D E S M O N D D A R B Y 11 -
The intersection of this region with the experi- mentally allowed regiona i s finite.
If Eq. (4.2) i s not t rue , [i .e. , leptons t ransform under the extra U, group, ] i t becomes m o r e diffi- cult to find the allowed range for the p a r a m e t e r s C, and C , , but other s imple c a s e s d o exist; e.g., in the singlet-doublet model [case (a ) of Sec. 111, Eqs. (6.1), (6.2), and (6.3) a r e c o r r e c t providing
This implies that p, = 1 s o that C, = h. Clearly, solutions exist for which C, h a s reasonable values.
The recently discovered resonancess a r e usually assumed to be 1- states . However, the angular distribution of the decay products, especially in the leptonic channels, i s uncertain. If the A', of this theory w e r e to be identified a s one of the recent resonances, the decay angular distribution i s determined by the ra t io of the couplings to the vector and axial-vector cur ren ts Zy,e and Zy,y,e. These couplings come f rom the t e r m X, (g,Zy,e +gAZy,~ ,e ) in the Lagrangian (4.1), s o that the coupling constants g , and gA a r e gi en by, using (3.9), (3.10), (3.1 I ) , and (6.3),
In the c a s e discussed in Sec. IV, p, = p, = 0; it i s interesting to observe that the simple choice p 2 = $ implies o!,/o!, = 0, which would give a (1 + cos2p) distribution for the electrons in the decay Xu- e'e-. The identification i s , of course , purely speculative, but it i s nevertheless interesting that such a possibility exis ts .
V11. OTHER REPRESENTATIONS
With case (a), the doublet-singlet representat ion, and case (c), the two-doublet-one-singlet r e p r e - sentation, the cancellation of the effective local interaction for A S = 1 neutral semileptonic pro- c e s s e s cannot be achieved naturally.
While we a r e making an Abelian extension of the SU2@ U, model, the question a r i s e s about yet another model based on the Abelian extension to SU,@ UU,@ U,B U,. In this situation the proportion- ality of neutral leptonic cur ren ts i s no longer nec- e s s a r y f o r the cancellation of the effective neutral AS = 1 semileptonic interaction. However, it i s necessary, for the elimination of the en t i re s t rangeness changing neutral cur ren t , that a t l eas t two of the th ree neutral vector bosons a r e de- generate ,...