Triangle anomaly free weak interaction with two neutral currents
C V A V B C H A N D R A R A J U
KSM, Mining Engineering Complex, (OU), Kothagudem 507 106, India MS received 11 August 1984
Abstract. Considering that the neutral interaction is free of triangle anomalies we derive an expression for the most general neutral interaction with two neutral currents. We show that the Bargers version is a special case. We also determine the interaction whenever Z, D are mass eigen states and show that this differs from the Barger's version in an essential way.
Keywords. Left-right model; anomaly free neutral interaction; Bargers ver~aa.
PACS No. 12.30, 12.10
In this note we derive an expression for the weak interaction Hamiltonian with two neutral currents. We consider the theoretical requirement that this interaction Hamiltonian be free of triangle anomalies (Adler 1969; Bell and Jackiw 1969).
Let the two neutral currents come from an SU (2) L x SU(2)~ x U(1) gauge group. Let J3L' J3a andj~be the neutral currents o f this group with the associated gauge b o s o m W3L, W3a and W r. The corresponding gauge constants are g L, OR and g y. The interaction in the neutral sector is given by,
H NC = g t W~t + g , W3.J ,, + g y Wrj r"
O)
The electromagnetic c u r r e n t Jem must he equal to J3L +J3a + J r and in general the other two currents are Ji = J3L + a~J3~ + bjJr, where i = 1, 2. Denoting the corresponding physical vector bosons as Aem and (Zt, Z2) the neutral interaction is given by,
Hsc -- eAemJem + 01 Z t j t + o2Zzj2,
(2)
where e is the electromagnetic coupling constant, and gl, 02 are the weak coupling constants. Using the expansions of Aem and Zi and the orthonormality relations among them we easily find that,
e2/g2r--(l--XL--Xa), bt = ( K + a t a 2 ) / ( K +a2), x t = e2/o2t = ( K + a t a 2 ) / ( l - a l ) ( l - a 2 ) = Kx~, b2 = (K + at a2)/(K + at ),
e 2 (K + a z ) e2(K + aj)
012 = xL(1 - a t ) ( a 2 - a l ) ' and g, z = XL(1 --a2)(al --a2)"
(3)
In the above we have defined K through the relation g~ = K g~. Whenever K = 1, it is a left-right symmetric model, otherwise it is a left-right asymmetric model. By eliminating L657
L658 C V A V B Chandra Raju
b~ and Jr the weak part of the Hamiltonian is given by, (1 - al ) (a2 JzL - K Jz~) Hint = g t Z t weak (K + a2)
+ g2 Z2 (1 - a2) (ax JzL-- K JzR) (K + a l )
where Jz,~ = J3L- xLJm and JzR = J3s - xRjem"
Let (Z, D) be vector bosons with the following mass squared matrix,
(4)
where g2 z is an overall constant and it will be defined shortly. The above matrix in (5) can be diagonalized by an orthogonal matrix R (~)
= ( cos~0 sin ~'~,
\ - s i n ~b c o s ~ J where
tan 2~ = 2B/C - A. (6)
We can always rewrite (4) in terms of (Z, D) bosons, which are not mass eigen states, by rotating (Zs, Z2) basis through R (~,) to (Z, D) basis. In other words,
(Zst=R(~)(ZD).Z2 (7)
The purpose of this rotation is to see that the total interaction of Z with JzR is zero. We therefore introduce (7) into (4) and require that the total interaction of Z with JzR he zero. This simple requirement yields the following expressions for the various quantities.
Hint = gzJzL Z + gzD(flJzL + (~ + ~)Jzx), (8)
where
and
( 1 - a l ) ( a 2 - a l ) c o s O , (9)
gz = 01 ( K + a 2 )
a 2 tan ~b + ) cot ~b, (10)
t~ = (a2 - a , ) (a2 - a ,
(~,+/~--
- - K ( t a n ¢ + c o t ~ , ) . 0 1 )(a2 - al) We will obtain (8), if and only if,
gl ( 1 - a t ) ( K + a l )
t a n O = 02 (1 - a 2 ) ( K +a2)" (12)
In general ,4, B and C of (5) are functions of the various VEV'S, /~ and (0c +/~) as discussed by Barger et al (1982a, b). If,4, B, C are known then tan ~ is known because of (6). Usually ,4, B, C are not known. Our expression (8) for /-/int is free of triangle " " w e a k
anomalies. But gz, {3 and (~ +/~) here are all functions ofax, a2 and K. Moreover tan ~, is
also a function o f these three variables only. Barger et al (1982, 1983) express Oz, fl a n d (a + fl) as functions o f only two variables x L and x~. Barger et al consider ~ to be an independent variable (see Barger 1983). In order to obtain Bargers expressions for Oz, [3 and (~ + fl) we first find an expression for a2 from (3) in terms ofxL, K and al. This gives
(K - x L + al xL) (13)
a2 = (1 - x L ) a I + x L "
F r o m (3), (12) and (13) we also note that,
y = tan2~b = o~/g~ [(1 - x L ) a I + x t . ] s
= ( K _ x L _ K x L ) . (14)
Equation (14) can be used to determine a~. We have,
al = ( - x t + p ) / ( 1 - xL), ( 1 5 )
where p2 = y ( K - x L - K x L ) . (16)
In the expansion o f their Z-boson, Barger et al (1982, 1983) do not use W3a. This means that whenever we put a~ = 0 in our expressions we should obtain expressions for fl, (a + fl) and g z identical to theirs. Whenever p = x L, a I = 0 and for this value o f p from (16) we find that
t a n s ¢, = xL x ~ / ( l - x ~ - xR). (17)
Choosing the positive sign for the square root o f 17, and with ai - - 0 and a2 = - ( K - x L ) / x L, we find that
and
~/2 /2, (18)
O z - - e / x t ( l - x L ) 1
# = (x~x~W 2
( I _ x L _ x ~ ) I / 2 = t a n ~ , (19)
x[/2 (1 - x L)
(oz -~- f l ) = XIRI 2 (1 - - X L - - XR) 1/2" (20)
All these expressions agree with those o f Barger et al. Equation (19) shows that ¢, is not an independent variable. The (Z, D) bosons o f (8) for this situation (i.e. whenever (18) to (20) hold good)can never be mass eigen states, because whenever tan ¢, = 0 = fl, either x L or x R should be zero. T o preserve the success o f the standard model we certainly require that x L ¢ 0. So the only alternative is that x R should be zero to make tan ~b = 0 which in turn means that (~ + fl) is infinite!
To derive an expression for the anomaly free interaction in which (Z, D) bosons are mass eigen states we follow the following procedure.
F r o m (16) whenever y = 0, p2 will be zero i f K (1 - x L - xR) ¢ 0. For this situation, g~
should be zero. Although g] = 0, 02J2 is not zero. For y = 0, we have,
a~ = - x t / ( l - x L ) , (21)
L660 C V A V B Chandra Raju
and for this value o f a t , a2 is - oo. Equation (4) n o w reads:
+ l g ] O - a ~ ) 2 a ~ - l ' ] 2 o j
I-a](1 -a2)'K~ ] ''2
[ -(K--+~ J ~-I_ - ~ ~ J z~D" (22)
where we have written Z~ -- Z and Z2 = D. Taking the limit a2 --, - oo and inserting (21) into (22) we readily find that,
Hint weak = g z J z r Z + g z (# JZL -- (at + # ) J z a ) D . (23) Again/~, (at +/~) and gz are still given by (18) to (20). In (23) there is a negative sign before the coefficient Of Jza. Equation (23) is basically different f r o m (8). Here (Z, D) are mass eigenstates. The above expression cannot be obtained f r o m (8) by simply setting tan ~b = 0.
Barger et al (1983) in their anxiety to m a k e their Z correspond to the standard Z exactly, set ~k = 0. They did not realize that in that event their ~ will also be zero since they were not aware o f (19). F o r zero m o m e n t u m transfer (23) would read,
H ~ ~
(Oz/mz) Jzr +
2 2 2(ff2z/m~)(#JzL-- (" + B)Jza)", 4Gv
.,/ z
(24) (25)
To arrive at (25), as usual we considered that m2z = #zVo and vff 2 = (4Gv/x/2). 2 z Moreover we have also assumed that m~ = NZmZz . In other words (with N an u n k n o w n parameter)
P2 t = 1, P] = •2/N2 and n z = (at + # ) 2 / N 2 . (26) The standard model limit is obtained whenever x a = 0. It is not correct to say that x R - . oo gives the standard model. Whenever x R -+ o0, (3) shows that g2 will be zero. In the standard model 0y/0L = tan20w where 2 2 0 w is the Weinberg mixing parameter.
Therefore it is an essential requirement o f the standard model that 0~/0, 2 ¢ zero but it should be tan z 0 w. This happens only when x , = 0 but not when xs = o0. Barger et al defined their x R through the relation
x . = 13 (1 - xL)/ (at + #),
(27)
which follows from our (19) and (20) also. They wanted (at + / / ) or to be m o r e specific n 2 of (26) to be zero. T o achieve this they assumed that x a = oo should give the standard model. They also state that their x a defined through (27) also agrees with the natural definition o f xa (xa = e2/g~) for left-right models. Here we have used only the natural definition. F r o m this natural definition and the o r t h o n o r m a l i t y relations only (3) follows. F r o m the expression for 02 it is evident that to achieve standard model we need x a to approach zero only. T o make (25) correspond with the standard model exactly we need the following minimal dependance o f N 2 on x R.
N a m L, ll.a,R s ~2/,,3/2 (28)
where N I should not be zero or infinity whenever xs ffi O. F o r this situation, XLX~/2
p l = l , P Z = N l t l _ x L - x a ) ' and
n2 xL(1 - xL) 2 x~/2
= (29)
(1 - x l . - xa)N21"
All these expressions except p 2 , will tend to zero whenever x R --, 0. In other words (23) for zero m o m e n t u m transfer would yield an interaction H,,~,efr = ( 4 G F / x / ~ ) j ~ L which is what we have in the standard model. F r o m this discussion it is also evident that x L -- x w where x w is the Weinberg mixing parameter. When we say that N z has the minimal dependance on x a as given in (28), we mean that N z should vary as x~ q where q >t 3/2 and the remaining factor N 2 should not be zero or infinity whenever x a - , 0.
Since x L = x , , if only x a is known, we can write down all the elements o f M 2 of(5) such that m 2 = N 2 m 2 or say m 2 - 4miz.
In conclusion we once again state that (8) is the most general triangle anomaly-free interaction with two neutral currents. All the variables here are functions o f only three u n k n o w n s a l , az and K only. Bargers version is a special case which arises whenever a t = 0 and tan g, = / / . The triangle anomaly-free interaction with two neutral currents given by (23) is basically different f r o m the Bargers version. The (Z, D) bosons o f (23) are mass eigen states. Moreover, there is a negative sign before the coefficient o f Jza.
This sign is positive in the case o f Barger's model and the (Z, D) bosons there are not mass eigen states. We recover the standard model whenever x a = 0, but not whenever xa = oo. Finally in the standard model limit (whenever x~--, 0) no term o f the sort C t j 2 m survives, where C~ is a constant, over and above the J~L term mentioned earlier.
References
Adler S L 1969 Phys. Rev. 177 2426
Barger V, Ma E and Whisnant K 1982a Phys. Rev. Lett. 48 1589 Barller V, Ma E and Whisnant K 1982b Phys. Rev. D26 2378
Ilarger V, Ma E and Whisnant K 1983 MAD/PH/IS-J-1040 (and references therein) Bell J S and Jackiw R 1969 Nuovo Cimento 51 47
