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Does the decay mode K+ → π+ + axion already rule out the axion?
Volume 74B, number 4, 5
PHYSICS LETTERS
17 April 1978
D O E S THE D E C A Y M O D E K + ~ n + + A X I O N A L R E A D Y R U L E O U T THE A X I O N ?
J. KANDASWAMY, Per SALOMONSON i and J. SCHECHTER Physics Department, Syracuse University, Syracuse, N Y 13210, USA
Received 27 February 1978
The branching ratio F(K + ~ 7r+ + axion)/U(K+ ~ all) is predicted to be of the order of 10 -5. Such a relatively high value already violates an experimental upper bound deduced by several authors. Some discussion of the formalism we use is given and a possible way out is mentioned.
One test, which has been proposed by Wilczek [1], Goldman and Hoffman [2], and Weinberg [3], for the axion [4] a is the decay process K + ~ ~r+a. The claim is made [2] that the experiment to search for K + -+ 7r+uV already gives an experimental limit P(K + ~ 7r+a)/I'(K + --> all) ~ 2.7 × 10 - 7 .
(1)
We have found that the process K + --> 7r+a is to be expected theoretically at the level of 10 -5 , thereby giving some more evidence against the existence of at least the simplest weak interaction model with an axiom According to the formalism developed by the authors [5] (and independently in a slightly different form by Weinberg [3] ) the axion field contains small mixtures o f the flavorless pseudoscalars according to a
~'~
b
(2)
... - z.~ PbOb ,
where the quantities Pb are given in the appendix. The ~bb are the usual tensor components of the flavorless pseudoscalars. If one neglects SU(3) mixing (not too bad an approximation and not critical anyway for our purposes) the mixtures ofTr 0, r~, ~', r/" are given by
Prr°=(1/~v/2)(Pl--P2)'
Pr~=(1/V/-6)(Pl+P2--2P3) '
PTf=(1/V/3)(Pl+P2+P3) '
P~"=P4"
(3)
The amplitude for K + -+ n+a is then P~r° amp(K+ ~ 7r+Tr0) + Pn amP(K+ ~ 7r+rT) + ....
(4)
since we have assumed the p's to be small. Now as is well known the amplitude for K + -+ 7r+Tr0 is very heavily suppressed by the A I = I/2 rule. Hence the second term completely dominates the first. We may easily estimate the strength o f the second term by using an effective weak hamiltonian and SU(3) symmetry. Denoting qb as the traceless 3 X 3 matrix of pseudoscalars and defining the spurion d =
0
,
1
we have the effective non-leptonic hamiltonian which obeys the 2~I = 1/2 rule and is CP invariant:
1 Permanent address: Institute of Theoretical Physics, Fack, S-402 20, G6teborg 5, Sweden. 377
Volume 74B, number 4, 5
PHYSICS LETTERS
17 April 1978
HNLeff= i Gef f (Tr([S]q~qbdpc5 ) - Tr(~qb[S]qbcJ)},
(5)
where [] is the d'alembertian operator. As discussed [6] a long time ago, eq. (5) satisfies all the current algebra zero four-momentum limits. It is, of course, crucial to have derivative couplings since the whole thing vanishes if the d'alembertian is not present. Eq. (5) is unique to second order in derivatives. From eqs. (4) and (5) we immediately find T(K +-+ 7r+a) _ Pr~ (2m2 + m2 - 3"2) j m2 - /-12 2 - vP• 2' T(Ks0 -+ v+Tr-) 2V'-f rn2k_m~r m2_m~r
(6)
where/a is the axion mass. The reason the (small) P,r term appears in eq. (6) is that, with derivative coupling in the effective lagrangian the A I = 1/2 rule only holds exactly if we set rn 2 = p2. Roughly, the desired branching ratio is then F(K + -+ v+a)/P(K + -+ all) ~
40p 2.
(7)
With a typical choice of parameters we find (see appendix) p,~ = - ( 1 . 0 0 tan 7 + 4.87 cot 3') X 10 -4,
(8)
3' being the new weak interaction angle. From eq. (8) the minimum value of Ip~l is seen to be 4.41 X 10 -4. This leads to the theoretical prediction r ( K + -+ rr+a)/r(K + -+ all) ~ 7.8 X 10 - 6 ,
(9)
which clearly contradicts eq. (1). Actually if the weak interaction hamiltonian is taken to be an effective one transforming as the 20-dimensional representation of SU(4) the situation remains essentially the same. In this case the mixing of the axion with 7/' and r/" should also be taken into account. An interesting feature of this mixing is that it does not, as one might first expect, decrease for the heavier r/'s, but remains of the same order of magnitude. This is because the axion couplings to heavier quarks increase with the quark mass. So even though it is not clear from an experimental point of view that 2_Qdominance has any validity, we should try to strengthen our conclusion by calculating in this model too. We may write the effective SU(4) hamiltonian as
HNL ' "3 "a .3 -4 + 2/2c~/3ce)+h.c., eff ' = Geff(]ac~]2~ 214c~12~
j:c='
c~ b +~[(% x%G¢c
+%)~
1
~G)] + .-.
(10)
a O, where S a are the scalar fields.) One £mds in(The currents in eq. (10) are taken from the sigma model. % = (Sa) stead of eq. (6) 1 T(~K+~Tr+a~) = ' v " 2 ( a l 1 (al +a2)mk2 (Ctl + a a ) m 2 I T(Ks0 -+ 7r+Tr-) -~ (Pl - P3) + m 2 - m 2 (P2 - Pl) + a2.) 2a4P 4 - 2 a l P 1 -t -m2- --rnTr ~--15.7tanT-3.4cot3'lX
10 - 4
(11)
(forN=2).
In the case of eq. (11) no rigorous statement can be made since the tan 3' and cot 3, terms have opposite signs. However if we choose 3' to suppress the axion-nucleon nucleon coupling constant in order to explain the null results of the reactor experiment [3] we get (requiring P~r ~ 0), tan23' = 6.6 which leads to F(K + -+ 7r+a)/r(K + ~ all) --~ 1.7 X 10 - 4 . The precise numbers depend somewhat on the exact choice of current-algebra parameters but our conclusion remains the same. TakingN = 3 (see appendix) rather than N = 2 does not affect our conclusion either. We stress that the violation of the experimental bound is by at least an order of magnitude and does not depend on any special choice of parameters. The problem of obtaining natural strongP and T conservation thus must probably be considered again. One 378
Volume 74B, number 4, 5
PHYSICS LETTERS
17 April 1978
possible way of achieving this might be to imagine that nature for some reason requires the parameter sin 23' to be extremely small. Then according to the mass formula (A6) the axion mass could be as large as we like. One might imagine an axion in the charmonium or upsilon range (or even higher) which might mix with the r~c etc. The difficulty with this proposal is that it looks like a heavy axion which could lead to an unacceptable distortion of the hadron mass spectrum. The question is a little complicated algebraically so we shall report any progress in this direction elsewhere. Appendix. In an earlier note [5] we described the derivation of the axion mixing formalism using a generalized sigma model. Essentially the same derivation can be given using conventional current algebra techniques. We consider the 6 currents.
d~ = }i ((~¢~0)+0 - 0 + q)~b},
dX = ½i {(cbo~X)+X - X+ oDe,X),
pa = iglaTc~Tsqa,
(A1)
where 0 and X are the two Higgs doublets and @~ is the SU(2) X U(1) covariant derivative. All the currents in eq. (A1) would be conserved were it not for the presence of the following terms in the effective s t r o n g - w e a k - e m lagrangian: - U {det Ct(1 + 75)q + det C:l(1 - 75)@ + (1/2Xl){~[
m2c°s 0cCt2( 1 + ")'5) 41 +m2sin 0c~12( 1 +T5)42 (A2)
- m3sin 0cgl3(1 + 75)41 - m3c°s 0cC13(1 + 75)42] + h.c.} + ( 1 / 2 X 2 ) { x T i r 2 [m lCtl(1 +75)41 +m4c14(1 +YS)ff2] +h.c.}, where ( ) 41 = cos 0cq 2 ql + sin 0cq 3 ,
( q4 ) 42 = --sin 0cq 2 + cos 0cq 3 "
The first term in eq. (A2) is an effective one which breaks the UA(1 ) symmetry of the strong interactions. Denoting the six currents in eq. (A1) by JBa we can find the squared mass matrix of the corresponding six flavorless mesons by the current-algebra formula
FAFe(M2)A8= ~ fd3x d3y (0[ [JA(x,
0), [ j B ( y , 0),/2(0)1 ] [0)+ (,4 o B ) ,
(A3)
where we write F A = (X 1 , X2, F 1 , F 2, F3, F4). Xl = (00), X2 = (X0) and F 1 = F 2 = F~ "" rGr0. F 3 and F 4 are expected to be of the order of F~. Carrying out the commutations in eq. (A3) (noting that, for example, [~0(x, t), ~0(0, t)] = 2i83(x) with our normalization) we find for the r.h.s, ofeq. (A3) m2 + m3 0 0 -2N 2 -2~ 3 0
ma - ma(C:taqa)o,
0 ml +m4 2r~1 0 0
0 2ml 4(~ 1 + /7) 4/~ 4/J
2~4
4/~
--2m2 0 4U 4(v~2 + l.~) 4/7
4U
-2~3 0 4/7 4/7 4(~3 + U)
4/7
0 2r~4 4/7 4/7 4U
4(m4 +/7)
U - U ({det C:l(1 + 75)q + det q(1 - 75)@)0.
) '
(A4)
The generalization ofeq. (A4) from the case of 2 quark doublets to N quark doublets (2Nflavors) is obvious. Note that eq. (A4) is the same as eq. (11) of ref. [5] with the substitutions 2 A a % = - m a ( q a q a ) 0 (each a),
--
1
% - 7Fa,
ffa = U.
(A5)
Thus we can take over the results of ref. [5] to write (in the N doublet case) the formula for the axion mass 379
Volume 74B, number 4, 5
PHYSICS LETTERS
1
u = lstn 23`t
17 April 1978
1
-~-
ma(~taqa)o_
(A6)
'
a,,~()t2+~2)-l/2{_X2(~)+)k1(~)}_~b
pb(pb"
(A7)
Pb = 2-3/4G1F/2Fb [ - N ( t a n 3` + cot 3') + (cot 3`)eb } [mb ( F q b q b ~ + ~a lfm~(qaqa)o )
(AS)
where eb = +1 for - 1 / 3 charge quarks and e b = - 1 for +2/3 charge quarks. Also tan 3 ` - Xl/X 2.
(A9)
Note that ~t ~ 0 as expected when any of the "quark masses" m a -> 0 or when ff -+ 0. Numerically, the dominant terms in eq. (A6) are the 1/rn 1 and 1/m 2 terms. Weinberg [3] has independently presented a special case ofeq. (A6) corresponding to (7-1 __>0, m21 --, 0 (a >14), F a = Fn (all a), and (Ctaqa) 0 = (Ctlq 1) (all a). For calculating amplitudes of physical processes the quantities Ob are the important ones. Note that to a good approximation
Pb "~ 2-3/4Gl/2Fb (c°t 3`)eb'
(b >/3).
(A10)
A typical choice of parameters is F3/F 1 = F4/F 1 = (Ct3q3)0/(~llql)0 = (Ct4q4)0/(~tlql)0 = 1.73;
F 1 = F 2 = m~o;
(qlql)o = (q2q2)o;
2rn3/(m 1 + m 2 )
=
4 ml(Cllql)0 = - 0 . 1 5 5 m ro; 36.2;
2m4/(m 1 + m 2 ) = 748;
4 m2(q2q2) 0 = -0.345 m rO; U
-----
-4.9
(All)
4
/T/,tr 0 •
With eq. (A11) we find the mixing coefficients: Pl "~ - ( 0 . 9 3 tan 3' + 3.67 cot T) X 10 -4,
P2 2 ( - - 1 . 6 5 tan 3' + 1.09 cot 7) X 10 - 4 ,
P3 ~ (4.68 cot 3' - 0 . 0 6 tan 7 ) × 10-4,
P4 ~ 4 . 7 4 X 10 -4 tan')',
On ~--(0.51 tan 7 - 3 . 3 7
fin ~- - ( 1 . 0 0 tan 3' + 4.87 cot 7) X 10 -4,
cotT) X 10 -4 ,
(A12)
for N = 2 and Pl = - ( 2 . 7 7 tan 7 + 5.51 cot 3') × 10-4,
P2 = ( - 2 . 4 7 tan T + 0.26 cot 7) X 10 -4,
P3 = ( - 0 . 0 9 tan T + 4.64 cot T) × 1 0 - 4 ,
/94 =4.73 tan3`X 10 -4 ,
p~ = - ( 0 . 2 1 tan 7 + 4.08 cot 3') X 10 - 4 ,
fin = - ( 2 . 0 6 tan 7 + 5.93 cot 7) X 10 - 4 ,
(A13)
f o r N = 3. One of us (P.S.) is supported by the Swedish Atomic Research Council, under contract #301-026. The others are supported in part by the U.S. Department of Energy, under contract #EY-76-S-02-3533. [1] [2] [3] [4] [5] [6] 380
F. Wilczek, Phys. Rev. Lett. 40 (1978) 279. T. Goldman and C.M. Hoffman, Phys. Rev. Lett. 40 (1978) 220. S. Weinberg, Phys. Rev. Lett. 40 (1978) 223. R.D. Peccei and H.R. Quinn, Phys. Rev. D16 (1977) 1791. J. Kandaswamy, P. Salomonson and J. Schechter, Mass of the axion, Syracuse Univ. report SU-4211-109. J. Schechter, Phys. Rev. 161 (1967) 1660.
PHYSICS LETTERS
17 April 1978
D O E S THE D E C A Y M O D E K + ~ n + + A X I O N A L R E A D Y R U L E O U T THE A X I O N ?
J. KANDASWAMY, Per SALOMONSON i and J. SCHECHTER Physics Department, Syracuse University, Syracuse, N Y 13210, USA
Received 27 February 1978
The branching ratio F(K + ~ 7r+ + axion)/U(K+ ~ all) is predicted to be of the order of 10 -5. Such a relatively high value already violates an experimental upper bound deduced by several authors. Some discussion of the formalism we use is given and a possible way out is mentioned.
One test, which has been proposed by Wilczek [1], Goldman and Hoffman [2], and Weinberg [3], for the axion [4] a is the decay process K + ~ ~r+a. The claim is made [2] that the experiment to search for K + -+ 7r+uV already gives an experimental limit P(K + ~ 7r+a)/I'(K + --> all) ~ 2.7 × 10 - 7 .
(1)
We have found that the process K + --> 7r+a is to be expected theoretically at the level of 10 -5 , thereby giving some more evidence against the existence of at least the simplest weak interaction model with an axiom According to the formalism developed by the authors [5] (and independently in a slightly different form by Weinberg [3] ) the axion field contains small mixtures o f the flavorless pseudoscalars according to a
~'~
b
(2)
... - z.~ PbOb ,
where the quantities Pb are given in the appendix. The ~bb are the usual tensor components of the flavorless pseudoscalars. If one neglects SU(3) mixing (not too bad an approximation and not critical anyway for our purposes) the mixtures ofTr 0, r~, ~', r/" are given by
Prr°=(1/~v/2)(Pl--P2)'
Pr~=(1/V/-6)(Pl+P2--2P3) '
PTf=(1/V/3)(Pl+P2+P3) '
P~"=P4"
(3)
The amplitude for K + -+ n+a is then P~r° amp(K+ ~ 7r+Tr0) + Pn amP(K+ ~ 7r+rT) + ....
(4)
since we have assumed the p's to be small. Now as is well known the amplitude for K + -+ 7r+Tr0 is very heavily suppressed by the A I = I/2 rule. Hence the second term completely dominates the first. We may easily estimate the strength o f the second term by using an effective weak hamiltonian and SU(3) symmetry. Denoting qb as the traceless 3 X 3 matrix of pseudoscalars and defining the spurion d =
0
,
1
we have the effective non-leptonic hamiltonian which obeys the 2~I = 1/2 rule and is CP invariant:
1 Permanent address: Institute of Theoretical Physics, Fack, S-402 20, G6teborg 5, Sweden. 377
Volume 74B, number 4, 5
PHYSICS LETTERS
17 April 1978
HNLeff= i Gef f (Tr([S]q~qbdpc5 ) - Tr(~qb[S]qbcJ)},
(5)
where [] is the d'alembertian operator. As discussed [6] a long time ago, eq. (5) satisfies all the current algebra zero four-momentum limits. It is, of course, crucial to have derivative couplings since the whole thing vanishes if the d'alembertian is not present. Eq. (5) is unique to second order in derivatives. From eqs. (4) and (5) we immediately find T(K +-+ 7r+a) _ Pr~ (2m2 + m2 - 3"2) j m2 - /-12 2 - vP• 2' T(Ks0 -+ v+Tr-) 2V'-f rn2k_m~r m2_m~r
(6)
where/a is the axion mass. The reason the (small) P,r term appears in eq. (6) is that, with derivative coupling in the effective lagrangian the A I = 1/2 rule only holds exactly if we set rn 2 = p2. Roughly, the desired branching ratio is then F(K + -+ v+a)/P(K + -+ all) ~
40p 2.
(7)
With a typical choice of parameters we find (see appendix) p,~ = - ( 1 . 0 0 tan 7 + 4.87 cot 3') X 10 -4,
(8)
3' being the new weak interaction angle. From eq. (8) the minimum value of Ip~l is seen to be 4.41 X 10 -4. This leads to the theoretical prediction r ( K + -+ rr+a)/r(K + -+ all) ~ 7.8 X 10 - 6 ,
(9)
which clearly contradicts eq. (1). Actually if the weak interaction hamiltonian is taken to be an effective one transforming as the 20-dimensional representation of SU(4) the situation remains essentially the same. In this case the mixing of the axion with 7/' and r/" should also be taken into account. An interesting feature of this mixing is that it does not, as one might first expect, decrease for the heavier r/'s, but remains of the same order of magnitude. This is because the axion couplings to heavier quarks increase with the quark mass. So even though it is not clear from an experimental point of view that 2_Qdominance has any validity, we should try to strengthen our conclusion by calculating in this model too. We may write the effective SU(4) hamiltonian as
HNL ' "3 "a .3 -4 + 2/2c~/3ce)+h.c., eff ' = Geff(]ac~]2~ 214c~12~
j:c='
c~ b +~[(% x%G¢c
+%)~
1
~G)] + .-.
(10)
a O, where S a are the scalar fields.) One £mds in(The currents in eq. (10) are taken from the sigma model. % = (Sa) stead of eq. (6) 1 T(~K+~Tr+a~) = ' v " 2 ( a l 1 (al +a2)mk2 (Ctl + a a ) m 2 I T(Ks0 -+ 7r+Tr-) -~ (Pl - P3) + m 2 - m 2 (P2 - Pl) + a2.) 2a4P 4 - 2 a l P 1 -t -m2- --rnTr ~--15.7tanT-3.4cot3'lX
10 - 4
(11)
(forN=2).
In the case of eq. (11) no rigorous statement can be made since the tan 3' and cot 3, terms have opposite signs. However if we choose 3' to suppress the axion-nucleon nucleon coupling constant in order to explain the null results of the reactor experiment [3] we get (requiring P~r ~ 0), tan23' = 6.6 which leads to F(K + -+ 7r+a)/r(K + ~ all) --~ 1.7 X 10 - 4 . The precise numbers depend somewhat on the exact choice of current-algebra parameters but our conclusion remains the same. TakingN = 3 (see appendix) rather than N = 2 does not affect our conclusion either. We stress that the violation of the experimental bound is by at least an order of magnitude and does not depend on any special choice of parameters. The problem of obtaining natural strongP and T conservation thus must probably be considered again. One 378
Volume 74B, number 4, 5
PHYSICS LETTERS
17 April 1978
possible way of achieving this might be to imagine that nature for some reason requires the parameter sin 23' to be extremely small. Then according to the mass formula (A6) the axion mass could be as large as we like. One might imagine an axion in the charmonium or upsilon range (or even higher) which might mix with the r~c etc. The difficulty with this proposal is that it looks like a heavy axion which could lead to an unacceptable distortion of the hadron mass spectrum. The question is a little complicated algebraically so we shall report any progress in this direction elsewhere. Appendix. In an earlier note [5] we described the derivation of the axion mixing formalism using a generalized sigma model. Essentially the same derivation can be given using conventional current algebra techniques. We consider the 6 currents.
d~ = }i ((~¢~0)+0 - 0 + q)~b},
dX = ½i {(cbo~X)+X - X+ oDe,X),
pa = iglaTc~Tsqa,
(A1)
where 0 and X are the two Higgs doublets and @~ is the SU(2) X U(1) covariant derivative. All the currents in eq. (A1) would be conserved were it not for the presence of the following terms in the effective s t r o n g - w e a k - e m lagrangian: - U {det Ct(1 + 75)q + det C:l(1 - 75)@ + (1/2Xl){~[
m2c°s 0cCt2( 1 + ")'5) 41 +m2sin 0c~12( 1 +T5)42 (A2)
- m3sin 0cgl3(1 + 75)41 - m3c°s 0cC13(1 + 75)42] + h.c.} + ( 1 / 2 X 2 ) { x T i r 2 [m lCtl(1 +75)41 +m4c14(1 +YS)ff2] +h.c.}, where ( ) 41 = cos 0cq 2 ql + sin 0cq 3 ,
( q4 ) 42 = --sin 0cq 2 + cos 0cq 3 "
The first term in eq. (A2) is an effective one which breaks the UA(1 ) symmetry of the strong interactions. Denoting the six currents in eq. (A1) by JBa we can find the squared mass matrix of the corresponding six flavorless mesons by the current-algebra formula
FAFe(M2)A8= ~ fd3x d3y (0[ [JA(x,
0), [ j B ( y , 0),/2(0)1 ] [0)+ (,4 o B ) ,
(A3)
where we write F A = (X 1 , X2, F 1 , F 2, F3, F4). Xl = (00), X2 = (X0) and F 1 = F 2 = F~ "" rGr0. F 3 and F 4 are expected to be of the order of F~. Carrying out the commutations in eq. (A3) (noting that, for example, [~0(x, t), ~0(0, t)] = 2i83(x) with our normalization) we find for the r.h.s, ofeq. (A3) m2 + m3 0 0 -2N 2 -2~ 3 0
ma - ma(C:taqa)o,
0 ml +m4 2r~1 0 0
0 2ml 4(~ 1 + /7) 4/~ 4/J
2~4
4/~
--2m2 0 4U 4(v~2 + l.~) 4/7
4U
-2~3 0 4/7 4/7 4(~3 + U)
4/7
0 2r~4 4/7 4/7 4U
4(m4 +/7)
U - U ({det C:l(1 + 75)q + det q(1 - 75)@)0.
) '
(A4)
The generalization ofeq. (A4) from the case of 2 quark doublets to N quark doublets (2Nflavors) is obvious. Note that eq. (A4) is the same as eq. (11) of ref. [5] with the substitutions 2 A a % = - m a ( q a q a ) 0 (each a),
--
1
% - 7Fa,
ffa = U.
(A5)
Thus we can take over the results of ref. [5] to write (in the N doublet case) the formula for the axion mass 379
Volume 74B, number 4, 5
PHYSICS LETTERS
1
u = lstn 23`t
17 April 1978
1
-~-
ma(~taqa)o_
(A6)
'
a,,~()t2+~2)-l/2{_X2(~)+)k1(~)}_~b
pb(pb"
(A7)
Pb = 2-3/4G1F/2Fb [ - N ( t a n 3` + cot 3') + (cot 3`)eb } [mb ( F q b q b ~ + ~a lfm~(qaqa)o )
(AS)
where eb = +1 for - 1 / 3 charge quarks and e b = - 1 for +2/3 charge quarks. Also tan 3 ` - Xl/X 2.
(A9)
Note that ~t ~ 0 as expected when any of the "quark masses" m a -> 0 or when ff -+ 0. Numerically, the dominant terms in eq. (A6) are the 1/rn 1 and 1/m 2 terms. Weinberg [3] has independently presented a special case ofeq. (A6) corresponding to (7-1 __>0, m21 --, 0 (a >14), F a = Fn (all a), and (Ctaqa) 0 = (Ctlq 1) (all a). For calculating amplitudes of physical processes the quantities Ob are the important ones. Note that to a good approximation
Pb "~ 2-3/4Gl/2Fb (c°t 3`)eb'
(b >/3).
(A10)
A typical choice of parameters is F3/F 1 = F4/F 1 = (Ct3q3)0/(~llql)0 = (Ct4q4)0/(~tlql)0 = 1.73;
F 1 = F 2 = m~o;
(qlql)o = (q2q2)o;
2rn3/(m 1 + m 2 )
=
4 ml(Cllql)0 = - 0 . 1 5 5 m ro; 36.2;
2m4/(m 1 + m 2 ) = 748;
4 m2(q2q2) 0 = -0.345 m rO; U
-----
-4.9
(All)
4
/T/,tr 0 •
With eq. (A11) we find the mixing coefficients: Pl "~ - ( 0 . 9 3 tan 3' + 3.67 cot T) X 10 -4,
P2 2 ( - - 1 . 6 5 tan 3' + 1.09 cot 7) X 10 - 4 ,
P3 ~ (4.68 cot 3' - 0 . 0 6 tan 7 ) × 10-4,
P4 ~ 4 . 7 4 X 10 -4 tan')',
On ~--(0.51 tan 7 - 3 . 3 7
fin ~- - ( 1 . 0 0 tan 3' + 4.87 cot 7) X 10 -4,
cotT) X 10 -4 ,
(A12)
for N = 2 and Pl = - ( 2 . 7 7 tan 7 + 5.51 cot 3') × 10-4,
P2 = ( - 2 . 4 7 tan T + 0.26 cot 7) X 10 -4,
P3 = ( - 0 . 0 9 tan T + 4.64 cot T) × 1 0 - 4 ,
/94 =4.73 tan3`X 10 -4 ,
p~ = - ( 0 . 2 1 tan 7 + 4.08 cot 3') X 10 - 4 ,
fin = - ( 2 . 0 6 tan 7 + 5.93 cot 7) X 10 - 4 ,
(A13)
f o r N = 3. One of us (P.S.) is supported by the Swedish Atomic Research Council, under contract #301-026. The others are supported in part by the U.S. Department of Energy, under contract #EY-76-S-02-3533. [1] [2] [3] [4] [5] [6] 380
F. Wilczek, Phys. Rev. Lett. 40 (1978) 279. T. Goldman and C.M. Hoffman, Phys. Rev. Lett. 40 (1978) 220. S. Weinberg, Phys. Rev. Lett. 40 (1978) 223. R.D. Peccei and H.R. Quinn, Phys. Rev. D16 (1977) 1791. J. Kandaswamy, P. Salomonson and J. Schechter, Mass of the axion, Syracuse Univ. report SU-4211-109. J. Schechter, Phys. Rev. 161 (1967) 1660.