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Generalized gauge fields and applications to weak interactions
ANNALS
OF
PHYSICS:
Generalized
67, 145-157 (1971)
Gauge Fields and Applications TUAN
to Weak
Interactions
Wu CHEN
Department of Physics, New Mexico State University, L.as Cruces, New Mexico Received August 18, 1970
Yang-Mills’ field is generalized to possess a nontrivial scalar part. The most general transformations for such a field under the 3-parameter isotopic gauge transformation is obtained. Using this generalized gauge field, a gauge invariant Lagrangian is constructed within the framework of the quark model. Interactions for spin-l as well as for spin-O are generated. As a further application a weak interaction theory mediated by the generalized gauge (boson) field is formulated. The entire weak interactions are generated in two halfs; the hadron-boson interaction is generated according to YangMills’ trick using the generalized gauge field and the other half (boson-lepton, etc.) is then generated by making use of the scalar part of the gauge fields according to the conventional pion gauge principle. The effective Lagrangian is then found to be mediated by the effective propagators which fall off as pWe at high momenta; the unitarity of the theory can thereby be insured. Universality in weaker sense than the usual one is applied to the intermediate bosons; our theory for p-decay then reduces to Cabibbo’s at low energy.
1. INTRODUCTION Is is well known that the invariance of (matter) Lagrangian under local gauge transformation can be maintained by introducing spin-l gauge fields [l]. The interactions between spin-l fields and matter fields are then generated. This powerful principle has been used extensively in recent years not only to generate interactions but also to uncover the underlying principles behind conservation laws or symmetry groups [l].l Not too long ago another kind of gauge principle was introduced by Gell-Mann and Levy [3] in which spin-0 fields instead of spin-l are used as gauge fields. The interactions between spin-0 and lepton fields (or currents) are thereby generated. More importantly, the hypothesis of partially conserved axialvector currents (PCAC) in weak interactions can be explained in a natural way.2 In this paper we would like to further generalize the existing gauge principles 1 A good account on this subject can be found in the book by J. Bernstein [2]. 2 See Refs. [2 and 31; also G. Kramer and W. F. Palmer [4].
145 595/67/r-10
146
CHEN
by considering more general gauge fields: Four-vector fields with nontrivial scalar parts3 Our purpose is twofold: In analogy to the method of Yang-Mills’ we wish to generate interactions for spin-0 as well as for spin-l fields; moreover, we wish to use the spin-0 parts of four vector fields to generate weak interactions closely following Gell-Mann and Levy’s method. Allowing the explicit presence of spin-0 fields in transformed gauge fields under local gauge transformation, the generalized gauge field is developed in next section. The gauge invariant Lagrangian is then constructed in Section 3. The rest of the paper is then devoted to an application of our idea to weak interactions where intermediate bosons are considered as the generalized Yang-Mills’ fields. Local gauge transformations without parity conservation are considered in Section 4; there the boson-hadron interactions are also constructed. In Section 5 Gell-MannLevy’s method is then applied to introduce boson-lepton interactions. Finally, to complete the weak interaction Lagrangian, the neutral boson-current term is constructed in Section 6 to account for nonleptonic weak interactions. In Section 7 a weaker form of universality in the spirit of Cabibbo’s [5] is applied to the coupled boson fields to reduce the number of parameters involved. In addition to satisfying all the usual selection rules, the weak interactions so generated are found to have the important feature: It yields the effective weak Lagrangian mediated by weak propagators which fall off as p-z at high momenta. Some discussions are followed. 2. GENERALIZED
YANG-MILLS’
FIELDS
Consider the Hermitian fields of scalar (designated by bJ and vector (designated by Bip) belonging to the octet representations of SU(3). Under the rotation in isotopic space they transform according to the rules b ---f b’ = FbS,
(2-l) (2-2)
Bu --+ B’u = S-IBuS, where b = bihi , BP EZ&‘&
(i = l,..., S),
with hi the usual 3 x 3 SU(3) matrices, and S, a 3 x 3 unitary matrix with determinant unity, is given in general by s = ei=j%, (2-3) where 0~~(j = l,..., 8) are constants. s Gauge fields Au in Yang-Mills’ theory always satisfy the supplementary %A,, = 0, so that the scalar part of AH does not exist. See Ref. [l].
condition,
viz.
GENERALIZED
GAUGE
FIELDS
AND
147
APPLICATIONS
Equations (2-l) and (2-2) with S independent of x, y, and z are nothing but the precise statement of SU(3) properties of b and B”. The question we like to raise in this section is then to ask what is the most general change the field B” can suffer if S is a function of x, y, and z. We shall look for such a generalization of (2-2) within the following criteria: 1. It is compatable with Eq. (2-2) for constant S. 2. It has at most first derivatives. 3. It can only be at most linear in fields BJ‘ or b. Note that the scalar field b is allowed to appear in our generalization; this is to obtain more general transformation than the usual local gauge transformation; moreover there is no a priori reason to ignore the spin-0 part in a four vector field. The most general form can easily be seen to be B” + B’u = S-lB”S
+ fiS-W’s
+ &Wb@S
+ &%!P1bS,
(2-4)
where 5 and 5 are constants. Note that every additional term in Eq. (2-4) contains LPS or aUS1 which vanishes if S is a constant matrix. The Hermiticity of BP implies that and t* = 5 51* = 5, * Moreover, the vanishing of the trace of B’u says that [I = 5, . Thus we conclude that the most general form for B’” is (with real c and 5) B”1 = S-lB%
+ i&!-WS
+ &S-lbauS
+ aW1bS).
(z-5)
Equation (2-5) gives the generalized transformation property of the vector field Bu under a local gauge transformation. We recognize that, with the exception of the last term in Eq. (2-5), B” transforms like Yang-Mills’ field. The change of the vector field under the infinitesimal gauge transformation can be written down easily from Eq. (2-5). We have, after some manipulation B’u = BP - i[q B”J - &%
+ i&3%,
b]
(2-6)
B’” = B” - +ol - ig{[ol, B”] - y[&x,
b]}.
e-7)
or, after redefining the constants, we have4:
4 The constants g and y in Eq. (2-7) are related to [ and 1 according to g = l/f and y = 1; moreover, 01in ELq. (2-7) differs from oi in Eq. (2-6) by a factor of g.
148
CHEN 3. THE INVARIANT LAGRANGIAN
The Lagrangian invariant under the local gauge transformation (2-5) can be achieved in a straightforward manner if one observes that Eq. (2-5) can be rewritten as B’L” - @ub’ = S-l(Bu
- &Fb) S + i&-9@S,
(3-l)
where b’ = S-lbS. We see that the combination B” - @b, in fact, transforms exactly like Yang-Mills’ field. The invariant Lagrangian can thus be constructed in terms of this combination according to the conventional way. For convenience, let us define W’1 = Bfi - {@b.
The kinetic part of the invariant Lagrangian LYKa = -&Tr(G@V)2
is then of the familiar - +Tr(Dub)2,
(3-2) form (3-3)
where G” = auw - i? W” - ig[ Wu, w’]
(3-4)
Dub = @b - ig[W”, b].
(3-5)
and
We have also included the kinetic part for the b-field in Eq. (3-3). Clearly, Eq. (3-3) alone gives a local gauge invariant theory for massless spin-l and spin-0 fields. For massive fields, the mass terms then break the local gauge invariance. The total Lagrangian PO with added common mass terms for Bu and b is given by
-% = %E - q
Tr(B’Q2 - $ Tr(b)2.
(3-7)
It should be noted that go still is an SU(3) invariant Lagrangian although it is not local gauge invariant. We shall see later that the mass terms in fact give rise to some interesting results. To complete the gauge invariant Lagrangian we shall consider the quark field q, 3 x 1 matrix belonging to the triplet representation of SV(3).5 Under the local gauge transformation q transforms according to the rule 4-4
‘ = s-lq.
(3-8)
6 One could indeed consider any matter fields instead of the quarks and carry out the following without much modification.
GENERALIZED
149
GAUGE FIELDS AND APPLICATIONS
The gauge invariant Lagrangian -E”, for q can be derived with the help of a gauge field. Using B” of previous sections, PQ is obtained through the replacement of 8‘ by DLLin the free Lagrangian of q, where Du = au -
igw”
(3-9)
= au - ig(B’* - (@b).
We have 6” = -Zjy”a,q
- Mqq + ig@“B,,q - ig&y”aubq.
(3-10)
Note that we have generated not only the interactions for spin-l particles but also the interactions for spin-0 particles. The equations of motion can now be easily written down from Eqs. (3-7) and (3-10). In particular for B” and b, we have (( 0 - m2) g”’ - a”a”} Bty + 2gfiik{(GY W,, + Dubi) bk - a,( Wj” W,“)} czz-igtjy”Aiq (3-l 1) and (0 - ~3 bi + &&{a,(Wj”b&
+ D,bjWku}
= --m2@~Bt,, .
(3-12)
From Eq. (3-l 1) and the relation
which follows easily from the equations of motion for the quarks, we note that the lowest order contribution to (quarks 1 BiM 1quarks) is of order g, whereas to (quarks 1bi / quarks> is of order gs. Thus, the interaction for bf is not so interesting for this case even though we are able to generate the coupling of bi to the vector current iqy@&q. In the following, we shall further generalize the above idea to include chiral gauge transformation and see a more interesting coupling of bi to axial vector currents.
4. INTERMEDIATE
BOSONS AND GENERALIZED
YANG-MILLS'
FIELDS
In previous sections we have restricted ourselves to the Iocal gauge transformations which leave the parities strictly unchanged. In this section we wish to apply the gauge principle we have developed so far to weak interactions. Since the parity
150
CHEN
is not conserved in weak interactions, it is more interesting to consider the unitary matrix S of the form s
=
pl+aYJcr
(4-l)
instead of Eq. (2-3), where a is a fixed constant and (II = h&x). Only a slight modification is needed in the procedure of constructing the invariant Lagrangian under Eq. (4-l). We assume that weak interactions are mediated by vector boson9 BLL as well as scalar boson’ b, which transform under the infinitesimal local gauge transformation* according to Eqs. (2-l) and (2-7), i.e., b -+ b’ = b - ig[a, b], B”+
(4-2)
BtU = B - a% - ig[a, Bu] + ic[a%, b].
(4-3)
While quark fields transform according to 4 + 4’ = (1 - ig(l + q&4
4.
(4-4)
Note that the function (II has no definite parity. It is a mixture of scalar and pseudoscalar functions. That this is the case can be seen from Eqs. (4-2))(4-4) and the fact that B” and b do not have definite parities. It would probably be asking too much to expect nature to respect the symmetry under Eq. (4-4) strictly. Even for weak interactions, we shall only expect the theory to be approximately invariant under Eqs. (4-2)--(4-4). Now the gauge invariant Lagrangian can easily be constructed. We shall adopt the point of view that Lagrangian is a (local) gauge invariant only when all of mass terms are absent; moreover, we shall see that establishing mass terms in fact is a natural symmetry breaking mechanism. The invariant Lagrangian for Bu and b without mass terms is obviously the same as in Section 3. In order to obtain the part for q, we need slight modification in our procedure in Section 3; we should replace au in the free Lagrangian for q by D”
= au - ig(1 + uy,) W”,
(4-5)
BThere are many intermediate vector boson models, for example, the recent model by G. Segre [6]. Earlier work can be founded in the references cited there. ’ There are also many weak interaction models mediated by scalar bosons. See, for example, N. Christ [7]. 8 Treating intermediate vector bosons as gauge fields can be found in many places, for example, by S. Okubo [8].
GENERALIZED
instead of (3-9). Putting terms, we have finally
GAUGE FIELDS AND APPLICATIONS
the above invariant
151
parts together with the usual mass
where (4-7) with -Y&(B~, b) given by Eq. (3-4), zrn = - $ and the quark-intermediate
Tr(Bu)P - f Tr(b)2 - Mjq
boson interaction Lagrangian =%a = m41
Clearly &n symmetry.
+ =&
(4-g)
dip,-, is given by
+ ayJ wqu .
(4-9
is local gauge invariant while the mass part =.%&breaks the
5. MODEL OF WEAK INTERACTIONS BETA DECAY LAGRANGIAN Customarily the weak interaction model mediated by vector bosons can be written down just by inserting the bosons between current-current couplings. A more sophisticated way is to employ the assertion of field-current identity to generate the Lagrangian [9]. Here we shall use a different approach9 closely following the work of Gell-Mann and Levy [3] based on fundamental (gauge) principle to achieve a formalism which has the definite advantage over the usual one due to its less divergence difficulty. Since quark-boson interactions have already been generated based on local gauge symmetry in the previous section, we need only to construct the interaction between the bosons and leptons to complete the beta decay Lagrangian. To do this, we start with the current-current form, i.e., PBel = g’l~j~’ + h.c., (5-l) where 1“ is the usual lepton current given by
111= 331 + yd e + C’y@(l+ y5) p 9 A primitive work on this idea is briefly reported by the author [lo].
(5-2)
152
CHEN
and j,$, is the linear sum of the positively charged components responsible for the transformationlO
of the current j$’
(5-3) where vt are constants. The current ji@ can easily be obtained from the total quarkboson Lagrangian (Eq. (4-6)): Au=-=
a9
rn”< - g Biu + 1 Dub,). ( m”5
(5-4)
This current is not conserved. In fact, its divergence is given by (5-5) However the matrix elements of a, jiu between hadron states is given by (hadrons I a,jiu I hadrons) = f
(hadrons I bi I hadrons) + O(g2)
(5-6)
since the contribution of the 2nd term in (5-5) is of the order O(g3). Moreover, one sees easily from the equation of motion for bi that the first-order contribution (which is also the lowest order) in g to (hadrons 1bi I hadrons) is purely pseudoscalar. Thus, even though the current ji’ has mixed parities and is not conserved, its vector part is conserved (i.e., CVC) while the divergence of its axial vector part is proportional to the pseudoscalar field bi in analogy to the PCAC hypothesis. It is also interesting to note that our Eq. (5-4) is essentially the field-current identity asserted by Lee [9]. Using n+, K+, etc. to denote the W(3) states, we have for the positively charged components Of ji’: 1
*IL Jn+ - -d/zg
[
B,’ - iB,@ + ---& D%
*Ii JK+ - -$-g
[B4u - iB$ + &.
- W],
(5-7)
D”(b4 - ib&].
And the current j&, in Eq. (5-l) is thus given by
A+)= 01; jc++ a,'ji+
.
lo It is important to note that we require Wp, instead of BP, to remain unchanged.
(5-9)
GENERALIZED
GAUGE FIELDS AND APPLICATIONS
153
In summary, we have the beta decay interactions by
+ gzu[al (B;+ + &
qhv+) + 01~(q+ + &
q)++)/ + h.c.9 (5-W
where hiu are hadronic currents given by hill = iqyu’(1 + ay&q. 6. MODEL
OF WEAK
INTERACTIONS,
(5-l 1) CONTINUED
In order to have a complete theory of weak interactions, we must add some extra terms to account for AZ = 4 strangeness changing nonleptonic weak interactions. For this purpose, there are two well known ways: one is to couple the neutral boson to neutral currents (so called schizon theory) [11], the other is through some enhancement mechanism [12]. Here we shall adopt the first way but proceed along the same principle as in the previous section. We assume that the strangeness charging interactions are due to the term CL.(o)
%--s =gvJG
>
(6-l)
where jf) is strangeness carrying neutral current and is given by the linear sum of neutral components ofj,“ of Eq. (5-4). That isll
with two real constants &’ and pZ’. The neutral hadron current Vu is chosen to be i@s(l + ay,) q, a scalar under SU(3). This indeed is the simplest possible form to insure AZ = 4 and AS < 1, for then we have -EB-S of (6-l) transform like an octet member. The relevant strangeness changing weak interactions thus become
+ h.c.
(6-3)
with real constants g, /31 , and /3Z. I1 In order to satisfy 41 = l/2 rule in a straightforward manner, only the currents with isotopic charge < l/2 are included. The current, B,p + l/cm8 Dfib, , is dropped since it will only make the constant p1 in Eq. (6-3) complex rather than real.
154
CHEN
7. EFFECTIVE LAGRANGIAN FOR WEAK INTERACTIONS A. Universality Based on two kinds of gauge transformations we have constructed the Lagrangians as general as possible; consequently, there are many free parameters involved. Here we would like to impose some conditions so as to reduce our Lagrangians to the one consistent with Cabibbo’s theory. We would like to impose the condition of universality on the vertices in a slight different sense from Gell-Mann’s but still in analogous to the photon field. The beauty of the intermediate boson model for weak interactions is recognized actually through the similarity between the vector boson fields and the electromagnetic field and customarily we assume that the boson fields couple to currents through a universal constant. Since, in fact, we are dealing with one or more intermediate bosons (instead of just one photon field in electrodynamics) we propose to modify the meaning of universality in the following way 12: The eflective coupling constants for diagonal current-current interactions should be the same. That is for a Lagrangian 9 = jlu(a$l,
+ a2B2p + . ..) + j2u(blBl,
+ b2B2, + .. 9,
we demand that aI2 + a22 + ... = b12 + b22 + *** .
It follows then that the constants in the Lagrangian
(5-10) are related by
a12 + a22 = 1,
(7-l)
a2 = 1
(7-2)
and those in (6-3) are related by 2/3,2 + p22 = 1. Moreover, applying the same universality bosons b, , we have from Eq. (5-lo),
principle
(7-3) to the spin-0 intermediate
k5)2= (*)2 + (%)’ Together with Eq. (7-l), we have the interesting result
5 = llm, la This indeed is along the same line as Cabibbo’s universality.
(7-5)
GENERALIZED
GAUGE
FIELDS
AND
155
APPLICATIONS
which says that the BP - b coupling constant is simply equal to the mass of Bp.13 The relation (7-l) clearly reduces our beta decay Lagrangian to the theory of Cabibbo’s. The relation (7-2) on the other hand gives
lal=
1,
(7-6)
yielding the conventional V - A form for the hadron currents, another wellestablished basic property of weak interactions. B. Efective Weak Lagrungians Making use of the relations (7-l)-(7-3) and (7-5) we can now sum up our weak interaction Lagrangian. We have
cos 0 (B,“’ + f
8&bn+) + sin 8 Bfc + A aybK+)) (
+ O(g2) + h.c.,
(7-7)
where 19and 8’ are arbitrary constants and O(g2) term is of the order g2 or higher and at least binomial in boson fields. Note the different signs jn front of b fields in coupling to hY and I“ (or VU) which give rise to a rather amazing second-order effective Lagrangian, viz.,
(Iw(xl) h;yx2))+ + sin 8 f
COS 8’
(I&,)
I s
(&(X1)
h;‘(x,))+ hvn(X2))+
G”“(x~
-
x2)
dX,
GUY& - x2) dx, GUY(X1
+ i sin 13’j
-
X2) dX2
- x2) dxz/,
(7-8)
I8 It is particularly interesting to note that theories based on Al - n mixed fields yield Al - T coupling constant to be precisely the mass of A, meson. See Ref. [13].
156
CHEN
where
G”“(x, - 4 = WW
W~)+)O - $ @Y k2?
m2 ’
((WJ &N+)o pa - ma m2 (k2 + ma)(k2 + p2)
k??
In Eq. (7-8), the first two terms describe leptonic weak interactions while the last two terms give nonleptonic strangeness changing weak interactions. All of interactions are of current-current form but mediated by the effective weak propagator G@”whose high-momentum behavior can easily be seen to fall off as k-2. As far as the low energy description of weak interactions is concerned, our theory indeed gives exactly the same results as the ordinary one and nothing more. However, the k-2 behavior of the propagator makes the present theory superior than the ordinary (IVB) theory whose weak propagator behaves like (k)O at high momenta. In fact, it attributes to overcoming all of the contradictions of ordinary weak interaction theory which arise from the use of unitary cut-off to estimate divergent expressions. The idea of properly incorporating scalar bosons to vector bosons so as to result in less singular weak propagators has been demonstrated through ad hoc models by Gell-Mann, Goldberger, Kroll and Low [14]. Since an excellent account on how the less singular propagators can do to improve weak interaction theory is given in their work and since our major concern here is to study the underlying principles, we shall be content with noting simply that k-2 behavior of the propagator undoubtly makes the weak interaction theory less divergent. The less singular behavior of effective propagators for semileptonic and strangeness changing nonleptonic interactions can easily be seen from Eq. (7-7) to be absent for diagonal weak interactions, i.e., the parity violating interactions among leptons or nonleptons. In fact, in those cases, the present model has exactly the same difficulty as the usual one. However, since those interactions are presumably completely dominated by interactions other than weak interactions, divergences in perturbation expansions in the weak coupling constant mean nothing other than that perturbation theory can not be applied in those cases. One could, nonetheless, break up V - A form and introduce different intermediate bosons for vector and axialvector hadron currents so as to preserve parity conservation in nonleptonic diagonal interactions and to have the less singular propagator even for the parity violating nonleptonic strangeness unchanging interactions. This would clearly mean more intermediate bosons and lack of symmetry in hadron and lepton currents. It should be noted that the gauge principle and the universality of bosons upon which the present formalism is based have reduced the number of parameters not more than those in the usual intermediate vector boson field (besides the
GENERALIZED
GAUGE
FIELDS
AND
APPLICATIONS
157
masses of scalar bosons which could be put to be equal to the masses of vector bosons). This is the main contrast to the models of Gell-Mann et al [14], where symmetry principles are not the main concern. Nevertheless, it perhaps is interesting to note that the propagator of their model I does reduce to ours if let cx-+ 0, even though their IVB’s (i.e., LQ) which in this limit is hard to understand have an apparent different structure. In closing we recall that our purpose of this work is mainly to introduce the generalized gauge fields with explicit scalar parts and to demonstrate their usefulness in weak interactions. In view of the above model which not only is consistent with existing experiments but also gives definite advantage over the usual theory, we hope, in addition to having achieved our purpose, that we have revealed some underlying principles behind weak interactions through the generalized gauge fields. Finally, we note that it is indeed not essentially to insure the nonexistence of scalar part in vector fields; contrarily the scalar part could play an important role in explaining physical interactions.
REFERENCES 1. C. N. YANG AND R. L. MILLS, Phys. Rev. 96 (1954), 191; R. UTIYAMA, Phys. Rev. 101 (1956), 1597; A. SALAM AND J. C. WARD, Nuovo Cimento 11 (1960), 568; S. L. GLASHOW AND M. GELL-MANN, Ann. Whys. (New York) 15 (1961), 437. 21 J. BERNSTEIN, “Elementary Particles and Their Currents,” W. H. Freeman and Co., San Francisco, 1968. 3. M. GELL-MANN AND M. LEVY, NUOVO Cimento 16 (1960), 705. 4. G. KRAMER AND W. F. PALMER, Phys. Rev. 182 (1969), 1490. 5. N. CABIBBO, Whys. Rev. Letfers 10 (1963), 531. 6. G. SEGRE, Phys. Rev. 181 (1969), 1996. Earlier work can be found in the references cited there. 7. N. CHRIST, Phys. Rev. 176 (1968), 2086 and the references cited there. 8. S. OKUBO, Ann. Whys. (New York) 49 (1968), 218. 9. T. D. LEE, Phys. Rw. 168 (1968), 1714. IO. T. W. CHEN, Nuovo Cimenfo Letter 3 (1970), 105. 11. T. D. LEE AND C. N. YANG, Whys. Rev. 119 (1960), 1410. 12. R. DASHEN, S. FRAUTCHI, M. GELL-MANN, AND Y. HARA, in “The Eightfold Way,” (M. Gell-Mann and Y. Ne’eman, Eds.), p. 254, W. A. Benjamin, Inc., New York, 1964, 13. T. W. CHEN AND R. E. PUGH, Phys. Rev. Letters20 (1968), 880; M. J. SWAG AND W. W. WADA, ibid21 (1968), 441; A. B~RNEL AND H. CAPRASSE, Nucl. Phys. B8 (1968), 65; also T. W. CHEN, Phys. Rev. 184 (1969), 1673. 14. M. GELL-MANN, M. L. GOLDBERGER, N. M. KROLL, AND F. E. Low, Whys. Rev. 179 (1969), 1518; See also G. SEGRE, Phys. Rw. 181 (1969), 1996.
OF
PHYSICS:
Generalized
67, 145-157 (1971)
Gauge Fields and Applications TUAN
to Weak
Interactions
Wu CHEN
Department of Physics, New Mexico State University, L.as Cruces, New Mexico Received August 18, 1970
Yang-Mills’ field is generalized to possess a nontrivial scalar part. The most general transformations for such a field under the 3-parameter isotopic gauge transformation is obtained. Using this generalized gauge field, a gauge invariant Lagrangian is constructed within the framework of the quark model. Interactions for spin-l as well as for spin-O are generated. As a further application a weak interaction theory mediated by the generalized gauge (boson) field is formulated. The entire weak interactions are generated in two halfs; the hadron-boson interaction is generated according to YangMills’ trick using the generalized gauge field and the other half (boson-lepton, etc.) is then generated by making use of the scalar part of the gauge fields according to the conventional pion gauge principle. The effective Lagrangian is then found to be mediated by the effective propagators which fall off as pWe at high momenta; the unitarity of the theory can thereby be insured. Universality in weaker sense than the usual one is applied to the intermediate bosons; our theory for p-decay then reduces to Cabibbo’s at low energy.
1. INTRODUCTION Is is well known that the invariance of (matter) Lagrangian under local gauge transformation can be maintained by introducing spin-l gauge fields [l]. The interactions between spin-l fields and matter fields are then generated. This powerful principle has been used extensively in recent years not only to generate interactions but also to uncover the underlying principles behind conservation laws or symmetry groups [l].l Not too long ago another kind of gauge principle was introduced by Gell-Mann and Levy [3] in which spin-0 fields instead of spin-l are used as gauge fields. The interactions between spin-0 and lepton fields (or currents) are thereby generated. More importantly, the hypothesis of partially conserved axialvector currents (PCAC) in weak interactions can be explained in a natural way.2 In this paper we would like to further generalize the existing gauge principles 1 A good account on this subject can be found in the book by J. Bernstein [2]. 2 See Refs. [2 and 31; also G. Kramer and W. F. Palmer [4].
145 595/67/r-10
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CHEN
by considering more general gauge fields: Four-vector fields with nontrivial scalar parts3 Our purpose is twofold: In analogy to the method of Yang-Mills’ we wish to generate interactions for spin-0 as well as for spin-l fields; moreover, we wish to use the spin-0 parts of four vector fields to generate weak interactions closely following Gell-Mann and Levy’s method. Allowing the explicit presence of spin-0 fields in transformed gauge fields under local gauge transformation, the generalized gauge field is developed in next section. The gauge invariant Lagrangian is then constructed in Section 3. The rest of the paper is then devoted to an application of our idea to weak interactions where intermediate bosons are considered as the generalized Yang-Mills’ fields. Local gauge transformations without parity conservation are considered in Section 4; there the boson-hadron interactions are also constructed. In Section 5 Gell-MannLevy’s method is then applied to introduce boson-lepton interactions. Finally, to complete the weak interaction Lagrangian, the neutral boson-current term is constructed in Section 6 to account for nonleptonic weak interactions. In Section 7 a weaker form of universality in the spirit of Cabibbo’s [5] is applied to the coupled boson fields to reduce the number of parameters involved. In addition to satisfying all the usual selection rules, the weak interactions so generated are found to have the important feature: It yields the effective weak Lagrangian mediated by weak propagators which fall off as p-z at high momenta. Some discussions are followed. 2. GENERALIZED
YANG-MILLS’
FIELDS
Consider the Hermitian fields of scalar (designated by bJ and vector (designated by Bip) belonging to the octet representations of SU(3). Under the rotation in isotopic space they transform according to the rules b ---f b’ = FbS,
(2-l) (2-2)
Bu --+ B’u = S-IBuS, where b = bihi , BP EZ&‘&
(i = l,..., S),
with hi the usual 3 x 3 SU(3) matrices, and S, a 3 x 3 unitary matrix with determinant unity, is given in general by s = ei=j%, (2-3) where 0~~(j = l,..., 8) are constants. s Gauge fields Au in Yang-Mills’ theory always satisfy the supplementary %A,, = 0, so that the scalar part of AH does not exist. See Ref. [l].
condition,
viz.
GENERALIZED
GAUGE
FIELDS
AND
147
APPLICATIONS
Equations (2-l) and (2-2) with S independent of x, y, and z are nothing but the precise statement of SU(3) properties of b and B”. The question we like to raise in this section is then to ask what is the most general change the field B” can suffer if S is a function of x, y, and z. We shall look for such a generalization of (2-2) within the following criteria: 1. It is compatable with Eq. (2-2) for constant S. 2. It has at most first derivatives. 3. It can only be at most linear in fields BJ‘ or b. Note that the scalar field b is allowed to appear in our generalization; this is to obtain more general transformation than the usual local gauge transformation; moreover there is no a priori reason to ignore the spin-0 part in a four vector field. The most general form can easily be seen to be B” + B’u = S-lB”S
+ fiS-W’s
+ &Wb@S
+ &%!P1bS,
(2-4)
where 5 and 5 are constants. Note that every additional term in Eq. (2-4) contains LPS or aUS1 which vanishes if S is a constant matrix. The Hermiticity of BP implies that and t* = 5 51* = 5, * Moreover, the vanishing of the trace of B’u says that [I = 5, . Thus we conclude that the most general form for B’” is (with real c and 5) B”1 = S-lB%
+ i&!-WS
+ &S-lbauS
+ aW1bS).
(z-5)
Equation (2-5) gives the generalized transformation property of the vector field Bu under a local gauge transformation. We recognize that, with the exception of the last term in Eq. (2-5), B” transforms like Yang-Mills’ field. The change of the vector field under the infinitesimal gauge transformation can be written down easily from Eq. (2-5). We have, after some manipulation B’u = BP - i[q B”J - &%
+ i&3%,
b]
(2-6)
B’” = B” - +ol - ig{[ol, B”] - y[&x,
b]}.
e-7)
or, after redefining the constants, we have4:
4 The constants g and y in Eq. (2-7) are related to [ and 1 according to g = l/f and y = 1; moreover, 01in ELq. (2-7) differs from oi in Eq. (2-6) by a factor of g.
148
CHEN 3. THE INVARIANT LAGRANGIAN
The Lagrangian invariant under the local gauge transformation (2-5) can be achieved in a straightforward manner if one observes that Eq. (2-5) can be rewritten as B’L” - @ub’ = S-l(Bu
- &Fb) S + i&-9@S,
(3-l)
where b’ = S-lbS. We see that the combination B” - @b, in fact, transforms exactly like Yang-Mills’ field. The invariant Lagrangian can thus be constructed in terms of this combination according to the conventional way. For convenience, let us define W’1 = Bfi - {@b.
The kinetic part of the invariant Lagrangian LYKa = -&Tr(G@V)2
is then of the familiar - +Tr(Dub)2,
(3-2) form (3-3)
where G” = auw - i? W” - ig[ Wu, w’]
(3-4)
Dub = @b - ig[W”, b].
(3-5)
and
We have also included the kinetic part for the b-field in Eq. (3-3). Clearly, Eq. (3-3) alone gives a local gauge invariant theory for massless spin-l and spin-0 fields. For massive fields, the mass terms then break the local gauge invariance. The total Lagrangian PO with added common mass terms for Bu and b is given by
-% = %E - q
Tr(B’Q2 - $ Tr(b)2.
(3-7)
It should be noted that go still is an SU(3) invariant Lagrangian although it is not local gauge invariant. We shall see later that the mass terms in fact give rise to some interesting results. To complete the gauge invariant Lagrangian we shall consider the quark field q, 3 x 1 matrix belonging to the triplet representation of SV(3).5 Under the local gauge transformation q transforms according to the rule 4-4
‘ = s-lq.
(3-8)
6 One could indeed consider any matter fields instead of the quarks and carry out the following without much modification.
GENERALIZED
149
GAUGE FIELDS AND APPLICATIONS
The gauge invariant Lagrangian -E”, for q can be derived with the help of a gauge field. Using B” of previous sections, PQ is obtained through the replacement of 8‘ by DLLin the free Lagrangian of q, where Du = au -
igw”
(3-9)
= au - ig(B’* - (@b).
We have 6” = -Zjy”a,q
- Mqq + ig@“B,,q - ig&y”aubq.
(3-10)
Note that we have generated not only the interactions for spin-l particles but also the interactions for spin-0 particles. The equations of motion can now be easily written down from Eqs. (3-7) and (3-10). In particular for B” and b, we have (( 0 - m2) g”’ - a”a”} Bty + 2gfiik{(GY W,, + Dubi) bk - a,( Wj” W,“)} czz-igtjy”Aiq (3-l 1) and (0 - ~3 bi + &&{a,(Wj”b&
+ D,bjWku}
= --m2@~Bt,, .
(3-12)
From Eq. (3-l 1) and the relation
which follows easily from the equations of motion for the quarks, we note that the lowest order contribution to (quarks 1 BiM 1quarks) is of order g, whereas to (quarks 1bi / quarks> is of order gs. Thus, the interaction for bf is not so interesting for this case even though we are able to generate the coupling of bi to the vector current iqy@&q. In the following, we shall further generalize the above idea to include chiral gauge transformation and see a more interesting coupling of bi to axial vector currents.
4. INTERMEDIATE
BOSONS AND GENERALIZED
YANG-MILLS'
FIELDS
In previous sections we have restricted ourselves to the Iocal gauge transformations which leave the parities strictly unchanged. In this section we wish to apply the gauge principle we have developed so far to weak interactions. Since the parity
150
CHEN
is not conserved in weak interactions, it is more interesting to consider the unitary matrix S of the form s
=
pl+aYJcr
(4-l)
instead of Eq. (2-3), where a is a fixed constant and (II = h&x). Only a slight modification is needed in the procedure of constructing the invariant Lagrangian under Eq. (4-l). We assume that weak interactions are mediated by vector boson9 BLL as well as scalar boson’ b, which transform under the infinitesimal local gauge transformation* according to Eqs. (2-l) and (2-7), i.e., b -+ b’ = b - ig[a, b], B”+
(4-2)
BtU = B - a% - ig[a, Bu] + ic[a%, b].
(4-3)
While quark fields transform according to 4 + 4’ = (1 - ig(l + q&4
4.
(4-4)
Note that the function (II has no definite parity. It is a mixture of scalar and pseudoscalar functions. That this is the case can be seen from Eqs. (4-2))(4-4) and the fact that B” and b do not have definite parities. It would probably be asking too much to expect nature to respect the symmetry under Eq. (4-4) strictly. Even for weak interactions, we shall only expect the theory to be approximately invariant under Eqs. (4-2)--(4-4). Now the gauge invariant Lagrangian can easily be constructed. We shall adopt the point of view that Lagrangian is a (local) gauge invariant only when all of mass terms are absent; moreover, we shall see that establishing mass terms in fact is a natural symmetry breaking mechanism. The invariant Lagrangian for Bu and b without mass terms is obviously the same as in Section 3. In order to obtain the part for q, we need slight modification in our procedure in Section 3; we should replace au in the free Lagrangian for q by D”
= au - ig(1 + uy,) W”,
(4-5)
BThere are many intermediate vector boson models, for example, the recent model by G. Segre [6]. Earlier work can be founded in the references cited there. ’ There are also many weak interaction models mediated by scalar bosons. See, for example, N. Christ [7]. 8 Treating intermediate vector bosons as gauge fields can be found in many places, for example, by S. Okubo [8].
GENERALIZED
instead of (3-9). Putting terms, we have finally
GAUGE FIELDS AND APPLICATIONS
the above invariant
151
parts together with the usual mass
where (4-7) with -Y&(B~, b) given by Eq. (3-4), zrn = - $ and the quark-intermediate
Tr(Bu)P - f Tr(b)2 - Mjq
boson interaction Lagrangian =%a = m41
Clearly &n symmetry.
+ =&
(4-g)
dip,-, is given by
+ ayJ wqu .
(4-9
is local gauge invariant while the mass part =.%&breaks the
5. MODEL OF WEAK INTERACTIONS BETA DECAY LAGRANGIAN Customarily the weak interaction model mediated by vector bosons can be written down just by inserting the bosons between current-current couplings. A more sophisticated way is to employ the assertion of field-current identity to generate the Lagrangian [9]. Here we shall use a different approach9 closely following the work of Gell-Mann and Levy [3] based on fundamental (gauge) principle to achieve a formalism which has the definite advantage over the usual one due to its less divergence difficulty. Since quark-boson interactions have already been generated based on local gauge symmetry in the previous section, we need only to construct the interaction between the bosons and leptons to complete the beta decay Lagrangian. To do this, we start with the current-current form, i.e., PBel = g’l~j~’ + h.c., (5-l) where 1“ is the usual lepton current given by
111= 331 + yd e + C’y@(l+ y5) p 9 A primitive work on this idea is briefly reported by the author [lo].
(5-2)
152
CHEN
and j,$, is the linear sum of the positively charged components responsible for the transformationlO
of the current j$’
(5-3) where vt are constants. The current ji@ can easily be obtained from the total quarkboson Lagrangian (Eq. (4-6)): Au=-=
a9
rn”< - g Biu + 1 Dub,). ( m”5
(5-4)
This current is not conserved. In fact, its divergence is given by (5-5) However the matrix elements of a, jiu between hadron states is given by (hadrons I a,jiu I hadrons) = f
(hadrons I bi I hadrons) + O(g2)
(5-6)
since the contribution of the 2nd term in (5-5) is of the order O(g3). Moreover, one sees easily from the equation of motion for bi that the first-order contribution (which is also the lowest order) in g to (hadrons 1bi I hadrons) is purely pseudoscalar. Thus, even though the current ji’ has mixed parities and is not conserved, its vector part is conserved (i.e., CVC) while the divergence of its axial vector part is proportional to the pseudoscalar field bi in analogy to the PCAC hypothesis. It is also interesting to note that our Eq. (5-4) is essentially the field-current identity asserted by Lee [9]. Using n+, K+, etc. to denote the W(3) states, we have for the positively charged components Of ji’: 1
*IL Jn+ - -d/zg
[
B,’ - iB,@ + ---& D%
*Ii JK+ - -$-g
[B4u - iB$ + &.
- W],
(5-7)
D”(b4 - ib&].
And the current j&, in Eq. (5-l) is thus given by
A+)= 01; jc++ a,'ji+
.
lo It is important to note that we require Wp, instead of BP, to remain unchanged.
(5-9)
GENERALIZED
GAUGE FIELDS AND APPLICATIONS
153
In summary, we have the beta decay interactions by
+ gzu[al (B;+ + &
qhv+) + 01~(q+ + &
q)++)/ + h.c.9 (5-W
where hiu are hadronic currents given by hill = iqyu’(1 + ay&q. 6. MODEL
OF WEAK
INTERACTIONS,
(5-l 1) CONTINUED
In order to have a complete theory of weak interactions, we must add some extra terms to account for AZ = 4 strangeness changing nonleptonic weak interactions. For this purpose, there are two well known ways: one is to couple the neutral boson to neutral currents (so called schizon theory) [11], the other is through some enhancement mechanism [12]. Here we shall adopt the first way but proceed along the same principle as in the previous section. We assume that the strangeness charging interactions are due to the term CL.(o)
%--s =gvJG
>
(6-l)
where jf) is strangeness carrying neutral current and is given by the linear sum of neutral components ofj,“ of Eq. (5-4). That isll
with two real constants &’ and pZ’. The neutral hadron current Vu is chosen to be i@s(l + ay,) q, a scalar under SU(3). This indeed is the simplest possible form to insure AZ = 4 and AS < 1, for then we have -EB-S of (6-l) transform like an octet member. The relevant strangeness changing weak interactions thus become
+ h.c.
(6-3)
with real constants g, /31 , and /3Z. I1 In order to satisfy 41 = l/2 rule in a straightforward manner, only the currents with isotopic charge < l/2 are included. The current, B,p + l/cm8 Dfib, , is dropped since it will only make the constant p1 in Eq. (6-3) complex rather than real.
154
CHEN
7. EFFECTIVE LAGRANGIAN FOR WEAK INTERACTIONS A. Universality Based on two kinds of gauge transformations we have constructed the Lagrangians as general as possible; consequently, there are many free parameters involved. Here we would like to impose some conditions so as to reduce our Lagrangians to the one consistent with Cabibbo’s theory. We would like to impose the condition of universality on the vertices in a slight different sense from Gell-Mann’s but still in analogous to the photon field. The beauty of the intermediate boson model for weak interactions is recognized actually through the similarity between the vector boson fields and the electromagnetic field and customarily we assume that the boson fields couple to currents through a universal constant. Since, in fact, we are dealing with one or more intermediate bosons (instead of just one photon field in electrodynamics) we propose to modify the meaning of universality in the following way 12: The eflective coupling constants for diagonal current-current interactions should be the same. That is for a Lagrangian 9 = jlu(a$l,
+ a2B2p + . ..) + j2u(blBl,
+ b2B2, + .. 9,
we demand that aI2 + a22 + ... = b12 + b22 + *** .
It follows then that the constants in the Lagrangian
(5-10) are related by
a12 + a22 = 1,
(7-l)
a2 = 1
(7-2)
and those in (6-3) are related by 2/3,2 + p22 = 1. Moreover, applying the same universality bosons b, , we have from Eq. (5-lo),
principle
(7-3) to the spin-0 intermediate
k5)2= (*)2 + (%)’ Together with Eq. (7-l), we have the interesting result
5 = llm, la This indeed is along the same line as Cabibbo’s universality.
(7-5)
GENERALIZED
GAUGE
FIELDS
AND
155
APPLICATIONS
which says that the BP - b coupling constant is simply equal to the mass of Bp.13 The relation (7-l) clearly reduces our beta decay Lagrangian to the theory of Cabibbo’s. The relation (7-2) on the other hand gives
lal=
1,
(7-6)
yielding the conventional V - A form for the hadron currents, another wellestablished basic property of weak interactions. B. Efective Weak Lagrungians Making use of the relations (7-l)-(7-3) and (7-5) we can now sum up our weak interaction Lagrangian. We have
cos 0 (B,“’ + f
8&bn+) + sin 8 Bfc + A aybK+)) (
+ O(g2) + h.c.,
(7-7)
where 19and 8’ are arbitrary constants and O(g2) term is of the order g2 or higher and at least binomial in boson fields. Note the different signs jn front of b fields in coupling to hY and I“ (or VU) which give rise to a rather amazing second-order effective Lagrangian, viz.,
(Iw(xl) h;yx2))+ + sin 8 f
COS 8’
(I&,)
I s
(&(X1)
h;‘(x,))+ hvn(X2))+
G”“(x~
-
x2)
dX,
GUY& - x2) dx, GUY(X1
+ i sin 13’j
-
X2) dX2
- x2) dxz/,
(7-8)
I8 It is particularly interesting to note that theories based on Al - n mixed fields yield Al - T coupling constant to be precisely the mass of A, meson. See Ref. [13].
156
CHEN
where
G”“(x, - 4 = WW
W~)+)O - $ @Y k2?
m2 ’
((WJ &N+)o pa - ma m2 (k2 + ma)(k2 + p2)
k??
In Eq. (7-8), the first two terms describe leptonic weak interactions while the last two terms give nonleptonic strangeness changing weak interactions. All of interactions are of current-current form but mediated by the effective weak propagator G@”whose high-momentum behavior can easily be seen to fall off as k-2. As far as the low energy description of weak interactions is concerned, our theory indeed gives exactly the same results as the ordinary one and nothing more. However, the k-2 behavior of the propagator makes the present theory superior than the ordinary (IVB) theory whose weak propagator behaves like (k)O at high momenta. In fact, it attributes to overcoming all of the contradictions of ordinary weak interaction theory which arise from the use of unitary cut-off to estimate divergent expressions. The idea of properly incorporating scalar bosons to vector bosons so as to result in less singular weak propagators has been demonstrated through ad hoc models by Gell-Mann, Goldberger, Kroll and Low [14]. Since an excellent account on how the less singular propagators can do to improve weak interaction theory is given in their work and since our major concern here is to study the underlying principles, we shall be content with noting simply that k-2 behavior of the propagator undoubtly makes the weak interaction theory less divergent. The less singular behavior of effective propagators for semileptonic and strangeness changing nonleptonic interactions can easily be seen from Eq. (7-7) to be absent for diagonal weak interactions, i.e., the parity violating interactions among leptons or nonleptons. In fact, in those cases, the present model has exactly the same difficulty as the usual one. However, since those interactions are presumably completely dominated by interactions other than weak interactions, divergences in perturbation expansions in the weak coupling constant mean nothing other than that perturbation theory can not be applied in those cases. One could, nonetheless, break up V - A form and introduce different intermediate bosons for vector and axialvector hadron currents so as to preserve parity conservation in nonleptonic diagonal interactions and to have the less singular propagator even for the parity violating nonleptonic strangeness unchanging interactions. This would clearly mean more intermediate bosons and lack of symmetry in hadron and lepton currents. It should be noted that the gauge principle and the universality of bosons upon which the present formalism is based have reduced the number of parameters not more than those in the usual intermediate vector boson field (besides the
GENERALIZED
GAUGE
FIELDS
AND
APPLICATIONS
157
masses of scalar bosons which could be put to be equal to the masses of vector bosons). This is the main contrast to the models of Gell-Mann et al [14], where symmetry principles are not the main concern. Nevertheless, it perhaps is interesting to note that the propagator of their model I does reduce to ours if let cx-+ 0, even though their IVB’s (i.e., LQ) which in this limit is hard to understand have an apparent different structure. In closing we recall that our purpose of this work is mainly to introduce the generalized gauge fields with explicit scalar parts and to demonstrate their usefulness in weak interactions. In view of the above model which not only is consistent with existing experiments but also gives definite advantage over the usual theory, we hope, in addition to having achieved our purpose, that we have revealed some underlying principles behind weak interactions through the generalized gauge fields. Finally, we note that it is indeed not essentially to insure the nonexistence of scalar part in vector fields; contrarily the scalar part could play an important role in explaining physical interactions.
REFERENCES 1. C. N. YANG AND R. L. MILLS, Phys. Rev. 96 (1954), 191; R. UTIYAMA, Phys. Rev. 101 (1956), 1597; A. SALAM AND J. C. WARD, Nuovo Cimento 11 (1960), 568; S. L. GLASHOW AND M. GELL-MANN, Ann. Whys. (New York) 15 (1961), 437. 21 J. BERNSTEIN, “Elementary Particles and Their Currents,” W. H. Freeman and Co., San Francisco, 1968. 3. M. GELL-MANN AND M. LEVY, NUOVO Cimento 16 (1960), 705. 4. G. KRAMER AND W. F. PALMER, Phys. Rev. 182 (1969), 1490. 5. N. CABIBBO, Whys. Rev. Letfers 10 (1963), 531. 6. G. SEGRE, Phys. Rev. 181 (1969), 1996. Earlier work can be found in the references cited there. 7. N. CHRIST, Phys. Rev. 176 (1968), 2086 and the references cited there. 8. S. OKUBO, Ann. Whys. (New York) 49 (1968), 218. 9. T. D. LEE, Phys. Rw. 168 (1968), 1714. IO. T. W. CHEN, Nuovo Cimenfo Letter 3 (1970), 105. 11. T. D. LEE AND C. N. YANG, Whys. Rev. 119 (1960), 1410. 12. R. DASHEN, S. FRAUTCHI, M. GELL-MANN, AND Y. HARA, in “The Eightfold Way,” (M. Gell-Mann and Y. Ne’eman, Eds.), p. 254, W. A. Benjamin, Inc., New York, 1964, 13. T. W. CHEN AND R. E. PUGH, Phys. Rev. Letters20 (1968), 880; M. J. SWAG AND W. W. WADA, ibid21 (1968), 441; A. B~RNEL AND H. CAPRASSE, Nucl. Phys. B8 (1968), 65; also T. W. CHEN, Phys. Rev. 184 (1969), 1673. 14. M. GELL-MANN, M. L. GOLDBERGER, N. M. KROLL, AND F. E. Low, Whys. Rev. 179 (1969), 1518; See also G. SEGRE, Phys. Rw. 181 (1969), 1996.