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ΔS = −ΔQ processes in asymptotically free gauge theories
Nuclear Physics B107 (1976) 300-320 North-Holland PubhshmgCompany
AS = - A Q PROCESSES IN ASYMPTOTICALLY FREE GAUGE THEORIES * D. BAILIN, D.R.T. JONES ** and A. LOVE School of Mathematical and Phystcal Sctences, Universtty o f Sussex, Brighton, England Received 14 January 1976
The possibility of an enhancement of the short-distance contribution to AS = -£xQ semileptonic processes is investigated m the context of asymptotically free gauge theories of the strong interactions
1. Introduction
Asymptotically free theories of the strong interactions allow reliable calculations of the short-distance contribution to decay amplitudes to be made. In particular, the IAII -- ½ rule for non-leptonic decays [2,5], various rare decay modes of kaons [3], and the non-leptonic decays of charmed particles [4], have been investigated. In the present paper, AS --- - A Q semileptonic decays are discussed. The search for "charm" during the past year has stimulated fresh interest in AS = - A Q processes. Assuming the absence of a hadronic current with such quantum numbers, the discovery of any such process is interpreted as evidence that the transition has proceeded through a real charmed intermediate state. However, we shall be concerned here with genuine higher-order weak effects (1.e. those which do not proceed through real intermediate states) in the decays of known hadrons. We shall estimate not only the most likely levels at which the decays are expected to occur in the most fashionable model of the weak interactions, but also the highest levels at which they could occur using only what we know for certain about the weak currents. Thus we assume that the Cabibbo current ~ has the form ~ = cos 0 C ~ + sin 0 C S~,
(1)
where J~, S~ are respectively strangeness conserving, non-conserving currents. We assume that 9 a and the known leptonic current La are coupled to the field W~ of a
* Research supported in part by the Science Research Council under grant B/RG/5935-6. ** Present address: Department of Theoretical Physics, 12 Parks Road, Oxford, England. 300
D Bathn et al. / ~xS = - A Q processes
301
charged vector boson with (semi-weak) coupling constant f, _
+
~ I - f ( 9 ,~ + L~) Wa + h.c.,
(2)
in the framework of a renormahzable theory of the weak interactions. In addition, we assume that there is no strangeness changing neutral current, as indicated by experiment. None of these assumptions is controversial. Then it follows that the ~S = - A Q processes occur first in order f 4 . The matrix element for the process
(where A and B are hadron states with S A - S B = +1), is given by 9~ = f 4 sm 0 C cos 2 0 C ~-Tx(1 - 75) v~PXU(q)
×fax dy Aa~fx - y )
(BI T{J+(x)S-~(y)J~ )l A ) ,
(3)
where q = ~ + v is the m o m e n t u m carried away by the leptons. (We use the same letter to denote the particle, its m o m e n t u m and, where appropriate, its spinor.)P and A denote the W propagator in m o m e n t u m and coordinate space, respectively. Thus the d~S = - A Q processes are controlled by the connected part of the operator
c ; = f ax dy
(4)
or its hermltian conjugate. The operators J+~(x) and S~0') above are connected by the W-propagator, at least, so the overall connectedness requires J ~ to be connected by some internal hadron exchange. Thus we may dwide the (connected) contribu+ tions to Gu into three classes. (i) Those in which the W-propagator is not part of a closed loop, i.e. the momentum of the virtual W is fixed by the external momenta. (ii) Those in which the W-propagator ts part of a closed loop, but J~ is outside of that loop. (111) All others.
2. Long-range contributions The contribution G (1) of class (i) to (BI G u+ IA ) is evidently G (1) = - t ( b l l J + [ al)ec~¢(al -
bl)fdy e
t(al-bl)Y
t ~QL~I.r
X (b21r{s~-O')./~ } [ a 2 ) - i ( b ' l l S ~ l a l ~ r
×fax e'<-<)x(bll T{J;(x)L+ IC,
tol-al)
t
(50
D. Badm et al. / AS = -AQ processes
302 where
IA) = la 1) lao) = [a'l ) la~ ),
(5b)
IB) = Ib 1) Ib 2) = Ibm)Ibm).
(5c)
The momenta a 1 - b 1 and a'1 - b~ are determined by the external momenta, so for the decay processes at least, the T-product matrix elements are presumably well estimated by their Born terms. Some of these are illustrated m fig. 1 for the process K 0 --> n+~ - ~Q. Plainly, these represent the truly "long-range" contributions to G u all internal lines have momenta determined by the external momenta and are of order mH, a hadronic mass. Thus, x,y, x - y ~ m~ 1. So G (1) is determined by known, or at least measurable, matrix elements of the weak currents between low-mass hadron states. For example, we estimate the contribution from the diagram in fig. lb to be
G(1)=(Tr+lJ+lO)e'~(n)(OIS~lK+)m2 #
! lr-- m2g
( K + I J + [K o) /a
i =/~KTr P(~O(Tr)lr~ m 2 _ m 2 FK(q2 ) (K + 70.
=f~rfKm2 (m 2 _ ~2"~ FK(q2) (K + zr)u, Wt K
(6)
"'it j
where F K is the electromagnetic form factor of the K +. The matrix element for the AS = +AQ process ~0 _> rr+ £- O~
(7)
is and
c/~ 0 = f 2 sin 0C£~X(1 -- 75) vt pXU(q) (rr+l S + [E0)
(8a)
0r+ i Su IFo>= g+(q2)( K + 7r)u + g_ (q2)qu "
(8b)
The parameter x, measuring AS = - A Q amplitudes relative to those with AS = +AQ, is defined as the ratio of the spin-one parts in (8) and (3). The contribution of dia-
s./ ,,.St~j
K*
S~J' ~2.1~
s.Af rr÷
K*
(b) (o) Fig. 1. Long-range contributions to cat (K° ~ ~r+~-b'~)
D. Baihn et aL / aS = _AQ processes
303
gram lb IS seen to be FK (q2)
f•fK m2 x ( l b ) = f2 cos20c m 2 , ( ~ -2.. _~ m2~ W~ K "'Tr J
g+(q2)
"" ~/~-G cos2 0C flrfKm2(m 2 - m2}-l~r= 1.4 X 10 - 8 .
(9)
The pion pole diagram in fig. la looks more promising for a larger effect, and indeed since the pole is in the physical region, the decay may actually proceed via the production of a real rr- : K 0 ~ lr+rr -
I , ~-~.
(10)
Thus the transition wdl be observed as a first-order weak effect. We are concerned with the higher-order weak effects, of course, i.e. the contributions when q2 4= m 2. These are presumably characterized by a denominator of order m 2 rather than m 2, but in any case cannot contribute to the spin-one part of (8): they vanish if we neglect rn~.
3. Short/long-range effects The contributions characterized by class (11) reqmre that j2(x) and S~(y) are connected by internal hadrons as well as the virtual W, so that we have an internal loop. The "large" mass of the W ensures that the dominant contribution in this class is when the loop m o m e n t u m k ~ mw, i.e. x - y ~ m~vt which is "short". Since J~- is outside of the loop nothing forces x, y to be close to 0, so to goodapproximation we need consider only the contributions when Xo,Y 0 >>0 or 0 >> Xo, YO" Writing the short-distance expansion [5]
T{J÷(x)SSCv))x-y~O
n
C
(11)
the contribution G u(2) to (B I G~ I A) is evidently C (2)u =~n f d x d y A a ~ ( x - y ) C ~ n ( X - y
f
)(B[T(On(x)J +~.}[A)
.
As before, the second integral is well-approximated by retaining only low-mass
(12)
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D Baihn et al. / AS = - A Q processes
K* xil~"
Io)
K,
K+
0.
lx+
K°
0.
j~
÷
t'-
l~e
(b)
(c)
Fig. 2. Long/short-range contributions to ~ (K° --, n+~-ff~). exchanges between O n and J ~ . Some of these are illustrated in fig. 2 for the process K 0 ~ rr+£- ~Q. Diagrams 2a and 2b are the short-dxstance analogues of la and lb, while 2c is a contribution only permitted by short-distance effects. The operators O n are those which provide the short-distance contribution to the ordinary nonleptonic decays, and if we are sure we know about them we may estimate the diagrams accordingly. However we take a more conservative standpoint (for the present). We observe that the contributions from classes (i) and (ii) consist o f a transition allowed by the ordinary non-leptonic Hamiltonian succeeded or preceded (at a long distance) by a semlleptomc transition. So we may reliably estamate the t o t a l contributions from these two classes by taking the required amplitudes from "experiment". As before we ignore the spin-0 contributions in fig. 2a. The contribution from figs. lb and 2b is f 2 sin 0 C cos 0 tG(lb) + G! 2b)) = ( ~ + I ~ [ K C~ ~ . . . . _
+> ~ m~z
(K+[ J ; I K°> m2K
-i FK(q2 ) (Tr+I~NLIK+) (K + lr)u m~ - m~
- (rr+n'-[C3~NLIKO) (K + n')u m 2 - m2~r s '
(13)
using symmetric soft pion theory. Thus the contribution to x is
x/ lf.I
Ix(lb +2b)l - - cot 0 C I(zr+Tr-19gNLIK0)l m2 _ m2 = 1.4 X 10 - 6 .
(14)
D. Batlin et al / AS = -AQ processes
305
The contribution from fig. 2c may be evaluated in the same way and combined with the result (14). It yields a total contribution to x which is proportional to (Tr+Tr0I~NL IK+ ), which itself is known experimentally to be a factor of about 20 smaller than (lr+rr - I~NLIK0). Because of this cancellation, which derives from the observed IMI ---½ rule m non-leptomc de~ays, we should really proceed to discuss the contributions from higher mass exchanges: p, K* . . . . . However we know nothing experimentally about their non-leptonlc amplitudes, and we prefer to take (14) as a generous upper bound on their contributions. More reahstically we should expect them to be down by a factor of 4 or so on account of their larger mass denominators. In summary, then the net contribution from classes (i) and (il) to x satisfies Ix(1 +2)1 <~ 1.4 × 10 - 6 ,
(15a)
and probably Ix(1 +2)1 "~ 4 × 10 . 7 .
(15b)
The extra order of magnitude which we have gamed over the long-range contribution (9) derives simply because the non-leptonlc amplitudes are known experimentally to have this feature.
4. Short-range effects
We turn finally to the short-range contributions (fii) to Gp. These are illustrated in fig. 3. The m o m e n t u m k of the virtual W flows also through hadrons connecting J~+ to J ~ and J ; to S f . As before, the dominant contribution is when k ~ m w so x,y ~ m~¢I . Thus we are concerned with the contribution when all three currents are separated by short distances. This is controlled by the expansion of the threecurrent product:
T { J+(x) S~(y) J~ } x,y'~O ~n K~flun(X' y) Tn (0).
(16)
In terms of the " k n o w n " quark fields (p, n, X) we know that J ~ and S ; certainly have the pieces Ja+ = i07~(1 - 75 ) n ,
(17a)
S f = ~,#(1 - ? 5 ) p .
(17b)
K° ~ ~ Fig. 3. Short-range contributions
1"1"+ to ~
(K° ~ 7r+£-F£).
306
D. Bailin et al. / AS = - AQ processes
Then it is easily seen that the leading operators Tn(0 ) must be four-quark operators of rank 1 or 3, e.g. Tn = PTo(1 - 75) n XTo( 1 - 75) a m ,
(lS)
or contractions/permutations of these. Thus T n has (mass) dimension D n = 7, and in free field theory this implies that Kaaun(X,y ) has the scahng property K t~un(~,-~)x__" ~ x 9 - D n K o ~ u n ( X , y ) .
(19)
Consequently the truly short-distance contribution G (3) to (BI G~[ A) has the property G(3)CAmw) ~ "'3-DnG(3)tmu ~ W""~'
(20)
as noted by Roy [6]. Hence, since D n = 7, G( 3 ) ~ m w 4 ( B I T I A ) ,
(21)
where T u is constructed from T n by contractions. For the process K 0 ~ rr+£ - O~, the required matrix element has the form (;r+ [ T z l K ° >~ ta4(K + rr) u ,
(22)
where/a 0 is a mass parameter. Thus the free field estimate of this short d~stance contributxon to x is x(3) ~ f2 cos 2 0 COa4/m 4 ) = ~
G cos 2 0 C . 4 r n ~ .
(23)
We may estimate/a 0 from the known amplitude for K ~ 2rr. A similar free field calculation for this process yields c/g (K -+ 2n) ~ X/~2G sin 0 C cos 0 C (21rl O n IK) "V/~21Gsin 0 C cos 0 C/.103 ,
(24)
where O n is given in (11) and is a four-quark operator of dimension 6. Then from the decay rate for K 0 ~ lr+rr - we find /a0 ~ 607 MeV,
(25)
and substituting into (22) gives x ( 3 ) ~ 1.1 X 10 - 8 ,
(26)
D. Baihn et al / AS = -AQ processes
307
since it is known experimentally that m w ~> 10 GeV.
(27)
Of course, this estimate of x(3) depends crucially upon the value (25) for #0 derived assuming Ks0 -+ 7r+zr- is a "typical" amplitude. Had we used the IAI[ = a amphtude C~(K+ -+ 7r+lr0) the resulting value o f # 0 (219 MeV) gives a much lower estimate (1.8 × 10-1°). Thus (26) assumes that the I A/I = ~2 amplitudes are suppressed (and therefore atypical) rather than the IA/I = ½ amphtudes being enhanced. We shall return to this point later. If one believes, on the evidence from the deep inelastic scattering experiments, that Bjorken scaling is really true, with no deviations, then (26) is a reasonable estimate. It would indicate that the dominant contributions to x are an order of magnitude larger than (26), as we found in the previous section, and controlled by the characteristics of the ordinary non-leptonic Hamiltonian rather than the short-distance structure of the three-current product that we have been discussing here. However, there is some evidence of scaling violation [7], and the most popular belief is that these are logarithmic deviations, as forecast by the asymptotically free gauge theories [1]. In this case D n would continue to have its free field value 7, but (20) would be modified by the appearance of (calculable) powers of In k: G(3)(Xmw ) --- X-4(1 + k In X)pn G(3)(mw ) .
(27)
For such theories, then, the short-range contribution to x is x(3) ~ / ~ G
mw IPn #---~4 cos20 C m 2 (1 + k i n ms ]
(28)
The mass/a appearing in (28) is different from the/a 0 in (23), because in this model the K ~ 2rr amplitude (24) is also modified [2] by (different) powers of 1 + k ln(mw/ms). Consequently U3 = 03(1 + k In x)q,
(29)
and our estimate (23) is modified.
__1 a4 [
mwlPn-~q
x(3) ~ X/~2Gcos 2 0 C m 2 \,1 + k in ms !
(30)
For an SU(N) colour group with quarks in the N representation [1 ]
g2 k=
8n 2
g2 b=
247r2
(1 I N - 2 F ) ,
(31)
where F is the number of quarks (p, n, ~, ...). We "know" that F~> 3, N = 3 from
308
D. Badin et al / AS = - A Q processes
rt0 ~ 3'7, and it IS customary to assume g2/41r ~ 1 with the choice m s ~ 1 GeV, although this is determined in pnnciple by the "observed" scaling deviations. Thus our previous upper bound (26) is modified: x ( 3 ) ~ 1.1 X 10 -8 X (4.3) p n - ~ q
(32)
So provided Pn - ~q >~ 2.5, this upperbound would be comparable with our earlier estimate (15b), and if Pn - ~q ~> 4 we might even entertain the possibility that these structure dependent contributions were dominant. Further, such values of the exponent are not unreasonable. Calculations of the predicted scaling deviations in the structure function moments for electro- and neutrino-production [8,9] give powers of order 1 for quite low moments, and arbitrarily large for high enough moments. In the following section we shall perform the anomalous dimension calculations necessary to see whether either of these possibilities is actually realized.
5. Anomalous dimension calculations We have already noted that the short-distance expansion (16) of the three-current product is dominated by operators of the form (18) in free field theory. Because of the (colour) gauge invariance of the asymptotically free theories we shall be concerned with the gauge covariant generalizations of the operators (18). These may be constructed by replacing the derivative ~r by its covariant form D r: Dr = ~r - igRBABr '
(33)
where g is the coupling constant of the gauge field theory, R B are the matrices representing the colour group G s in the representation to which the quark fields have been assigned, and A~ are the gauge field operators. We shall consider only the colour groups G s = SU0 V) with the quarks in the fundamental, N-dimensional, representation. There are then two gauge covariant generalizations associated with each free field operator. In the case of (18) these are: /53"o(1 -- 3'5)InX3'o(1 - 3'5)I(Drn),
(34a)
iC~o(1 - 3'5)RBn)t3'a (1 -- 3'5)RB(Dr n)"
(34b)
We have suppressed the colour indices on the quark fields and on the colour matrices I and R. Before proceeding further we introduce a more compact notation. We associate the fields/5 and Xin (34) with the creation of quarks having momenta P and K, respect:rely. The two fields n, coupled to/5 and X, are associated with the destruction of quarks having momenta p, k, respectively. Then the D r in (34) is associated with - i ( k r + g R B A B). Our notation for the operators (34) makes explicit only the Lorentz radices, the field upon which the covariant derivative acts, and which of the two
D. Batlm et al. / AS = -AQ processes
309
colour combinations (34a) or (34b) is revolved. Precisely we define:
Ikr(po ) =- i/37p(1 - 3,5 )In~-3,a(1 - 3,5 ) I ( D n ) ,
(35a)
Rk.(po) = ip3,p(1 - 3'5)RBn~3,o(1 - 75)RB(D-*rn ) .
(35b)
Evidently there are six more basic operators: two for each covariant derivative of the other three fields. Thus we define
Ipr(P°) =--tP3'o(1 - 3,5)I(D-'*zn) A--Ta(1 - 3'5 ) I n '
(35c)
Rpr(o°) - iPTo(1 - 7J) R B(D-*~n) X-7o(1 - "/5)RBn'
(35d)
IKr(PO ) =- -t/33,0 (1 - 3,5)In(kDr) 3,o(1 - 3,5)n,
(35e)
RKr(po ) = - i ~ p ( 1 - 3,5) R BnC/~D~r T) 3,~(1 - 75)RBn ,
(35 0
I P (po) = -,(pD~-tr) 3'p(1 - 75)In~3"o(1 - 3'5)In,
(35g)
RPr(po ) =- -i(pD~-'fr)3,p(1 - 75) RBn~'3'a(1 -- 3'5)RBn.
(35h)
To Incorporate the implications of the Fermi statistics of the quarks we shall have occasion to Fierz reorder expressions of the above type. We therefore work with linear combinations of the I and R operators which facilitate this. We define
Sqr(po) ~ - ~ -
Iqr(po ) + 2Rqr(po) ,
(36a)
Aqz(pa ) = - ~
Iqr(pO ) - 2Rq~(po),
(36b)
where q = p, k, P, K. The combinations S, A are respectively symmetric, antisymmetric under Fierz reordering of the colour labels. We consider first the total divergences
SAr(po ) --S(p + k - e - K)r (po),
(37a)
AAz(po) =-A(p + k - P - K)r (po) .
(37b)
The matrix elements of these operators are fixed by the matrix elements of the rank-2 operators S(po) and A(po) which are defined analogously to (36):
S(po) =
N+I N
A(po) =- ~ 7Y
I(po) + 2R(po) ,
(38a)
I(.po) - 2R(po),
(38b)
310
D. B a i h n e t al. / A S = - AxQ p r o c e s s e s
where I(0o) - ~o(1
- vs)In~o(1
-
v5)I.,
(38c)
R(0o) =/37o(1 - 7)R BnXTo(1 - 75)RBn.
(38d)
We may Fierz reorder S(0o) and A(0o), using the well known v-matrix Fierz identities, and find
S(0o) = -½ [S(0o) + S(op) - gooS(otet) + i(poo43) S(a3)] ,
(39a)
A(0o) = +½[A(0o) + A(op) -gooA(ua) + i(0eoe3) A ( ~ ) ] ,
(39b)
where (poa3) denotes e~ . Thus we deduce the following useful identities: po
S ~oo ) =-S(0a) + S(op) = ½goo S(a~),
(40a)
S [po] =S(0o) - S(oO) = -t(0oa3) S(o43),
(40b)
A (as) = 0 ,
(40c)
A [pa ] = +i(0oo43) A(oJJ).
(40d)
The anomalous dimensions of these operators are evaluated by calculating the renormalization constants of the associated inserted Green functions. The order g2 contributions to these are shown in fig. 4. We work m the Landau gauge and are concerned only with the divergent pieces. We find S(0o)-~ l ~ 2 1 n - ~
( + -
(--2)(goag~-g3og~o)(SA(O43)),
(41)
where ~ is a renormalization mass and (AS(O~))is the zeroth order matrix element. The anomalous dimension of the multiplicatively renormalized operator 0 is defined as
O(ln Zo) g2 = a o + O(g4) 7o = a(lngt) 16~r2
(42)
where Z o is the renormalization constant. Using (39) we find that the following operators 01 (t = 1..... 4) are eigenvectors with anomalous dimension (g2/16n2)dol
{po),
oi = s ( 0 p ) ,
s[oo],
A
doi = 6 (N-N 1) ,
2 (IV-N 1) ,
2 (N+N1) ,
A [po], 2(N+N 1)
(43)
Each of these is associated with a logarithmic factor of the form given in (28) with
Po = -do/2b , where b is defined by (31).
(44)
D. Badm et al. / AS = - AQ processes
311
p
n
n
/
Fig. 4 Inserted Green functions m single-looporder.
The calculation of the other anomalous dimension eigenvectors and eigenvalues proceeds analogously. Since we are primarily interested in the eigenvalues, rather than the precise form of the eigenvectors, we need only consider inserted Green functions in which the inserted operator carries away zero momentum. This is because the total divergence operators, S A r ( p o ) and AAT(po ) defined in (37), are multiplicatively renormalized, as we have seen, and do not "mix" with the other basic operators. This makes the following calculations considerably simpler than they would otherwise have been. We have to calculate not only the analogues of the diagrams shown in fig. 4, with S ( p o ) replaced by SqT(po), but also the "gauge diagrams" in which the virtual gauge field is coupled to the vertex via the - i g R a A B piece o l D r in (33). These are shown in fig. 5. Despite the simplification obtained by assuming momentum conservation, p + k = P + K in the diagrams, the analogues of (41) for the operators (35) are still very complicated. Further simplification is achieved by the use of operator "equivalences", analogous to (40). They are obtained by the use of Fermi statistics and Fierz reordering together with the equations of motion: (45) We have described the relationships thus obtained as "equivalences", rather than
Fig. 5. "Gauge diagram" contnbunons to inserted Green functions m single-looporder.
D. Baihn et aL /
312
~, =
-AQ processes
identities, because some of them depend upon momentum conservation for their validity, and as such are equivalent only when taken between states of equal momenta. The relations derived are summarized in the appendix. The basic calculations involve the evaluation of the divergent pieces of the inserted Green functions corresponding to the operators Skr(po), Akr(PO), SPr(po) and AP~(po). To find the divergent pieces of the Green functions inserted with Spz(po) we apply the "mirror operation" to the expression derived from Skr(Pa); this IS defined as the combined interchanges p *-* k, P *-*K, p *-* o and left 3'-index ~ right 3,-index. The same operation applied to the contributions from the remaining three of the above operators yields the contributions from Apz(po ), SKz(po) and AKT(PO). (These last two are obtainable from the previous calculations by momentum conservation, an any case.) The results of these computations yield (divergent) expressions which are linear combinations of the (zeroth order) matrix elements of 35 operators; this should be compared with the four operators appearing on the right of (41), after use of (40). Rather than presenting these "raw" calculations we shall do a little preliminary "processing". We present first the results for the rank-one operators, obtamed by contracting two of the indices of the rank-three tensors (35). It follows from the equivalences given in the appendix that there are just four independent rank-one tensors Skz(pp ), A kr(pp), S(P- k)r(pp), A [pP(rp)+ k p(p T)]. For these operators we find:
Skr(pP)
1-+-
A[pO(rp)+kO(p,)lj
g2 lnA2 16Tr2
/~2
N-1
(~l)2j
(SkT(PP))
(46)
X
:A [pP(rp) + kP(pr)]
These yield two eigenvectors with anomalous dimensions (g2/167T2) d (1), where d(1) _= 2 [2N2 + 7 N - 7 + (4N 4 + lON3+6N2-80N+ 64) 1/2] -
(47)
3N
Similarly,
S(P- k)z(PP)I_
g2 In
AkT(PP)
)
16"2
A2 p2
I ( N - 1)(2N+7) 3N 32-(N- 1)
~(N+ 1) _ (N+ 1)(2N-7) 3N
"" (48)
X
(Ak~.(pp))
D. Badin et al / aSS = -2xQ processes
313
give elgenvalues with d~2) = 3 ~ [2N2 - 7 -+N(4N 2 + 21) 1/2 ] .
(49)
Next we turn to the rank-three operators. Since the rank-one operators mix only with themselves, as is apparent from (46) and (48), we may slmphfy the analogous equations for the rank-three operators by deleting the rank-one matrix elements from the right-hand sides. As noted earlier, this procedure will not affect the eigenvalues although the corresponding eigenvectors will only be correct up to the terms we have deleted. The results given in the appendix show that the independent rankthree operators are
s ( p - k)~ {po },
s(p + k),[po],
s(e- p)~.[oo],
S ( P - k)rlPel,
A(p +k), {po },
A ( e - K)¢ {p o },
A(p+ k)TDo] ,
A(P-p)r[pa] .
With the exception ofS(P-k)r[PO], each of these gwes rise to three operators by cyclic permutation of p, o, r. However, the equivalence (A.18) shows that only two permutations of S ( P - k)r [po] are independent, so we shall find just 23 rank-three eigenvectors of anomalous dimension. With the above noted simplificahons, we find g2 l n A 2 / N - 1
S{p-k)r{PO}-+16rr2
112 [ 6N [(8-2N)(S(p-k)r{P°})
- (N+2) (S(p-k)o {or}) - (N+2) (S(p-k)p {or}) + (2N + 10) (S(p + k)o[pr ] ) + (2N + 10) (S(p + k)o[or ]) ]
+ ~N+ 1 [_2 (A(P_ K)r {PO }) _ (A(P_ K)o ~ r }) _ (A(P- K)p {or}) + 2(A(p + k)o[Pr]) + 2(A(p + k)o [or] )] } ,
g2 lnA_~2 {~_ /~2
S(p +k)r[Pr ] -+ 16rr2 - (N+2)
(50a)
[_6(N+3)(S(p+k)r[po])
(S(p+k)a[pr ]) + ( N + 2 ) (S(p+k)p[Or])]
+N+I 6 [2Ot(p +k)r[pO l) + ( A ( P - K)o for}) - ( A ( P - K)o {or})
- (A(p + k)o[pr ]) + (A(p + k)p[Or])] ], )
(50b)
314
D. Baihn et al. / AS = -LxQ processes g-_~_2lnA2 (N__6S~I [2(S(p+k)r[po ]) A(p + k)r[po ] ~ 16rr2 02 +- - N+I
+ (S(p + k)p[o~']}] + ~
[ - 6 ( N - 3) (A(p + k)z[po ])
- ( N - 2) (A(p + k)o [pz]) + (A (t9 + k)p [o~] )] }
(50c)
#
g2 A2 { U 6 1 A ( P - K ) r {po} --* 16rr--~-ln U~- .
[-2(S(p--k)r (po))
- (S(p- k)o {pT }) - (S(.p- k)p {o'r }) + 2(S(p + k)o [07"]) N+I + 2(S(p + k)p[or])] + ~ - (N-2)
(A(P-K)o{pr)) -
[-(2N+ 8) ( A ( P - K)T {po}) (N+ 2)
+ ( 2 N - 10) (A(p + k) ° [pr] ) + ( 2 N - 10) (A(p + k)o [or])] }.
(50d)
/
Evidently the 12 operators arising here may be diagonalized separately from the remaining 11. First we sum over cyclic permutations of the three indices in the above equations. This immediately yields the "cyclic" eigenvalues d(3) - - -4 IN2 + 1 + N(N 2 +3) 1/2 ] 3N '
(51a)
2 [2N 2 - 7 + N(4N 2 + 21) 1/2 ] = d~2) . d (4) - -~-
(5 lb)
The equality of d_+ (4) and d+(2) can be understood with the aid of the equivalences given in the appendLx. For example, cychc perms.
S(p + k)z[po ] = -(rpoX) (/~al3X)S(p + k) u [0~ l t(zpoX) S ( P - k)x((~e0 ,
(52)
using (A.2), (A.3) and (A.16). An analogous relationship holds for Y. A(p + k)z[po ]. Thus both of these "cyclic" operators is proportional to a rank-one operator, which implies the equality (5 lb) of their eigenvalues. The remaining 8 operators dealt with in (50) split into two groups of 4 which may be separately diagonalized. We find that both of these sets yield the same (quartic) characteristic equation, although we have not understood why. We are able to factorize the quartic into quadratics and obtain
D. Baihn et al. / AS = -AQ processes
315
the following four elgenvalues: d(.5) =___~1 [4N2_ 12N- 5 + (16N 4 - 6N3-96N2+30N+225)1/2], -
(53a)
3N
d~6) ~
1
[4N2+ 12N-5 + (16N4+6N3-96N 2 -30N+225) 1/2] .
(53b)
The remaining calculations yield: g2 ln A2 {_ _~_ (N_ l ) (N + 2) (S(P_ k)r[PO]) S(P- k)z[po] -~ 167r 2 /a2 + ~ (N + 1) [(A(19+ k)o {.or}) - Gt (p + k)o {or })] } ,
(54a)
A(p+k)r{O°}~ -1-g26n ~ InA2U-}- {N31-- [ N+I + (S(P- k)p[or])] + ~ [-6(N- 1) (A(p+k) r {po}) + (2 -At) Gl(p + k)a {pr)) + (2 - A r) (A(.p + k)o {or})] } ,
(54b)
S(P- p),[po] ~ g 2 lnA2 { [-2(2N+ 7)
N-1 6N
+ N+ ~ 1 [4(A(P-p)~[Ool)+ (A(P-P)o[OrI>-l } A(P-P)r[P°]~ gl@n21n-~ { ~ @
(54c)
[4(S(P-P)r[P°')
N+I - +
( N - 2) (A(P- p). [pr] ) + ( N - 2) (A(P- p)p[or])] } .
(54d)
As before we consider the cychc operators first. The one associated with (54a) as zero. (54b) gives an eigenvector with eigenvalue 2 (iV+ I) (4N- 5). d (v) = -~-
(55)
316
D. Bathn et al. / AS
= -AQ
processes
(54c, d) yield the cyclic eigenvalues d(_.+8) =d~ 1)
(56)
for the same sort of reason as illustrated m (52). The remaining operators spht into four sets of two operators and each pair yields the same two eigenvalues. d (9) =---/ [4N 2 + 2 N - 5 + ( 1 6 N a - 2 N 3 - 1 2 N 2 - 2 N + 1) 1/2] -
3N
(57)
-
It is apparent from (44) and (28) that we seek the operator with the largest value of - d . For the case of physical interest N = 3, and the maximum value of - d is given by eq. (53a) with the minus sign: d(__ 5) ~ - 3 . 2 4 .
6.
(58)
Conclusions
The calculations of sect. 5 on the short-range contribution to AS = --AQ decays show that the dominant operator in the expansion (16) of the three-current product is the operator given in (58a). It is associated with a logarithmic power of the form (28) with P=-
d(_s } 2b = 0 " 1 8 '
(59)
since b = 9 in the theory with which we are presently concerned (N = 3, F = 3). In the same theory it has been shown [2] that the value of q is q = 0.44.
(60a)
(4.3) p--~ q = 0.55,
(60b)
Thus
and we see from (32) that the free field estimate (26) is n o t enhanced in the asymptotically free theories. If anything, it is suppressed slightly. With the value (60b) the upper bound (32) becomes x(3) ~ 0.61 X 10-8 ,
(61)
which is at least one order of magnitude below the non-short-range contribution (15b). Further, the value (61) is the maxLmum that could posmbly arise using k n o w n currents and asymptotic freedom. The upper bound derived from the presently favoured Weinberg-Salam model [10] of the weak and electromagnetic interactions is surely lower than (61). This further suppression derives from two sources. In the first place, the mass of the W-boson certainly has a larger lower bound than (27). In
D. Batlin et al / AS = - A Q processes
317
fact m w ~> 37 GeV.
(62)
In the second place, the model employs the GIM mechamsm [11] to ehminate strangeness changing neutral currents, as a result of which the Cabibbo current acquires additional pieces deriving from the charmed quark. In fact, ~ + = PT~(1 - 7 5 ) ( n cos 0 C + X sin 0 C ) , + gTa(1 - 3 ' 5) ( - n sin 0 C + X cos 0C).
(63)
The extra piece revolving the charmed quark c contributes to the ~xS = - A Q processes, also m order sin 0 C cos 2 0 C, and in the hmit that the p and c quarks have equal masses it precisely cancels the short-&stance contribution evaluated in the previous section. Since m_F 4: m_ our free field estimate (23) is suppressed by a factor (m 2 - m 2 ) / m 2 , since m W is the only mass with which we can scale in the short distance limit. In the asymptotically free gauge theories this mass term is associated with further logarithmic factors [ 12],
2 m2 ( l + k l n
( m p ~ _ c) m2W
mw]r
ms !
~ 6 X l O -5
(64) '
where the numerical bound is derived [3] using m c ~ 2 GeV, m w ~ 100 GeV, r = - ~ . With this value of m w a realistic estimate of the short range effects would lead to a value at least six orders of magnitude smaller than (61). Thus the unambiguous (negative) prediction of the asymptotically free theories is that the AS = - A Q process K 0 -+ 7r+e-~ will not be observed before the 10 - 6 level obtained in (15). In the event that such processes are observed at a higher level we should have to conclude that the short-range contributions were exhibiting anomalous dimension effects far beyond anything explainable by the asymptotically free theories. We have confined our numerical estimates of the AS = - A Q effects to the K 0 --, 7r transition. This is because the neutral kaon system affords us the unique opportunity to measure the AS = - A Q amplitude, vta interference effects [13]. Clearly, at the level of effect we are concerned with, such experiments are far more feasible than rate measurements for processes such as 2;+ ~ n e+Ve. A precisely similar analysis may also be applied to IAYI = 2 semileptonic processes. We predict that these will occur primarily via non-leptonic decay preceded or succeeded by a AY = -+1 semileptomc decay. Unlike AS = - A Q processes, these can only be detected by absolute rate measurements. We are grateful to Probir Roy for introducmg us to his work. One of us (D.B.) thanks Professors A.N. Mitra, S.H. Pahl and V. Singh for hospltahty at Delhi University, Indian Institute of Technology, Bombay and the Tata Instnute o f Fundamental Research, Bombay.
D. Baihn et al / AS = -AQ processes
318
Appendix There are a number of"equivalences" between tensor operators obtained by the use of Flerz reordering together with the equations of motion of eq. (45) (to eliminate operators less singular at short distance than the leading dimension 7 operators) and "momentum conservation" (corresponding to neglecting total divergences, for the reason discussed m the text.) They are as follows. For the S-type rank-one operators,
(Up)kp = ~p)Ktj = pp (P/J) = Pp (P/J) = 0 ,
(A. 1)
Pp(lap) = P ( l a p ) = ½Pu(pp) ,
(A.2)
(pla)k ° = ( o u ) K ° = ½Ku(pp) ,
(A.3)
ku(pp ) = pu(pp) ,
(A.4)
kx (X,upTr ) (Trp) = Px (XIJpTr) (~rp) = p o (llp ) ,
(A.5)
KX(XIIp~ ) (Trp) = px(X/~pTr) 0rp) = - k o ( p ~ ) .
(A.6)
For the A-type rank-one operators,
(po )kp = (lap )Kp = p o (pla)
=
P(OU) = 0,
(A.7)
(pu)kp = (ptl)Kp = ½[ko(p/~) + ppOap) - pu(pp)] ,
(A.8)
poOap) = Po(/ap) = ½[k0(pU ) + p o ~ p ) + pu(pp) ] ,
(A.9)
P (pp) = Ku(pp ) = 0 ,
(A.10) /
p , (pp) = - k ( p p ) ,
(A. 1 1)
kX(XI.Lp~T) (Trp) = Px(X/~pTr) (rip) = -Pp(UP),
(A. 12)
Kx(X/zpcr) 0rp) = px(X/lp~T) 0zp) = k ° (p/l).
(A. 13)
For the S-type rank-three operators the necessary equivalences may be derived from the following, P {e43) = ½god3P(pp) ,
(A.14)
and similarly for Ku, Pu(o~po) (op) = [~a] P ,
(A.15)
D. Badin et al. / AS = -AQ processes
319
and similarly for Ku, lc,u(o~po ) (op) = PU [/3oz]
(A.16)
,
and similarly with k ++ p, Ka(o~Uo ) (pp) = --2
K u [/3a] ,
cychc perms
(A.17)
and similarly for P with a change of sign on the right-hand side, cychc perms
(p-k)u [eg] =0,
(A.18)
po(~po)(p~) = p~(~u) -
p~(~),
ka(o~PO) (UP) = ka(u~) - k~Oaa) .
(A.19)
(A.20)
For the A-type rank-three operators the necessary eqmvalences are derived from the following relations
P ( o ~ p o ) (op)
= -P~
[/3oil ,
(A.21)
Pu '
(A.22)
and similarly for Ku, ku(o~3po ) (op)
= - [/3a]
and similarly wath k ++ p, {cq3} (k - p), = g ~ o k ( p p ) ,
(A.23)
Kx (34apo) (op) = px(XUpo) (op ) = kp (pU) ,
(A.24)
PxGqlpo) (op) = kx(~upo ) (op) = -po(up) ,
(A.25)
(p - K)u [043] = 0 ,
(A.26)
Pa(o~po) (PU) = p~(~3U) - pt3((~),
(A.27)
Po(e.[3po) Cop) = P~(aU)
Pa(/3U),
(A.28)
ko(o43PO) (UP) = k (p~) - k~(ua) ,
(A.29)
Ka(o43po) (UP) = Kf3~a) - K~(UJ3) ,
(A.30)
cychc perms.
-
320
D. Baihn et al. / A S = - A Q processes
also
2 [t~a] ( K - p)~, = (p + k),~ ~ u ) - (p + k)~ ( ~ ) , + g~. [k ~oa) + p.(~o)l - g~, [k.(ot3) + p.(~o)]
.
References [1] H.D. Politzer, Phys. Rev. Letters 30 (1973) 1346; D. Gross and F. Wilczek, Phys. Rev. Letters 30 (1973) 1343. [2] G. Altarelh and L. Maaani, Phys. Letters 52B (1974) 351, M.K Gadlard and B.W. Lee, Phys. Rev. Letters 33 (1974) 108. [3] M.K. Gadlard and B W. Lee, Phys. Rev D10 (1974) 897; D.V. Nanopoulos and G G. Ross, Phys. Letters 56B (1975) 279, A.I Va.mshtem et al Moscow preprmt ITEP-44 (1975), M.K. Gafllard, B.W. Lee and R.E. Schrock, CERN preprlnt, TH-2066 (1975) [4] G. Altarelh, N. Cablbbo and L. Maianl, Rome preprmt ISS P 75•4 (1975); J. Ellis, M.K. Galliard and D.V Nanopoulos, CERN preprmt TH-2030 (1975) [5] K. Wilson, Phys. Rev. 179 (1969) 1499 [6] P Roy, Phys. Rev D5 (1972) 1180. [7] Y. Watanabe et al., Phys. Rev. Letters 35 (1975) 898, C. Chang et al., Phys. Rev. Letters 35 (1975) 901. [8] D. Gross and F Wdczek, Phys. Rev. D8 (1973) 3633, D9 (1974) 980; H. Georgl and H.D. Pohtzer, Phys. Rev D9 (1974) 416. [9] D. Badm, A. Love and D.V Nanopoulos, Nuovo Cxmento Letters 9 (1974) 501. [10] S. Wemberg, Phys Rev. Letters 19 (1967) 1264, A. Salam, in Elementary particle physics, ed. N. Svartholm (1968) [11] S L. Glashow, J. Ihopoulos and L. Maxam, Phys. Rev D2 (1970) 1285. [12] G . ' t Hooft, Nucl. Phys B61 (1973) 455, S. Welnberg, Phys. Rev. D8 (1973) 3497. [13] K. Klemknecht, 17th lnt Conf. on high-energy physics, London, 1974.
(A.31)
AS = - A Q PROCESSES IN ASYMPTOTICALLY FREE GAUGE THEORIES * D. BAILIN, D.R.T. JONES ** and A. LOVE School of Mathematical and Phystcal Sctences, Universtty o f Sussex, Brighton, England Received 14 January 1976
The possibility of an enhancement of the short-distance contribution to AS = -£xQ semileptonic processes is investigated m the context of asymptotically free gauge theories of the strong interactions
1. Introduction
Asymptotically free theories of the strong interactions allow reliable calculations of the short-distance contribution to decay amplitudes to be made. In particular, the IAII -- ½ rule for non-leptonic decays [2,5], various rare decay modes of kaons [3], and the non-leptonic decays of charmed particles [4], have been investigated. In the present paper, AS --- - A Q semileptonic decays are discussed. The search for "charm" during the past year has stimulated fresh interest in AS = - A Q processes. Assuming the absence of a hadronic current with such quantum numbers, the discovery of any such process is interpreted as evidence that the transition has proceeded through a real charmed intermediate state. However, we shall be concerned here with genuine higher-order weak effects (1.e. those which do not proceed through real intermediate states) in the decays of known hadrons. We shall estimate not only the most likely levels at which the decays are expected to occur in the most fashionable model of the weak interactions, but also the highest levels at which they could occur using only what we know for certain about the weak currents. Thus we assume that the Cabibbo current ~ has the form ~ = cos 0 C ~ + sin 0 C S~,
(1)
where J~, S~ are respectively strangeness conserving, non-conserving currents. We assume that 9 a and the known leptonic current La are coupled to the field W~ of a
* Research supported in part by the Science Research Council under grant B/RG/5935-6. ** Present address: Department of Theoretical Physics, 12 Parks Road, Oxford, England. 300
D Bathn et al. / ~xS = - A Q processes
301
charged vector boson with (semi-weak) coupling constant f, _
+
~ I - f ( 9 ,~ + L~) Wa + h.c.,
(2)
in the framework of a renormahzable theory of the weak interactions. In addition, we assume that there is no strangeness changing neutral current, as indicated by experiment. None of these assumptions is controversial. Then it follows that the ~S = - A Q processes occur first in order f 4 . The matrix element for the process
(where A and B are hadron states with S A - S B = +1), is given by 9~ = f 4 sm 0 C cos 2 0 C ~-Tx(1 - 75) v~PXU(q)
×fax dy Aa~fx - y )
(BI T{J+(x)S-~(y)J~ )l A ) ,
(3)
where q = ~ + v is the m o m e n t u m carried away by the leptons. (We use the same letter to denote the particle, its m o m e n t u m and, where appropriate, its spinor.)P and A denote the W propagator in m o m e n t u m and coordinate space, respectively. Thus the d~S = - A Q processes are controlled by the connected part of the operator
c ; = f ax dy
(4)
or its hermltian conjugate. The operators J+~(x) and S~0') above are connected by the W-propagator, at least, so the overall connectedness requires J ~ to be connected by some internal hadron exchange. Thus we may dwide the (connected) contribu+ tions to Gu into three classes. (i) Those in which the W-propagator is not part of a closed loop, i.e. the momentum of the virtual W is fixed by the external momenta. (ii) Those in which the W-propagator ts part of a closed loop, but J~ is outside of that loop. (111) All others.
2. Long-range contributions The contribution G (1) of class (i) to (BI G u+ IA ) is evidently G (1) = - t ( b l l J + [ al)ec~¢(al -
bl)fdy e
t(al-bl)Y
t ~QL~I.r
X (b21r{s~-O')./~ } [ a 2 ) - i ( b ' l l S ~ l a l ~ r
×fax e'<-<)x(bll T{J;(x)L+ IC,
tol-al)
t
(50
D. Badm et al. / AS = -AQ processes
302 where
IA) = la 1) lao) = [a'l ) la~ ),
(5b)
IB) = Ib 1) Ib 2) = Ibm)Ibm).
(5c)
The momenta a 1 - b 1 and a'1 - b~ are determined by the external momenta, so for the decay processes at least, the T-product matrix elements are presumably well estimated by their Born terms. Some of these are illustrated m fig. 1 for the process K 0 --> n+~ - ~Q. Plainly, these represent the truly "long-range" contributions to G u all internal lines have momenta determined by the external momenta and are of order mH, a hadronic mass. Thus, x,y, x - y ~ m~ 1. So G (1) is determined by known, or at least measurable, matrix elements of the weak currents between low-mass hadron states. For example, we estimate the contribution from the diagram in fig. lb to be
G(1)=(Tr+lJ+lO)e'~(n)(OIS~lK+)m2 #
! lr-- m2g
( K + I J + [K o) /a
i =/~KTr P(~O(Tr)lr~ m 2 _ m 2 FK(q2 ) (K + 70.
=f~rfKm2 (m 2 _ ~2"~ FK(q2) (K + zr)u, Wt K
(6)
"'it j
where F K is the electromagnetic form factor of the K +. The matrix element for the AS = +AQ process ~0 _> rr+ £- O~
(7)
is and
c/~ 0 = f 2 sin 0C£~X(1 -- 75) vt pXU(q) (rr+l S + [E0)
(8a)
0r+ i Su IFo>= g+(q2)( K + 7r)u + g_ (q2)qu "
(8b)
The parameter x, measuring AS = - A Q amplitudes relative to those with AS = +AQ, is defined as the ratio of the spin-one parts in (8) and (3). The contribution of dia-
s./ ,,.St~j
K*
S~J' ~2.1~
s.Af rr÷
K*
(b) (o) Fig. 1. Long-range contributions to cat (K° ~ ~r+~-b'~)
D. Baihn et aL / aS = _AQ processes
303
gram lb IS seen to be FK (q2)
f•fK m2 x ( l b ) = f2 cos20c m 2 , ( ~ -2.. _~ m2~ W~ K "'Tr J
g+(q2)
"" ~/~-G cos2 0C flrfKm2(m 2 - m2}-l~r= 1.4 X 10 - 8 .
(9)
The pion pole diagram in fig. la looks more promising for a larger effect, and indeed since the pole is in the physical region, the decay may actually proceed via the production of a real rr- : K 0 ~ lr+rr -
I , ~-~.
(10)
Thus the transition wdl be observed as a first-order weak effect. We are concerned with the higher-order weak effects, of course, i.e. the contributions when q2 4= m 2. These are presumably characterized by a denominator of order m 2 rather than m 2, but in any case cannot contribute to the spin-one part of (8): they vanish if we neglect rn~.
3. Short/long-range effects The contributions characterized by class (11) reqmre that j2(x) and S~(y) are connected by internal hadrons as well as the virtual W, so that we have an internal loop. The "large" mass of the W ensures that the dominant contribution in this class is when the loop m o m e n t u m k ~ mw, i.e. x - y ~ m~vt which is "short". Since J~- is outside of the loop nothing forces x, y to be close to 0, so to goodapproximation we need consider only the contributions when Xo,Y 0 >>0 or 0 >> Xo, YO" Writing the short-distance expansion [5]
T{J÷(x)SSCv))x-y~O
n
C
(11)
the contribution G u(2) to (B I G~ I A) is evidently C (2)u =~n f d x d y A a ~ ( x - y ) C ~ n ( X - y
f
)(B[T(On(x)J +~.}[A)
As before, the second integral is well-approximated by retaining only low-mass
(12)
304
D Baihn et al. / AS = - A Q processes
K* xil~"
Io)
K,
K+
0.
lx+
K°
0.
j~
÷
t'-
l~e
(b)
(c)
Fig. 2. Long/short-range contributions to ~ (K° --, n+~-ff~). exchanges between O n and J ~ . Some of these are illustrated in fig. 2 for the process K 0 ~ rr+£- ~Q. Diagrams 2a and 2b are the short-dxstance analogues of la and lb, while 2c is a contribution only permitted by short-distance effects. The operators O n are those which provide the short-distance contribution to the ordinary nonleptonic decays, and if we are sure we know about them we may estimate the diagrams accordingly. However we take a more conservative standpoint (for the present). We observe that the contributions from classes (i) and (ii) consist o f a transition allowed by the ordinary non-leptonic Hamiltonian succeeded or preceded (at a long distance) by a semlleptomc transition. So we may reliably estamate the t o t a l contributions from these two classes by taking the required amplitudes from "experiment". As before we ignore the spin-0 contributions in fig. 2a. The contribution from figs. lb and 2b is f 2 sin 0 C cos 0 tG(lb) + G! 2b)) = ( ~ + I ~ [ K C~ ~ . . . . _
+> ~ m~z
(K+[ J ; I K°> m2K
-i FK(q2 ) (Tr+I~NLIK+) (K + lr)u m~ - m~
- (rr+n'-[C3~NLIKO) (K + n')u m 2 - m2~r s '
(13)
using symmetric soft pion theory. Thus the contribution to x is
x/ lf.I
Ix(lb +2b)l - - cot 0 C I(zr+Tr-19gNLIK0)l m2 _ m2 = 1.4 X 10 - 6 .
(14)
D. Batlin et al / AS = -AQ processes
305
The contribution from fig. 2c may be evaluated in the same way and combined with the result (14). It yields a total contribution to x which is proportional to (Tr+Tr0I~NL IK+ ), which itself is known experimentally to be a factor of about 20 smaller than (lr+rr - I~NLIK0). Because of this cancellation, which derives from the observed IMI ---½ rule m non-leptomc de~ays, we should really proceed to discuss the contributions from higher mass exchanges: p, K* . . . . . However we know nothing experimentally about their non-leptonlc amplitudes, and we prefer to take (14) as a generous upper bound on their contributions. More reahstically we should expect them to be down by a factor of 4 or so on account of their larger mass denominators. In summary, then the net contribution from classes (i) and (il) to x satisfies Ix(1 +2)1 <~ 1.4 × 10 - 6 ,
(15a)
and probably Ix(1 +2)1 "~ 4 × 10 . 7 .
(15b)
The extra order of magnitude which we have gamed over the long-range contribution (9) derives simply because the non-leptonlc amplitudes are known experimentally to have this feature.
4. Short-range effects
We turn finally to the short-range contributions (fii) to Gp. These are illustrated in fig. 3. The m o m e n t u m k of the virtual W flows also through hadrons connecting J~+ to J ~ and J ; to S f . As before, the dominant contribution is when k ~ m w so x,y ~ m~¢I . Thus we are concerned with the contribution when all three currents are separated by short distances. This is controlled by the expansion of the threecurrent product:
T { J+(x) S~(y) J~ } x,y'~O ~n K~flun(X' y) Tn (0).
(16)
In terms of the " k n o w n " quark fields (p, n, X) we know that J ~ and S ; certainly have the pieces Ja+ = i07~(1 - 75 ) n ,
(17a)
S f = ~,#(1 - ? 5 ) p .
(17b)
K° ~ ~ Fig. 3. Short-range contributions
1"1"+ to ~
(K° ~ 7r+£-F£).
306
D. Bailin et al. / AS = - AQ processes
Then it is easily seen that the leading operators Tn(0 ) must be four-quark operators of rank 1 or 3, e.g. Tn = PTo(1 - 75) n XTo( 1 - 75) a m ,
(lS)
or contractions/permutations of these. Thus T n has (mass) dimension D n = 7, and in free field theory this implies that Kaaun(X,y ) has the scahng property K t~un(~,-~)x__" ~ x 9 - D n K o ~ u n ( X , y ) .
(19)
Consequently the truly short-distance contribution G (3) to (BI G~[ A) has the property G(3)CAmw) ~ "'3-DnG(3)tmu ~ W""~'
(20)
as noted by Roy [6]. Hence, since D n = 7, G( 3 ) ~ m w 4 ( B I T I A ) ,
(21)
where T u is constructed from T n by contractions. For the process K 0 ~ rr+£ - O~, the required matrix element has the form (;r+ [ T z l K ° >~ ta4(K + rr) u ,
(22)
where/a 0 is a mass parameter. Thus the free field estimate of this short d~stance contributxon to x is x(3) ~ f2 cos 2 0 COa4/m 4 ) = ~
G cos 2 0 C . 4 r n ~ .
(23)
We may estimate/a 0 from the known amplitude for K ~ 2rr. A similar free field calculation for this process yields c/g (K -+ 2n) ~ X/~2G sin 0 C cos 0 C (21rl O n IK) "V/~21Gsin 0 C cos 0 C/.103 ,
(24)
where O n is given in (11) and is a four-quark operator of dimension 6. Then from the decay rate for K 0 ~ lr+rr - we find /a0 ~ 607 MeV,
(25)
and substituting into (22) gives x ( 3 ) ~ 1.1 X 10 - 8 ,
(26)
D. Baihn et al / AS = -AQ processes
307
since it is known experimentally that m w ~> 10 GeV.
(27)
Of course, this estimate of x(3) depends crucially upon the value (25) for #0 derived assuming Ks0 -+ 7r+zr- is a "typical" amplitude. Had we used the IAI[ = a amphtude C~(K+ -+ 7r+lr0) the resulting value o f # 0 (219 MeV) gives a much lower estimate (1.8 × 10-1°). Thus (26) assumes that the I A/I = ~2 amplitudes are suppressed (and therefore atypical) rather than the IA/I = ½ amphtudes being enhanced. We shall return to this point later. If one believes, on the evidence from the deep inelastic scattering experiments, that Bjorken scaling is really true, with no deviations, then (26) is a reasonable estimate. It would indicate that the dominant contributions to x are an order of magnitude larger than (26), as we found in the previous section, and controlled by the characteristics of the ordinary non-leptonic Hamiltonian rather than the short-distance structure of the three-current product that we have been discussing here. However, there is some evidence of scaling violation [7], and the most popular belief is that these are logarithmic deviations, as forecast by the asymptotically free gauge theories [1]. In this case D n would continue to have its free field value 7, but (20) would be modified by the appearance of (calculable) powers of In k: G(3)(Xmw ) --- X-4(1 + k In X)pn G(3)(mw ) .
(27)
For such theories, then, the short-range contribution to x is x(3) ~ / ~ G
mw IPn #---~4 cos20 C m 2 (1 + k i n ms ]
(28)
The mass/a appearing in (28) is different from the/a 0 in (23), because in this model the K ~ 2rr amplitude (24) is also modified [2] by (different) powers of 1 + k ln(mw/ms). Consequently U3 = 03(1 + k In x)q,
(29)
and our estimate (23) is modified.
__1 a4 [
mwlPn-~q
x(3) ~ X/~2Gcos 2 0 C m 2 \,1 + k in ms !
(30)
For an SU(N) colour group with quarks in the N representation [1 ]
g2 k=
8n 2
g2 b=
247r2
(1 I N - 2 F ) ,
(31)
where F is the number of quarks (p, n, ~, ...). We "know" that F~> 3, N = 3 from
308
D. Badin et al / AS = - A Q processes
rt0 ~ 3'7, and it IS customary to assume g2/41r ~ 1 with the choice m s ~ 1 GeV, although this is determined in pnnciple by the "observed" scaling deviations. Thus our previous upper bound (26) is modified: x ( 3 ) ~ 1.1 X 10 -8 X (4.3) p n - ~ q
(32)
So provided Pn - ~q >~ 2.5, this upperbound would be comparable with our earlier estimate (15b), and if Pn - ~q ~> 4 we might even entertain the possibility that these structure dependent contributions were dominant. Further, such values of the exponent are not unreasonable. Calculations of the predicted scaling deviations in the structure function moments for electro- and neutrino-production [8,9] give powers of order 1 for quite low moments, and arbitrarily large for high enough moments. In the following section we shall perform the anomalous dimension calculations necessary to see whether either of these possibilities is actually realized.
5. Anomalous dimension calculations We have already noted that the short-distance expansion (16) of the three-current product is dominated by operators of the form (18) in free field theory. Because of the (colour) gauge invariance of the asymptotically free theories we shall be concerned with the gauge covariant generalizations of the operators (18). These may be constructed by replacing the derivative ~r by its covariant form D r: Dr = ~r - igRBABr '
(33)
where g is the coupling constant of the gauge field theory, R B are the matrices representing the colour group G s in the representation to which the quark fields have been assigned, and A~ are the gauge field operators. We shall consider only the colour groups G s = SU0 V) with the quarks in the fundamental, N-dimensional, representation. There are then two gauge covariant generalizations associated with each free field operator. In the case of (18) these are: /53"o(1 -- 3'5)InX3'o(1 - 3'5)I(Drn),
(34a)
iC~o(1 - 3'5)RBn)t3'a (1 -- 3'5)RB(Dr n)"
(34b)
We have suppressed the colour indices on the quark fields and on the colour matrices I and R. Before proceeding further we introduce a more compact notation. We associate the fields/5 and Xin (34) with the creation of quarks having momenta P and K, respect:rely. The two fields n, coupled to/5 and X, are associated with the destruction of quarks having momenta p, k, respectively. Then the D r in (34) is associated with - i ( k r + g R B A B). Our notation for the operators (34) makes explicit only the Lorentz radices, the field upon which the covariant derivative acts, and which of the two
D. Batlm et al. / AS = -AQ processes
309
colour combinations (34a) or (34b) is revolved. Precisely we define:
Ikr(po ) =- i/37p(1 - 3,5 )In~-3,a(1 - 3,5 ) I ( D n ) ,
(35a)
Rk.(po) = ip3,p(1 - 3'5)RBn~3,o(1 - 75)RB(D-*rn ) .
(35b)
Evidently there are six more basic operators: two for each covariant derivative of the other three fields. Thus we define
Ipr(P°) =--tP3'o(1 - 3,5)I(D-'*zn) A--Ta(1 - 3'5 ) I n '
(35c)
Rpr(o°) - iPTo(1 - 7J) R B(D-*~n) X-7o(1 - "/5)RBn'
(35d)
IKr(PO ) =- -t/33,0 (1 - 3,5)In(kDr) 3,o(1 - 3,5)n,
(35e)
RKr(po ) = - i ~ p ( 1 - 3,5) R BnC/~D~r T) 3,~(1 - 75)RBn ,
(35 0
I P (po) = -,(pD~-tr) 3'p(1 - 75)In~3"o(1 - 3'5)In,
(35g)
RPr(po ) =- -i(pD~-'fr)3,p(1 - 75) RBn~'3'a(1 -- 3'5)RBn.
(35h)
To Incorporate the implications of the Fermi statistics of the quarks we shall have occasion to Fierz reorder expressions of the above type. We therefore work with linear combinations of the I and R operators which facilitate this. We define
Sqr(po) ~ - ~ -
Iqr(po ) + 2Rqr(po) ,
(36a)
Aqz(pa ) = - ~
Iqr(pO ) - 2Rq~(po),
(36b)
where q = p, k, P, K. The combinations S, A are respectively symmetric, antisymmetric under Fierz reordering of the colour labels. We consider first the total divergences
SAr(po ) --S(p + k - e - K)r (po),
(37a)
AAz(po) =-A(p + k - P - K)r (po) .
(37b)
The matrix elements of these operators are fixed by the matrix elements of the rank-2 operators S(po) and A(po) which are defined analogously to (36):
S(po) =
N+I N
A(po) =- ~ 7Y
I(po) + 2R(po) ,
(38a)
I(.po) - 2R(po),
(38b)
310
D. B a i h n e t al. / A S = - AxQ p r o c e s s e s
where I(0o) - ~o(1
- vs)In~o(1
-
v5)I.,
(38c)
R(0o) =/37o(1 - 7)R BnXTo(1 - 75)RBn.
(38d)
We may Fierz reorder S(0o) and A(0o), using the well known v-matrix Fierz identities, and find
S(0o) = -½ [S(0o) + S(op) - gooS(otet) + i(poo43) S(a3)] ,
(39a)
A(0o) = +½[A(0o) + A(op) -gooA(ua) + i(0eoe3) A ( ~ ) ] ,
(39b)
where (poa3) denotes e~ . Thus we deduce the following useful identities: po
S ~oo ) =-S(0a) + S(op) = ½goo S(a~),
(40a)
S [po] =S(0o) - S(oO) = -t(0oa3) S(o43),
(40b)
A (as) = 0 ,
(40c)
A [pa ] = +i(0oo43) A(oJJ).
(40d)
The anomalous dimensions of these operators are evaluated by calculating the renormalization constants of the associated inserted Green functions. The order g2 contributions to these are shown in fig. 4. We work m the Landau gauge and are concerned only with the divergent pieces. We find S(0o)-~ l ~ 2 1 n - ~
( + -
(--2)(goag~-g3og~o)(SA(O43)),
(41)
where ~ is a renormalization mass and (AS(O~))is the zeroth order matrix element. The anomalous dimension of the multiplicatively renormalized operator 0 is defined as
O(ln Zo) g2 = a o + O(g4) 7o = a(lngt) 16~r2
(42)
where Z o is the renormalization constant. Using (39) we find that the following operators 01 (t = 1..... 4) are eigenvectors with anomalous dimension (g2/16n2)dol
{po),
oi = s ( 0 p ) ,
s[oo],
A
doi = 6 (N-N 1) ,
2 (IV-N 1) ,
2 (N+N1) ,
A [po], 2(N+N 1)
(43)
Each of these is associated with a logarithmic factor of the form given in (28) with
Po = -do/2b , where b is defined by (31).
(44)
D. Badm et al. / AS = - AQ processes
311
p
n
n
/
Fig. 4 Inserted Green functions m single-looporder.
The calculation of the other anomalous dimension eigenvectors and eigenvalues proceeds analogously. Since we are primarily interested in the eigenvalues, rather than the precise form of the eigenvectors, we need only consider inserted Green functions in which the inserted operator carries away zero momentum. This is because the total divergence operators, S A r ( p o ) and AAT(po ) defined in (37), are multiplicatively renormalized, as we have seen, and do not "mix" with the other basic operators. This makes the following calculations considerably simpler than they would otherwise have been. We have to calculate not only the analogues of the diagrams shown in fig. 4, with S ( p o ) replaced by SqT(po), but also the "gauge diagrams" in which the virtual gauge field is coupled to the vertex via the - i g R a A B piece o l D r in (33). These are shown in fig. 5. Despite the simplification obtained by assuming momentum conservation, p + k = P + K in the diagrams, the analogues of (41) for the operators (35) are still very complicated. Further simplification is achieved by the use of operator "equivalences", analogous to (40). They are obtained by the use of Fermi statistics and Fierz reordering together with the equations of motion: (45) We have described the relationships thus obtained as "equivalences", rather than
Fig. 5. "Gauge diagram" contnbunons to inserted Green functions m single-looporder.
D. Baihn et aL /
312
~, =
-AQ processes
identities, because some of them depend upon momentum conservation for their validity, and as such are equivalent only when taken between states of equal momenta. The relations derived are summarized in the appendix. The basic calculations involve the evaluation of the divergent pieces of the inserted Green functions corresponding to the operators Skr(po), Akr(PO), SPr(po) and AP~(po). To find the divergent pieces of the Green functions inserted with Spz(po) we apply the "mirror operation" to the expression derived from Skr(Pa); this IS defined as the combined interchanges p *-* k, P *-*K, p *-* o and left 3'-index ~ right 3,-index. The same operation applied to the contributions from the remaining three of the above operators yields the contributions from Apz(po ), SKz(po) and AKT(PO). (These last two are obtainable from the previous calculations by momentum conservation, an any case.) The results of these computations yield (divergent) expressions which are linear combinations of the (zeroth order) matrix elements of 35 operators; this should be compared with the four operators appearing on the right of (41), after use of (40). Rather than presenting these "raw" calculations we shall do a little preliminary "processing". We present first the results for the rank-one operators, obtamed by contracting two of the indices of the rank-three tensors (35). It follows from the equivalences given in the appendix that there are just four independent rank-one tensors Skz(pp ), A kr(pp), S(P- k)r(pp), A [pP(rp)+ k p(p T)]. For these operators we find:
Skr(pP)
1-+-
A[pO(rp)+kO(p,)lj
g2 lnA2 16Tr2
/~2
N-1
(~l)2j
(SkT(PP))
(46)
X
:A [pP(rp) + kP(pr)]
These yield two eigenvectors with anomalous dimensions (g2/167T2) d (1), where d(1) _= 2 [2N2 + 7 N - 7 + (4N 4 + lON3+6N2-80N+ 64) 1/2] -
(47)
3N
Similarly,
S(P- k)z(PP)I_
g2 In
AkT(PP)
)
16"2
A2 p2
I ( N - 1)(2N+7) 3N 32-(N- 1)
~(N+ 1) _ (N+ 1)(2N-7) 3N
"
X
(Ak~.(pp))
D. Badin et al / aSS = -2xQ processes
313
give elgenvalues with d~2) = 3 ~ [2N2 - 7 -+N(4N 2 + 21) 1/2 ] .
(49)
Next we turn to the rank-three operators. Since the rank-one operators mix only with themselves, as is apparent from (46) and (48), we may slmphfy the analogous equations for the rank-three operators by deleting the rank-one matrix elements from the right-hand sides. As noted earlier, this procedure will not affect the eigenvalues although the corresponding eigenvectors will only be correct up to the terms we have deleted. The results given in the appendix show that the independent rankthree operators are
s ( p - k)~ {po },
s(p + k),[po],
s(e- p)~.[oo],
S ( P - k)rlPel,
A(p +k), {po },
A ( e - K)¢ {p o },
A(p+ k)TDo] ,
A(P-p)r[pa] .
With the exception ofS(P-k)r[PO], each of these gwes rise to three operators by cyclic permutation of p, o, r. However, the equivalence (A.18) shows that only two permutations of S ( P - k)r [po] are independent, so we shall find just 23 rank-three eigenvectors of anomalous dimension. With the above noted simplificahons, we find g2 l n A 2 / N - 1
S{p-k)r{PO}-+16rr2
112 [ 6N [(8-2N)(S(p-k)r{P°})
- (N+2) (S(p-k)o {or}) - (N+2) (S(p-k)p {or}) + (2N + 10) (S(p + k)o[pr ] ) + (2N + 10) (S(p + k)o[or ]) ]
+ ~N+ 1 [_2 (A(P_ K)r {PO }) _ (A(P_ K)o ~ r }) _ (A(P- K)p {or}) + 2(A(p + k)o[Pr]) + 2(A(p + k)o [or] )] } ,
g2 lnA_~2 {~_ /~2
S(p +k)r[Pr ] -+ 16rr2 - (N+2)
(50a)
[_6(N+3)(S(p+k)r[po])
(S(p+k)a[pr ]) + ( N + 2 ) (S(p+k)p[Or])]
+N+I 6 [2Ot(p +k)r[pO l) + ( A ( P - K)o for}) - ( A ( P - K)o {or})
- (A(p + k)o[pr ]) + (A(p + k)p[Or])] ], )
(50b)
314
D. Baihn et al. / AS = -LxQ processes g-_~_2lnA2 (N__6S~I [2(S(p+k)r[po ]) A(p + k)r[po ] ~ 16rr2 02 +
+ (S(p + k)p[o~']}] + ~
[ - 6 ( N - 3) (A(p + k)z[po ])
- ( N - 2) (A(p + k)o [pz]) + (A (t9 + k)p [o~] )] }
(50c)
#
g2 A2 { U 6 1 A ( P - K ) r {po} --* 16rr--~-ln U~- .
[-2(S(p--k)r (po))
- (S(p- k)o {pT }) - (S(.p- k)p {o'r }) + 2(S(p + k)o [07"]) N+I + 2(S(p + k)p[or])] + ~ - (N-2)
(A(P-K)o{pr)) -
[-(2N+ 8) ( A ( P - K)T {po}) (N+ 2)
+ ( 2 N - 10) (A(p + k) ° [pr] ) + ( 2 N - 10) (A(p + k)o [or])] }.
(50d)
/
Evidently the 12 operators arising here may be diagonalized separately from the remaining 11. First we sum over cyclic permutations of the three indices in the above equations. This immediately yields the "cyclic" eigenvalues d(3) - - -4 IN2 + 1 + N(N 2 +3) 1/2 ] 3N '
(51a)
2 [2N 2 - 7 + N(4N 2 + 21) 1/2 ] = d~2) . d (4) - -~-
(5 lb)
The equality of d_+ (4) and d+(2) can be understood with the aid of the equivalences given in the appendLx. For example, cychc perms.
S(p + k)z[po ] = -(rpoX) (/~al3X)S(p + k) u [0~ l t(zpoX) S ( P - k)x((~e0 ,
(52)
using (A.2), (A.3) and (A.16). An analogous relationship holds for Y. A(p + k)z[po ]. Thus both of these "cyclic" operators is proportional to a rank-one operator, which implies the equality (5 lb) of their eigenvalues. The remaining 8 operators dealt with in (50) split into two groups of 4 which may be separately diagonalized. We find that both of these sets yield the same (quartic) characteristic equation, although we have not understood why. We are able to factorize the quartic into quadratics and obtain
D. Baihn et al. / AS = -AQ processes
315
the following four elgenvalues: d(.5) =___~1 [4N2_ 12N- 5 + (16N 4 - 6N3-96N2+30N+225)1/2], -
(53a)
3N
d~6) ~
1
[4N2+ 12N-5 + (16N4+6N3-96N 2 -30N+225) 1/2] .
(53b)
The remaining calculations yield: g2 ln A2 {_ _~_ (N_ l ) (N + 2) (S(P_ k)r[PO]) S(P- k)z[po] -~ 167r 2 /a2 + ~ (N + 1) [(A(19+ k)o {.or}) - Gt (p + k)o {or })] } ,
(54a)
A(p+k)r{O°}~ -1-g26n ~ InA2U-}- {N31-- [
(54b)
S(P- p),[po] ~ g 2 lnA2 { [-2(2N+ 7)
N-1 6N
+ N+ ~ 1 [4(A(P-p)~[Ool)+ (A(P-P)o[OrI>-
(54c)
[4(S(P-P)r[P°')
N+I -
( N - 2) (A(P- p). [pr] ) + ( N - 2) (A(P- p)p[or])] } .
(54d)
As before we consider the cychc operators first. The one associated with (54a) as zero. (54b) gives an eigenvector with eigenvalue 2 (iV+ I) (4N- 5). d (v) = -~-
(55)
316
D. Bathn et al. / AS
= -AQ
processes
(54c, d) yield the cyclic eigenvalues d(_.+8) =d~ 1)
(56)
for the same sort of reason as illustrated m (52). The remaining operators spht into four sets of two operators and each pair yields the same two eigenvalues. d (9) =---/ [4N 2 + 2 N - 5 + ( 1 6 N a - 2 N 3 - 1 2 N 2 - 2 N + 1) 1/2] -
3N
(57)
-
It is apparent from (44) and (28) that we seek the operator with the largest value of - d . For the case of physical interest N = 3, and the maximum value of - d is given by eq. (53a) with the minus sign: d(__ 5) ~ - 3 . 2 4 .
6.
(58)
Conclusions
The calculations of sect. 5 on the short-range contribution to AS = --AQ decays show that the dominant operator in the expansion (16) of the three-current product is the operator given in (58a). It is associated with a logarithmic power of the form (28) with P=-
d(_s } 2b = 0 " 1 8 '
(59)
since b = 9 in the theory with which we are presently concerned (N = 3, F = 3). In the same theory it has been shown [2] that the value of q is q = 0.44.
(60a)
(4.3) p--~ q = 0.55,
(60b)
Thus
and we see from (32) that the free field estimate (26) is n o t enhanced in the asymptotically free theories. If anything, it is suppressed slightly. With the value (60b) the upper bound (32) becomes x(3) ~ 0.61 X 10-8 ,
(61)
which is at least one order of magnitude below the non-short-range contribution (15b). Further, the value (61) is the maxLmum that could posmbly arise using k n o w n currents and asymptotic freedom. The upper bound derived from the presently favoured Weinberg-Salam model [10] of the weak and electromagnetic interactions is surely lower than (61). This further suppression derives from two sources. In the first place, the mass of the W-boson certainly has a larger lower bound than (27). In
D. Batlin et al / AS = - A Q processes
317
fact m w ~> 37 GeV.
(62)
In the second place, the model employs the GIM mechamsm [11] to ehminate strangeness changing neutral currents, as a result of which the Cabibbo current acquires additional pieces deriving from the charmed quark. In fact, ~ + = PT~(1 - 7 5 ) ( n cos 0 C + X sin 0 C ) , + gTa(1 - 3 ' 5) ( - n sin 0 C + X cos 0C).
(63)
The extra piece revolving the charmed quark c contributes to the ~xS = - A Q processes, also m order sin 0 C cos 2 0 C, and in the hmit that the p and c quarks have equal masses it precisely cancels the short-&stance contribution evaluated in the previous section. Since m_F 4: m_ our free field estimate (23) is suppressed by a factor (m 2 - m 2 ) / m 2 , since m W is the only mass with which we can scale in the short distance limit. In the asymptotically free gauge theories this mass term is associated with further logarithmic factors [ 12],
2 m2 ( l + k l n
( m p ~ _ c) m2W
mw]r
ms !
~ 6 X l O -5
(64) '
where the numerical bound is derived [3] using m c ~ 2 GeV, m w ~ 100 GeV, r = - ~ . With this value of m w a realistic estimate of the short range effects would lead to a value at least six orders of magnitude smaller than (61). Thus the unambiguous (negative) prediction of the asymptotically free theories is that the AS = - A Q process K 0 -+ 7r+e-~ will not be observed before the 10 - 6 level obtained in (15). In the event that such processes are observed at a higher level we should have to conclude that the short-range contributions were exhibiting anomalous dimension effects far beyond anything explainable by the asymptotically free theories. We have confined our numerical estimates of the AS = - A Q effects to the K 0 --, 7r transition. This is because the neutral kaon system affords us the unique opportunity to measure the AS = - A Q amplitude, vta interference effects [13]. Clearly, at the level of effect we are concerned with, such experiments are far more feasible than rate measurements for processes such as 2;+ ~ n e+Ve. A precisely similar analysis may also be applied to IAYI = 2 semileptonic processes. We predict that these will occur primarily via non-leptonic decay preceded or succeeded by a AY = -+1 semileptomc decay. Unlike AS = - A Q processes, these can only be detected by absolute rate measurements. We are grateful to Probir Roy for introducmg us to his work. One of us (D.B.) thanks Professors A.N. Mitra, S.H. Pahl and V. Singh for hospltahty at Delhi University, Indian Institute of Technology, Bombay and the Tata Instnute o f Fundamental Research, Bombay.
D. Baihn et al / AS = -AQ processes
318
Appendix There are a number of"equivalences" between tensor operators obtained by the use of Flerz reordering together with the equations of motion of eq. (45) (to eliminate operators less singular at short distance than the leading dimension 7 operators) and "momentum conservation" (corresponding to neglecting total divergences, for the reason discussed m the text.) They are as follows. For the S-type rank-one operators,
(Up)kp = ~p)Ktj = pp (P/J) = Pp (P/J) = 0 ,
(A. 1)
Pp(lap) = P ( l a p ) = ½Pu(pp) ,
(A.2)
(pla)k ° = ( o u ) K ° = ½Ku(pp) ,
(A.3)
ku(pp ) = pu(pp) ,
(A.4)
kx (X,upTr ) (Trp) = Px (XIJpTr) (~rp) = p o (llp ) ,
(A.5)
KX(XIIp~ ) (Trp) = px(X/~pTr) 0rp) = - k o ( p ~ ) .
(A.6)
For the A-type rank-one operators,
(po )kp = (lap )Kp = p o (pla)
=
P(OU) = 0,
(A.7)
(pu)kp = (ptl)Kp = ½[ko(p/~) + ppOap) - pu(pp)] ,
(A.8)
poOap) = Po(/ap) = ½[k0(pU ) + p o ~ p ) + pu(pp) ] ,
(A.9)
P (pp) = Ku(pp ) = 0 ,
(A.10) /
p , (pp) = - k ( p p ) ,
(A. 1 1)
kX(XI.Lp~T) (Trp) = Px(X/~pTr) (rip) = -Pp(UP),
(A. 12)
Kx(X/zpcr) 0rp) = px(X/lp~T) 0zp) = k ° (p/l).
(A. 13)
For the S-type rank-three operators the necessary equivalences may be derived from the following, P {e43) = ½god3P(pp) ,
(A.14)
and similarly for Ku, Pu(o~po) (op) = [~a] P ,
(A.15)
D. Badin et al. / AS = -AQ processes
319
and similarly for Ku, lc,u(o~po ) (op) = PU [/3oz]
(A.16)
,
and similarly with k ++ p, Ka(o~Uo ) (pp) = --2
K u [/3a] ,
cychc perms
(A.17)
and similarly for P with a change of sign on the right-hand side, cychc perms
(p-k)u [eg] =0,
(A.18)
po(~po)(p~) = p~(~u) -
p~(~),
ka(o~PO) (UP) = ka(u~) - k~Oaa) .
(A.19)
(A.20)
For the A-type rank-three operators the necessary eqmvalences are derived from the following relations
P ( o ~ p o ) (op)
= -P~
[/3oil ,
(A.21)
Pu '
(A.22)
and similarly for Ku, ku(o~3po ) (op)
= - [/3a]
and similarly wath k ++ p, {cq3} (k - p), = g ~ o k ( p p ) ,
(A.23)
Kx (34apo) (op) = px(XUpo) (op ) = kp (pU) ,
(A.24)
PxGqlpo) (op) = kx(~upo ) (op) = -po(up) ,
(A.25)
(p - K)u [043] = 0 ,
(A.26)
Pa(o~po) (PU) = p~(~3U) - pt3((~),
(A.27)
Po(e.[3po) Cop) = P~(aU)
Pa(/3U),
(A.28)
ko(o43PO) (UP) = k (p~) - k~(ua) ,
(A.29)
Ka(o43po) (UP) = Kf3~a) - K~(UJ3) ,
(A.30)
cychc perms.
-
320
D. Baihn et al. / A S = - A Q processes
also
2 [t~a] ( K - p)~, = (p + k),~ ~ u ) - (p + k)~ ( ~ ) , + g~. [k ~oa) + p.(~o)l - g~, [k.(ot3) + p.(~o)]
.
References [1] H.D. Politzer, Phys. Rev. Letters 30 (1973) 1346; D. Gross and F. Wilczek, Phys. Rev. Letters 30 (1973) 1343. [2] G. Altarelh and L. Maaani, Phys. Letters 52B (1974) 351, M.K Gadlard and B.W. Lee, Phys. Rev. Letters 33 (1974) 108. [3] M.K. Gadlard and B W. Lee, Phys. Rev D10 (1974) 897; D.V. Nanopoulos and G G. Ross, Phys. Letters 56B (1975) 279, A.I Va.mshtem et al Moscow preprmt ITEP-44 (1975), M.K. Gafllard, B.W. Lee and R.E. Schrock, CERN preprlnt, TH-2066 (1975) [4] G. Altarelh, N. Cablbbo and L. Maianl, Rome preprmt ISS P 75•4 (1975); J. Ellis, M.K. Galliard and D.V Nanopoulos, CERN preprmt TH-2030 (1975) [5] K. Wilson, Phys. Rev. 179 (1969) 1499 [6] P Roy, Phys. Rev D5 (1972) 1180. [7] Y. Watanabe et al., Phys. Rev. Letters 35 (1975) 898, C. Chang et al., Phys. Rev. Letters 35 (1975) 901. [8] D. Gross and F Wdczek, Phys. Rev. D8 (1973) 3633, D9 (1974) 980; H. Georgl and H.D. Pohtzer, Phys. Rev D9 (1974) 416. [9] D. Badm, A. Love and D.V Nanopoulos, Nuovo Cxmento Letters 9 (1974) 501. [10] S. Wemberg, Phys Rev. Letters 19 (1967) 1264, A. Salam, in Elementary particle physics, ed. N. Svartholm (1968) [11] S L. Glashow, J. Ihopoulos and L. Maxam, Phys. Rev D2 (1970) 1285. [12] G . ' t Hooft, Nucl. Phys B61 (1973) 455, S. Welnberg, Phys. Rev. D8 (1973) 3497. [13] K. Klemknecht, 17th lnt Conf. on high-energy physics, London, 1974.
(A.31)