fulltext

Global view of PCAC

M. D. SCADRON

Physics Department, University of Arizona - Tucson, AZ 85721, USA

(ricevuto il 2 Giugno 1997; approvato il 7 Ottobre 1997)

Summary. — When combined with current algebra, the notion of partial conservation of axial currents (PCAC) is quite predictive. In fact, when this PCAC is extended to PCAC consistency for multiple pion or kaon states, the above procedure is in excellent agreement with data for strong, electromagnetic and nonleptonic weak interactions over a wider range of energies even above 1 GeV. Usually physicists are wary of invoking PCAC notions far from the soft-pion limit.

PACS 11.40.Ha – Partially conserved axial-vector currents. PACS 11.40 – Currents and their properties.

PACS 11.30.Rd – Chiral symmetries.

1. – Introduction

In this paper we survey the extremely useful property of partial conservation of axial currents (PCAC) as it applies to the pseudoscalar mesons (p, K , h8) when

interacting strongly, electromagnetically and weakly with other mesons and with baryons.

In the excellent early reviews of current algebra and of PCAC [1], usually one or at most two pions are taken soft mainly for strong interactions and always at very low energies. However, in a later survey [2], rapidly varying poles are taken into account for strong, electromagnetic (em) and weak interactions and then PCAC when combined with current algebra appears to be valid even when the decaying pion momentum is not so small.

Recently the results of the (tedious) rapidly varying pole procedure are reproduced by the (much simpler) PCAC consistency approach as applied to K2 p, K3 p and, e.g., D K2p weak decays [3] and to hK3p em decays [4]. Now that the data are accurately

known even for heavier meson decays, one can make a global study of PCAC consistency and test its validity over the entire range of interactions—strong, electromagnetic and weak and for lower mass and perhaps even for higher mass particles (and pion momentum).

In sect. 2 we study strong interaction transitions involving one or more pions. First we test PCAC involving one soft pion combined with current algebra in the chiral limit and in the chiral-broken world. Then we remind the reader how the PCAC

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field-theoretic equation is manifested in various chiral Lagrangian models. Next we review the successful predictions of this model-independent current algebra-PCAC scheme once rapidly varying pole terms, s terms and the symmetry for multipion final states are taken into account.

In sect. 3 we review PCAC applied to electromagnetic (em) interactions, both of the em current-current type and also of the nonphotonic (Coleman Glashow (CG) tadpole [5]) type. Then we combine this CG tadpole approach with Dashen’s observations [6] concerning PCAC for electromagnetic transitions, and (successfully) apply it to em mass splittings. We next extend this scheme to DI 41 charge symmetry breaking em interactions such as ap0

NHemNhb and ar0NHemNvb and (consistently) apply

the former to h3 p decays by symmetrizing PCAC over the three final-state pions

(PCAC consistency).

Finally in sect. 4 we study PCAC for nonleptonic weak decays. First we review how to link the charge-current algebra to the chiral nonleptonic weak Hamiltonian. Then we apply this current algebra together with PCAC consistency to charged K Kpp, D K K–p, D K K–K, D Kpp and BK D–p weak decays. In the latter four cases, PCAC is

not expected to be too accurate because of the larger pion decay momenta, yet we will show that PCAC consistency continues to agree reasonably well with data.

We give our global overview of PCAC in the concluding sect. 5. 2. – PCAC for strong interactions

Given the definition of the pion decay constant ( fpB 93 MeV ) via the pion to vacuum matrix element of the axial-vector current a 0 NAmiNpjb 4idijfppm, the original Nambu [7] version of conserved axial currents for m2

p4 0, ¯Ai4 0, was extended to the field-theoretic linear sigma model [8] PCAC equation

¯Ai

4 fpmp2fp. (1)

Combining (1) with the model-independent Gell-Mann charge-current algebra [1, 2], [Qi_{, V}

mj] 4ifijkVmk, [Qi, Amj] 4ifijkAmk, (2a)

[Qi

5, Vmj] 4ifijkAmk, [Q5i, Amj] 4ifijkVm, (2b)

one can test the validity of PCAC.

Specifically, first consider the pion matrix elements of the vector current definition of the pion form factor ap 8i

NVmNj pkb 4ifijkFp(q2)(p 81p)m, where q24 (p 8 2 p)2 and

Fp(q24 0 ) 4 1. Reducing in the final-state pion and letting p 8 K 0 while using the current algebra (2b) then gives

ap 8iNVmNj pkb K (2i/fp)a 0 N[Q5i, Vmj] Npkb (3a)

4 (2i/fp) ifijkifppm. (3b)

Comparing (3b) with the above pion form factor version then requires

Fp(p24 mp2) K1 . (3c)

This demonstrates the compatibility of PCAC with the pion form factor normalization

867 masses. It also justifies the PCAC reduction step (3a) without making reference to the LSZ representation in coordinate space or to the neglect of surface terms on which such PCAC reduction is usually based. However, if the reader is more comfortable by always carrying out the LSZ formalism, then the single soft-pion reduction step in (3a) can be treated as a short-hand mnemonic.

As for the chiral-broken world, even when mpc0, the PCAC reduction typified by (3a) leads to a consistent picture. Invoking the covariant form of the Heisenberg equation of motion, i¯A 4 [Q5, H] for Hamiltonian density H, GMOR in ref. [9] showed

that the single PCAC reduction yields

mp24 ap 8 NHNpb K (2i/fp)a 0 N[Q5, H ] pb

(4a)

4 ( 1 /fp)a 0 N¯ANpb 4mp2, (4b)

which is clearly a consistent statement. The simple mnemonics (3a) and (4a) apply only to single soft-pion reduction. In sect. 3 and 4, we will again use the PCAC reduction step (3a) or (4a) with H in (4a) replaced by Hemand Hw, respectively.

Given the linear s model (LsM) chiral-breaking tadpole Lagrangian density [8, 10] LLsM

tad 42 fpmp2fs, (5)

straightforward use of the Lagrange equation of motion converts (5) to the field-theoretic PCAC eq. (1). However, eq. (1) is in fact model-independent. For example, the Cronin [11] exponential Lagrangian

(

)

still leads to the PCAC operator eq. (1) (linear in the pion field). This was stressed by Cronin in ref. [11].

Single soft-pion predictions such as the Goldberger-Treiman relation, the Adler consistency condition for p8 Kp8, and the Fubini-Furlan-Rossetti theorem for g8K

p8 always appear to match experiment [1, 2]. Also many combined current algebra-PCAC (model independent) predictions are in agreement with data. But when two pions are taken soft together, this requires accounting for pole, current commutator, and s terms [1, 2], as in the Adler-Weisberger sum rule [12] for p8 Kp8, the KSRF relation [13] for r Kpp and Weinberg’s pp expansion for ppKpp [14] and for Kl4semileptonic weak decays [15].

In particular, the Weinberg pp amplitude due to current algebra-PCAC is [14, 1, 2]

Tabcd4 A(s , t , u) dabdcd1 A(t , u , s) dacdbd1 A(u , t , s) daddbc, (6a)

with A(s , t , u) K (s2m2

p) /fp2, etc. This result is also recovered in the chiral-limiting LsM [10] as one might expect. However, away from the chiral limit, the tree-level LsM predicts [16] in the s-channel

A(s , t , u) 42 2l

y

z

g

hg

h

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for the LsM coupling [8, 10] l 4 (m2

s2 mp2) /2 fp2. Then the Weinberg I 40 s-wave pp scattering length prediction [14] app( 0 )4 7 mp/32 pfp2B 0.16mp21 increases by 23%, more in line with data to

app( 0 )NLsMB

g

h

p/ms2B 0.046 with [16] msA 650 MeV. This (simple) tree order LsM predic-tion ( 6 d) also follows numerically from the (complicated) O(p4_{) chiral perturbation}

theory approach [17].

3. – PCAC for electromagnetic interactions

First we follow Coleman and Glashow (CG [5]) and split up the nonphotonic electromagnetic (em) Hamiltonian density Hem into its photonic current-current loop

and CG DI 41 tadpole parts,

Hem4 HJJ1 Htad3 .

(7a)

Here the first em current-current Hamiltonian in (7a) is formally given by the time-ordered LSZ operator HJJ4 i 2e 2



while the quark mass matrix muu–u 1mdd

– d 1mss–s 4cH81 c 8 H3 has the D I 41 tadpole part Htad3 4 c 8 H34 (mu2 md)(u–u 2d – d) /2 . (7c)

This CG decomposition in (7) successfully explains all ground-state pseudoscalar (P), vector (V), octet baryon (B) and decuplet baryon (D) em mass splittings with empirical 2% em to semistrong strengths [18, 19]

(c 8 /c)PB 0.020 , (c 8 /c)VB 0.022 , (c 8 /c)BB 0.0172 , (c 8 /c)DB 0.0179 .

(8)

To find the first ratio in (8), it is convenient to scale the meson mass splittings to the observed [20] mass differences (squared)

(Hem)Dp4 Dmp24 mp12 m_p20_{B 0.00126 GeV}2,

(9a)

(Hem)DK4 DmK24 mK212 m_K20_{B 20.00399 GeV}2.

(9b)

Also one knows from (7a) that (with, e.g., (Hem)Dp4 (Hem)p12 (H_em)_p0

)

(Hem)Dp4 (HJJ)Dp1 (Htad3 )Dp4 (HJJ)Dp1 0 4 Dmp2, (10a)

(Hem)DK4 (HJJ)DK1 (Htad3 )DK4 DmK2.

869 The zero in (10a) is due to SU(3): (H3

tad)DpA d3334 0. Subtracting (10a) from (10b) then

yields

(H3

tad)DK4 DmK22 Dmp21 (HJJ)DK2 (HJJ)Dp.

(10c)

To compute the latter difference (HJJ)DK 2Dp in (10c), we now invoke Dashen’s PCAC

analysis [6]. Reducing in the final-state p or K meson, this yields in the soft momentum limit,

(HJJ)p14 (H_JJ)_K1, (H_JJ)_p0_{4 (HJJ})_K0_{4 (HJJ})_K–0_{4 0 .}

(11)

Thus (HJJ)DK 2Dp in (10c) vanishes due to Dashen’s PCAC results (11), leading to the

kaon tadpole scale

(H3

tad)DK4 DmK22 Dmp2B 2 0.00525 GeV2, (12a)

as found from the observed em mass differences (9). Then the 2% Coleman-Glashow em to semistrong pseudoscalar (P) ratio in (8) is [19]

(c 8 /c)P4 2

g

h

obtained using SU(3) for l3 _{and l}8 _{operators combined with the CG tadpole-Dashen}

PCAC scale (12a).

We note that the near equality of the four c 8 /c 2% ratios in (8) and (12b) suggests that we have implicitly generated the observed pseudoscalar mass splitting ratio DmK2/Dmp2B 2 3.17 as found in eqs. (9). If instead one ignores the Coleman-Glashow tadpole component of Hemwhen computing the (then total) Dashen contribution to the

pseudoscalar mass splittings, the predicted ratio DmK2/Dmp2NDashenA 1.9 is far from the

observed ratio 23.17, as already noted [21]. This just means that the Coleman-Glashow tadpole component of Hemmust be included together with Dashen’s PCAC analysis of HJJ meson matrix elements if one wishes to make reasonable contact with em mass splitting data.

Next one can examine the DI 41 charge symmetry breaking em transitions ap0

NHemNh , h 8 b and ar0NHemNvb. Since v is almost purely nonstrange (NS), we

construct via SU(3) the NNSb 4Nu–u 1d–db /k_{2 to NI41, I}34 0 b 4 N u–u 2d

–

db /k2 pseudoscalar meson transition [19, 22]

ap0NHemNhNSb 4DmK22 Dmp2B 2 0.00525 GeV2 (13a)

from the empirical mass splittings in (9). By construction we require the same SU(6) vector mass splitting value [19]

ar0NHemNvb B DmK*2 2 Dmr24 DmK22 Dmp2B 2 0.00525 GeV2. (13b)

The latter r0

-v mixing transition is in good agreement with the best v Kr0

K 2 p measurement finding [23]

ar0NHemNvb 4 2 0.0045 6 0.0005 GeV2.

(13c)

Given the SU(3) transition (13a), one can compute the mixed h–p0_{transition as}

ap0NHemNh 4 cos fPap0NHemNhNSb B2 0.0039 GeV2,

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using the theoretical mixing angle relative to the NS–S basis [18, 19]

fP4 arctan

y

z

or the analog empirical mixing angle [24] fP4 41 7 6 2 7. This h-p0 transition (14) can

then be used in the PCAC consistency [4] reduction of h3 p:

NMh3 p0N f Na 3 p0NH_emNhb N B ( 3 /2 f_p2) Nap0NH_emNhb N B 0.68 .

(16a)

In eq. (16a) the implied doubly reduced charge commutator [Q3

5, [Q53, Hem] ] becomes Hem because of the cyclic identity [25] on HJJ combined with Gell-Mann’s algebra of quark densities (see, e.g., refs. [1, 2]) for H3

tad. The factor of 3 in (16a) comes from

reducing in two of the pions as in Weinberg’s pp PCAC expansion (6a). The d in (16a) is due to the mismatch of Feynman amplitudes with Bose symmetry [3], but it also follows by including h and p0_{poles [4]. Finally, folding in the phase space factor of 816 eV, the}

PCAC amplitude (16a) predicts an h3 pdecay rate

G(h3 p0) 4 (816 eV )NM_{h3 p}0N2B 377 eV ,

(16b)

which is in excellent agreement with the experimental h3 p0rate [20]

G(h3 p0) 4383635 eV .

(16c)

We suggest that the above PCAC consistency scheme characterized by (16a) is a direct generalization of Weinberg’s [15] multiple soft-pion approach for strong interactions but now extended to electromagnetic (and later weak) interactions. Compare this accurate (and simple) PCAC consistency approach to h3 p decays in

eqs. (16) with the much more detailed and complicated chiral perturbation theory (ChPT) scheme predicting [26] an h3 p0decay rate A100 eV.

Finally, we note that the universal scale [19] of these meson em transitions 20.005 GeV2in (12a), (13a), (13b) and (13c), are all linked via PCAC of Dashen to the

h3 p0PCAC amplitude (16a) via (14). Such universality has also been extended to all P,

V, B, D em mass splittings [19].

4. – PCAC for nonleptonic weak decays of mesons

First we invoke the chiral current algebra relation needed for all nonleptonic weak decays driven by a Hamiltonian built up from V-A left-handed weak currents:

[Q 1Q5, Hw] 40 or [Q5, Hw] 42 [Q, Hw] .

(17)

We shall refer to (17) as the weak chirality relation. For simplicity we consider only two-body weak decays involving a charged decaying meson, in order to isolate only a single final-state phase shift. Unlike em decays when the PCAC-induced charge Q5

acted on Hemin sect. 3, here in the weak chirality relation (17) it is more convenient to

let the chiral charge Qi _{act back onto the SU(3) meson states via Q}i NPjb 4

ifijk_NPkb.

To begin, we study the DI 4 (3O2) K1

K p1p0 _{Cabibbo angle-suppressed weak}

decay. PCAC Consistency requires that we reduce in both final state pions, but in successive order

(

)

871

Fig. 1. – a) W pole graph for ap1_NH

wNK1b. b) W exchange graph for ap0NHwNK0b.

reduction as in eq. (3a) or (4a) along with the weak chirality relation (17), one then finds [3] (18) _Nap1_p0 NHwNK1bN4

N

N

N

N

Note that the isospin structure of (18) automatically “filters out” the large DI 41/2 components, leaving the PCAC Consistency amplitude (18) as purely DI 43/2, just as is needed. Consequently in the following analysis of this DI 43/2 K1

2 p(and later the D2 p1)

decay we shall not bother to consider the DI 41/2 graphs since they must cancel out in the final result.

To evaluate (18) we use the standard form of the nonleptonic weak Hamiltonian density

Hw4 (GF/2k2)(J†J 1JJ†)

(19a)

for Fermi coupling GFB 11.66 3 1026GeV22. Sandwiching meson states ap1N and

NK1b around Hwin (19a), the W pole graph of fig. 1a) for fK/fpB 1.22 and fpB 93 MeV, leads to the reduced matrix element [3, 27]

ap1

NHwNK1bW pole4 (GFOk2) s1c1 fpfKmK2B 0.47 3 1028GeV2.

(19b)

This W pole amplitude in fig. 1a) and (19b) is a straightforward q–q relativistic Feynman calculation following from the vacuum saturation of eq. (19a).

On the other hand, the PCAC-reduced matrix element ap0

NHwNK0b in (18) is

controlled by the W exchange ( Wx) graph of fig. 1b). For qqq baryons such an analog W-exchange analysis was worked out by Riazuddin and Fayyazuddin in the nonrelativistic limit [28]. In the same spirit the W-exchange graph of fig. 1b) for q–q mesons has a nonrelativistic weak (parity-conserving) Hamiltonian density

HWx4 2 (GF/k2) s1c1(a1b21 b1a2)( 1 2s1Qs2) d3(r) ,

(20a)

where here a and b respectively convert a d into a u quark and an s– into a u–-quark. Then sandwiching K0_{and p}0 _q–_{q states around H}

872

one estimates this Wx transition as (20b) ap0

NHwNK0bWx4 2 (GF/k2) s1c1k4 mKmp/2 Q 4 cK( 0 ) cp( 0 ) 4

4 2 (GF/ 2 ) s1c1( 4 /3 ) mpfpmKfKB 2 0.12 3 1028GeV2.

Here the spin factor in (20a), ( 1 2s1Qs2) 4422s2for s 4 (1O2)(s11s2) generates the

factor 4 in the first line of (20b) when operating on the S 40 pseudoscalar meson state. When passing to the second line of (20b) we have used the quark model relation between the wave function at the origin and the pseudoscalar (P) decay constant [29],

cP( 0 ) 4kmP/6 fP.

Finally, adding ap1

NHwNK1bW polefrom (19b) to ap0NHwNK0bWxin (20b), the PCAC

consistency K1

2 pamplitude in (18) is found to have a magnitude for fpB 93 MeV,

(21a) _Nap1_p0 NHwNK1b NB 1 2 fp [ (0.47) 2k2 Q 0.12 ] 31028_GeV2 B 1.6 3 1028_{GeV .}

This estimate (21a) is near the experimental K1

K p1p0amplitude [20] Nap1p0

NHwNK1b Nexpt4 mKk8 pG/q 4 (1.834 6 0.007)31028GeV .

(21b)

Although the current algebra-PCAC prediction (21a) is only 10% less than the experimental amplitude (21b), one might still question the nonrelativistic estimate in (20) for the tightly bound (relativistic) pion. This justification is because the W-exchange contribution in (20) is small and any modest relativistic corrections to (20b) will be negligible in (21a).

We stress that the above PCAC consistency procedure for K1

pp decay requires only the PCAC-reduced graphs of fig. 1 as needed for reduced transitions ap1

NHwNK1b and ap0NHwNK0b. These graphs of figs. 1 (and similarly for

fig. 2-5), are not model-dependent factorization-type graphs for the larger amplitude ap1_p0

NHwNK1b. Instead, our model dependence is buried in our first PCAC

consistency step eq. (18).

Alternatively applying the W emission (vacuum saturation) procedure directly to the K1

K p1p0 _{amplitude [30] without the PCAC step eq. (18), one finds that (21a)}

becomes 1.9 31028_{GeV, within 4% of the observed amplitude in (21b). However for the}

heavier meson D and B weak decays, our PCAC Consistency procedure will better approximate the data and the corresponding W-exchange graphs will be more accurate in the nonrelativistic limit.

Fig. 2. – a) W pole graph for ap1_NH

wNDs1b. b) W exchange graph for aK

–₀

873 Specifically, we next consider the Cabibbo angle-enhanced D1

K K–0p1_{weak decay.}

Weak chirality (17) together with the PCAC Consistency extension of (18) then requires [3, 31]

ap1_K–0

NHwND1b 4 (i/k2 fK)[ap1NHwNDs1b 1 ( fK/fp)aK –₀

NHwND0b] .

(22)

The larger PCAC-reduced matrix element ap1

NHwNDs1b is determined by the W pole

graph of fig. 2a), giving the amplitude analogous to (19b) but on the decaying D-meson mass shell [3, 27, 31]: ap1 NHwNDs1bW pole4 (GF/k2) c12fpfDsm 2 DB 0.42 3 1026GeV2, (23a)

for [32] fDs/fpB 1.8 and c1B VudB 0.975. Likewise, the smaller PCAC-reduced matrix

element aK–0

NHwND0b is now reasonably estimated by the (nonrelativistic) W-exchange

graph of fig. 2b), leading to the amplitude analogous to (20b): aK–0

NHwND0bWx4 2 (GF/k2) c12( 4 /3 ) mKfKmDfDB 2 0.17 3 1026GeV2

(23b)

for [32] fD/fpB 1.7. Adding (23a) and (23b) in the PCAC consistency DKp1 amplitude (22)

then yields the magnitude [31] (24a) _Nap1_K–0

NHwND1b NB (1/k2 fK)[ 0.42 2 (1.22)(0.17) ]31026GeV2B

B 1.32 3 1026GeV . This PCAC Consistency prediction (24a) is in excellent agreement with the observed amplitude [20]

Nap1K–0

NHwND1b Nexpt4 mDk8 pG/q 4 (1.32 6 0.14)31026GeV .

(24b)

Now we examine Cabibbo angle-suppressed D1

K K–0K1_{weak decay. Reducing in}

both kaons in succession according to PCAC Consistency, this amplitude has magnitude [3, 31] for fKB 113 MeV,

(25a) _NaK1_K–2 NHwND1b N4 (1/k2 fK) NaK1NHwNDs1b N4 4 GF 2 s1c1fDsm 2 DB 0.73 3 1026GeV . (25b)

Here we have used the W-pole reduced matrix element depicted in fig. 3 analogous to (23a) again evaluated on the D mass shell for fDsB 1.8 fp[32]. This predicted PCAC

Fig. 3. – W pole graph for aK1_NH wNDs1b.

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Fig. 4. – a) W pole graph for ap1_NH

wND1b. b) W exchange graph for ap0NHwND0b.

consistency D1

K K– amplitude (25b) is also quite close to observation [20]

NaK1K–0

NHwND1b Nexpt4 mDk8 pG/q 4 (0.73 6 0.08)31026GeV .

(25c) As with D1

K K–0p1_{in (24), the PCAC D}1

K K–0K1_{amplitude (25) is within 2% of the}

data because a W-exchange (nonrelativistic) graph for tightly bound pions is absent, whereas it causes a 10% deviation from data in the K1

K p1p0 decay analysis in (21).

Hence we might expect the Cabibbo angle-suppressed D1

K p1p0_{D I 43/2 weak}

decay PCAC consistency amplitude likewise to deviate from the data by up to 10% because both final state pions are q–q tightly bound and a W-exchange nonrelativistic estimate as for K1

K p1p0_{should not be completely accurate for D}1

K p1p0_{. This is}

indeed the case.

Specifically, the weak chirality-PCAC consistency amplitude for D1

ppis ap1_p0

NHwND1b 4 (2i/2 fp)[ap1NHwND1b 2k2ap0NHwND0b] ,

(26)

which is manifestly D I 43/2, as required. The W-pole reduced matrix element ap1

NHwND1b depicted in fig. 4a) by analogy with (19b) is

ap1

NHwND1bWx4 2 (GF/k2) s1c1fpfDmD2B 29.3 3 1028GeV2,

(27a)

the minus sign in (27a) due to GIM mechanism [33]. Also the W-exchange matrix element ap0

NHwND0b of fig. 4b) is

ap0

NHwND0bWx4 (GF/2 ) s1c1( 4 /3 ) mpfpmDfDB 0.6 3 1028GeV2,

(27b)

in analogy with (20b). Then the PCAC consistency D1

ppamplitude (26) is Nap1p0 NHwND1b NB 1 2 fp [9.3 2k2 Q 0.6] 31028_GeV2 B 0.45 3 1026GeV , (28a)

about 18% greater than the experimental amplitude [20] Nap1p2

NHwND1b Nexpt4 mDk8 pG/q 4 (0.3860.05)31026GeV .

(28b)

As was the case for K1

2 p, the small W-exchange nonrelativistic estimate of (27b) is only a

rough approximation for a tightly bound q–q pion.

Finally we consider the heaviest q–q bottom B-meson weak decay B1

K D–0p1_{. Here}

875

Fig. 5. – W exchange graph for aD–0

NHwNB0b.

using (17) predicting the B1

Dpamplitude magnitude Nap1D–0 NHwNB1b N4 (1/fp) NaD –₀ N[Q5p2, Hw] NB1b N4 1 k2 fp NaD–0NHwB0b N . (29)

Since there is no rapidly varying pole term in this case, this PCAC estimate (29) for B1 Dp

(even though ppis not near zero) may prove to be reasonable. We now show that this is indeed the case.

Specifically, the fig. 5 W-exchange reduced matrix element aD–0

NHwNB0b in (29)

should be accurately determined by the nonrelativistic estimate in analogy with (23b)

NaD–0NHwNB0b NWx4 (GF/k2) VudVbc( 4 /3 ) mDfDmBfBB 0.093 3 1026GeV2.

(30)

Substituting (30) back into the PCAC amplitude (29) then leads to the current algebra-PCAC prediction

Nap1D–0

NHwNB1b N4 (1/k2 fp)( 0.093 31026GeV2) B0.7131026GeV , (31a)

which is not too distant from the observed value [20]

Nap1D–0_NHwNB1b Nexpt4 mBk8 pG/q 4 (0.83 6 0.10)31026GeV .

(31b)

Finally, three body weak decays involving pions, K3 pand D3 pcan also be computed

using the PCAC Consistency prescription [3]. Like electromagnetic decay h3 p0 in

eqs. (16), all four K3 p decays computed via PCAC Consistency are in good agreement

with data [4, 34].

Specifically, PCAC Consistency predicts the D I 41/2 ratio [3, 35]

N

N

with the D I 41/2 Hamiltonian (large) scale dividing out in (32a). The factor of 3/2 in (32a) is the same as for h3 p0em decay in (16a), namely the 3 corresponds to reducing in

the K3 p0 pions consistently. The 1/2 follows from the mismatch of this Feynman

amplitude with Bose symmetry [3] (or from K0 _{and p}0 _{rapidly varying pole}

876

PCAC Consistency result (32a) is not far removed from the experimental amplitude ratio [20]

N

N

In fact, removing the small 10% D I 43/2 reduction of the K3 p0amplitude [2] increases

the measured ratio (32b) to 7.9 GeV21_{, then in closer agreement with the PCAC}

Consistency DI 41/2 prediction (32a). 5. – Conclusion

In this paper we have surveyed the application of partial conservation of axial currents (PCAC) in conjunction with the charge algebra of currents as it applies to strong interactions, electromagnetic interactions and to nonleptonic weak interactions.

In sect. 2 we reviewed PCAC for strong interactions. Reducing in just one pion was first tested when one pion four-momentum became soft with the second pion on mass shell. Then successful predictions when two pion momenta became soft were reviewed, provided pole terms, s terms and successive PCAC steps (such as in the Weinberg pp expansion [14]) were taken into account.

Next in sect. 3 we surveyed electromagnetic interactions involving PCAC with the Coleman-Glashow (CG) tadpole folded in at the outset. Pseudoscalar SU(2) mass splittings are then understood by combining CG tadpoles with Dashen’s PCAC procedure. The latter 2% ratio of electromagnetic to semistrong mass breaking (c 8 /c)P

can be extended to vector mesons and to octet and decuplet baryons again by including CG tadpoles and using SU( 3 ) symmetry [18, 19]. Without accounting for the CG tadpoles, however, ref. [21] concludes that the Dashen PCAC prediction DmK24 Dmp2is “not phenomenologically robust”. Also, for h3 p decays, PCAC again is extremely

accurate when i) all the tedious pole terms are taken into account or ii) PCAC Consistency is invoked [4]. The latter is the natural extension of Weinberg’s [14] PCAC

pp expansion, but extended to nonphotonic electromagnetic transitions.

Lastly, in sect. 4 we studied the five two-body weak nonleptonic decays K1

pp, DK–1p,

D_K–1

K, Dpp1, BD–1p, all from the perspective of PCAC Consistency. Note that Kpp1 and Dpp1 must be pure DI 43/2 transitions, thus avoiding the much larger DI41/2 enhanced terms as in, e.g., K0

2 p. In fact these above five weak decays can be treated in a uniform

manner by employing the PCAC consistency, with PCAC-reduced charged W poles and neutral W-exchange diagrams. Once the above quark pole graphs are taken into account, there is little extra rapid variation of pion momentum. Thus it should not be surprising that such PCAC-reduced weak amplitudes are always in approximate agreement with experiment, even though the final-state pion momentum is sometimes quite substantial. We ended sect. 4 by reviewing the PCAC consistency ratio of

K3 p0/K_{2 p}0 and comparing the similar PCAC consistency reduction of K_{3 p}0 weak decay

relative to h3 p0electromagnetic decay.

* * *

The author is grateful for conversations with N. PAVER, RIAZUDDIN and R. KARLSEN.

877 R E F E R E N C E S

[1] WEISBERGERW. I., Brandeis Summer School Lectures 1967, edited by M. CHRETIENand S. SCHWEBER (Gordon and Breach, New York) 1967; ADLERS. L. and DASHENR. F., Current

Algebra (Benjamin, New York) 1968; BJORKENJ. D. and NAUENBERGM., Ann. Rev. Nucl.

Sci., 18 (1968) 229.

[2] SCADRONM. D., Rep. Prog. Phys., 44 (1981) 213.

[3] KARLSENR. E. and SCADRONM. D., Phys. Rev. D, 45 (1992) 4108, 4113.

[4] CHOUDHURY S. R., KARLSEN R. E. and SCADRON M. D., Int. J. Mod. Phys. A, 11 (1996) 271.

[5] COLEMANS. and GLASHOWS., Phys. Rev. B, 134 (1964) 671. [6] DASHENR. F., Phys. Rev., 183 (1969) 1245.

[7] NAMBUY., Phys. Rev. Lett., 4 (1960) 380.

[8] GELL-MANNM. and LE´VYM., Nuovo Cimento, 16 (1960) 705.

[9] GELL-MANNM., OAKESR. and RENNERB., Phys. Rev., 175 (1968) 2195 (GMOR).

[10] DE ALFARO V., FUBINI S., FURLAN G. and ROSSETTI C., Currents in Hadron Physics (North-Holland, Amsterdam) 1973, Ch. 5 (DFFR).

[11] CRONINJ., Phys. Rev., 161 (1967) 1483, see p. 1487.

[12] ADLERS. L., Phys. Rev. Lett., 14 (1965) 1051; WEISBERGERW. I., Phys. Rev. Lett., 14 (1965) 1047.

[13] KAWARABAYASHI K. and SUZUKI M., Phys. Rev. Lett., 16 (1966) 255; RIAZUDDIN and FAYYAZUDDIN, Phys. Rev., 147 (1966) 1071 (KSRF).

[14] WEINBERGS., Phys. Rev. Lett., 17 (1966) 616. DFFR in ref. [10], pp. 324-327, show explicitly how the LsM s, t, u narrow width s poles cancel against the pp lead term l/4, again resulting in the Weinberg pp model-independent amplitude in eqs. (6).

[15] WEINBERGS., Phys. Rev. Lett., 17 (1966) 336.

[16] KOP. and RUDAZS., Phys. Rev. D, 50 (1994) 6877. The s-meson has now been reinstated in the PDG tables of ref. [20] and also by TORNQVISTN. and ROOSM. , Phys. Rev. Lett., 76 (1996) 1575 as f0(400-900), with average s mass ms4 650 MeV. Also see SVECM., Phys. Rev. D, 53

(1996) 2343; HARADAM., SANNINOF. and SCHECHTERJ., Phys. Rev. D, 54 (1996) 1991. [17] GASSERJ. and LEUTWYLERH., Phys. Lett. B, 125 (1983) 321, 325; in Ann. Phys., 158 (1984)

142, Appendix B, these authors construct a pseudo-LsM which does not include the quartic

pp contact term l/4 and so does not recover the Weinberg [14] pp PCAC amplitude

(s 2m2

p) /fp2as found in ref. [10], pp. 324-327.

[18] JONESH. F. and SCADRONM. D., Nucl. Phys. B, 155 (1979) 409; FUCHSN. H. and SCADRONM. D., J. Phys. G., 15 (1989) 957.

[19] COONS. A. and SCADRONM. D., Phys. Rev. C, 51 (1995) 2923.

[20] PARTICLEDATAGROUP(R. M. BARNETTet al.), Phys. Rev. D, 54 (1996) 1.

[21] DONOGHUEJ. F., HOLSTEINB. R. and WYLERD., Phys. Rev. D, 47 (1993) 2089.

[22] See, e.g., GOURDINM., Unitary Symmetries and their Application to High Energy Physics (North-Holland, Amsterdam) 1967, pp. 99-101.

[23] COONS. A. and BARRETTR. C., Phys. Rev. C, 36 (1987) 2189. [24] BRAMONA. and SCADRONM. D., Phys. Lett. B, 234 (1990) 346. [25] SUTHERLANDD., Phys. Lett., 23 (1966) 384.

[26] GASSER J. and LEUTWYLER H., Nucl. Phys. B, 250 (1985) 539; ANISOVICH A. V. and LEUTWYLER H., Dispersive analysis of the decay h K3p, hep–ph/9601237; see CRYSTAL BARRELCOLLABORATION(C. AMSLERet al.), Nuovo Cimento, 107 (1994) 2589 for pointing out that ChPT predicts an h3 prate that is only 25% of data.

[27] See, e.g., KARLSENR. E. and SCADRONM. D., Phys. Rev. D, 44 (1991) 2192.

[28] RIAZUDDINand FAYYAZUDDIN, Phys. Rev. D, 18 (1978) 1478; 19 (1978) 1630E. For q–q mesons, the minus sign of H_Wxin (20) is due to the anti-fermion wavefunction. RIAZUDDIN, private communication.

878

[29] VANROYENR. and WEISSKOPFV. F., Nuovo Cimento A, 50 (1967) 617; 51 (1967) 583; JACKSON J. D., in Proceedings of the 1976 SLAC Summer School on Particle Physics, SLAC–198, 1976; KRASEMANNH., Phys. Lett. B, 96 (1980) 397.

[30] SCADRONM. D., Phys. Rev. D, 29 (1984) 1375; Charm 2000 Workshop, pp. 407-409, edited by D. M. KAPLANand S. KWAN(Fermilab) 1994; Mod. Phys. Lett. A, 10 (1995) 2733.

[31] KARLSENR. E. and SCADRONM. D., Mod. Phys. Lett. A, 6 (1991) 543.

[32] DOMINGUEZC. and PAVERN., Phys. Lett. B, 197 (1987) 423; MENDELR. R. and TROTTIERH. D., Phys. Lett. B, 231 (1989) 312; ROSNERJ. L., Phys. Rev. D, 42 (1990) 3732; ABADAA. et al.,

Nucl. Phys. B, 376 (1992) 172; BERNARDC. W. et al., Nucl. Phys. B (Proc. Suppl.), 26 (1992) 384; AOKIS. et al., Prog. Theor. Phys., 89 (1993) 131.

[33] GLASHOWS., ILIOPOULOSJ. and MAIANIL., Phys. Rev. D, 2 (1970) 1285 (GIM).

[34] CHOUDHURYS. R. and SCADRONM. D., Nuovo Cimento A, 108 (1995) 289; Phys. Rev. D, 53 (1996) 2421.

[35] CALLANC. G. and TREIMANS. B.; Phys. Rev. Lett., 16 (1966) 197; HARAY. and NAMBUY.,