- Email: [email protected]
The ΔI = 12 rule in the chiral quark model
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
16 August 1984
THE A I = 1 / 2 RULE IN THE CHIRAL Q U A R K M O D E L
Andrew G. C O H E N and Aneesh V. M A N O H A R 1 Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 7 May 1984
We calculate the one-loop renormalization group equations for the operators responsible for AS = 1 weak decays, and demonstrate that the renormalization group scaling enhances the A I = l / 2 decays and is in reasonable agreement with experiment.
A long-standing problem in low-energy hadronic physics is the enhancement of A I = 1 / 2 weak decays over those that change isospin by 3/2. For example, F ( K ° ~ 7 r % r - ) / F ( K + --+ rr%r °) = 450,
accord with experiment. Thus we obtain a simple explanation of the AI = 1 / 2 rule We will use an effective chiral quark theory [3] to describe low energy physics. The model is predicated on the assumption that the scale which characterizes chiral symmetry breakdown, Axs B, is somewhat larger than the confinement scale, AQCD. In fact, there are several indications that Axs B is about 1 GeV [4]. In this case, the effective lagrangian in the intermediate region has fundamental quark, gluon, and goldstone boson fields. The quarks have a mass of approximately 350 MeV due to the breakdown of chiral symmetry. As discussed in ref. [3], the Q C D interactions are weakened in this effective theory, allowing us to describe baryons as bound states of non-relativistic chiral quarks. The lagrangian is
because the K + decay is a pure A I = 3 / 2 process, whereas the K ° decay involves both A I = 1 / 2 and A I = 3 / 2 amplitudes. There is a similar enhancement in the non-leptonic hyperon decays. Wilson [1] was among the first to suggest that this might be a dynamical enhancement due to the strong interactions, although our inability to calculate hadronic matrix elements has prevented us from verifying this conjecture. Several authors have shown that hard gluon corrections to the effective weak hamiltonian produce an effect which goes in the right direction, but is too small to explain the observed enhancement [2]. Since then, many sources of additional enhancement have been proposed, but none of these seems very attractive. In this paper, we will attempt to show that we can improve our estimate of the matrix elements of the weak hamiltonian by the use of an effective low energy theory of Goldstone bosons (pions) interacting with chiral quarks. By scaling the AS = 1 weak decay operator in this effective theory, we obtain an enhancement of the A I = 1 / 2 process in
gA = 0.75,
1 Junior Fellow, Harvard Society of Fellows.
G~, is the field strength tensor for the gluons, and we have dropped higher order terms, which
~ = ~ ( i D + lff)~p + g.~A~ys~p - m~/~p + ¼f2tr 0,~0 ~52. -
(1/2g z) tr G~,~G~'~ (1)
-}-'~AS= 1,
where V~ = ½(~*0.~ + ~0~ t ) , = ei~r/f,
0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
A~½i(~*O~ - ~0.~*),
~ = e2i~/f,
f = 93 MeV.
481
V o l u m e 143B, n u m b e r 4, 5, 6
PHYSICS LETTERS
are suppressed by inverse powers of Axs B. We always work in the S U ( 3 ) x SU(3) limit, so all pseudo-goldstone boson mass terms are neglected. The AS = 1 lagrangian is ~0 a s = l = pm~/~ thg;~p+ om~/li thl~'ystp
+ l~kf2 tr ha, Ea"E*
+ ¼afZTj~J'(ZO~,S,t)'/(s,a~'z* ) ~,
(2)
where h=
interpret Xexp, a e x p a s X ( # = 2 m . ) , a(/x = 2m=) and Pcxp as p(m,,). In order to relate these experimental parameters to the full theory above AxsB, we must scale these coefficients up to the b o u n d a r y between the two theories. The one-loop renormalization group equations can be calculated from the graphs shown in fig. 1. The results are (in MS), #d)~/d/x=-(3/¢rZ)g~(rn2/fz)p=-2.450,
(4)
tsda/dlx = 0,
(5)
I~dp/dl~ = (1/8"rrZ ) [152g2 ( m 2 / f 2 ) ~. 0 0
3 2(m2/f2)p] ~gA
,
T1~=T12t=T),2= T32' = ½, T 2 2 = T322-
--(2Ots/~)p
= 0.043X --0.344p,
and
1 2.
X and a are the coefficients of the A I = 1 / 2 and A I = 3 / 2 meson operators, and p and o are the coefficients of the &I = 1 / 2 two-quark operators. The factors of f and m have been introduced to make them dimensionless. (There are obviously no AI = 3/2 two-quark operators.) The ellipses represent higher dimension operators, whose effects are small. These include four-quark operators, as well as two-quark one-pion operators and operators involving the symmetry breaking mass term M. We find the parameters ~, a, and p by calculating the rates for K-decay and s-wave hyperon nonleptonic decay and fitting to experiment. This gives )t~p = 3.2 X 10 7, Pexp = 1 X 10-7.
aex p =
(6)
with m = 350 MeV, and a~ = 0.3. The important point is that chiral quarks have a large effect on the pion sector, whereas pions have a relatively mild effect on the quark sector. We have set the pseudo-goldstone masses to zero. Including them would not renormalize X, a, and P, but induce higher dimension operators involving powers of the SU(3) breaking mass matrix M. These are suppressed because they are of higher dimension. The running of parameters in the AS = 0 part of the lagrangian is of higher oder, and we will treat these parameters as fixed when we integrate the renormalization group equations. If we had not constructed an effective low energy theory with chiral quarks, the only scaling would
/ I
------@------
f--,~.
\
1 x 10 -8, tf A
(3)
The contribution of o to hyperon decays is suppressed by a power of p/m because of the 3'5Hence o is not determined from experiment. However, it turns out that renormalization scaling does not cause it to mix with )t, a, and p, so its value is irrelevant. As in any effective theory, when we quote values for the coefficients of operators we must ask, at what scale? For the hyperon decays, this scale is characterized by the pion mass, rn,~, and for the kaon decays by 2rn.. We therefore 482
16 A u g u s t 1984
~ 1 _-__/____
f
N
/
t
/ I -- "~N kv
k
Fig. 1. G r a p h s c o n t r i b u t i n g to the o n e - l o o p r e n o r m a l i z a t i o n g r o u p equations. T h e • stands for an insertion of p or o; the x an insertion of ~; a n d the zx an insertion of a.
Volume 143B, n u m b e r 4, 5, 6
PHYSICS LETTERS
come from soft gluons. This effect would have been large since a s becomes infinite at low energy. However this calculation is untrustworthy since perturbation theory is completely unreliable. We hope that by constructing the effective theory of chiral quarks we have traded the effects of soft gluons for a constituent chiral quark mass. Since the gluons do not couple to the mesons, we expect gluon effects to be negligible in the meson sector, and hence we do not include the running of a s in our scaling. If a s is truly small in the effective theory then it will not run much between m . and Axs ~, so we might as well hold it fixed. The greatest uncertainty in our calculation lies in the matching between the full theory above Axs B and our effect lagrangian. We therefore choose to scale the experimental numbers up to Axs B, and c o m p a r e with the theoretical estimate at Axs B. a does not scale. The results for O and )~ are plotted in fig. 2. We find
8;
P(AxsB)=6.5X10 a(AxsB)=l×10
X(Axs,)_7.5X10-8:
8.
(7)
N o t e that X(AxsB)/X(2m=) is sensitive to the exact value of X / p and Axs B. [In fact the eigenvalues of the system of eqs. (4), (6) are - 0 . 1 7 + 0.27i so that p and X have d a m p e d oscillatory behavior as one scales up in/1.] N o w that we know how to scale the effective theory up to Axs n, we must worry about the
),, p x 10 7
16 August 1984
matching condition, which we estimate using dimensional analysis. The AS = 1 weak lagrangian in the full theory is
= ( c r / ¢ 2 ) cos
×sinOc[e+(l~)O+ +c_(Iz)O],
(8)
where 2 0 + = dv~(1 - ~'5)ufiv,(a - Ys)s
_+ dr"(1 - "~5)sfi~',(1 - Vs)u.
(9)
N o t e that O_ is pure A I = 1 / 2 , while O+ contains both A I = 1 / 2 and A I = 3 / 2 pieces, c+ and ¢_ are both 1 at M w, and are renormalized due to hard gluon effects. At/~ = 1 GeV [2], c+ (1 GeV) = 0.74,
e_ (1 GeV) = 1.84,
for AQc D = 100 MeV. Equating matrix elements of 0+ between offshell mesons with a p 2 _ (1 GeV) 2, we find
p2)~/2 =
(GF/2V/2) sin 0c cos
Ocf2p2,
so that )~--3X10-8c
=5.5×10
-8
(10)
Similarly, a=2×10-8c+=1.5x10
8,
(11)
where the factor of 2 / 3 is a C l e b s c h - G o r d a n coefficient because O+ is 2 / 3 ( A I + 3 / 2 ) and 1 / 3 ( A I = 1/2). Similarly, the matching condition for p is most easily obtained by taking the matrix element of the weak hamiltonian between ds and a one-pion state. Then
pm/2f= (GF/2V~) sin 0c cos Ocmfc_, or
p=1.6×10
I m'tr
I 2m~
I "lGeV
Fig. 2. Plot of )~ and p versus renormalization scale kt.
z.
8c ~ 3 × 1 0
-8 .
(12)
C o m p a r i n g (10)-(12) with (7), we see that they agree to within the errors of the calculation. T h o u g h we cannot take the values too seriously, it is encouraging that they go in the right direction, and have the right magnitude to explain the observed effect. We therefore need no magical en483
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
h a n c e m e n t to e x p l a i n the A I = 1 / 2 rule. W h a t a b o u t p e n g u i n o p e r a t o r s ? T h e coefficients of the p e n g u i n o p e r a t o r s at 1 G e V are - 0 . 1 so they have to be e n h a n c e d b y 200 to explain the A I = 1 / 2 rule. I n the chiral q u a r k theory, the p e n g u i n s induce the same t w o - q u a r k o p e r a t o r s as the other A S = 1 operators. H e n c e the o n l y enh a n c e m e n t of the penguins can c o m e f r o m the m a t c h i n g condition. The c o n v e n t i o n a l v a c u u m insertion a p p r o x i m a t i o n a p p e a r s to give a very large m a t r i x element for the p e n g u i n operator. This a p p r o x i m a t i o n gives a m a t r i x element which is inversely p r o p o r t i o n a l to q u a r k masses. But this piece of the m a t r i x element is certainly n o t present, b e c a u s e it does n o t vanish as mq ~ 0. [ A n y SU(2)L octet SU(3)R singlet A S = 1 o p e r a t o r in the chiral l a g r a n g i a n vanishes linearly as mq ~ 0 since it has at least two derivatives]. Thus the v a c u u m insertion a p p r o x i m a t i o n is n o t evidence for p e n g u i n e n h a n c e m e n t . It is possible that
484
16 August 1984
p e n g u i n o p e r a t o r m a t r i x elements are enhanced, b u t our analysis shows that they c a n n o t d o m i n a t e the A I = 1 / 2 amplitude. If they did, we would get too large an amplitude. Thus we can say n o t h i n g a b o u t c'/E. W e would like to t h a n k H. Georgi a n d J. Polchinski for helpful discussions. This research is s u p p o r t e d in p a r t b y the N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t No. PHY-82-15249.
References [1] K.G. Wilson, Phys. Rev. 179 (1969) 1499. [2] M.K. Gailtard and B.W. Lee, Phys. Rev. Lett. 33 (1974) 108; G. Altarelli and L. Maiani, Phys. Lett. 52B (1974) 351; F. Gilman and M.B. Wise, Phys. Rev. D20 (1979) 2392. [3] A. Manohar and H. Georgi, Nucl. Phys. B234 (1984) 189. [4] J.F. Donoghue, E. Golowich and B.R. Holstein, University of Massachusetts preprint IMHEP-191.
PHYSICS LETTERS
16 August 1984
THE A I = 1 / 2 RULE IN THE CHIRAL Q U A R K M O D E L
Andrew G. C O H E N and Aneesh V. M A N O H A R 1 Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 7 May 1984
We calculate the one-loop renormalization group equations for the operators responsible for AS = 1 weak decays, and demonstrate that the renormalization group scaling enhances the A I = l / 2 decays and is in reasonable agreement with experiment.
A long-standing problem in low-energy hadronic physics is the enhancement of A I = 1 / 2 weak decays over those that change isospin by 3/2. For example, F ( K ° ~ 7 r % r - ) / F ( K + --+ rr%r °) = 450,
accord with experiment. Thus we obtain a simple explanation of the AI = 1 / 2 rule We will use an effective chiral quark theory [3] to describe low energy physics. The model is predicated on the assumption that the scale which characterizes chiral symmetry breakdown, Axs B, is somewhat larger than the confinement scale, AQCD. In fact, there are several indications that Axs B is about 1 GeV [4]. In this case, the effective lagrangian in the intermediate region has fundamental quark, gluon, and goldstone boson fields. The quarks have a mass of approximately 350 MeV due to the breakdown of chiral symmetry. As discussed in ref. [3], the Q C D interactions are weakened in this effective theory, allowing us to describe baryons as bound states of non-relativistic chiral quarks. The lagrangian is
because the K + decay is a pure A I = 3 / 2 process, whereas the K ° decay involves both A I = 1 / 2 and A I = 3 / 2 amplitudes. There is a similar enhancement in the non-leptonic hyperon decays. Wilson [1] was among the first to suggest that this might be a dynamical enhancement due to the strong interactions, although our inability to calculate hadronic matrix elements has prevented us from verifying this conjecture. Several authors have shown that hard gluon corrections to the effective weak hamiltonian produce an effect which goes in the right direction, but is too small to explain the observed enhancement [2]. Since then, many sources of additional enhancement have been proposed, but none of these seems very attractive. In this paper, we will attempt to show that we can improve our estimate of the matrix elements of the weak hamiltonian by the use of an effective low energy theory of Goldstone bosons (pions) interacting with chiral quarks. By scaling the AS = 1 weak decay operator in this effective theory, we obtain an enhancement of the A I = 1 / 2 process in
gA = 0.75,
1 Junior Fellow, Harvard Society of Fellows.
G~, is the field strength tensor for the gluons, and we have dropped higher order terms, which
~ = ~ ( i D + lff)~p + g.~A~ys~p - m~/~p + ¼f2tr 0,~0 ~52. -
(1/2g z) tr G~,~G~'~ (1)
-}-'~AS= 1,
where V~ = ½(~*0.~ + ~0~ t ) , = ei~r/f,
0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
A~½i(~*O~ - ~0.~*),
~ = e2i~/f,
f = 93 MeV.
481
V o l u m e 143B, n u m b e r 4, 5, 6
PHYSICS LETTERS
are suppressed by inverse powers of Axs B. We always work in the S U ( 3 ) x SU(3) limit, so all pseudo-goldstone boson mass terms are neglected. The AS = 1 lagrangian is ~0 a s = l = pm~/~ thg;~p+ om~/li thl~'ystp
+ l~kf2 tr ha, Ea"E*
+ ¼afZTj~J'(ZO~,S,t)'/(s,a~'z* ) ~,
(2)
where h=
interpret Xexp, a e x p a s X ( # = 2 m . ) , a(/x = 2m=) and Pcxp as p(m,,). In order to relate these experimental parameters to the full theory above AxsB, we must scale these coefficients up to the b o u n d a r y between the two theories. The one-loop renormalization group equations can be calculated from the graphs shown in fig. 1. The results are (in MS), #d)~/d/x=-(3/¢rZ)g~(rn2/fz)p=-2.450,
(4)
tsda/dlx = 0,
(5)
I~dp/dl~ = (1/8"rrZ ) [152g2 ( m 2 / f 2 ) ~. 0 0
3 2(m2/f2)p] ~gA
,
T1~=T12t=T),2= T32' = ½, T 2 2 = T322-
--(2Ots/~)p
= 0.043X --0.344p,
and
1 2.
X and a are the coefficients of the A I = 1 / 2 and A I = 3 / 2 meson operators, and p and o are the coefficients of the &I = 1 / 2 two-quark operators. The factors of f and m have been introduced to make them dimensionless. (There are obviously no AI = 3/2 two-quark operators.) The ellipses represent higher dimension operators, whose effects are small. These include four-quark operators, as well as two-quark one-pion operators and operators involving the symmetry breaking mass term M. We find the parameters ~, a, and p by calculating the rates for K-decay and s-wave hyperon nonleptonic decay and fitting to experiment. This gives )t~p = 3.2 X 10 7, Pexp = 1 X 10-7.
aex p =
(6)
with m = 350 MeV, and a~ = 0.3. The important point is that chiral quarks have a large effect on the pion sector, whereas pions have a relatively mild effect on the quark sector. We have set the pseudo-goldstone masses to zero. Including them would not renormalize X, a, and P, but induce higher dimension operators involving powers of the SU(3) breaking mass matrix M. These are suppressed because they are of higher dimension. The running of parameters in the AS = 0 part of the lagrangian is of higher oder, and we will treat these parameters as fixed when we integrate the renormalization group equations. If we had not constructed an effective low energy theory with chiral quarks, the only scaling would
/ I
------@------
f--,~.
\
1 x 10 -8, tf A
(3)
The contribution of o to hyperon decays is suppressed by a power of p/m because of the 3'5Hence o is not determined from experiment. However, it turns out that renormalization scaling does not cause it to mix with )t, a, and p, so its value is irrelevant. As in any effective theory, when we quote values for the coefficients of operators we must ask, at what scale? For the hyperon decays, this scale is characterized by the pion mass, rn,~, and for the kaon decays by 2rn.. We therefore 482
16 A u g u s t 1984
~ 1 _-__/____
f
N
/
t
/ I -- "~N kv
k
Fig. 1. G r a p h s c o n t r i b u t i n g to the o n e - l o o p r e n o r m a l i z a t i o n g r o u p equations. T h e • stands for an insertion of p or o; the x an insertion of ~; a n d the zx an insertion of a.
Volume 143B, n u m b e r 4, 5, 6
PHYSICS LETTERS
come from soft gluons. This effect would have been large since a s becomes infinite at low energy. However this calculation is untrustworthy since perturbation theory is completely unreliable. We hope that by constructing the effective theory of chiral quarks we have traded the effects of soft gluons for a constituent chiral quark mass. Since the gluons do not couple to the mesons, we expect gluon effects to be negligible in the meson sector, and hence we do not include the running of a s in our scaling. If a s is truly small in the effective theory then it will not run much between m . and Axs ~, so we might as well hold it fixed. The greatest uncertainty in our calculation lies in the matching between the full theory above Axs B and our effect lagrangian. We therefore choose to scale the experimental numbers up to Axs B, and c o m p a r e with the theoretical estimate at Axs B. a does not scale. The results for O and )~ are plotted in fig. 2. We find
8;
P(AxsB)=6.5X10 a(AxsB)=l×10
X(Axs,)_7.5X10-8:
8.
(7)
N o t e that X(AxsB)/X(2m=) is sensitive to the exact value of X / p and Axs B. [In fact the eigenvalues of the system of eqs. (4), (6) are - 0 . 1 7 + 0.27i so that p and X have d a m p e d oscillatory behavior as one scales up in/1.] N o w that we know how to scale the effective theory up to Axs n, we must worry about the
),, p x 10 7
16 August 1984
matching condition, which we estimate using dimensional analysis. The AS = 1 weak lagrangian in the full theory is
= ( c r / ¢ 2 ) cos
×sinOc[e+(l~)O+ +c_(Iz)O],
(8)
where 2 0 + = dv~(1 - ~'5)ufiv,(a - Ys)s
_+ dr"(1 - "~5)sfi~',(1 - Vs)u.
(9)
N o t e that O_ is pure A I = 1 / 2 , while O+ contains both A I = 1 / 2 and A I = 3 / 2 pieces, c+ and ¢_ are both 1 at M w, and are renormalized due to hard gluon effects. At/~ = 1 GeV [2], c+ (1 GeV) = 0.74,
e_ (1 GeV) = 1.84,
for AQc D = 100 MeV. Equating matrix elements of 0+ between offshell mesons with a p 2 _ (1 GeV) 2, we find
p2)~/2 =
(GF/2V/2) sin 0c cos
Ocf2p2,
so that )~--3X10-8c
=5.5×10
-8
(10)
Similarly, a=2×10-8c+=1.5x10
8,
(11)
where the factor of 2 / 3 is a C l e b s c h - G o r d a n coefficient because O+ is 2 / 3 ( A I + 3 / 2 ) and 1 / 3 ( A I = 1/2). Similarly, the matching condition for p is most easily obtained by taking the matrix element of the weak hamiltonian between ds and a one-pion state. Then
pm/2f= (GF/2V~) sin 0c cos Ocmfc_, or
p=1.6×10
I m'tr
I 2m~
I "lGeV
Fig. 2. Plot of )~ and p versus renormalization scale kt.
z.
8c ~ 3 × 1 0
-8 .
(12)
C o m p a r i n g (10)-(12) with (7), we see that they agree to within the errors of the calculation. T h o u g h we cannot take the values too seriously, it is encouraging that they go in the right direction, and have the right magnitude to explain the observed effect. We therefore need no magical en483
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
h a n c e m e n t to e x p l a i n the A I = 1 / 2 rule. W h a t a b o u t p e n g u i n o p e r a t o r s ? T h e coefficients of the p e n g u i n o p e r a t o r s at 1 G e V are - 0 . 1 so they have to be e n h a n c e d b y 200 to explain the A I = 1 / 2 rule. I n the chiral q u a r k theory, the p e n g u i n s induce the same t w o - q u a r k o p e r a t o r s as the other A S = 1 operators. H e n c e the o n l y enh a n c e m e n t of the penguins can c o m e f r o m the m a t c h i n g condition. The c o n v e n t i o n a l v a c u u m insertion a p p r o x i m a t i o n a p p e a r s to give a very large m a t r i x element for the p e n g u i n operator. This a p p r o x i m a t i o n gives a m a t r i x element which is inversely p r o p o r t i o n a l to q u a r k masses. But this piece of the m a t r i x element is certainly n o t present, b e c a u s e it does n o t vanish as mq ~ 0. [ A n y SU(2)L octet SU(3)R singlet A S = 1 o p e r a t o r in the chiral l a g r a n g i a n vanishes linearly as mq ~ 0 since it has at least two derivatives]. Thus the v a c u u m insertion a p p r o x i m a t i o n is n o t evidence for p e n g u i n e n h a n c e m e n t . It is possible that
484
16 August 1984
p e n g u i n o p e r a t o r m a t r i x elements are enhanced, b u t our analysis shows that they c a n n o t d o m i n a t e the A I = 1 / 2 amplitude. If they did, we would get too large an amplitude. Thus we can say n o t h i n g a b o u t c'/E. W e would like to t h a n k H. Georgi a n d J. Polchinski for helpful discussions. This research is s u p p o r t e d in p a r t b y the N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t No. PHY-82-15249.
References [1] K.G. Wilson, Phys. Rev. 179 (1969) 1499. [2] M.K. Gailtard and B.W. Lee, Phys. Rev. Lett. 33 (1974) 108; G. Altarelli and L. Maiani, Phys. Lett. 52B (1974) 351; F. Gilman and M.B. Wise, Phys. Rev. D20 (1979) 2392. [3] A. Manohar and H. Georgi, Nucl. Phys. B234 (1984) 189. [4] J.F. Donoghue, E. Golowich and B.R. Holstein, University of Massachusetts preprint IMHEP-191.