Physics Letters B 313 ( 1993 ) 197-202 North-Holland PHYSICS LETTERS B

Chiral-symmetry and strange four-quark matrix elements

X i a n g d o n g Ji Center for Theorettcal Physics, Laboratory for Nuclear Sctence and Department of Phystcs, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Recexved 22 February 1993, revised manuscript received 21 June 1993 Editor: H Georgl

We consider the matrix elements of the left-handed flavor-conserving four-quark operators in the nucleon and plon states Using chlral symmetry, we derive relationships among these matrix elements We argue that the A /= ½ rule of hyperon and kaon non- leptonlc weak decay lmphes a possible large strange-quark content m the nucleon and p~on

The nucleon's strange content has received consid- erable attention since the EMC measurement of the proton's spin structure function g~(x) [ 1 ]. Many more deep-inelasnc and (quasi)-elastlc experiments are now in running or have been proposed to make further measurement of strange-quark matrix ele- ments [2]. Study of the nucleon's non-valence de- grees of freedom helps us to understand better the role played by the sea o f Quantum Chromodynam~cs (QCD) . It also provtdes clues for more realistic model-building and for finding better approxima- tions to solve QCD.

In th~s letter we discuss the strange four-quark ma- trix elements in the nucleon and plon states. Our d~s- cusslon is monvated by a recent paper by Kaplan [ 3 ], who showed that the well-known A I = ½ rule o fhype- ron non-loptonic weak decay lmphes a large strange four-quark matrix element, (PlaLyuSL.qL~'UdL[N). Our goal here is to elaborate on his result from the relatlonshtps among four-quark matrix elements im- plied by chxral symmetry, and to show that the van- 1shrug strange-quark matrix elements are incompati- ble with the M = ½ rule. Then we extend our discussion to the plon and argue that accommodauon of both the A / = ½ rule and vanishing strange-quark matrix elements requires a drasncally different valence quark

~" This work is supported in part by funds provtded by the US Department of Energy (DOE) under contract #DE-AC02- 76ER03069

model from what we have constructed. If there ts a trail of plon large strange content here, this may be the first evidence that the plon has an intricate flavor structure.

Let us consider the following tensor o f four-quark operators:

T~ = ~ #,yU( 1 -?5)qjqkYu ( 1 -75) ql, ( 1 )

which will be denoted by (q, qj)(~tkqZ), or simply q, q~q~qz. The radices t, j, k, l run through hght-quark flavors u, d, and s. Color indices, unless specified ex- phcttly, are coupled to singlets m the quark pairs (l, j ) and (k, l). The tensor is symmetric under simula- tneous exchange o f / a n d k, and j and 1, and thus sym- me tnzmg t and k and antl-symmetrizmg j and l, or vice versa, yield a null result. We have therefore a to- tal of 3 4 - 3 × 6 × 2 = 45 independent tensor compo- nents. The symmetrized tensor T~k) ) has 6 X 6 = 3 6

T Dk] components and the antlsymmetrlzed tensor - t i n has 3 × 3 = 9 components. The former contains 27, 8, and 1 representanons of SU ( 3 ) r and the latter 8 and 1 representations.

The construction of the operators belonging to dtf- ferent SU ( 3 ) r representanons lS standard. Here we present the result in order to specify the normaliza- tion we adopt. For 27, we subtract away the trace of the symmetric tensor,

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Volume 313, number 1,2 PHYSICS LETTERS B 26 August 1993

l ( ' T ' ( m k ) t m t k m t J (ml) (~J -~- T{mjl(~l -'~ V{~,I + .T("k) ,-',, 1 m n t k t k + g6T{mnl( 6,61 -.I-a16 , ) . (2)

For the symmetric and ann-symmetric octets we define

O~,A=(q,~T"q,~)(qpqp)+(GT~q~)(qpq,~) , (3)

where the Gell-Mann matrices T a are normalized ac- cording to Tr TAT°= ½~ab. The repeated a en fl in- dices denote summation over solor, and the flavor indices are coupled in each bracket imphcilty. For symmetric and anti-symmetric singlets we define

Ss.A = (q,q~) (qeqa) + (q-qa) (qpq-) - (4)

Any tensor component in eq. (1) can be decom- posed into a sum of operators belonging to these five different representations. For dmgonal hadron ma- trix elements, we are interested in operators with no net flavor change. There are nine of them: auau, dddd, #sgs, audd, augs, dd~s, addu, asgu, and ds~d. Their decomposit ion into different representations is straightforward,

- - - u u 2 3 2 uuuu= T"~ + 3Os + ~ 08 + ~Ss '

aada= - ~ ~o~ + 5 - ~ o~ + ~s~ Taa- 5

gsgs= -'~ 5 - ~ Og + ± S T s s - 12 s

½ ( audd + addu ) - - ~a l - T u~ + g - ~ o l + h & ,

1 ½ (ddgs+dsgd) -a~ 08 + ~Ss = r ~s - ~o O ~ - T d - ~

1 ½(ssau+suas)- - s ~ + ~ o ~ - o~+~Ss - T~u ~ ,

1 ½ (auda-aadu) = ~ o~ + ~ & ,

1 (aaes-asea) = - h o ~ - ~ ox + ~SA,

1 ½(esau-suas)=~o~- ~--~ o~+fi&. (5)

Likewise, we can decompose the As= 1 non-leptonlc weak decay hamiltonian density,

GF Yi'-= ~ &c, 4(du)(as) . (6)

However, this hamiltonian density is defined at the scale of Mw, the mass of the W boson. Since we are interested in the non-perturbative part of the four- quark matrix elements, we have to run down the scale using a renormalizatlon group equation. The calcu- lation is standard and a recent reference shows [ 4]

x~22 ( -au +0 '2806+ '7 ,~(/~=1 G e V ) = - - s ~ c ~ 2.8T,s

-- 3 .640 6+17 + 0.003Qs --0.01 Q6 ) . (7)

where

Q5 = d~Tu( 1-75)s~qpTu( 1 + 7s)@,

Q6 = d~Yu( 1 - 75)s~q~yu( 1 +Ys)q~, (8)

are generated from the penqum diagram and are in- teresting due to their distinct chiral structure.

We now consider the matrix elements of those fla- vor-conserving operators in the nucleon and pion states. [Since the operators are scale-dependent, we assume to work at the scale of 1 GeV. ] To do that, we first map these quark operators onto operators con- raining baryon and meson fields with the same flavor symmetry. We use the non-linear representations for mesons and baryons employed in ref. [ 5 ]: the Gold- stone bosons octet is represented by a 3 × 3 matrix.

£=exp( 2in/f,d , (9)

where

1 ~=~

r 1 1 It°+ ~+

x/2 ~ "

n - -- n ° + ~ r /

K - K °

and the baryon octet is represented by

K +

K o

2

(10)

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Volume 313, number 1,2 PHYSICS LETTERS B 26 August 1993

B=

22 S ° + ~66 A X+

1 s

P

N

2

(11)

Under chlral transformation, we have

Z--, LZR * ,

¢=v/Z--, L~U*= U~R* ,

B--, UBU*. (12)

In constructing effective operators, we keep only leading terms m chlral perturbatton expansion. For 27, we have

7 TM ~)) ---, a B ~ ) ) , ( 13 )

where the tensor B is

B(,k) = ((fl~t)j (?,B~t) ~ ' (14) Ol)

and for 8's

O~,A---~Fs,ATr B[~*T"~, B]

+ Ds,ATr/~[~*T~, B] + , ( 15 )

and finally for l 's

SS,A ~ & a T r / ~ B . ( 16 )

The seven parameters, a, FS.A, Ds,A, and Ss, a deter- mine all matrix elements of the four-quark operators between the baryon octet states plus an arbitrary number of Goldstone bosons.

Using the above mapping, we calculate the matrix elements o f the flavor-conserving operators between the proton states,

( PI ~ u a u l e )

= - 3 a + ½ ( F s + D s ) + ~ ( 3 F s - D s ) +~2ss,

( PlddaTd[P)

' ( Fs + Ds) + ~ ( 3 F s - D s ) + ~ & , = 2 ~ a - - 3

( PlgsgslP) = - 3 a - ~ (3Fs - D s ) + ~ s , , (17)

( P I ½ ( audd + addu ) I e )

= - ~ o a + ~ o ( 3 F s - D s ) + ~ a S s ,

( PI ½ (ddgs+ dsyd) I P )

= - + o a - ~o (Fs + O s ) - {o ( 3 F s - D s ) + f i s s ,

( P I ½ ( Ysau + euas ) I e )

- 4~a+ ~ (Fs + D s ) - ~o (3Fs - D s ) + hs~,

(PI ½ (auc ld -adc lu ) IP) = ~ (3FA --DA) + ~Sa,

( PI ½ (ddgs-dsYd) IP)

=--¼(FA + D A ) - - ~ ( 3 F A - - D A ) + ~ S a ,

( P[ ½ ( gsau-~uas) IP)

=¼(FA+DA)--~(3FA - - D A ) + ~2S a .

(17 cont 'd)

The coefficients in front of the lnvanant parameters are related to the SU(3 ) Clebsch-Gordan coefficients.

I f we assume that five stange quark matrix ele- ments in eq. ( 1 7 ) vanish, we Immediately derive five relations among seven lnvarlant parameters,

FA = --DA = I S a ,

a = -- ½s s ,

F s = ~ S s , Ds=¼Ss (18)

Here we have taken s~ and s~ as independent. The rest four non-strange matrix elements can be expressed in terms of these two parameters,

( P l a u a u l P ) = ~s~ ,

( P l d d d d l e ) = 0 ,

( P I ½ ( a u d d - addu ) I e ) = JSa,

( PI ½ ( audd+ addu) IP) = ~s~. (19)

The vanishing of the four-d-quark matrix element is a little suprising, but it can be simply interpreted in valence quark models in which there is only one d quark and thus the two-body matrix element must vanish. However, our result is independent of va- lence quark models and is directly linked to the SU(3) symmetry and vanishing strange-quark ma- trix elements. The relation between the four-u-ma- trix element and the ud-summetric matrix element (the last one m eq (19) ) ~s simply a consequence of isospin symmetry.

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We argue, however, that the pattern of four-quark matrix elements shown in eqs. (18) and (19) is in- consistent with the data on As= 1 hyperon non-lep- tonic decay. To show this, we map the hamiltonlan in eq. (7) onto an operator with meson and baryon fields and use it to calculate the hyperon decay rates. Here we neglect the contributions of the penqum op- erators because of their small coefficients (a more careful analysis was made in ref. [ 3 ] ). From fitting to experimental data [ 6 ], we have

2 m,f,~ 0 . 0 7 F s - 0.91FA = 1 . 4 1 / - - ,

2Sl C1

0 . 0 7 D s - 0.91DA = - 0.58~/m~f~ (20) 2Sl C 1 '

and a is negligible. Here r/is a phase factor, and time- reversal symmetry restricts r/to + 1. From eq. (18), the size o f a restricts Fs and Ds to be small, i .e, the symmetric octet is also strongly suppressed if all the strange quark matrix elements vanish Then from eq. (20), we find the ratio between FA and DA IS --2.4, which contradicts with F A / D A = - 1, which IS im- plied by eq. ( 18 ).

Thus some strange matrix elements must be large. From eq. (17) and the fact that a is small, we have

-~ ( f s + D s ) = ( r [ ( a u - d d ) ~ s + ( a s g u - d s d u ) ] P ) ,

( F A + D ,, ) = < P I ( au - d d ) es - ( aseu - dsdu ) I e ) • (21)

Combining these with eq. (20), we have

(PI - 0 . 5 6 ( a u - d d ) g s + 1.26( a s g u - d s g d ) IP )

=0.82q m~f~ (22) 2Sl C 1 '

which is the matrix element obtained by Kaplan [ 3 ], except for an isospln rotation.

In the rest o f the letter, we show that the above hne of discussion can be extended to the pion's matrix elements, although some arguments for the conclu- sion are less tight. In the Goldstone boson sector, we have the following mapping for the four quark matrix elements:

T,,k) _ f ~ 6,) --+a ~--~=/~{J~, ) , (23/

where the tensor P~)) IS defined as

p(,k) ( X O u X , ) { ( X O u X , ) ~ (24) 01 ) =

and

f~ O~,A --~ b~,a ~ Tr [ T ~ OuXO uX* ] , f r l

Ss f 2 A--~Ss a ..-77 Tr[0v Z0uZ* ] , (25) ' ' m ~

where we introduce five invariant parameters, a, bs.a, and ss,. Using these, we obtain the following four- quark matrix elements o f the pion:

( n ° ] auau[ n ° ) = - A a + ~ob~ + As~ ,

< =° I,sesl ~° > = - £ a - , ~ b ~ + , q s ~ ,

1 3 - - - 1 L _ [ _ ~ 4 S s , (no I l ( a u d d + addu ) I n ° ) = ~ u - r ~v~

( n°1½ ( auxs + aseu ) l n° ) = l a - li~b~ + ~4s~

(n°1½ ( a u d d - a d d u ) ln ° ) = l ba + hsa ,

( n o [ ½ ( a u e s - a s g u ) L n ° ) = - ~ 4 b a + h S a . (26)

The matrix elements of three other operators, dddd, dd~s, and dsgd, are redundant because of lSOspin symmetry.

If we assume all strange quark matrix elements vanish, we derive three relations among five invar- iant parameters,

ba=2S a b~=-a=2s~. (27)

Three non-strange matrix elements are

( 7~° l auau l n° ) = ½ ss ,

(n°1½ ( audd+ addu ) l n ° ) = - ~s~ ,

(no 1½ ( a u d d - addu ) l n° ) = ] sa . (28)

The first two matrix elements are again related by isospin rotation.

We now argue that eqs. (27) and (28) and the k I = ½ rule imply either a nonconventional valence quark model or a large strange content in the pion. From the neutral and charged K non-leptonic decay, we can extract the strengths o f the 27 and 8 [5 ]:

a = 1.4× 10-sq GeV 4 ,

0 .07b~- 0.91ba = 2 . 1 4 × 10-4// ' GeV 4 . (29)

To be more precise, the second equation above shall include the penquin contributions which are poten-

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tially important in the meson due to the speceal chiral structure o f Q5 and 0 6 [ 7 ]. Since their precise size at 1 GeV is uncertain, we neglect It temporarily. Cou- pling eq. (29) with eq. (27) we deduce

ss= - 0 . 7 × 10-s t /GeV 4 ,

sa = - 1.2X 10 - 4 r ] ' G e V 4 , (30)

and so ISa/Ssl ~ 17. Or, in terms of a ratio of non- strange matrix elements,

( nI addul n ) - 1 . (31)

X= ( nlauddln)

One way to estimate the penguin contributions is to consider a chlral theory m the large Arc (the num- ber of colors) limit. It was shown in ref. [ 8 ] that the corresponding effectxve operator for Q6 is

~f~ ) 4 m 2 m 2 f : f : Tr[T"OuSOuX*] (32) -- -- 1 m~ rn~

This contributes a term of 5.3 X 10-6GeV4 to the left- hand side of the second equation in (29). This is clearly too small to bring b~ and ba to a same size. It is known that the large Nc method is not always reli- able [ 8 ]nowever, for the penguin to explain the A / = ½ rule, the realisuc penguin matrix element must be 40 times the large Nc result. A recent lattice calculauon shows that the large Arc estimate is correct in order o f magmtude at least for the K- , n matrix element [ 9 ]. In light o f this, we take eq. (31) as qualitatxvely true under the assumpuon about the strnage matrix elements.

This result, however, contradicts various valence- quark models for the pion. For instance, in the non- relativistic quark model, we have,

Z = - ~ . ( 3 3 )

The same result can also be obtained in the vacuum lnsertton approximation, in which the matrix ele- ment of four-quark operators are calculated by in- serting a physical vacuum m the middle [7]. In the MIT bag model, we find

1 ; ( j~ + j ~ - ~o,~'~ '~, Z= 3 f(]o2 +j~)2 , (34)

where Jo and j~ are upper and lower components of the bag wave function. Clearly, for anyjo andj~, g is larger than zero, but smaller than ~. Thus either dif-

ferent four-quark operators m these models acquire different renormahzatlon constants, or the strange degrees o f freedom must be added exphcitly, or both.

If the above discrepancy implies a large strange content of the pton, it is the first such evidence. To support this claim, let us consider the matrix ele- ments of a few famdar strange quark operators in the plon. The matrix elements o f type ( 0 lYFs[ n °) clearly vanish due to lsospin symmetry. The matrix ele- ments of type (n°lYFs [ n °) vanish exactly except for F = I . However, the chlral symmetry predicts (n°lgsl n°) =0 up to high-order terms in the chlral expansion. The argument goes like this: using the first- order perturbation theory we calculate the masses of Goldstone bosons, for instance [ 10 ],

m,=2 (nlmuau+mddd+msgS]n) (35)

On the other hand, we can calculate these masses from the axial current two-point function:

2 1 mr = - -;s( mu + rod) (01 ~U] 0 ) . J ~

(36)

Matching these two results, we have

<n°lgsln°> =0 . (37)

Thus, it seems difficult to find evidence that the strange quark content is large in the pion.

The A I = ½ rule is a solid experimental fact. To translate this mto a statement about the strange con- tent of the nucleon and plon ~s not entirely straight- forward. The contributions of penguin operators, particularly in the meson sector, must be calculated in a more reliable way, although they seem negligible at the scale we consider. Our analysis in the effective theory is made only at the tree order in chiral pertur- bation, and higher order corrections could be large desptte the folklore that they are typically at the 30% level. Furthermore, m the plon case we do not know the real value o f z and the available methods for eval- uating it do not explain the A I = ½ rule themselves. Finally, I should emphasize the scale dependence of the strange content. It ~s possible that large strange matrix elements at the scale of 1 GeV become negh- glble at a scale, say, 0.2 GeV. However, besides the untrustworthy perturbation evolution, no one knows the scale dependence of hadron matrix elements at low energy.

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Volume 313, number 1,2 PHYSICS LETTERS B 26 August 1993

I thank A. Manohar and S. Sharpe for discussions on the penguin contributions.

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