Volume 212, number 2 PHYSICS LETTERS B 22 September 1988

M A S S - S P L I T T I N G S IN T H E BARYON O C T E T A N D T H E N U C L E O N e - T E R M I N LATTICE QCD ~

S, G O S K E N , K. SCHILLING, R. S O M M E R Physics Department, University of Wuppertal, l)-5600 Wupperta[ t, Fed. Rep. Germany

K.-H. M U T T E R ~ and A. PATEL Theory Division, CERN, C[-[- 1211 Geneva 23, Swi*:zerland

Received 16 May ~988; revised manuscript received 14 July 1988

We calculate in the valence approximation the F and D parameters governing the mass-split*ings in the ba..~,on octe1. The use of an improved fermion action enables us to go down in quark mass to the SU (3) symmetric point. The F parame*er is consistent with its experimental value. On the other hand, the D parameter turns out to be large and shows a strong dependence on the quark mass. This is evidence that ~he Z-A mass-splitting is not a first-order SU (3) fiavour symmet~ breaking effect. We also estimate the contribution of the valence quarks to the nucleon 6-~erm.

The program of calculating static hadron proper- ties via the s imulat ion o f latt ice Q C D has been in- tensely pursued in recent years. Fol lowing the s tandard de te rmina t ion of hadron masses from two- poin t correlators, the natura! next step is to measure three-point correla t ion functions and to extract had- ronic matr ix elements therefrom. The matr ix ele- ments of scalar densit ies provide a s imple example o f this type. They te!l us how the hadron masses shift as the bare quark mass is change& In case o f the baryon octet, there are three such parameters character is ing the mass shifts. The F and D parameters control the strengths of the f lavour SU (3) breaking effects within the baryon octet, while the nucleon c- term measures the effect of changing the mean quark mass. a ap- pears in the analysis of N o n - n u c l e o n scattering da ta as weil, and has been a subject o f much controversy lately [ I ].

In a recent paper [ 2 ], Maiani , Mart inel i i , Paciello and Tagi iemi have per formed a first computa t ion o f F, D and er wi thin the lat t ice Q C D f ramework with

Work supported in part by Deutsche Forschungsgemeinschaft grant Sehi 257/t-2.

i On 1cave of absence from Physics Department, University of Wuppertal, D-5600 Wuppertal, Fed. Rep. Germany.

2 i 6

Wilson fermions. Their results Iook fairly encourag- ing as far as the D / F ratio is concerned, whereas the absolute size o f the indiv idual terms differs by about a factor o f two f rom experiment . R should be men- t ioned, however, that these first es t imates were ob- ta ined at rather large quark masses, corresponding to the large pion to rho mass-ratios (0.89, 0.84, 0.73) ~.

It has turned out that with the s tandard Wilson ac- tion, it is very difficult to get away f rom the static l imit into the chiral domain . This s i tuat ion ca..-', be improved by using effective fermion act ions which are der ived from block-diagonal izat ion [3] . We found in our previous spect rum calculat ions [ 4 ] that in the blocking schemes with scale 2 and v ,~ geome* tries one can approach the chiral 1trait as ciose as M J .9/0=0.45. This mot iva ted us to extend the compu- tat ions ofrefo [ 2 ] to smaller quark masses.

Throughout , we assume exact isospin symmet ry and fNlow the nota t ion of ref. [ 2]. To begin with, let us recall the mass term., in the cont inuum hamiltonian:

m = rn,, = t a d , ~ = ½ ( 2 m + m ~ ) . (1)

~ We thank G. Martineiii for their raw data on :iG/M~.

Assuming first-order effects of the SU(3) flavour symmetry breaking, the mass-splittings within the baryon octet are conventionally parameterized in terms of two reduced matrix elements (evaluated in the SU (3) symmetric limit), F and D:

(B,, i ~2j ~v I B~ ) = F T r ( B ~ [2~, B2] )

+ D Tr (BI {,~.,, B2}). (2)

The quantities F and D are related to the matrix ele- ments of scalar densities according to

F = ( A - C ) / 2 , D = ( A - 2 B + C ) / 2 ,

A= ( p l f m l p ) , B = ( p l d d l p } ,

C= ( p j s s l p ) • (3)

The nucleon (r-term can be written in terms of these matrix elements as

(r= m( A + B ) = m( 3 F - D + 2C)

= ~.~ +2inC. (4)

From the masses of the baryon octet, one estimates

D/F~ - 0 . 3 2 ,

eva1 ~/N 3MN(Mg-MN) M~, - 2(M~c-M~) ~1.32, (5)

where M8 is the average mass of the baryon octet. We compute the physical quantities involving sca-

lar densities using the operator structure mq~/Omq so as to avoid renormalization Z-factors. For our choice of lattice action [4], we define the Wilson quark mass as

mq = in[ I + } ( l / r e - I/x~) ] . (6)

Following ref. [2 ], we extract the matrix element of a scalar density sandwiched between proton states from

E~ ( ~ (x, t) [ E_Xiq (z) I ~ (o, o) ) R , ( t ) = 2 ~ ( ~ ( x , t ) l ~ ( 0 , 0 ) >

t !arge ~ ' const. + f (P lClq lp) t . (7)

Here the LHS denotes the ratio between a three-point correlation function and the proton propagator,

PHYSICS LETTERS B 22 September t 988

I~) {hi

Fig. 1. The two different contractions of tile three-point correla- tion function appearing in the scalar density matrix element of a baryon. The second diagram is understood to mean only the con- nected part of the correlation function.

whereas ~ denotes the proton operator. The extra factor in front of the matrix element, f - 8rnq/O( 1/x), converts our lattice results to the con- tinuum normatisation.

The two contractions that occur in the three-point correIation function are shown in fig. I. Only the first graph given there is relevant for the computation of the non-singlet quantities F and D. The second graph contributes to a, but we drop it ( C = 0) in the spirit of the valence approximation and arrive at (rva,, ~2

The present computation is based on the 47 quenched configurations at fl= 6.3 on a 243;< 48 lat- tice that we used previously to estimate the hadron spectrum [4]. The background gauge fields were blocked twice with ,~/3 geometry. The quark propa- gators were evaluated on the resulting blocked con- figurations of size 83)<16 with the improved fermionic action and the finite-size reduction tech- nique described in ref. [4]. The three-point correIa- tions appearing in eq. (7) were computed with the method explained in ref. [ 2]. For the numerical in- versions, we used the conjugate residual and (when necessary) the conjugate gradient aigorithms. The values of the hopping parameter ~: and the number of inversions at each of them are compiled in table 1. These values were selected in order to cover the range M,,/Moe [0.45, 0.98].

In figs. 2 and 3 we show the "raw data" for the ra- tios Ru and R~-2R,~ for x values of 0.1345 and 0.1360. The figures demonstrate the quality of the signal as given by the linear behaviour of the data in the following ranges:

4 < t < 7 for the proton,

ll-<<t~<13 for the N * ( ½ - ) .

~2 The discrepancy between a extracted from pion-nucleon scat- tering data and a,,~l of eq. (5) leaves open the possibility that the contribution we have dropped, ~r-cr~,~, may be sizeable.

Volume 212, number 2 PHYSICS LETTERS B 22 September !988

Tab!e i The number of quark propagators at various ~ values in our data sample. Also iisted are the N*(½-) masses in !attiee units and the ?Gv,dMN mass-ratios.

*c Number of MN*a M N....~* propagators MN

0.1270 5 3,57(11) 1.i9(5) 0.1290 5 3.24(15) 1.23(8) 0.I345 46 2.13(5) i.44(3) 0.13525 46 2.01(4) 1.58(5)

1.90(27) 1,88(27) 0.1360 94 0.1362 94 -- -

O u r errors were es t imated with the j ackkni fe method .

T.he ma t r ix e l emen t s o f the scalar densi t ies in b o t h

the ½ + and ½ - oc te ts (cf, eq, (3 ) ) are ex t rac ted f r o m

the assoc ia ted siopes. T h e y are shown in fig. 4 as a

f unc t i on o f ~;Sq. N o t e that the b e h a v i o u r o f f and D is

d i s t inc t ly d i f fe ren t in the regions o f Iarge and smal l

qua rk masses. Both F and D increase in abso lu te

m a g n i t u d e at smai ie r qua rk masses, D m u c h m o r e

rapid ly than F, T h e D/F ra t io is p lo t t ed in fig, 5 as a

24 r | ® R~

20 ~ ~ R,~ - 2 R~

20

.o i og~-

E

- I 0 -

-2O -

- 5 0

- 4 0

• R.

- i

2 4. 5 8 ~ o - - - q ~ - - T ~

Fig. 3. Same as fig. 2 but at ;<=0.!360.

8

4

o o

0 ,,,Lz~ ~ -

e

2 4 6 8 t 0 12 14

Fig. 2. The ratios Ru and R..,-2Ra at a:..=0.1345. The two fits cor- respond to the ½ + and ~ - bad/on octet states.

E i f unc t i on o f M,~/Mo toge ther wi th the p rev ious results

i o f refo [ 2 ], F o r c o m p a r i s o n we h a v e also i nc luded the

qua rk m o d e l p red ic t ion [ 5 ] (wi th m ~ b e i n g the con- s t i tuent qua rk mass ) :

F o n d D m a t r i x e l e m e n t s

kt >> i '~ < p i O u t p >

- 4 t - I o <N°t Ou IN'> -", <p i ~u - 2 Ed

=6 A <Nol Ou - 2 ~d IN"

I __ . .L .... ~ . I 0.1 0.2 0.3 0.4 0.5 0.8

Nq, q

Fig. 4, The Fond D matrix dements of the ~+ and ½- ba.,~'on octets as functions of the !attice bare quark mass. The a~ow marks the position of the SU(3) symmetric point.

Volume 212, number 2 PHYSICS LETTERS B 22 September 1988

NUCLEON SCALAR DENSITY RATIOS

o8i ,hi .... k ,o -~ +~o quark rnodei eq. 7

0.6 b 0.5 0.8

o.4

, , ' IT+

° i o.2 °.3 oI~ o.! °.8 °.7 °.8 0.g ~. M , , / M~

Fig. 5. The D/F ratio as a function of MJMp. The insert shows the same data on a log scale. The arrow marks the position of the SU (3) symmetric point.

D / F = - 3x / (2 + x ) ,

x = (223 MeV/merf) 3 . (8 )

The figures show that our signals are good enough to bring us down to the S U ( 3 ) symmetr ic poin t g % F and consequent ly the E - N mass-spli t t ing can be es- t imated, and they agree with their exper imenta l val- ues, Both D and D / F turn out to be too large - by

~3 We estimated ~ in two ways: using/~, 9 and K masses gives ~= 0.136 i 7, while r~, N and K masses produce Je= 0.13605. Our -value for the critical hopping parameter is g~=0.13650.

roughly a factor of two - compared to the values in- ferred from the baryon octet masses (c£ eq. ( 5 ) ) . However , the rap id var ia t ion of D in the range mq < m~ implies that the Z - A mass-spli t t ing cannot be t reated as a f i rs t-order SU (3) f lavour symmetry breaking effect. A realistic es t imate o f this mass-spli t- ting would therefore necessitate combining quark propagators o f unequal masses.

Fig. 4 also indicates that the D and F matr ix ele- ments in the ½ - octet are smaller than their counter- parts in the ½ + octet. However, the quali ty of the da ta is not good enough to allow for a predic t ion at the

219

Volume 212, number 2 22 September t988

NUCLEON SIGMA TERM

\ - i

# J

,,.2 L

J

J

® ref° 2

- this work

~x derived from exp-

.d O

J

f i o. a i l L ' ~ i

0.2 0.4 0.6 018

M~/ M~ Fig. 6. The value of o,~IMN/M{ as a function of M,ffMp. The result in eq. ( 5 ) is pierced at the SU (2) symmetric point.

Our results in the range 0,13 58 < ~c< 0,1362 [ 4 ! yield (A + B ) ~ 5 or a~,~.>~,4N/M~ ~ 1.2, a little smaller than but nov inconsistent with the calcu!ation reproved here,

Both in figs. 5 and 6 we note a substantial decrease in the statist ical errors with respect to the ones ofref . [2 i . We stress that only a minor part of this error reduct ion is due to the present increase in statistics and the finite°size reduct ion technique. The major improvemen t comes from a bet ter overlap between our blocked proton operators and tk,,e physical pro° ton wave-funct ion (by a factor of the order i 00) .

To summarize , we have calculated F, D and a~,:~ down to the point MJM~= 0.45° Regarding F u n d D, this vatue is sma!t enough to avoid the nuisance of extrapola t ions from large quark masses. Ceterum censemus, the s imulat ion should be repeated with dynamica l quarks.

We are grateful to Ph° de Forc rand for allowing us to use his configurations. The computa t ions were per formed on the CRAY XMP-48 of tk, e H L R Z in Jfilich, We apprecia te the great suppo.~ and the efiq- ciency o f the staff o f finis inst i tut ion.

References

SU (3) symmetr ic point . For further reference, we have included in table 1 the measured N * ( ~ - ) masses, de te rmined f rom fits in the range i 0 < t ~< 12.

The Mr,.,/MN mass-rat io extrapolates to approxio mutely the correct value.

Finally, we show in fig, 6 our results for the nu- cleon ~-term plot ted in the combina t ion a,.~MN/M~, which is more suitable for extrapo!at ion to the chiral limit, Our data is consistent wi.:h the result in eq, (5) , with the possibi l i ty o f a rise at small quark masses. Assuming (A + B) to be approx imate ly independen t of the quark mass close to the chirai l imit , ~ can also be ob ta ined from the slope o f MN versus i /~c

[ I ] j. Denoghue and C. Nappi, Phys. Lett. B !68 (1986) !05; V. Bernard, R. Jaffe and U. Meissner, M~T preprint MIT- CTP-1547 (I987).

[2] L, M#.ani, G. Martinelli, M. Paciell.o and B. Taglienti, Nuc!. Phys. B 293 (1987) 420.

[3] K.-H. Mt~tter and K. Schii[ing, NucL Phys. B 230 [FS!0j (i984) 275; Ph. de Forcrand, A. K6nig, K.-H. Mfitter, K. Sci~iiiing and R. Sommer, in: Lattice gauge theory '86, eds. H. Satz, L Harfity and J. Potvia (P!enum, New York, i987).

[4] Ph. de Forcrand, R. Gusts, S~ Gasken, K.-H. Mfitter, A. Patei, K. Schilling and R. Sommer, Phys. Lett. B 200 (1988) 143; R. Sommer Ph de Forcrand, S ~. Giisken K.-H. Matter, A, Pate1, K. Schilling and R. Gupta, in: Fie!d theo~- or,. lattice: LAT87, eds. A. Billoire et aI., Nuc!. Phys. B (Prec. Supp!. ) 4 (1988) !47.

[5] S~ One, Phys. Roy. D I7 (i978) 888.