The πN sigma term — an evaluation using staggered fermions

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PROCEEDINGS SUPPLEMENTS

The

Nuclear Physics B (Proc. Suppl.) 34 (1994) 376-378 North-Holland

Sigma term - an evaluation using staggered fermions *t

R. Altmeyer a M. GSckeler b R. Horsley b'~ E. Laermann ° and G. Schierholz ~'b aDeutsches Elektronen-Synchrotron DESY, Notkestraf3e 85, D-22603 Hamburg, G e r m a n y bHSchstleistungsrechenzentrum HLRZ, c/o Forschungszentrum Jiilich, D-52425 Jiilich, G e r m a n y CFakult~t fiir Physik, Universit/~t Bielefeld, D-33501 Bielefeld, G e r m a n y A lattice calculation of the :rN sigma term is described using dynamical staggered fermions. Preliminary results give a sea term comparable in magnitude to the valence term.

1. T H E O R E T I C A L

DISCUSSION

The 7rN sigma term, ~ N , is defined as that part of the mass of the nucleon coming from the expectation value of the up (u) and down (d) quark mass terms in the QCD Hamiltonian,

(r~N = m ( N l ~ u + ddlN),

(1)

where we have taken these quarks to have equal current mass (= m). Other contributions to the nucleon mass come from the chromo-electric and chromo-magnetic gluon pieces and the sea terms due to the s quarks. Experimentally this m a t r i x element has been measured from low energy rrN scattering, [1]. A delicate extrapolation to the chiral limit [2] gives a result for the isospin even amplitude of E / f ~ with E = C%N, from which the 7rN sigma term m a y be found. The precise value obtained this way has been under discussion for m a n y years. For orientation we shall just quote a range of results from later analyses of C%g 56MeV, [3], down to 45MeV, [4]. To estimate valence and sea contributions to ~ N , classical current algebra analyses assume octet dominance and make first order perturbation theory about the SUF(3) flavour symmetric Hamiltonian. This gives

,r~N

.~

m ( N t ~ u + d d - 2gslN)sy.~

de] =

val sea O'rg "{- O'TrN,

2. M E A S U R I N G

(2)

~r. N

We now turn to our lattice calculation. We have generated configurations using dynamical staggered fermions on a 163 × 24 lattice at j3 = 5.35, m -= 0.01 (plus some larger masses), [5]. Practically there are several possibilities open to us for the evaluation of the matrix element. The easiest is simply to differentiate the shift operator $4 giving

mOMN I~

+2m(NlgslNLum *The MTc Collaboration. ~Talk presented by R. Horsley.

where we have first assumed that the nucleon wave function does not change much around the symmetric point. We then subtract and add a strange component. At the symmetric point the u and d quarks each have equal valence and sea part, while the s quark matrix element only has a sea component. Thus in the first term the sea contribution cancels, justifying the definitions given in eq. (2). Using first order perturbation theory for the baryon mass splittings C%g~al m a y be calculated to give Crry ,at ~ 25MeV and a SO O'8TcN r ~-~ 31 "" 20MeV. This in turn means that m s ( N l g s l N ) ~ 400 ,-- 250MeV, which would indicate a sizeable portion of the nucleon mass (938MeV) comes from the strange quark contribution.

= ( N I m ~ x I N ) = Cr=N,

(3)

(the Feynman-Hellmann theorem). Thus we need to measure MN for different masses m and numerically estimate the gradient. This leads to o'~rg

0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSD1 0920-5632(94)00287-6

~

ll.9(8)rnlm=0.01 ~

0.12(i),

(4)

R. Altmeyer et a l . / The 7rN sigma term - an evaluation using staggered fermions

giving (rrN/MN ~ 0.15, which is to be compared with the experimental result of C%N/MN 0.06 ~ 0.05. The numerical result is much larger, but presumably this simply indicates that we have used much too large a quark mass in our simulation (m RGI ~-. 35MeV, [5]). At present we, and everybody else, cannot avoid doing this. We now wish to separately estimate the valence and sea contributions. This is technically more complicated and involves the evaluation of a 3point correlation function, [6], (B = baryon)

377

results. At present we have not done this, but just checked that separate fits give consistent results. In Fig. 1 we show preliminary results for t = 8. 0,25

0.20

0.15

0.10

0.05

O.OC

C(t; -r) = (B(t)m2x(v)Bw (0)>

(5) - .05.

(W = wall source). This m a y be diagrammatically sketched as the sum of two terms:

-.10.

-.15

-.20

-.25 '

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Figure 1. The ratio of the 3-point connected correlation function to the 2-point correlation function against r, with a fit from 2 to 6. (The dotted lines just extend the fit to neighbouring points.) (Up until now we have only evaluated about a quarter of our available configurations.) We find (a) is the connected (or valence) term; the nucleon quark lines are joined with the mass insertion. (b) represents the disconnected (or sea) term. For the connected piece we have fixed t to be 8 or 9 and then evaluated C(t; 7") as a function of v. The appropriate fit function is, with #~ = ~ exp ( - M e ) , (A, ~h = --1 denoting the parity partner to the nucleon, N, ~g = +1),

e,~=N,A

(½T >> t >> r >> 0). AeZ and Me are known from 2-point correlation functions. We see that when = / ? = N we have the m a t r i x element that we require: (NIm2xIN). However there are other terms which complicate the fit:
~°~ tYlr N ~-~ 0.08(2),

(7)

with about the same result for the A matrix element. The cross term is smaller, roughly 0.02. Finally we have a t t e m p t e d to estimate the disconnected term using a stochastic estimator, [7]. (100 sets of Gaussian r a n d o m numbers were used.) One can improve the statistics by summing the 3-point correlation function over T; this is then equivalent to a differentiation of the 2point function with respect to m. This gives

C(t; r) ~ t ~ r

A~<~Irn2xl~>~,

(8)

a=N,A

(½T >> t >> 0). In Fig. 2 we show the disconnected 3-point correlation function. As expected the quality of the data is poor - optimistically we see a slope. A simple linear fit gives a.N'~ ~ 0.05(4).

(9)

378

R. Altmeyer et al./ The 7rN sigma term - an evaluation using staggered fermions

2.5

2,0

1.5

1.0

tttt t -0.~

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ACKNOWLEDGEMENTS

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sible at the present time. This should be the ultimate goal of lattice simulations, as other recent theoretical results, [4], have hinted that perhaps the strange quark content of the nucleon is not as large as supposed, previous results being explained by a combination of factors, such as Z ¢ O'rN and higher order corrections to first order perturbation theory. (Indeed there are already tantalising lattice indications, [12,13].) In conclusion we would just like to say that lattice results at present are generally in qualitative agreement with other theoretical and experimental results. However much improvement is required to be able to make quantitative predictions.

'

24

Figure 2. The ratio for the disconnected correlation function. To improve the signal we have averaged over the points t and T - t.

This work was supported in part by the DFG. The numerical computations were performed on the HLRZ Cray Y-MP in Jiilich. We wish to thank both institutions for their support.

3. D I S C U S S I O N

REFERENCES

We see that roughly eq. (4) is consistent with eqs. (7,9) and that ~,~N/a~N / ~a~ ~ 1.5, which is to be compared with the experimental value of about 2.2 ,-- 1.8. Although we can draw no firm conclusions at present our result tentatively indicates that the valence term is slightly larger than the sea term. Comparing our results with other work obtained using dynamical fermions, [8] uses 2 fiavours of Wilson fermions and finds for the ratio 2 ,-, 3, which indicates a somewhat larger sea component. On the other hand, [9] using results from [10] for 2 staggered flavours finds 1.5 ~ 2.0, while [11] has ,,, 2.0. These, like our result, seem to be lower than for the Wilson fermion case. We would also like to emphasise that although lattices can give a first principle calculation, at present one is not able to do this. Technically the fit formulae that we employ, eqs. (6, 8) are true only for the complete correlation function. The best way to circumvent this problem and others is to make simulations at different strange quark masses; differentiation as in eq. (3) would then give directly r n s ( N l g s l N ) , the strange content of the nucleon. However this calculation is not fea-

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(1991) 1062. 11. C. Bernard, et. al., IUHET-232, 1993. 12. S. Gfisken et. al., Phys. Lett. B212 (1988) 216. 13. R. Sommer, l r - N Newsletter Vol.4/5 (Karlsruhe, 1991).