Hadron masses from the MTc collaboration

3SO

Nuclear Physics B (Proc, Suppl.) 20 ~t,~J]} :tV~-3~:l

HADRON MASSES FROM THE MT: COLLABORATION E. LAERMANN HSchstleistungsrechenzentrum c/o Forschungszentrum JGlich, D-5170 JSlich, FRG with R.ALTMEYER, K.D.BORN, M.GOCKELER, R_HORSLEY, WJBES, T.F.VVALSH and P M Z~,W~S (MT~ Collaboration) Preliminary results of the MT~ co|laboration on the hadron spectrum in full QCD 'w~th four ~ax~orsof st~gge~e~ fermions are presented. Tb.e calculation is being carried out on a ]6 s × 24 ]~ttice at 3 = ~.L~ a~d 5.~-~3~[or a quark mass ma = 0.01. using a Hybrid Monte Carlo algorithm. Some emphasis is p~t on Che i~vestigat~D~ of flavor symmetry restoration. The results ~re compared with a quenched ana|ysis at 3 ~a~ues beb~-en 5.7 and 6.0, a range where the lattice spacing roughly corresponds to the unquenched simu|ati~n. INTRODUCTION In this report preliminary results on the hadron spectrum in full QCD with four flavors o f staggered fermions are presented. This work is part o f the ~4T~ project in which hadron masses and finite temperature effects are being calculated at the same fl and m values on ]63 × 24 and 163 × 8 lattices, l ' 2 The simulation at the critical temperature for N~ ~ 8, ~ = 5.15, is supplemented by an analysis o f the hadron spectrum at a ~ value o f 5-35 somewhat deeper ~n the continuum region. In all cases the quark mass is fixed at m = 0.01 in lattic~ unit~ Some emphasis is put on the investigation o f flavor symmetry restoration by ;aalyzing non-local ~- and p operators in addition to the local fields. The hadron masses in full QCD are then cornpared to the results o f a rough quenched analysis at values between 5.7 and 6.0 where data has been scarce so far. In this range o f ~ values the quenched lattice spacings are expected to roughly coincide with the spacings in current ~mulations o f full QCD. A comparison o f results from full QCD simulations with quenched ones at about the same physical parameters may reveal the effects of including dynamical quarks. This comparison seems essential especially in order to assess the problem of screening in the static quark potential. The potential as well as the quenched simulation will be presented in some detail in the talk by K.D.Born 3. 1.

2_ SOME ?,SPECTS OF THE SIMULATION We have chose~ to work w~th four f.avc~rs~f staggeJed fenm~ons because of theoretical dart~o h;gher s e n s i t i v ~ t o the effects of dynamica| quarks and the opportunity to employ ~n exact algorithm, the Hybrid Monte Carlo4" 5 At our lattice size, if; 3 × 2t, and the quark m~L~ o f raa = 0.0] we are forced to a time step size o f d r = 0 - ~ 1 to obtain reasonable accep tance rates o f ~ 70~. "Nrth 75 molecular dynamics steps, a trajectory is 0.3 t i m e units long, where our normalization coincides with that o f e.g. the LANL group 6 and is 1 / ~ 2 times that o f ref.5. We have been fairly conservative in selecting the stopping criter~n in the conjugate gradient inversion solver, settling for (.lftAfX _ <~)2/~2 = 10-zo after a series o f tests before the production runs2. The program has been optimized for the CRAY Y-MP8/832 at the HLRZ JGlich leading to a 1 processor performance o f 237 MFLOPS and a speedup o f 7.0 on 8 CPUs7. Since we started our runs at ma = 0.01 in April (this year) we have accumulated about 6000 CPU hours so that the present results must be considered preliminary. A considerable amount o f time has gone into therrealizing at #c- Starting from an equilibrated configuration at ( ~ , m a ) = (5.35 / 0.025) it took about 400 trajectories to reach a stable value in the plaquette, Fig.l, as well as in the quark condensate. In particular the last 2 % increase in the plaquette variable seems to be gained very slowly. Correspondingly, the equilibrated plaquette exhibits a correlated behavior in the form of waves < 200 trajectories long. As the number

0920-5632/91/$3.50 (~) Elsevier Science Publishers B.V. (North-Holland)

381

E. Laermann et a I . / tiadron masses from the 3IT¢ collaboration

I l i l l l i l l l l

q 0

i

.

~J

. . . . .

d

r O

l

2

3

4

6

~

7

B

0

10 11 I~

T-rain

F~ure I: Time e~mlatim~of the ptaglUette at ;J= 5.15. state) ~ t o o f get,rated tra~c~_~]es is smal~ a c~Telatio, a n a ~ sis sees o , l y the high fieque~-y compommt or I ~ h about 10. The stochastic estimate* for the quark condensate does not reproduce ~ prommaced character, it requires IKmever a s~rm'iady ~ themml~zadon time. At ~ = 5~3~ we have s~rtnd from e a d ~ pseuclofernfion runs at the same set o f p a r a m e t e r . Quark and espedally the plaquette seem t o remain stable at the PF values. So far we could analyze about 40 o f the ct:mfigurations ~ored on tape aL 5.15, and 15 at 5.35. These configurations were separated by 10 trajectories and, jud~ng from the plaquette are presumaBly not decorrelated. Apart from blocking confl~urat~ns at 5.15 we v~re not yet able t o carry out the detailed statistical error analys~s we'd wish for. H o ~ v e r in the hLsto~ o f the pion propagator at various time separations we did not detect long waves extending over 200 trajectories (20 configurations) as in the plaquette but observe a corrdation over 4 to 6 configurations. Keeping the not yet satisfactory status in mind we proceed to present our resuRs on the hadron masses. Quite clearly, our quoted errors reflect rather the attempt to estimate systematic errors than representing well founded statistical ones.

3.

t i e ~ mo~r:,at~ a~ T ~ +

~/~.

LOCAL HADRONS

In o m ~ s t romd o r auatys~ ~ ~ com:e~ tra'~d em the metal local operatm5 for = , p and nedeoe, la order to e x t r ~ t the ~ from ~ c o m b t m e f u m t m e s we ~ ~ ~o st~ (~ We Mted the fall protmgators~m~l~ome or. ~J~mum dine seeamd~ from ~

sm~c~ (i) Alter-

of the ~ were ~ e d im zero ~ f~s to o1~ tam dFedSve masses at m c . - ~ ti~e ~ . The t ~ o apl~aches are compared m -------------------~ for ~ e (ndm~tedly 5est) ~ of" the ~ a t 5.15. ~ t h ¢ 5 ~ sequence flatteos o f f akeady at small dine sep~r a t ~ 5ecause these 5ts t a ~ a c ~ o u t ~ the data at large t, both methods e s ~ l l y appr~c5 the same plateau values. Our quoted errors t l y to ~ u t for the fluctuations in the plateaux. At fl = 5.15 these approaches work weft for the z and So some e~ent for the nudeon. The local p propagator /s rather noisy in the center and already in that respect hehaves very similar to the quenched results at/3 = 5.7 (see also Ref.lO). W~- therefore const:~ned the fits to the flanks o f the propagator so that at least a dear upper bound on the p mass can be derived. At 5.35 the statistical quality o f the data is much

E. Laermann et al./Had~gn masses from the M T , co]]aSoration

382

better despite the smaller number of configuration~ analyzed. Again, this conforms with the experiences drawn from quenched simulations. At this /~ value effective mass plots can be generated for all chanr,els. The most important feature appears in the p 5.15

5.35

0.303(2) I 0.255(5)

iN

0.g3(5)

0.00(5)

1.20(4)

1.05(5)

Table 1: Masses from local operators channel in which a light state emerges at large time separations. VVhether this state has to he identified with the genuine p or perhaps a ~r~ state with large (negative) final state interaction energy =can not be decided .lt the present time (see also ref.8). At 5.15 the data is too noisy at large t as to help resolve this alternative. 4.

NON-LOCAL HADRONS AND FLAVOR SYMMETRY In addition to the local operators we analyzed correlation functions o f all operalors in which quark and anti-quark are separated by one link. These fall into three classes: (i) One set o f NLT operators, aon-kmai in time, is given generically (gauge links omitted) by

o ( ~ , 0 ~ ~ ( ~ ,t ) x ( ~ , ~ + 1) - ~ ( ~ , t + 1)x(~,t) These operators project onto the same states as local ones. Correlation functions can therefore be restricted to mixtures o f local and N I T operators. The advantage of those o~rrelators is the dynamical suppression of the contributions from parity partners11 so that in the ~/¢z~ channel, for example, the p pattide is much easier to isolate. Particularly interesting is the ~/0 ++ channel which for local operators is completely dominated by a 0 ++ state. Only the LT-NLT correlation can project out the 7r state. (ii) A different set o f NLT operators, generically

o ( ~ , 0 ~ ~(~,Ox(~,~ + ~) + ~(~,~ + 1)x(~, 0 projects onto states with opposite (lattice) charge conjugation. It is therefore necessary to study genuine NLT-NLT correlations. The states contributing

are no, at, b1 and the conserved U ( t ) vector charge. Correlations in this class generally show a steep decrease in time with only one state cont~buting except for the vector charge which is fiat in time. (iii) The third set of operators analyzed con~s~ o{ so-called NLS operators in which one of the qcarks is s~ifted by one lattice unit in the spatial d~rection, averaging over the three possible directions is under stood. These operators project onto states w~th new flavor quantum numbers and zre thus important to address the issue o f flavor symmetry rest=a~on {or staggered fe~rnions beyond the limited menu accessible by local operators. In summary we are awe to investigate 4 of 7 d~ferent ~ anti 5 out o f 11 di~erent p mult~plets- |naH channels the statistical quality o f the s~gnal is comparable to the local one. In Figs.3 we sumrn~6ze onr result¢ for those channels. In the # sector tlavor symmetry appears fairly ~ell sa~, at least within the present error bars_ I~¢markabfy e ~ u g h , the local p at 5.15 is the heaviest state where however the fit value should be considered an upper bound to the mass. Correspondingly there k a disorepancy t o the LT-NLT p altimugh both masses must be equal. The NLS p states show a slihfit tendency to be somewhat iib~ter than the standard p. From the local p / a I channel we were not awe to extract a mass. in the ; sector we see a major improvement when comparing the data at 5.15 with results at 5.35. In both r ~ s the (lattice) Goldstone ~r is the lightest state. Also the LT-NLT T agrees with the local one as it must. The local ~ / O ~ channel is completely dominated by the 0-H- state so that no ~r mass from this channel can be quoted. We do however obtain a mass for the • from the LT-NLT correlation function. Remarkably, all the ~" masses which do not correspoed to the spontaneously broken continuous chiral symmetry conserved on the lattice agree among each other though being heavier than the Goldstone ~r. In addition to investigating flavor symmetry we analyzed the energy of states with definite momentum in order to verify the dispersion relation (2sinh ~ ) 2 = M 2 4- ~-~(9.sin _~):z i where Pi is the lattice momentum Pi = 97~n/Ari. As

383

E. Laermann et aL / Hadron masses from ~he MT~ ¢ollaboratio~ ,e.

O It

It

i

It

i

l

I

1

It

it

p = 5.15

i

!

p=

It

5.15

'I c~

m

O

0

I

~

I

.~

I

!

$

o !

|

i

I

I

,@=5.35

o. N

,N

I

I

~

e

[

|

I

|

# - 5.35

m 0

qD

c~ m 0

ql,

f

0

m

m.

-f

X

X

0

0

o

o. O

I

|

!

2

I

I

I

4

fl

I

I

I

7

6

C~

I

I

I

2

I

I

l

4

S

II

I

|

2'

I

Figure 3: Flavor symmet~ in the p and ~r sector, in thepse~tm, ie~lthalf, dumne~ 1 amd 2damtetlkekxa~ p/J~ and #[az resp_,channel 4 and 5 the c ~ n ~ LT-NLT cureei,,~ns, channels 7 and 8 are the NLS states e ' / ~ and p ' / ~ . F~r tbe ~-, I and 2 ¢xynrespond to the local ~ / 0 + - and £-/0++ olp~cata~ 4 amd[5 t l ~ ¢ ~ , ~ 1 ~ LT-NLT ~ . 7 and 8 the NLS states ~'/0 ÷ - and ~'/~oindicated in Fig.4 this relation seems reasonably well satisfied ~ , b i n the error ban. S. COMPARISON TO QUENCHED RESULTS In order to assess possible effects of dynamical quarks we carried out a rough quenched simulation at fl between 5.7 and 6.0. The data allows us to bridge the gap in quenched staggered p masses between 5.7 and 6.0. As already noted above, the statistical quality alone suggests fl shifts around Aft > 0.5. This expectation is confirmed in Fig.5 where p and nucleon masses are compared with our quenched results3. Although the fl shift is not the same for both particles,

present en~f l~rs do mot a I o ~ a d e ~ ~ aboat qnaStat~re c h a ~ e s ~ b codd ~ L ~ t e f~0m the i n d u s m o f dynamical q~lks_ On tbe otlb~ IL~KI. ~ h k at # ----5.15 tbe N / p v~]o tmRs oot q ~ seall, value o f about 1.75 at # = 5~15. ~ t ~ s result indkates a peculiar feature of the p of reHec~sjust an unfortunate ~ in the non-universal scaling behavior of nucleon resp.p has to be left open at the present stage.

E, Laermann et aL./ Hadron masses from the MT~ collaboration

384

o

i

.~.15

p:

i

#= 6.~

t

/,/° .//I

/ . ....

/

/

I

p,N

mass

%

/

/

o

0.0

I

I

I

0.,2

0.4

0.6

p--squared

--

0.0

I

I

O.I~

0.4

s

I ~

p-squared

Figure 4: Dispersion relation for the ~, the intercept is taken from the local ~r mass at zero momentum. ACKNOWLEDGEMENTS We thank the scientific coundl of the HLRZ for the generous grant of the significant amount of CRAY Y-M P time which was necessaryfor these studies. VVe would also like to thank the staff of the computing center at JElich for their support, notably N.Attig, S.Knecht, W.Nagel and LWollschi~ger. REFERENCES 1. MTc Collaboration: R.Gavai et aL, Phys.Lett. 8241 (1990) 567; B.Petersson, these Proceedings; A.Irb~ck, these Proceedings. 2. E.Laermann et oJ. (114T~Collaboration), Nucl. Phys. B (Proc. Suppl.) 17 (1990) 436. 3. K.D.Born et aL (MTc Collaboration), these Proceedings. 4. S.Duane et aL, Phys. Lett. 195B (1987) 216. 5. S.GotUieb et aL, Phys. Rev. D35 (1987) 2531. 6. R.Gupta, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 562. 7. S.Knecht et al., Parallel Comput. 15 (1990) 3. 8. R.Altmeyer cL aL, report UMN-TH-838/90 and Phys. Rev., in print. 9. K.M.Bitar et aL (HEMCGC Collaboration), report FSU-SCRI-90-98. 10. P.Bacilieri eL al. (APE Collaboration), Nucl. Phys. B343 (1990) 228. 11. R.Gupta eL aL, Phys. Rev, D36 (1987) 2813.

!

!

II.4

!

!

~

~IB

I

e-,,O

I

62

6.4

# Figure 5: p and N masses compared to quenched results. • represent p masses, [] are the N masses multiplied by 2 to graphically disentangle p and N. The symbols -I- and x denote data taken from the literature (see ref.3). The full line is meant to guide the eye, the dashed one is the asymptotic scaling prediction for zero flavors.