Nucl ear Physi cs B ( Pr oc. Suppl . ) 26 ( 1992) 337- 340 Nor t h- Hol l and ~_ [ ~) ' U ~' 1 11 MATRI X ELEMENTS WI TH WI LSON FERMI ONS Raj an GUPTA T- 8, MS- B185, Los Al amos Nat i onal Labor at or y, Los Al amos, NM87545, I n col l abor at i on wi t h D. Dani el , USA G. Ki l cup, A. Pasel and S. Shar pe . Hi ghl i ght s of t he r esul t s f or t he spect r um, meson decay const ant s f a and f 1 , t he chi r al par amet er s Mq and 1v' , and t he Kaon B Par amet er ar e pr esent ed . The cal cul at i on was done usi ng 35 quenched 163 x 40 l at t i ces at 0 = 6 . 0 usi ng Wupper t al and Wal l smear ed sour ces . We show t hat smear ed sour ces i mpr ove t he si gnal si gni f i cant l y, consequent l y we ar e abl e t o i mpr ove t he qual i t y of r esul t s f or a number of t he phenomenol ogi cal l y i nt er est i ng quant i t i es. v 1 . I NTRODUCTI ON I n t hi s t al k I summar i ze r esul t s obt ai ned wi t h Wi l son f er mi ons usi ng smear ed sour ces . The det ai l s ar e gi ven i n Ref s . [ 1] and [ 2] , and I use t he same not at i on t o br i ef l y descr i be t he new devel opment s . The cal cul at i on was done usi ng t wo val - à L 0 . 9 - 0 NL ues of t he quar k mass cor r espondi ng t o a pi on mass of 660 and 540 MeV. Per i odi c boundar y condi t i ons wer e used i n al l f our di r ect i ons i n cal cul at i ng t he quar k pr opagat or s . I n t he anal ysi s we onl y consi der hadr ons made of degener at e quar ks, so our r esul t s have max i mumval i di t y i n t he l i mi t of SU( 3) f l avor symmet r y . I r r at i m I ca 21 Ear ~ V 0. 7 f - L. I. p ALS NLS 0. 6 0 10 20 Separ at i on 2. SURPRI SES I N HADRON SPECTRUM 10 0 t Fi g. 1 . Compar i son of t he ef f ect i ve mass f or t he nucl eon and A obt ai ned usi ng Wupper t al ( LS) and wal l sour ce The t wo mot i vat i ons f or usi ng non- l ocal sour ces pr opagat or s ( LW) . ar e ( a) t o i mpr ove t he over l ap wi t h t he desi r ed l owest st at e and ( b) t o i mpr ove t he si gnal in t he l ong- t i me behavi or of t he 2- poi nt cor r el at or s . The r esul t s of our compar at i ve st udy ar e t hat t hi s i mpr ovement i s achi eved bot h wi t h t he Wupper t al sour ce ( we used a smear i ng r adi us of 4- 4. 5 l at t i ce uni t s) and wi t h wal l sour ce pr opagat or s . Ther e ar e cl ear pl at eaus i n t he ef f ect i ve mass pl ot s f or t he 7r and p channel s and one can ext r act t he asympt ot i c mass est i mat e wi t h conf i dence . The pl at eaus i n t he bar yon channel s ( Nu- 0920- 5632/ 92/ $05 . 00 01992 - El sevi er Sci ence Publ i sher s B. V Al l r i ght s r eser ved . 338 R. Gupt a et al . / Mat r i x el ement s wi t h Wi l son f er mi ons cl eon and ®) ext end over 6- 8 t i me- sl i ces wi t h each of t he t wo sour ces and one get s what l ooks l i ke a convi nci ng si gnal , however , compar i ng t he r esul t s f r om t he t wo sour ces exposes a di st ur bi ng f eat ur e . The t wo est i mat es ar e si gni f i cant l y di f f er ent ; t he wal l sour ce est i mat e ar e 1- 3 st andar d devi at i ons l ower as shown i n Fi g. 1 . I t i s not yet known whet her t hi s di f f er ence i s pr esent because nei t her est i mat e i s asympt ot i c or due t o a sour ce dependent f i ni t e vol ume ef f ect . Ther ef or e, we advocat e f ur t her t est s usi ng di f f er ent sour ces, oper at or s and boundar y condi t i ons t o det er mi ne how best t o ext r act asympt ot i c mass est i mat es f or bar yons . 0. 4 0. 3 T+ 4- 4 0. 2 X o t hi s wor k X Ref . 4 + Ref . 5 1 1 1 1 1 11 _1 1 0. 1 0 0. 2 i k d) u f1 111 0. 4 0. 6 0. 8 1 11- 1 MPs / MV Fi g. 2. Compar i son of l at t i ce r esul t s f or t he decay const ant f v 1 wi t h phenomenol ogi cal est i mat es . 3. MESON DECAY CONSTANTS 4. CHI RAL PARAMETERS mq and ~i P To cal cul at e decay const ant s usi ng quar k pr opagat or s wi t h non- l ocal sour ces one needs t o cal cul at e bot h smear ed- l ocal and smear ed- smear ed cor r el at or s . The r el evant mat r i x el ement can t hen be ext r act ed f r om t he ampl i t ude of t he t wo 2- poi nt cor r el at or . As shown i n Ref . [ 1] ( see Eq . Wi t h Wi l son f er mi ons t he chi r al par amet er s mq and ( ) m9=o have t o be ext r act ed f r om combi nat i ons of 2- poi nt cor r el at or s . I n Ref . 1 ( see Eqs . 7 . 4- 7 . 10) we descr i be t he di f f er ent combi nat i ons of cor r el at or s t hat can be used and show 5. 2) , one can combi ne t he t wo cor r el at or s i n a number of r el at ed ways, and t he di f f er ence be- t hey gi ve consi st ent r esul t s . To get t ween t he r esul t s i s a measur e of t he syst emat i c er r or s . We used f our such combi nat i ons t o de- l i mi t of r esul t s obt ai ned at f i ni t e mq i s r el i abl e . t er ni ae f , , and t wo f or f v , and f i nd t hat t he r esul t s ar e consi st ent - - t he var i at i on bet ween we show t hat a l i near ext r apol at i on t o t he chi r al To compar e l at t i ce r esul t s wi t h cont i nuumval ues one has t o i ncl ude t he r enor mal i zat i on const ant s ZA, Zp and ZS, whi ch ar e not wel l det er mi ned Fur t her mor e, at Lat t i ce9l new r esul t s f r om t he ( est i mat es made usi ng per t ur bat i on t heor y can be uncer t ai n by a f act or of up t o 2) . Pr esent t hr ee gr oups APE [ Ref . 5] , QCDPAX [ Ref . 6] , l at t i ce est i mat es ar e not i n agr eement wi t h phe- and us wer e f ound t o be i n good agr eement . A nomenol ogi cal val ues ; we f i nd t hat m, comes out maj or i mpr ovement over pr evi ous cal cul at i ons i s a f act or of 2- 3 t oo smal l whi l e seen f or f v ; t he new r esul t s ar e i n much bet t er agr eement wi t h exper i ment al dat a as shown i n Fi g; 2. r espondi ngl y a f act or of 2- 3 t oo l ar ge. I ncl udi ng t he Z f act or s si gni f i cant l y r educes t he di f f er ence, met hods i s compar abl e t o t he st at i st i cal er r or s . i s cor - however , we cannot yet ascer t ai n how much of R. Gupt a et al . / Mat r i x ekr nent s wi t h «s on f er mi ons t he r emai ni ng di scr epancy i s due t o quenchi ng . 339 i zat i on const ant s much mor e pr eci sel y, or bet t er onl y an appr oxi mat i on t o t he oper at or wi t h t he desi r ed cont i nuum behavi or . One can el i mi nat e bot h a and / 3 at a f i xed va1L& of MK by usi ng st i l l use an i mpr oved act i on f or whi ch t he Zs t ake on val ues cl oser t o 1. 0 and have much smal l er t he non- per t ur bat i oe met hod of moment umsubt r act i on [ Ref s. 2, 8] . For exampl e, by cal cul at i ng O( a) ar t i f act s. I n pr evi ous cal cul at i ons one has f ound a l ar ge di f f er ence bet ween est i mat es obt ai ned usi ng Wi l - t he mat r i x el ement of Of or t wo di f f er ent val ues of p and t aki ng t he di f f er ence one get s son and st agger ed f er mi ons . The good news i s MK( P) - MK( 0) t hat on compar i ng r esul t s f or Wi l son f er mi ons = ( - 1 + yK) MK( E( p) - MK) + . . . . I n any case we need t o det er mi ne t he r enor mal - ( 3) wi t h t hose f r om st agger ed f er mi ons [ Ref . 3] we f i nd consi st ency once one t akes i nt o account t he I n pr act i ce we cal cul at e si zabl e syst emat i c and st at i st i cal er r or s i n t he t wo est i mat es . For t hi s compar i son we used t he same set of l at t i ces at Q= 6 . 0 and used smear ed pr opagat or s i n bot h cases . BK E( p) BK( p) - MK BK( 0) E( p) - MK ( 4) at each val ue of oc, wher e by BK( p) we mean t he r at i o of t he mat r i x el ement t o i t s VSA val ue, bot h cal cul at ed on t he l at t i ce at f i ni t e momen- 5. KAON B PARAMETER I n or der t o ext r act BK ( f or backgr ound and phenomenol ogi cal i mpl i cat i ons see t al k by Mar t i nel l i , Ref . 7) we wi sh t o cal cul at e t he mat r i x el ement ( ME) t umt r ansf er . For t hi s met hod t o be vi abl e t her e shoul d be a si gnal i n t he cor r el at or s at f i ni t e moment um, and p shoul d be smal l such t hat t he quar t i c and hi gher t er ms negl ect ed i n Eq . 2 ar e smal l . Thi s met hod does not r emove t he t hi r d uUp1 ysi cal coef f i ci ent - 1 ; we hope t hai c, i n a f ut ur e cal cul at i on, usi ng Ôwi t h an i mpr oved act i on wi l l MK( P) = ( N( P = 0) I ( sy zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Ld) ( gyi Ld) I K( P) ) . ( 1) t ake car e of i t . I n chi r al per t ur bat i on t heor y i t behaves as MK( P) The cal cul at i on of t he mat r i x el ement s usi ng ' ^ t KPK * PR wher e - yK = 8/ 3 f KBK, and pK and pK ar e t he on- shel l f our - moment a of t he ext er - smear ed sour ces i s done as f ol l ows : a wal l sour ce at t = 0 pr oduces a zer o moment umkaon whi ch nal st at es, so t hat PK * PR = MK M K + ( p) 2 . Unf or t unat el y, wi t h Wi l son f er mi ons chi r al sym- pr opagat e f or a t i me t , at whi ch poi nt t he oper at or i nser t s moment um p, and t he r esul t i ng met r y i s br oken expl i ci t l y by t he r t er m and t he K wi t h moment um p t hen pr opagat es t he r emai ni ng ( N=- t ) t i mesl i ces unt i l i t i s dest r oyed by a Wupper t al sour ce . As a consi st ency check we show, i n Ref . [ 1] , t hat t he Wupper t al sour ce cor - expansi on t akes t he f or m MK( P) =a+, QMK+( y +YK) PK' PK+ . . . , ( 2) wher e t he t er ms pr opor t i onal t o a, Q and y ar e unphysi cal cont r i but i ons and suppr essed by one power of t he l at t i ce spaci ng a. Usi ng t he per - r el at or s have si gni f i cant over l ap wi t h t he l owest f ew moment a al l owed on t he l at t i ce, and t he zer o moment umwal l sour ce pi on " r eaches" t he r egi on t ur bat i vel y i mpr oved oper at or O ( see Eq . 2. 3 i n Ref . 2) shoul d r educe t he l at t i ce ar t i f act s, but i n whi ch t her e i s a si gnal f or t he non- zer o moment um kaon . An exampl e of t he qual i t y of t he wi l l not el i mi nat e t hem compl et el y because i t i s si gnal f or BK usi ng Ôi s shown i n Fi g. 3. R. Gupt a et al . I Mat r i r el ement s wi t h Wdsonf mni ons We compar e our Wi l son r esul t s wi t h St agger ed dat a at same , Q and r oughl y t he same pseudoscal ar masses as gi ven i n Ref . [ 9] . The est i mat es of BK ar e consi st ent , t hough t he er r or s i n t he Wi l son est i mat e ar e l ar ger by a f act or of 10, a l ar ge par t of whi ch i s due t o t he pr ocess of moment um subt r act i on . I n addi t i on, we ment i on t hat t he i ndi vi dual axi al - axi al and vect or vect or t er ms ar e al so compar abl e and f ur t her mor e, show si mi l ar chi r al behavi or [ Ref . 2] . To concl ude, we bel i eve t hat t he met hod of moment um subt r act i on wor ks and f ut ur e hi gh st at i st i cs cal cul at i ons usi ng an i mpr oved act i on wi l l al l ow us t o cal cul at e BK r el i abl y wi t h Wi l son f er mi ons al so . Fi g. 3. The expect ed si gnal f or Bh i s a const ant . For p = ( 1, 0, 0) t her e i s a si gnal onl y f or t > 22 wher e t he ef f ect i ve mass pl ot shows a pl at eau . ACKNOW LEDGEMENTS I n or der t o st udy t he ef f ect of t he mi xi ng due These cal cul at i ons wer e done usi ng t i me pr o- t o t he r t er mwe have anal yzed BO separ at el y f or vi ded by LANL, NERSC, PSC and SDSC Super - each of t he 16 gamma mat r i x st r uct ur es and t he comput er cent er s . We t hank DOE and NSF f or t hei r suppor t of t hese cal cul at i ons . t wo possi bl e cont r act i ons . We f i nd t hat t her e i s a l ar ge cancel l at i on of t he di f f er ent ME, consequent l y si nce t he coef f i ci ent of t he per t ur bat i ve cor r ect i on i s . .^ï 2. x 10 - 3 t he mi xi ng i nduces at Ref er ences most a 2 - 3%ef f ect . Si mi l ar l y, t he 1- l oop r enor mal i zat i on of t he 4- f er mi LL oper at or i s l ar gel y D. Dani el , R. Gupt a, G. Ki l cup, cancel l ed by Z2 needed t o r enor mal i ze t he VSA. Shar pe, LA- UR- 91- 3285 . Thus, over al l t he 1- l oop cor r ect i ons ar e 5 - 10%, i . e . compar abl e t o t he st at i st i cal er r or s i n i ndi vi dual ME. Our best est i mat es af t er per f or mi ng t he moment um subt r act i on ar e BK( t c = 0. 154) BK( K = 0. 155) = 0. 69( 25) , = 0 . 74( 27) . [ 2] R. Gupt a, D. Dani el , G. Ki l cup, A. Pat el and S. A. Pat el and S. Shar pe, LA- UR- 91- 3522 . [ 3] R. Gupt a et al , Phys . Rev. D43 ( 1991) 2003 . [ 4] R. Sommer , Nucl . Phys . B( Pr oc . Suppl . ) 9 ( 1989) 513. [ 5] E. Mar i nar i , t hese pr oceedi ngs . [ 6] [ 7] T. Yoshi e, t hese pr oceedi ngs . [ 8] G. Mar t i nel l i , Nucl . Phys . B( Pr oc . Suppl . ) 9 ( 1989) G. Mar t i nel l i , t hese pr oceedi ngs. 523. G. Ki l cup, S. Shar pe, R. Gupt a and A. Pat el Phys. Our dat a gi ve a - 0. 003, ß - 0. 02 and 7 + y K t i 0 . 015, whi ch shows t hat t he l at t i ce ar t i f act s ar e l ar ge . Rev . Let t . 64B ( 1990) 25 .