Tim: Tusday, 5: Room: Chsapak A Eignvalu Distributions of Quark Matri at Finit Isospin Chmical Potntial Prsntr: Yuji Sasai Tsuyama National Collg of Tchnology Co-authors: Grnot Akmann, Atsushi Nakamura and Ttsuya Takaishi July 5, 8 Lattic 8, Williamsburg
Abstract W compar ignvalu distributions of phas qunchd Lattic QCD and Random Matri Thory (RMT). W calculatd ign-valu distributions of quark matri on 8 3 4 lattic by N f = KS frmions. W prformd fittings btwn ths lattic data and RMT at coupling β= 5.3 and iso-vctor chmical potntial μa =.,.4773,. and. (wak non-hrmiticity) and thn find good agrmnt. Our data indicats that F π dcrass as th iso- vctor chimical potntial incrass.
. Rsarch situation at µ SU() Full [] SU(3) Qunch [] RMM LGT SU(3) Phas Qunch This talk SU(3) Full [] Osborn, Splittorfff & Vrbarrschot (5), Akmann & Bittnr (6) [] Akmann & Wttig (4) Finit baryon-numbr numbr dnsity in SU(3) Finit dnsity lattic QCD introducs chmical potntial μ quark matri dtrminant positiv, ral for μ= compl for μ numrical study bcoms difficult! 3
. Formulation Lattic calculation Frmion : Kogut-Susskind (Staggrd) Quark matri dtrminant is compl on may prform Mont Carlo simulation β Sg DU O Qunching masur O = q β S g DU N /4 Phas qunching masur O = N /4 N f = Phas qunch, SU(3), 8 3 4 lattic, β=5.3, ma=.5 f DU dt Δ O β f DU dt Δ β Calculatd ignvalus: all ignvalus (N C N V =644) in 98 configurations 4 th smallst ignvalus in 5, /, / 5, configurations S g S g
Random Matri Modl G.Akmann and G.Vrnizzi, 3 G.Akmann, 3 J.Osborn, 4 N f = Phas qunchd spctral dnsity in wak non-hrmiticity limit * K ( ) ( ) s( ξη, ) Nf = Nf = ρ ( ξ) = ρ ( ξ) * * Ks( ηη, ) Ks( ξξ, ) whr qunchd dnsity is givn by ( ) N R( ) f ξ = ξ 4α ( * ρ ( ξ) = ξ K ) K,. s ξ ξ 4πα 4α s ( ) * α t ( ) ( * ξξ, ) ξ ξ K dt I t I t = I ( z) J ( iz) 5
Bridg btwn LGT and RMM ξ = za V Σ = za π / d rscald ignvalu which is usd in RMM η = ma V Σ = ma π / d masurd ignvalu on th lattic rscald mass givn mass on th lattic α = ( μa) FV a: lattic spacing π givn chmical potntial on th lattic d: man lvl spacing V: lattic volum Σ: chiral condnsat F π : pion dcay const. π 6
Man lvl spacing d is vry vry important! Is d -dimnsional or -dimnsional spacing? It sms that w should think of d as -dimnsional spacing. µ= Banks Cashr formula y O πρ() π Σ= ψψ = = V Vd d Masur th man lvl spacing d btwn nighbor ignvalus. µ for th smallst 7ignvalus y O Projct ignvalus on y-ais y O Calculat th man lvl spacing d on y-ais 7
3. Comparison of RMM rsult and lattic data Eign-valu distribution function of Lattic ρ ( y, ) ρ (, y ) ddy = N 3 = 3 8 4 = 644 Our purpos again Spctral dnsity of RMM * K ( ) ( ) s( ξη, ) Nf = Nf = ( ) = ( ) * * Ks ( ηη, ) Ks( ξξ, ) ρ ξ ρ ξ W want to dtrmin paramtrs in which th lattic data rappar. 8
8 3 4 lattic, N f =, β=5.3, ma=.5, μa=. Calculat man lvl-spacing d, and rscal lattic data by it. µ a=. ξ d =.775 = za π / d rscald ignvalu which is usd in RMM 33 (5, configurations, th smallst 7 ignvalus) masurd ignvalu on th lattic Obtain th rscald mass η by d η Th arial viw is obtaind from 98 configurations. = ma π / d = 57.6 Ths valus ar dtrmind uniquly. 9
3 Put η and choos α suitably in RMM Choos α in ordr to match thos distribution latituds, paks and plataus on th ral and imaginary ais. Thn α =.68 is obtaind. µ a=. LGT 8 4 4 lattic, N f=, β=5.3, RMM N f =, α=.68, η=57.6 ma=.5, μa=. ξ = za π / d
μa=. 5 configurations histogram LGT phas qunch RMM phas qunch RMM qunch histogram LGT phas qunch RMM phas qunch RMM qunch Charts coincid without tuning of thos normalizations. Bcaus th phas ffct is small, it is difficult to know which of RMM graphs corrsponds to LGT graph. Fr paramtr is α only.
Distribution of th first 3 ignvalus in LGT st ignvalu nd ignvalu 3 rd ignvalu Tuning of paramtr α α= 58.58 α=.68 α=.78
µa=. 3 d =.84 α = ( μa) F π V =. No fr paramtr! Spctral dnsity of RMM ( N ( ) ( f = ) y α t * ρ ( ξ) = dt I ξ t I ξ t ) ξ = + iy histogram LGT full RMM full RMM qunch 5 configurations This statistics ar not so rich. Th first thr paks of LGT full ar vry wll in agrmnt with th on of RMM full. 3
Distribution of th first 3 ignvalus in LGT st ignvalu nd ignvalu 3 rd ignvalu 4
µa=.4773 d =.66 3 This arial viw is th almost sam on at µa=.. Th clos-up nar th origin has vry narrow distribution width. 5, configurations st pak y=.635 RMM N f =, α=.8,η=59.4 histogram LGT phas qunch RMM phas qunch RMM qunch This statistics ar not so poor It sms that only th first pak of This statistics ar not so poor. It sms that only th first pak of LGT is in agrmnt with RMM. 5
Distribution of th first 3 ignvalus in LGT st pak y=.635 st ignvalu nd ignvalu 3 rd ignvalu Tuning of paramtr α st pak y=.635 α=.7 α=.8 α=.9 6
µa=. d = 4.34, configurations 33 Lft arial viw is obtaind from 58 configurations. RMM N f =, α=.38,η=36. histogram LGT phas qunch RMM phas qunch RMM qunch Thr is phas ffct at μa=.. It sms that statistics ar still insufficint in ordr to know whthr th phas qunchd graph of LGT corrsponds to th sam graph of RMM 7
Distribution of th first 3 ignvalus in LGT st ignvalu nd ignvalu 3 rd ignvalu Tuning of paramtr α α= 8.8 α=.38 α=.48 8
4. Pion dcay constant F π α μ a = Fπ V β=5.3 μa α fit α fit /μa. confinmnt non non.4773 (β < β C = 5.397(9)) confinmnt. (β < β C =5.34()) confinmnt.8 6.8.68 6.8. (β > β C =5.98()).38.9 dconfinmnt β C is from Kogut and Sinclar (4). It sms that F π on β C or in dconfinmnt phas is smallr than F π in confinmnt phas. 9
5. Summary A) W hav th phas qunchd configurations that calculatd on 8 3 4 lattic. To analyz th distributions ib ti of th ignvalus, w compard th distributions with RMM calculations. l B) In cas of μa=., w hav th full QCD configurations that ar N f =, ma=.5. Thr is no fr paramtr. Th first thr paks of LGT qunch ar vry wll in agrmnt with th on of RMM qunch. C) In cas of μa=.4773,.,., it is possibl to fit th RMM graph to th LGT on by tuning only α paramtr.
E) W stimatd th variations of F π at μa=.4773,.,., it sms that F π at μa=.4773, 4773.(confinmnt phas) is largr than F π at μa=. (β >~β C, almost on β C or dconfinmnt phas). F) In futur work, w try to stimat of th variations of F π at μa=.7 7at which hβ is a littl smallr than β C.
Nutron star Backup slids
Chiral condnsat confinmnt µa=. phas qunch µa=. rwightd µa=. phas qunch µa=. rwightd µa=.5 phas qunch µa=.5 rwightd < ψψ dconfinmnt β C =5.86(3) β C =5.34() β C =5.98() >= V ( ma ) ln Z Th bllow graph hibits both of no phas cas and r-wightd cas. ψψ No phas : ar th avrags ovr 4 trajctoris ach trajctoris. R-wightd : dtδ is calculatd ach trajctoris. ψψ ar th avrags ovr 4 trajctoris Ths signs ovrlap mutually. Phass of ψψ ar factorizd. W can t confirm th phas ffct. 3
Polyakov lin < L >= Tr( Utt U t t3 Ut n t n) 3 W attmpt th similar considration to Polyakov lin. µa=. phas qunch µa=. rwightd µa=. phas qunch µa=. rwightd µa=.5 phas qunch µa=.5 rwightd confinmnt dconfinmnt Th ffct of r-wighting was not sn as wll as th cas of Chiral condnsat. W want to amin th ffct of r-wighting g with mor biggr μa. At β=5., CG dosn t convrg in th dnsity rgion byond μa=.8. β C =5.86(3) β C =5.34() β C =5.98() Dos CG work wll in th high dnsity rgion (almost μa=.)? 4
Phas Qunchd Chiral condnsat < ψψ >= V (ma) ln Z Polyakov lin < L >= Tr( Ut tu t t3 U 3 t n t n )?? As µa incrass, As µa incrass, chiral symmtry is rstor. confinmnt phas dconfinmnt phas < L >= p( ε) 5 T confinmnt phas (Why?)
Chiral condnsat πρ() π ψψ = = V Vd d μa d ψψ masurd masurd ψψ d..569-3.783.5-3.4773.66-3.768.44-3..775-3.7484.77-3. 4.34-3.646.668-3 6
Lattic calculation F l ti Formulation QCD Lagrangian Baryon numbr oprator ψ γ ψ 4 3 ˆ = d N ( ) a f a L i D m F F μ μν μ μν ψ γ ψ = + N f : flavors T 3 / Partition function N f S g Δ DU L d d D DU D Z = + = 4 / 4 3 ) (dt )] ( p[ ψ μψγ τ ψ ψ S g : gaug action Frmion matri (Kogut-Susskind (Staggrd)) 3 µ + U µ U ψ ψ } ) ( ) ( { ) ( } ) ( ) ( { ) ( ), ( ˆ ˆ 3 ˆ, ˆ,, 3 + + + = + + + + + + + = a a i i y i y i i y y U U y U U m y Δ i δ δ δ δ δ µ µ U 4 U 4 a } ) ( ) ( { ) ( 4, 4 4, 4 + + + y y y U U δ δ a : lattic spacing 7
R-wighting mthod O DU dt Δ = DU Z DU dt Δ / iθ / ( ) / β Sg dt Δ O = / iθ / / iθ / β Sg DU dt Δ O = / / β S = O iθ DU iθ / / dt Δ g DU DU dt Δ / dt Δ iθ / / O β S β S β S β S g g g g 8
μa=. Spctral dnsity of RMM * K (, ) ( ) ( ) s ξη N f = N f = ρ ( ξ ) = ρ ( ξ ) * * Ks( ηη, ) Ks( ξξ, ) α = μ FV π qunch dnsity For α<<. Kν ( ) π / p( ) ( ) N R( ) f ξ = ξ 4α * ρ ( ξ) = ξ K K s ( ξ, ξ ) 4πα 4α ξ = + iy π ξ R( ξ ) 4α 4α α t * ξ dt I ξ t I ξ 4πα ξ 4α t. y α α t * = dt I ξ t I ξ t πα y α t ( ) ( * δ ( ) dt I ) ξ t I ξ t ( ) ( ) ( ) ( ) α. 9