熱 ケ ー ジ場 理 論 の相 構 造 の解 析 的研 究
Chi r al Phas eTr ans i t i ons i nQ
ED
at Fi ni t eTem
per at ur e:
D
ys on- Sc hw
i nger Equat i onAnal ys i s i nt heReal Ti m
e
H
ar d- Ther m
al - LoopAppr oxi m
at i on
中 川 寿夫●
Hi s aoNakkagawa
教養部 教 授
Al t houghl ot s of ef f or t s havebeenmadet oundes t andt het emper at ur e- and/ or dens i t y-dependent phas et r ans i t i oni nt her mal QCD/ QED, wec annot haveyet t r ul yunder s t oodeven t her el at i onbet weent hec hi r al t r ans i t i onandt hec onf i nement - dec onf i nement t r ans i t i on. Begi nni ngof t her el at i vi s t i c heavyi onc ol l i s i onexper i ment s at BNL- BRI Ghas at t r ac t edan i nc r eas i ngi nt er es t i ns t udyi ngt hephys i c s i nt her mal QCD, t hus has gi venus anenc our agi ng t i met opr oc eedt of ur t her i nves t i gat i ons of t hemec hani s mof phas et r ans i t i oni nhot and dens egauget heor i es , es pec i al l yi nQCDandQED.
TheDys on- Sc hwi nger ( DS} equat i oni s pr ovent obeapower f ul t ool t oi nves t i gat ewi t h t heanal yt i c pr oc edur et hephas es t r uc t ur eof gauget heor i es , es pec i al l yi nt hevac uumgauge t heor i es [ 1, 2, 3] . However , wec annot s ayt hat , at f i ni t et emper at ur eand/ or dens i t y, t heDS equat i onanal ys es of c hi r al and/ or di - quar kc ondens at i onhavebeenc ar r i edout s uc c es s f ul l y.
I nt hepr ec edi ngDSequat i onanal ys es [ 4$] , t hel es s ons f r omvac uumt heor i es havebeen s os i mpl yappl i edt ot her mal t heor i es wi t hout c l os eexami nat i on. I nmos t anal ys es t hel adder appr oxi mat i onwas us edbys i mpl ynegl ec t i ngal l t heHTLef f ec t s [ 5, 6, 7] , or onl ybyt aki ngt he i mpr oper HTLef f ec t s i nt ot hegaugebos onpr opagat or [ $] . As ar es ul t t heyhavemi s s edt he es s ent i al c ont r i but i onof t her mal gaugef i el dt heor i es , es pec i a皿yt hei mpor t ant ef f ec t f r omt he " dynami c al l ys c r eened" magnet i c mode{ havi ngi ngener al amoment um
- dependent " mas s " ,
t houghbei ngmas s l es s i nt hes t at i c l i mi t ) . Manyanal ys es , byf i xi ngi nt heLandaugauge, i gnor edt hef er mi onwave- f 皿c t i onr enor ma』z at i onc ons t ant s ( WFRCs ) byt aki ngt hei r nai ve t r eeval ues [ 5, 6, 7] . Fur t her mor e, mos t anal ys es donei nt her eal t i mef or mal i s mdi dnot di s c us s
t hephys i c al f er mi anmas s 血mot i onΣRi t s el f of t her et ar dedpr opagat or 【7β 】, wi t ht he
negl ec t i onof i t s i magi nar ypar t s , t oget her wi t ht hei nac c ur at eus eof t hei ns t ant aneous exc hange{ I E) appr oxi mat i ont ot hegaugebos onpr opagat i on[ 7, 8] . Al l s uc hi mpr oper appr oxi mat i onmet hods havec aus edt henegl ec t i onof woul d- be- l ar gec ont r i but i ons t ot heDS equat i onot her wi s eexs i s t ed.
Thenwes houl ds er i ous l yas kwhet her wec oul dr el yont hepr evi ous r es ul t s of t heDS e叩at i ona皿al ys i s ont hec hi r al phas et r a皿s i t i onas t her eal c ons equenc es of t her mal gauge f i el dt heor i es . Cons i der i ngt het r oubl es i nt hepr evi ous a皿al ys es 【4- 8】ment i onedabove, we s houl dmakear e- anal ys i s bys t udyi ngt hehar d- t her mal - l oop{ HTL) r es ummedDSequat i oni n t her eal t i mef or mal i s m, t hus mi ght gi vi nganewunder s t andi ngont hephas es t r uc t ur eand t hemec hani s mof phas et r ans i t i oni nt her mal gauget heor i es .
Mai ni nt er es t of t hepr es ent i nves t i gat i onl i es i nc l ar i f yi ngwhat ar et hees s ent i al t emper at ur eef f ec t s t hat gover nt hephas et r ans i t i onandal s oi nf i ndi nghowwec anc l os el y t aket hes eef f ec t s i nt ot he" ker nel " of t heDSequat i on. Es s ent i al pr oc edur es of our anal ys i s c anbes ummer i s edas f ol l ows ;
i ) Fi r s t l yweus et her eal t i mec l os ed- t i me- pat h( RT- CTP) f or mal i s m[ 9] , ands t udyt he
phys i c al mas s f unc t i onERi t s el f not t heE1, , of t her et ar dedf er mi onpr opagat or , bec aus ewe ar ei nt er es t edi nbot ht her eal andi magi nar ypar t s .
i i } Sec ondl yweac c ur at el yt akei nt oour a皿al ys i s t hef ac t t hat ΣRi s t hemas s f unc t i onof " uns t abl e" quas i
- par t i c l ei nt her mal f i el dt heor i es
, t hus havi ngnon- t r i vi al i magi nar ypar t s as weUas non・t r i vi al WFRCs . Negl ec t i onof i magi nar ypar t s andnon・t r i vi al WFRCs ac t ua皿ygi ve
c ons t r ai nt equat i ons t obes ol veds i mul t aneous l y, t ot al l ydi s mi s s edi nt hepr ec edi nganal ys es . '
i i i } Thi r dl yandmos t i mpor t ant l y, devot i ngour at t ent i ont oc l os el yes t i mat i ngt he domi nant t emper at ur e- dependent c ont r i but i ons , wef oc us ons t udyi ngt heDSequat i onbei ng exac t upt oHTLappr oxi mat i on: Bot ht hegaugebos onpr opagat or s andt hever t exf 皿c t i ons ar edet er mi nedwi t hi nt heHTLr es ummat i on[ 10, 11, 12] , wi t hwhi c ht hegaugei nvar i anc eof t her es ul t at l eas t i nt hepe血r bat i vea皿al ys i s i s guar ant eed. Wi t ht heHTLr es ummedver t ex f unc t i ons [ 12] wec anexpl i c i t l ywr i t edownt heHTLr es ummedDSequat i on.
i v} Fi na皿y, t hegauge- par emet er dependent c ont r i but i onmus t bec ar ef u皿ys t udi ed wi t hout f i xi ngt hegaugei nt os omedef i ni t eones , s uc has t heLandaugauge.
Thet hi r dpoi nt l i s t edabovei s bet t er t obet akens t epbys t epi nt ot heac t ual anal ys i s of t heDSequat i on. I nt hepr es ent anal ys i s wepr es ent t her es ul t of our 丘r s t s t epi nves t i gat i on i ns t r ongQED; f oc us s i ngonwhat happens whenwet akei nt oac c ount exac t l yat l eas t t he HTLr es ummedgaugebos onpr opagat or s . Anal ys i s i nQCDandef f ec t s of f ul l yi nc l udi ngt he HTLr es ummedver t i c es wi l l bepr es ent edi nt hes epar at epaper [ 13] .
N
akkagaw
a: Chi r al Phas eTr ans i t i ons i nQ
ED
at Fi ni t eTem
per at ur e:
HTLappr oxi mat i onc anbeobt ai nedbyappl yi ngt hef o皿owi ngappr oxi mat i ont ot hef Ul l DS equat i on;
i ) r epl ac et hef ul l gaugebos onpr opagat or wi t ht heHTLr es ummedpr opagat or , and l l ) appr oxi mat et hef u皿ver t exf unc t i ons t ot heHTLr es ummedver t exf unc t i ons .
Theni nt heRT・CTP丘) mal i s mweget i nQEDt hedes i r edDSequat i on口21At z er o t emper at ur e, t hewavef unc t i onr enor mal i z at i onc ons t ant A( P) c oi nc i des wi t hB( P) andequal s t ouni t yi nt heLandaugauge, whi l eat f i ni t et emper at ur ei t i s not Appear enc eof t heHTL r es ummedver t exf unc t i ons t oget her wi t ht heHTLr es ummedgaugebos onpr opagat or s as s ur es t hat t heHTLappr oxi mat i oni s c ons i s t ent l yc ar r i edout i ns t udyi ngt heHTL r es ummedDSequat i on[ 12] , andguar ant ees t her es ul t bei nggaugei nvar i ant , at l eas t , i nt he ef f ec t i veper t ur bat i onr egi me. Negl ec t i onof t heHTLc ont r i but i ont ot hever t ez f unc t i on,
δ. P、 β γ, s i mpl ybr i ngs us t ot hel adder DSequat i onwi t ht heHTLr es ummedgaugebos on
pr opagat or . I t s i gni f i c ant l ys i mpl i f i es t hes t r uc t ur eof t heDSequat i ont obeexami nedt hus r educ i ngt het ec hni c al di f f i c ul t yt ohandl et heDSequat i oni t s el f Thepr i c et opayi s t ol os e t heas s ur enc eof gaugei nvar i anc eof t her es ul t s .
I nt hepr es ent anal ys i s as al r eadyment i onedabove, wei nves t i gat et hec ons equenc es of t hel adder ( poi nt ver t ex} DSequat i onwi t ht heHTLr es ummedgaugebos onpr opagat or . The DSequat i onobt ai ned, i s s t i l l qui t et ought obeat t ac kedf or c i ngus f ur t her appr oxi mat i ons f or t heanal ys i s t obeef f ec t i vel yc ar r i edout However , t heappr oxi mat i onmadeus eof mus t be c ons i s t ent wi t ht heHTLappr oxi mat i on, wi t hout mi s s i ngt hei mpor t ant t her mal ef f ec t s out of t heker nel of t heDSequat i on.
Her ei t i s wor t hnot i c i ngt hat t hei ns t ant aneous exc hange( I E} appr oxi mat i onf r equent l y us edi nt hepr ec edi nganal ys es 【6, 7, 81i s nat c ol 珂pat i bl ewi t ht heHTLappr oxi mat i oni nt he s t r i c t s ens e. I nt heenac t I Er 血mi t 止eHTLr es ummed廿ans ever s emas s 血not i onvani s hes and t het r ans ever s e( magnet i c ) modebec omes t ot al l ymas s l es s . Namel yt heI Eappr oxi mat i on di s c ar ds t hei mpor t ant t he皿al ef f ec t c or m皿gf r omt heLandaudampi ng, t hus di s mi s s i ngt he dynami c al s c r eeni ngof t hemagnet i c mode. Thi s c aus es t hef amous quadr at i c di ver genc eof t heRut her f or ds c at t er i ngc r os s s ec t i on, Ther eas onwhyi nt hepr evi ous anal ys es t hi s di ver genc edi dnot appear , i s t hat t hei magi nar ypar t of ERi s c ompl et el ynegl ec t edt her e f r omt hebegi nni ng, namel yt hat t heequat i onf or I mEai s t ot al l ydi s c ar ded.
血nc t i on, ' nRL{ K) , has adef i ni t et her mal mas s mg2∼( gT) 2, r epr es ent i ngt heDebyes c r eeni ng duet ot her mal f l uc t uat i on, t hus eveni nt heI El i mi t t hel ongi t udi nal modec ant akei nt o ac ount t hees s ent i al t her mal ef f ec t I nt hepr es ent anal ys i s t hegaugei s f i xedt ot heLandau
gauge( ξ=0) .
I t i s f ai r t onot et hat i nt hepoi nt ver t exl adder appr oxi mat i on, as al r eadyment i oned above, t hegaugei nvar i anc eof t her es ul t s i s s poi l ed. Tomaxi mal l yr es pec t t hegauge i nvar i anc e, wes houl ds ol vet heDSequat i ons wi t ht hec ons t r ai nt A( P) =1, whi c hguar ant ees
Z2=1, bei ngc ons i s t ent wi t ht heWar di dent i t yZ1=ZZ. Thi s c anbedone【161bys uc c es s i vel y adj us t i ngt hegauge- par amet er gi ns ol vi ngt heDSequat i ons .
Nowwes houl ds ol venumer i c al l yt heDSequat i ons wi t ht heI Eappr oxi mat i ont ot he l ongi t udi nal mode[ 1? ] . Res ul t of t hepr es ent anal ys i s s hows t het wof ac t s ; i ) Thec hi r al phas e t r ans i t i oni s of s ec ondor der , s i nc eaf er mi onmas s i s gener at edat ac r i t i c al val ueof t he t emper at ur eTor at t hec r i t i c al c oupl i ngc ons t ant awi t hout anydi s c ont i nui t y, andi i ) t he c r i t i c al t emper at ur eTc at f b【edval ueof αi s s i gni f i c ant l yl ower t hant hepr evi ous r es ul t s
[ 6, 7, 8] , namel yt her es t or at i onof c hi r al s ymmet r yoc c ur s at l ower t emper at ur et han pr evi ous l yexpec t ed. Thes ec ondf ac t s hows t hat i nt hepr evi ous anal ys es t hei mpor t ant
t emper at ur eef f ec t s ar enegl ec t edduet ot hei napr opr i at eappr oxi mat i ons .
Thec r i t i c al c oupl i ngc ons t ant αc as af unc t i onof T, andt hecr i t i c al t emper at ur eTc as a f unc t i onof a, c anbeal s odet er mi ned. Fr omt hes er es ul t s wec anes t i mat et hec r i t i c al c oupl i ngc ons t ant αc i nt hel i mi t T→0, αc ( T→0) , andt heval ueof c oupl i ngc ons t ant α wher et hec r i t i c al t emper at ur eTc bec omes z er o, α{ Tc =0) . Our r es ul t s hows t hat , as T bec omes s mal l er , t hec r i t i c al c ouphngc ons t ant αc al s obec omes s ma皿er ands eems t o c ons i s t ent l ydec r eas ef r omabovet ot hez er ot emper at ur er es ul t However , t hees t i mat ed
val ues , αc ( T→0) ∼2. 5andα{ Tc =0) 一 一2. 5, ar es i gni f i c ant l y且ar ger 血ant heval ueαc { T30) = n/ 3det er mi nedbyt heor et i c al anal ys es [ 1] of t heDSequat i onf or t hef er mi ons el f - ener gy
par t E( P) at z er ot emper at ur e, T=0, i nt hel adder appr oxi mat i oni nt heLandaugaugewi t h t het r eel evel phot onpr opagat or . Si nc et heHTLr es ummedgaugebos onpr opagat or ' G°' i s s i mpl yr educ edt ot het r eel evel phot onpr opagat or i nt hel i mi t TAO, wear enot qui t es ur e whys uc hdi 血er enc eemer ges .
Pr es ent r es ul t s how
s t hec or r ec t nes s of our r es ear c h- s t r at egy, nam
el y, t hei m
por t anc eof
t hef ul l H
TLr es um
m
edD
Sequat i onanal ys i s of t hec hi r al phas et r ans i t i onat f i ni t e
N
akkagaw
a: Chi r al Phas eTr ans i t i ons i nQ
ED
at Fi ni t eTem
per at ur e:
Ref er enc es
[ 1] T. Mas kawaandH. Nakaj i ma, Pr og. Theor . Phys . 52. 1326{ 1974} ; 54, $60( 1975) ; R. FukudaandT. Kugo, Nuc LPhys . 8117, 250( 1976) .
[ 27KYamawaki , M. BandoandKMat umot o, Phys . Rev. Let t . 56, 1335( 1986) ; K, 1, Kondo, H. Mi noandK. Ya皿awaki , Phys . Rev. D38, 2430〔1989) 。 L37W. A. Bar deen, C. N. LeungandS. T. Love, Phys . Rev. Let t . , 56, 1230{ 1986} ;
C. N. Leung, S. T. LoveandW. A. Bar deen, Nuc l . Phys . 8273, 649{ 1986) ; W. A. Bar deen, C. N. LeungandS. T. Love, Nuc l . Phys . 8323, 493{ 1989) .
[ 4] A. Bar duc c i , RCas al buoni , S. DeCur t i s , R. Gat t oandG. Pet t i ni , Phys . Rev. D41, 1610{ 199( } } . [ 5] S. K. Kang, WニH. KyeandJ . K. Ki m, Phys . Let t 8299, 358{ 1993) .
[ 67K. - LKondoandK. Yos hi da, I nt J . Mod. Phys . A10, 199( 1995) . L7] M. Har adaandA. Shi bat a, Phys . Rev. D59, 014010( 1998) .
[ 8] K. Fukaz awa, T. I nagaki , S. Mukai gawaandT. Mut a, Pr og. Theor . Phys . 105, 979( 2001) . [ 9] K. - C. Chou, ZニB. Su, BニL. HaoandL. Yu, Phys . Rep. 118, 1( 1985) ;
LV. Kel dys h, Zh. Eks p. Teor . Fi z . 47, 1515( 1964) [ Sov. Phys . J ETP20, 1018( 1965) 1 [ 107E. Br aat enandR. D. Pi s ar s ki , Nuc LPhys . 8337, 569( 199( } ) ; 8339, 310{ 199( } ) ;
J . Fr enkel andJ . C. Tayl or , Nuc LPhys . 8334, 199( 1990) . [ l l ] M. E. Car r i ngt on, andU. Hei nz , Eur . Phys . J , C1, 619( 1998) ;
HouDef U, M, E. Car r i ngt on, R. Kobes , andU. Hei nz , Phys . Rev. D61085013{ 2000) . [ 12] Y. Fueki , H. Nakkagawa, H. Yokot aandK. Yos hi da, Pr og. Theor . Phys . 107, 759( 2002) . L13] Y. Fueki , H. Nakkagawa, H. Yokot aandK. Yos hi da, wor ki npr ogr es s .
[ 14] V. V. Kl i mov, Sov. J . Nuc LPhys . 33, 934( 1981) ; Sov. Phys . J ETP55. 199( 1982} ; H. A. Wel don, Phys . Rev. D2g, 1394( 1982} ; Phys . Rev. D2g, 2789{ 1982} . [ 157H. A. Wel don, Ann. Phys . { N. Y. } 271, 141( 1999} .
[ 167Y. Fueki , H. Nakkagawa, H. Yokot aandK. Yos hi da, t oappear .
