What is Elliptic Flow?

Transcription

1 Wha is Ellipic Flow? To Trainor (for he STAR collaboraion) Monreal July, 7

2 Agenda Aziuh auocorrelaions Nonflow and inijes Quadrupole (flow) syseaics A-A eccenriciy odels Universal quadrupole rends Flow flucuaions Quadrupole y dependence Wha is ellipic flow? Trainor

3 Auocorrelaions and Power Specra n i i Q i= = ρ( φ) = r δ ( φ φ ) = u( φ ) π FT FT Q RT r u( φ ) n = i i i= π Q Q d + = + π [ π ] [ π ] π = ρ ( φ ) φ ρ( φ) ρ( φ φ ) cos( φ ) A power specru: Wiener-Khinchine heore aziuh auocorrelaion: Q = n r + n n r φ D rando walk ( ) cos( ) n r V V ρ ( φ ) δ ( φ ) cos( φ ) A arxiv: RT V = n( n ) v signal = + + π [ π ] = [ π ] correlaions Trainor 3

4 Convenional EP Flow Analysis single-paricle densiy: ρ( φ) = ρ + v cos( [ φ Ψr ]) = noe assues only sinusoidal flow coponens Ψ r : rue reacion plane no observable. Even-wise flow vecors:. Even-plane angle fro Q : 3. Enseble average: Q n = r u( φ ) i= an Qy Ψ = Qx obs v = cos( [ φ Ψ ]) i i 4. Correc for even-plane resoluion: v = v obs / cos( ( Ψ Ψr )) Trainor 4

5 v Relaion o Aziuh Auocorrelaion n V V ρ ( φ ) δ ( φ ) cos( φ ) A = + + π [ π ] = [ π ] V u u n v n, n ( φi ) ( φ j) = j i= n u( φ j) n j i Q = n u( φi ) V n i= Q V cos( [ φ Ψ ]) obs n V cos( [ Ψ Ψr ]) v v {EP} v {} Trainor 5 obs v aziuh auocorrelaion assues only flow coponens n Q = i u( φ j ) j i reove auocorrelaion (even-plane resoluion) - fro sandard ehod convenional flow analysis is a D auocorrelaion in disguise

6 ρ[] / ρ ref GeV Cenraliy Trends projecion ono φ flow = = 4 6 away-side ridge Pearson s covariance ν ρ j / ρ ref, σ η, σ φ A φ ref ref = ref sae-side peak n / = + cos( φ ) + π Trainor = π n 6 σ η σ φ peak apliude 4 6 inijes ν ρ ρ[] ρ[ ] ( ) = + cos( φ ) + ρ ρ ρ σ V subrac saisical reference ν NONFLOW

7 η φ η φ ρ / ρ ref ρ / ρ ref φ φ sar preliinary 3 4 ρ / ρ ref 3.3. φ ρ (, ).5 η φ ρ.4 ref η Modeling D Auocorrelaions Michael Daugheriy daa hisogras -3% cenral Au-Au GeV = sae-side peak η η ρ / ρ ref φ 4 large dipole ρ / ρ ref φ 4 Trainor David Keler odel fis ρ[] ρ ref η HBT, elecrons quadrupole η ρ / ρ ref φ ρ[] ρ sall ref addiional D Gaussian on η, negligible for cenral collisions η

8 v Quadrupole Cenraliy Syseaics N-N n v ρ[] π ρ ref 6 GeV 7 GeV GeV sar preliinary v {} 4 6 ν ρ[] / ρ ref D-proj- X[]- X[]-6 v-7 v{} v{4} v {4} GeV 6 GeV v {} sar preliinary 7 GeV s NN Trainor 8. ransfor 4 6 priary D easureens D auocorrelaion odel fis dashed curves: all have coon shape apliudes follow linear dependence on log( ν David Keler / GeV)

9 Quadrupole Energy Syseaics /ε ρ[]/ ρ ref (ν~6) *log( s NN / GeV) / log(/ ) sar preliinary 6 GeV 7 GeV GeV quadrupole s NN (GeV) r..s. flucuaions (GeV/c) J. Phys. G 33, 45 (7) p flucuaions inijes σ p:n (STAR) SSC correced uncorreced Φ p (CERES) s NN (GeV) differen physics coon energy rend suggess a coon underlying echanis Trainor 9

10 poin-like objecs acing a a disance N-N inbias: ineracing spheres ε A-A Eccenriciy poin-like nucleons acing a a disance iniu-bias inersecing spheres poin-like nucleon flucuaions 3 4 n par ε N-N paricipan opical poin nucleons glue ν poin-like nucleon srucure Minbias N-N ineracions are no poin-like objecs acing a a disance The W-S disribuion ay beer describe low-x glue we use he opical Glauber eccenriciy Trainor

11 Universal Cenraliy and Energy Trends /ε ρ[] / ρ ref - - GeV 6 GeV 7 GeV.45 n binary sar preliinary /ε n par / ρ[] / ρ ref 3 3 n bin (b, s NN ) n (b, s bin NN ) - - Au-Au GeV Au-Au 6 GeV Pb-Pb 7 GeV Cu-Cu GeV Cu-Cu 6 GeV.9 n binary sar preliinary Au-Au D correced v /ε sar preliinary hydro GeV 6 GeV 7 GeV LDL /S dn ch / dη universal rends represen all A-A syses for energies above GeV is his hydro-inspired fora relevan o daa? quadrupole represened by iniial condiions; no ediu properies, EoS, viscosiy, hydro v ε does no describe daa Trainor

12 Quadrupole ν and s NN Dependence Cenraliy dependence is universal Energy dependence consisen wih QCD Opical eccenriciy represens low-x glue Cobined rends reveal a universal relaion Ideal hydro v ε no observed in daa Wha is ellipic flow in N-N collisions? Trainor

13 Flow Flucuaions I: v {} v {4} v {} v {4} = v 4 r ( + r / ) r σ v nucl-ex/6 / v difference aribued o flow flucuaions v {} ~ D projecion of D auocorrelaion: quadrupole + inijes ρ[] / ρ ref D-proj- X[]- X[]-6 v-7 v{} v{4} v {4} GeV 6 GeV v {} sar preliinary 7 GeV 4 6 v {4} ~ quadrupole only: no inijes ν v {} v {4} enirely due o inijes flow flucuaions are no required by hese daa Trainor 3

14 Flow Flucuaions II: Flow Vecor does he flow-vecor disribuion reveal flow flucuaions? = power disribuion: dn ( qɶ n vɶ ) ( vɶ v) dv exp exp dq ɶ ɶ + g( ν, n) σ v qɶ exp + g( ν, n) + nσ inferred variance Paul Sorensen s alk v assue ean v = siplify o D assued flow flucuaions change of variance wih rando rack discard assued o reveal flow flucuaions q Q / n inijes only easured wih D auocorrelaions g / []/ π ρ ρref bu, g (inijes) varies linearly wih n for rando discard FF I iplies widh variaion is doinaed by inijes Trainor 4

15 Flow Flucuaions III: Eccenriciy for GeV inijes fied wih cos(φ ): σ v /v ~ (π.4 ν.5 / n v ) " σ " π.4ν.4ν v g / π =.4ν we also observe: v = = n v ρ[]/ ρref ν n bin.5 ε ν.5 ρ[]/ ρ.45n ε is-aking ref (o a few percen) σ bin " v " g /n.5 npar ε iplies sar preliinary D auocorrelaion D projecion inije correlaions 5 5 b (f) copare nucl-ex/6 bu hus, ε op.3. ε par. poin-nucleon flucuaions 5 5 b (f) Trainor 5 σ ε,par /ε ~ ( / n par ε ) n / par ε, par " σ " σ ε v σ v, par ε sees o confir v ε rue flucuaions ay be sall

16 Wha Do We Learn Fro v ( p )? Cooper-Frye Foralis: ρ ( ) dn / d exp( [ ]/ T ) = exp( [cosh( y ) ]/ T ) µ ρ(, β ) exp( [ p u ]/ T ) anh( y ) = β boos = exp( [ γ { β p } ]/ T ) = exp( [ γ {cosh( y ) β sinh( y )} ]/ T ) = exp( [cosh( y y ) ]/ T ) µ source velociy Trainor 6 black-body radiaion Cooper-Frye expression black-body radiaion fro a boosed source relaivisic ransforaions siple on y

17 Quadrupole y Dependence and Boos β ( φ) = β [] + β []cos([ φ Ψ ]) r { ( p ) T} ρ[] = exp γ β ( φ) / quadrupole coponen dφρ[]( ; β( φ)) cos([ φ r ]) π Ψ V γ β[] p ρ[] ( ; β[]) i π T Fourier apliude v ρ[] ( ; β[]) γ β[] i p ρ[] ( ; β []) + ρ T nonflow ρ oal V v γ β[] ρoal = ρ[] ( ; β[]) i π p p T boosed M-B specru v boosed specra on y iniu-bias even saple zero-viscosiy hydro: pions D. Teaney PRC 68, 3 boosed source siply described Trainor 7 y v / p V / p = ρ ref v / p zero-viscosiy hydro: pions p (GeV/c) y Phys. Rev. D 74, 36 (6). boos: M-B.9 y =. S Q.8 σ boos = y β []=.8.6 H.6.5 p K π (. GeV/c) M-B nucl-ex/4933 GeV y

18 Quadrupole y Dependence and Boos β ( φ) = β [] + β []cos([ φ Ψ ]) r { ( p ) T} ρ[] = exp γ β ( φ) / quadrupole coponen dφρ[]( ; β( φ)) cos([ φ r ]) π Ψ V γ β[] p ρ[] ( ; β[]) i π T Fourier apliude v ρ[] ( ; β[]) γ β[] i p ρ[] ( ; β []) + ρ T nonflow ρ oal V v γ β[] ρoal = ρ[] ( ; β[]) i π p p T boosed M-B specru v boosed specra on y iniu-bias even saple zero-viscosiy hydro: pions boosed source siply described Trainor 8 y v / p V / p = ρ ref v / p zero-viscosiy hydro: pions p (GeV/c) y Phys. Rev. D 74, 36 (6). boos: M-B.9 y =. S Q.8 σ boos = y β []=.8.6 H.6.5 p K π (. GeV/c) M-B nucl-ex/4933 GeV y

19 Boos Cenraliy Dependence v.5 v / p.4. v.5 v / p proons proons. proons. proons 7-8% % y = y =.7.5. v p (GeV/c) Phys. Rev. C 7, 494 (5) proons 3 p (GeV/c) v / p proons y =.4-5% y 3 y 3 p (GeV/c) a sall fracion of eied hadrons carries he quadrupole ~ 5-%? σ boos.3 3 boos vs cenraliy y ~.5 ν ν quadrupole hadrons eerge fro a rapidly expanding cylinder Trainor 9 y N-N iniu bias β =.38 onopole y

20 Quadrupole y Dependence Hydro predicions apply o he nueraor of v, no he denoinaor Can hydro predic booss for differen ulipoles? Is a hydro descripion required? peried? Is ellipic flow acually glue-glue scaering? Trainor

21 Suary D angular auocorrelaions separae flow (quadrupole) fro nonflow (inijes) Quadrupole rends depend only on iniial paron condiions, no on syse evoluion or EoS Accurae daa inconsisen wih hydro expecaion v ε Opical Glauber eccenriciy beer odels low-x glue Claied flow flucuaions are a anifesaion of inijes Quadrupole y dependence reveals a boos disribuion Radial boos and quadrupole are presen in N-N collisions Ellipic flow ay represen a novel anifesaion of QCD Trainor