QCD sum rules for ρ-meson at finite density or temperature

Transcription

1 QCD sum rules for ρ-meson at finite density or temperature Youngshin Kwon Informal group seminar September 8, 1 contents based on : YK., Procura & Weise [ Phys. Rev. C78, 553 (8) ] YK, Sasaki & Weise [ Phys. Rev. C81, 653 (1) ]

2 Outline Introduction & motivation QCD sum rules for ρ-meson, in vacuum and in medium Finite energy sum rules at finite density Finite energy sum rules at finite temperature Summary and outlook

3 Goal: reliable framework of in-medium QCD sum rules for vector mesons constraints for the in-medium spectral properties Motivation Spontaneous chiral symmetry breaking: quark condensate: qq Goldstone bosons: π, K, etc. pion decay constant: f π 9.4 MeV mass splitting of chiral partners (e.g. ρ(77)-a 1 (15)) Spectral function Ρ a s GeV Chiral symmetry restoration in nuclear medium: degenerate chiral partners modifications of hadron spectrum Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 3 / 6

4 Restoration scenarios in medium Pole mass shift: masses of parity partners degenerate in medium. moving toward each other or going to zero (Brown-Rho). "ρ".8 Brown & Rho [ PRL 66, 7 (1991) ] Dropping Masses? V [τ > nπ ν Width broadening: τ ] A [τ > (n+1)π ν.6 τ ] "a 1 " in-mediumρ(77) spectral + cont. functions are broadly distributed. pert. QCD the continuum a 1 (16) merges + cont. the broadened spectral distributions. -Im Π V,A /(πs) [dim.-less].4 Spectral Function. "ρ" Dropping Masses? "a 1 " ν τ ] "ρ" pert. QCD pert. QCD 1 3 s [GeV ] "a 1 " Mass Mass Figure 5: Left panel: vector and axialvector spectral functions as measured in hadronic Melting Resonances? τ decays [5] with model fits using vacuum ρ and a 1 strength functions supplemented by perturbative continua [7]; right panel: scenarios for the effects of chiral symmetry restoration"ρ" on the in-medium vector- and axial-vector spectral functions. pert. QCD 3 Youngshin Kwon "a 1 " QCD sum rules for ρ-meson at finite density or temperature 4 / 6 Spectral Function Spectral Function Spectral Function Mass Melting Resonances?

5 Dilepton spectroscopy Dilepton production in RHIC ( γ l + l ): EM probe with pure information of the hot and/or dense region dilepton emission in-medium vector-meson spectroscopy σ σ dropping mass broadening Θ η dn/dm per MeV In-In NA6 semicentral dn < ch >=14 dη all p T broadening dropping mass Vacuumρ cockt. ρ DD M (GeV) Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 5 / 6

6 General review of QCD sum rules (in vacuum)

7 General review of QCD sum rules Current correlation function: Π µν (q) = i d 4 x e iq x T j µ (x) j ν () isovector vector- and axialvector-currents: j ρ µ (ū = 1 γ µ u d γ µ d ), j µ A = (ū 1 γ µ γ 5 u d γ µ γ 5 d ) invariant correlator: Π(q ) = 1 3 g µνπ µν (q) Operator product expansion (quark & gluon d.o.f.) at large Q = q : 1π Q Q Π(Q ) = c ln µ + c 1 Q + c Q 4 + Spectral representation (hadronic d.o.f.) at resonance region: Π(q ) = Π() + q Π () + q4 π ds Im Π(s) s (s q iɛ) Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 7 / 6

8 General review of QCD sum rules Borel transformation: 1π Π() + ds R(s) e s/m = c M + c 1 + c M + c 3 M 4 + dimensionless spectral function: R(s) 1π s Im Π(s) Coefficients c n : c = 3 c = π ( ) 1 + α s π +, c1 m q : negligibly small ) αs π G ± 6π ( m u ūu + m d dd.5 ±.4 GeV 4 m π f π = (.11 GeV) 4 Ioffe [ PPNP 56, 3 (6) ] [ Gellman-Oaks-Renner ] Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 8 / 6

9 General review of QCD sum rules Borel transformation: 1π Π() + ds R(s) e s/m = c M + c 1 + c M + c 3 M 4 + dimensionless spectral function: R(s) 1π s Im Π(s) Coefficients c n : c = 3 c = π ( ) 1 + α s π +, c1 m q : negligibly small ) αs π G ± 6π ( m u ūu + m d dd c 3 ( qq) uncertain value Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 8 / 6

10 General review of QCD sum rules Borel transformation: 1π Π() + ds R(s) e s/m = c M + c 1 + c M + c 3 M 4 + dimensionless spectral function: R(s) 1π s Im Π(s) expand for s M and compare term by term Coefficients c n : c = 3 c = π ( ) 1 + α s π +, c1 m q : negligibly small ) αs π G ± 6π ( m u ūu + m d dd c 3 ( qq) uncertain value Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 8 / 6

11 Finite energy sum rules Hierarchy of finite energy sum rules for moments of R(s): th moment : 1 st moment : s s ds R(s) = s c + c 1 1π Π() ds s R(s) = s c c Spectral distribution (resonance + continuum): R(s) resonance R(s) = R ρ (s) θ(s s) + R c (s) θ(s s ) continuum R(s) = Assumption for vector channel; s 4π f π s s Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 9 / 6

12 Finite energy sum rules Hierarchy of finite energy sum rules for moments of R(s): th moment : 1 st moment : s s ds R(s) = s c + c 1 1π Π() ds s R(s) = s c c Spectral distribution (resonance + continuum): R(s) resonance R(s) = R ρ (s) θ(s s) + c θ(s s ) continuum R(s) = Assumption for vector channel; s 4π f π s s Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 9 / 6

13 Consistency with current algebra Identification of s with Λ CSB 4π f π : mass [GeV] K K 1..5 a 1 ρ π Dipole Axial Dipole Goldstone Boson Gap 4π f π KSRF relation Kawarabayashi & Suzuki [ PRL 16, 55 (1966) ] Riazuddin & Fayyazuddin [ PR 147, 171 (1966) ] Weinberg sum rules Weinberg [ PRL 18, 57 (1967) ] m a1 = m ρ = 4π f π Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 1 / 6

14 onsistency with current algebra Introduction QCD sum rules In-medium FESR Summary Relation between Consistency s with and current the chiral algebra symmetry breaking scale Identification of s with Λ CSB 4π f π : Schematic example in vacuum, leading terms: 4πf π R(s) resonance s R ρ (s) = 1π m ρ g δ(s m ρ) continuum s KSRF relation Kawarabayashi & Suzuki [ PRL 16, 55 (1966) ] Riazuddin assume: & Fayyazuddin [ PR 147, 171 (1966) ] Weinberg s =4π sumf π rules Weinberg [ PRL 18, 57 (1967) ] m a1 = m ρ = 4π f π s th moment: ds R ρ (s) th = moment: 3 s s m ρ = g ds Rfπ ρ (s) = 3 s s 1st moment: ds s R ρ (s) 1st = moment: 3 4 s s g = π ds sr ρ (s) = 3 4 s m KSRF relation ρ = g f + π g = π Weinberg sum rules ( m a1 4πf π ) Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 1 / 6

15 Vacuum sum rule analysis Input: R ρ (s) from chiral effective field theory + vector mesons (VMD) 4 CHAPTER 4. VECTOR MESONS AT FINITE DENSITY 1 8 resonance R s 6 4 Ρ continuum s GeV s ds R ρ (s) = s c + c 1 s ds sr ρ (s) = s c c Figure 4.: Vector-isovector spectral function in vacuum showing the ρ resonance and continuum parts as described in the text and compared to e + e π + π (ρ resonance region) and e + e n π data with n even [6, 7]. ds sr(s) s = 1.14 ±.1 GeV 4π f π m ρ ds R(s) =.78 ±.1 GeV within an error band determined by the uncertainties of the input summarized in Table 4.1 and Eq. (4.1). This test turns out to be successful. The detailed analysis of uncertainties performed with Eq. (4.3) for the first moment is shown in Fig The resulting s = 1.14±.1 GeV is within % of the empirical 4πfπ 1.16 GeV using the physical value fπ = 9.4 MeV of the pion decay constant. The postulate, Eq. (4.3) identifying s with the scale characteristic of spontaneously broken chiral symmetry, appears to be working quantitatively. The relation between first and the zeroth moment, s s ds srρ(s) = F(s) ds Rρ(s), (4.33) Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 11 / 6

16 Vacuum sum rule analysis Input: R ρ (s) from chiral effective field theory + vector mesons (VMD) 1 st moment GeV Ρ N l.h.s : sr s s r.h.s : s c c s ds R ρ (s) = s c + c 1 s ds sr ρ (s) = s c c s = 1.14 ±.1 GeV 4π f π m ρ ds sr(s) ds R(s) =.78 ±.1 GeV Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 11 / 6

17 Sensitivity to threshold modeling replace the Heaviside step function with a ramp function: R(s) = R ρ (s) θ(s s) + R c (s) W(s) with the weight function W(s) 4 CHAPTER 4. VECTOR MESONS AT FINITE DENSITY 1 W(x) = for x s 1 x s 1 s s 1 for s 1 x s 1 for x s R s resonance Ρ continuum s GeV No dependence on details of the threshold modeling Figure 4.: Vector-isovector spectral function in vacuum showing the ρ resonance and continuum parts as described in the text and compared to e + e π + π (ρ resonance region) and e + e n π data with n even [6, 7]. within an error band determined by the uncertainties of the input summarized in Table 4.1 and Eq. (4.1). This test turns out to be successful. The detailed analysis of uncertainties performed with Eq. (4.3) for the first moment is shown in Fig The resulting s = 1.14±.1 GeV is within % of the empirical 4πfπ 1.16 GeV using the physical value fπ = 9.4 MeV of the pion decay constant. The postulate, Youngshin Kwon QCD sum rules for ρ-mesoneq. at(4.3) finiteidentifying density or s temperature with the scale characteristic of spontaneously broken 1 / 6

18 Introduction QCD sum rules tion (11). A test can bebetween performed replacing the step In-medium FESR resonance and continuum Summary regi function by a ramp function to yield a smooth transition The step f between resonance and continuumr(s) region, follows: = Ras ρ (s) Θ(s s) + Rc (s limitthe s1 ste The s Using th 1 limit R(s) = Rρ (s) Θ(s +R (s), (7) s)the c (s) Wfunction, where weight W (s),lowest isusing de the Usi Sensitivity threshold modeling the FIG low thus involves a uniquelytodetermined function of s : respect to variations in the slope (s s1 ) 1 of the ramp (the lot Wstep (s) the son where the with weight Wfunction (s), that is defined as function Wfunction, (s),weight thus confirming the function! " ( ss for x the s c + 3 ε1 c parametrization of the continuum is not restrictive: the ds ( " F (s ) =!, (6) into the continuum1 produces values ramping + c1 function with smooth s c + 3 ε step ban } replace the Heaviside a ramp function: s1 the narrow for xw (x) sthan 1 = 1 %) x of s that fall within (less un for s1wit up to the estimated uncertainties in the quantities ci and of the step function approach. certainty band We s note s1 x s R(s) = R (s) θ(s s) + R (s) W(s) 1 εn (the largest error being associated with αs (s )). ρthe c the best fit to the empirical at this W (x)point = sthat for s1 x sspectral (8) ( squared mass given by m ρ = F (s ) ".611±.13 GeV function has s " 1 GeV (see Fig.1). It can be1 s for xs( ss 1 s1 or m ρ ".78 ±.1 GeV, is very close to the physical ρ concluded that the present sum rule analysis and the ob( s with theas weight function W(s) ds meson mass expected. In fact the canonical relation served quantitative agreement of the continuum thresh# 1The step for xfunction s. behavior is recovere m = s / = πf turns out to be satisfied again limit s ρ π old with the chiral gap 4πfπ do not depend on details of at the % level, demonstrating the smallness of the nextthe threshold modeling. limit 1 s. for W (x) in the to-leading QCD corrections and of the condensate The term step function behavior is srecovered Using the function W (s), the c. limit s1 s s #GeV$ modifi Sets ofi lowest moments of the spectru (s1 modified + two s )/ Sets of int Using the function Wsthe (s), sum rules for = the isfy bot ana to continuum threshold modeling 1.15 Sets os B. Sensitivity ) ( sspectrum isfy * both for x s1 the lowest two moments of the R(s) become b den 3 isfy ε c + c + ds R (s) = s ) * ρ ( The question arises whether the quantitatively success 1.14 s clu no depe x s1 ful identification of the continuum W(x) = forthreshold s1 x s with s the ds R (s) = s c + 3 ε + c 1π Π() (t ρ 1 tion chiral symmetry sbreaking of isthe s1 scale (i.e. the consistency of now 1.13 the QCD sum with current algebra results) (c Risρ (s rule analysis T ( s )) intr old. As now 1 for x sparametrizais influenced by the schematic step-function is now s1 old. * Asspe sha tion (11). A test can be performed replacing the step (c (Rsρ(s )) ds W (s) ), (9) 1.1 old. function by a ramp function to yield a smooth transition s 5 s 6 3 dem c + ε 1 c ds*srρ1(s) ) between resonance and continuum region, as follows: ( s! = Lag s Slopeof3 W!s" #GeV $ ( c + ε1 c R(s) = Rρ (s) Θ(s s) + Rc (s) W (s), (7)ds srρ (s) = vec of s (determined FIG. 3: Dependence from Eqs.(9-31)) (c Rρ (s ))calc ( sw (s) describing on the slope (s s1 ) 1 of the ramp function }where Nothe dependence details of asthe threshold modeling weight function, on W (s), is defined the onset of the continuum in the vacuum sum rule. The grey s1 cor (c (s ))of the result ds sw (s). (3) band indicates the uncertainty obtained Rρ range for x s1 with step function parametrization of the continuum. s1 Th x s1 Sets of intervals [s1, s ] are then determ (8) W (x) = for s1 x s s s1 isfythen bothdetermined sum rules so (9,3), and thema sc Sets of intervals [s1, s ] are as to sat RW 1 for x s. isfy both sum rules andsum the RULES scale s defined by 4. (9,3), IN-MEDIUM s1/ 6+ s in c YoungshinThe Kwon QCD sum forin ρ-meson 1 step function behavior is recovered forrules W (x) the at finite density or temperature

19 Finite energy sum rules at finite density

20 Medium-modification of the sum rules The existence of nuclear matter causes breaking Lorentz invariance: two invariant correlator: longitudinal and transverse parts. choosing a preferred reference frame of the medium (q = ), longitudinal and transverse correlators coincide. Π L (ω, q = ) = Π T (ω, q = ) Π(ω, q = ) New operators with spin appear due to the broken Lorentz invariance: the first moment of in-medium FESR involves twist- operator (e.g. qγ ν D µ q ) to be considered. Medium-dependence in the OPE side contributes only to the condensates: non-perturbative contributions in OPE appear to be clearly separated into the condensates. medium effects are non-perturbative. Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 14 / 6

21 Density-dependence of OPE Hatsuda & Lee [ Phys. Rev. C 46, R34 (199) ] Expectation value: vacuum ground state of nuclear matter O O O ρn = N O N In-medium coefficients: c n c n + δc n δc = 3π 4 7 M() N σ N A 1 M N ρ N density dependence of gluon condensate ( M () N.88 GeV) density dependence of quark condensate (σ N 45 MeV) first moment of parton distribution from DIS (A 1 1.4) Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 15 / 6

22 Spectral functions at finite density ρ-meson spectral functions in nuclear medium ( ρ N = ρ =.17 fm 3 ): 1 nuclear matter vacuum Ρ RΡ s 1 KKW RW KKW: SU(3) chiral dynamics with vector meson dominance Klingl, Kaiser & Weise [ Nucl. Phys. A64, 57 (1997) ] RW: particle-hole excitations ( (13)-h and N (15)-h)) Rapp & Wambach [ Adv. Nucl. Phys. 5, 1 () ] Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 16 / 6

23 Results for ρ-meson at finite density In vacuum: s 1.14 GeV 4π f π 1 st moment GeV Ρ N Ρ l.h.s RW r.h.s l.h.s KKW m s s ds s R(s) ds R(s) In-medium KKW spectrum: s 1. ±. GeV s s = f π.87 f π m m : BR-scaling In-medium RW spectrum: s 1.9 ±.1 GeV s s m m.96 Kwon, Procura & Weise [ PRC 78, 553 (8) ] Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 17 / 6

24 Finite energy sum rules at finite temperature

25 Temperature-dependence of OPE Hatsuda, Koike & Lee [ Nucl. Phys. B 394, 1 (1993) ] Thermal expectation value: O O T = In-medium coefficients: c n c n + δc n δc = A 1 Tr O exp( H/T) Tr exp( H/T) m πt dy m π/t y ( ) m π T e y 1 T-dependence of gluon condensate T-dependence of quark condensate first moment of parton distribution from DIS Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 19 / 6

26 Vector & axialvector mixing with temperature Mixing of vector and axialvector: R V (s, T) = R V (s, ) ( 1 ɛ (T) ) + R A (s, ) ɛ (T) R A (s, T) = R A (s, ) ( 1 ɛ (T) ) + R V (s, ) ɛ (T) Eletsky & Ioffe [ PRD 47, 383 (1993), PRD 51, 371 (1995) ] the mixing parameter ɛ(t) is given by the thermal pion loop: ρ a 1 pion ɛ(t) = f π d 3 k 1 ω (k) 3 e ω/t 1 where ω = k + m π. m π T 6 fπ At critical temperature where ɛ 1, R V and R A become identical. Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature / 6

27 Mixing of finite-width spectrum Spectral functions with finite decay width: 6 Ρ T T 16 MeV 6 T T 16 MeV RΡ s,t 4 Ra1 s,t 4 a Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 1 / 6

28 Mixing of finite-width spectrum Sum rule result for vector channel: GeV s 4Πf Π T m Ρ Average ρ-meson mass: m ρ = s s ds s R ρ (s) ds R ρ (s) Comparison with ChPT: f π (T) = f π ( 1 1 ɛ(t) ) Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature / 6

29 GeV Simple test beyond V-A mixing Dropping pole mass in addition to the V-A mixing: The simplest ansatz (zero width): Ρ meson R ρ (s, ) = Fρ δ ( ) s m ρ R 4ΠfΠ T a (s, ) = Fa δ ( ) s m a R ρ (s, T) = R ρ (s, ) (1 ɛ) + R a (s, ) ɛ s 1 w o dropping mass 1.9 s w dropping mass.85 Brown-Rho scaling hypothesis: ( m ρ m ρ 1 1 ɛ(t) ) better agreement : s = 4π f π (T) Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 3 / 6

30 About four-quark condensates Sum rules for th and 1st moments: RHS quantities are accurately determined (pqcd and leading condensates) Sum rules for nd moment: involving four-quark condensates s ds s R(s) = s3 3 + c 3 c 3 = 6π 3 α s (ūγ µγ 5 λ a u dγ µ γ 5 λ a d) + 9 (ūγ µλ a u + dγ µ λ a d) qγ µ λ a q q=u,d,s Ground state saturation (κ = 1) ( qγ µ γ 5 λ a q) = ( qγ µ λ a q) = 16 9 κ qq valid approximation? Always κ > 3 and large uncertainties. Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 4 / 6

31 Summary The sum rules for the lowest two moments of the ρ-meson spectral function involve perturbative contributions and only leading condensates as small corrections: accuracy both in vacuum and in medium Chiral gap scale: 4π f π meaningful both in vacuum and in-medium. For broad spectral distributions, mass shift vs. broadening discussion must be specified in terms of first moment. Brown-Rho scaling as a statement involving the lowest two moments in the window of low-mass enhancement. Further step: extension to nonvanishing three-momentum. Youngshin Kwon QCD sum rules for ρ-meson at finite density or temperature 5 / 6

32 Thank you for your attention!