Jochen Wambach. to Gerry Brown

Transcription

1 Real-Time Spectral Functions from the Functional Renormalization Group Jochen Wambach TU-Darmstadt and GSI Germany to Gerry Brown an inspring physicist and a great human being November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG

2 Real-Time Spectral Functions from the Functional Renormalization Group based on: K. Kamikado, N. Strodthoff, L. von Smekal and J. Wambach, arxiv: [hep-ph] R.-A. Tripolt, N. Strodthoff, L. von Smekal and J. Wambach, arxiv: [hep-ph] November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 2

3 Outline I motivation I theoretical setup: - Functional Renormalization Group (FRG) - analytic continuation from imaginary to real time I results for the O(4) model at T =0 I results for the quark-meson model at T > 0 and µ>0 I summary and outlook November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 3

4 Motivation I FRG well suited for the description of critial phenomena! phases of QCD matter I formulated in imaginary time! no spectral information (Minkowski space-time) I problem for real-time observables: analytic continuation - e.g. Lattice QCD: numerical reconstruction of real-time correlation functions (MEM), difficult if Euclidean data is not dense and precise enough I we use non-perturbative FRG flow equations for two-point correlation functions and perform the analytic continuation on the level of the flow equations November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 4

5 Functional Renormalization Group a primer partition function: (scalar field (x)) Z Z [j] =e W [j] = [D ] e S[ ]+R d 4 x (x)j(x) generating functional: W [j] = Z [D ] e S[ ] = h (x)i '(x) j(x) Z [0] j=0 two-point correlation function: (Euclidean) 2 W [j] = h (x) (y)i h (x)ih (y)i G(x, y) j(x) j(y) j=0 effective action: ((Legendre transform of W) Z ['] = W [j]+ d 4 x '(x)j(x) stationarity condition: and thermodynamic potential: ['] ' '='0 = 0;! (T )= T V ['0] November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 5

6 Functional Renormalization Group Wilsonian coarse graining at given resolution scale k split into low- and high-frequency modes: (x) = qapplek (x)+ q>k (x) Z Z! Z [j] = [D ] qapplek [D ] q>k e S[ ]+R d 4 x j ; lim Z k [j] =Z [j] k!0 {z } =Z k [j] regulator R k (q): Z Z lim Rk (q) = 0 k [j] = [D ] e S[ ] S k [ ]+ R d 4 x j k!0 Rk (q) = lim k! effective action: S k [ ]= 2 Z! k ['] = ln Z k [j]+ d 4 x '(x)j(x) Z d 4 q (2 ) 4 ( q)r k (q) (q) {z } acts like a mass term m k S k ['] k interpolates between k = (no fluct.) and k = 0(full quantum action) lim k! k ['] =S[']; lim k!0 k ['] = November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 6 [']

7 Functional Renormalization Group flow equation C. Wetterich (993) flow equation for the effective k k ['] = 2 k R k h (2) k + R k i 2 (2) k (q) = k ['] '( q) '(q) [C. Wetterich, Phys. Lett. B30 (993) 90] November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 7

8 O(4) Model flow equation effective action: ( Local Potential Approximation ) ' =(',..., ' 4)=(~, ) Z k ['] = d 4 x 2 (@µ')2 + U k (' 2 ) c ; ' 2 = ' i' i = 2 + ~ 2 (2) (2) k,ij (q) = k, (q) ' i' j ij + (2) ' 2 k, (q) 'i'j ' 2 flow equation for the effective k U K = I +3I ; I i = 2 Trq k R k (q) for one gets R k (q) =(k 2 q 2 ) (k 2 q 2 ) i (2) k,i (q)+rk(q) I i = k E i, with E = p k 2 +2U 0 ; E = p k 2 +2U 0 +4U 00 ' 2 ; U '='0 etc. November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 8

9 O(4) Model flow equations for 2-point functions taking two functional derivatives of the flow equation for k yields approximation: 3 4 (3)! k ijm ' m ' j ' ; (4)! k i ijmn ' n ' m ' j ' i ensures that truncation is consistent with effective k (2) k, (p = 0) = 2@k U0 k (2) k, (p = 0) = 2@ k U 0 k +4@ k U 00 ' 2 and yields Nambu-Golstone boson in the chiral limit November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 9

10 O(4) Model analytic continuation I first solve flow equation for the effective k U k I substitute p 0 by continuous real frequency! (2),R (2),R k,j (!) = lim k,j (p 0 = i(! + i ),~p = 0); for j =,!0 I then solve flow equations k (2),R (2),R k, k k at global minimum of U k!0 I finally, spectral functions are given by discontinuity of the propagators, i.e. j(!) = Re (2),R j (!) (2),R Im j (!) 2 + Im 2 ; (2),R j (!) (2),R (2),R j (!) = lim k,j (!) k!0 November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 0

11 Results for O(4) Model in vacuum Re (2),R (!) and Im (2),R (!) pion: sigma: Re Γ (2) Im Γ (2) Re Γ (2) Im Γ (2) Γ (2) [Λ 2 ] Γ (2) [Λ 2 ] ω [MeV] ω [MeV] [K. Kamikado, N. Strodthoff, L. von Smekal and J. Wambach, arxiv: [hep-ph]] November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG

12 Results for O(4) Model in vacuum spectral functions sigma and pion spectral functions physical pion mass sigma and pion spectral functions chiral limit 0 ρ π ρ σ 00 ρ π, m π =37 MeV ρ σ, m π =37 MeV ρ π, m π =6 MeV ρ σ, m π =6 MeV 0 ρ [Λ -2 ] ρ [Λ -2 ] e-05 e ω [MeV] e ω [MeV] [K. Kamikado, N. Strodthoff, L. von Smekal and J. Wambach, arxiv: [hep-ph]] November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 2

13 Spectral Functions in a Thermal Medium Quark-Meson Model effective action: Z k [,, '] = n d 4 + h( + i~ ~ 5) µ (@µ')2 + U k (' 2 ) c o I effective low-energy model for QCD with two flavors I describes spontaneous and explicit chiral symmetry breaking I flow equation for the effective action: November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 3

14 Spectral Functions in the Medium phase diagram, masses and order m = 0 MeV 500 ms 400 my mp fp T = 0 MeV 500 ms my mp fp [R.-A. Tripolt, N. Strodthoff, L. von Smekal and J. Wambach, arxiv: [hep-ph]] November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG

15 Spectral Functions in the Medium flow of the two-point functions I quark-meson vertices given by (2,) = h, (2,) ~ = ih 5 ~ I meson vertices from scale-dependent effective potential: (0,3) i j m, (0,4) i j m n November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 5

16 Results for Quark-Meson Model spectral functions at µ -2 UV D T=0 MeV r p 2 : 0! 2: 0! 3: 0! 3 4: 0! 5: 0! : 0! r s [R.-A. Tripolt, N. Strodthoff, L. von Smekal and J. Wambach, arxiv: [hep-ph]] November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 6

17 Results for Quark-Meson Model spectral functions at µ -2 UV D T= MeV r p 2 : 0! 2: 0! 3: 0! : 0! 5: 0! 6: 0! r s [R.-A. Tripolt, N. Strodthoff, L. von Smekal and J. Wambach, arxiv: [hep-ph]] November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 7

18 Results for Quark-Meson Model spectral functions at µ -2 UV D T=50 MeV r p : 0! 2: 0! 3: 0! 4: 0! 5: 0! 6: 0! r s [R.-A. Tripolt, N. Strodthoff, L. von Smekal and J. Wambach, arxiv: [hep-ph]] November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 8

19 Results for Quark-Meson Model spectral functions at µ -2 UV D T= UV D T=250 MeV : 0! r p r s r p r s 2: 0! 3: 0! : 0! 5: 0! 6: 0! [R.-A. Tripolt, N. Strodthoff, L. von Smekal and J. Wambach, arxiv: [hep-ph]] November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 9

20 Temperature Evolution animation November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 20

21 Results for Quark-Meson Model spectral functions at finite -2 UV D r p 2 m=200 MeV : 0! 2: 0! 3: 0! 3 4: 0! 5: 0! : 0! r s [R.-A. Tripolt, N. Strodthoff, L. von Smekal and J. Wambach, arxiv: [hep-ph]] November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 2

22 Results for Quark-Meson Model spectral functions at finite -2 UV D m=292 MeV : 0! r p r s 2 3 2: 0! 3: 0! 4: 0! 5: 0! : 0! [R.-A. Tripolt, N. Strodthoff, L. von Smekal and J. Wambach, arxiv: [hep-ph]] November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 22

23 Results for Quark-Meson Model spectral functions at finite -2 UV D m=292.8 MeV : 0! r p r s 2 3 2: 0! 3: 0! 4: 0! 5: 0! : 0! [R.-A. Tripolt, N. Strodthoff, L. von Smekal and J. Wambach, arxiv: [hep-ph]] November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 23

24 Results for Quark-Meson Model spectral functions at finite -2 UV D m= UV D m=400 MeV : 0! r s r p r p r s 2: 0! 3: 0! 3 4: 0! 5: 0! 6: 0! [R.-A. Tripolt, N. Strodthoff, L. von Smekal and J. Wambach, arxiv: [hep-ph]] November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 24

25 Summary and Outlook I presented a tractable method to obtain hadronic spectral functions at finite T and µ from the FRG I involves analytic continuation from imaginary to real frequencies at level of flow equations for 2-point functions I results reveal complicated structure for in-medium spectral functions I inclusion of finite external spatial momenta will allow for calculation of transport coefficients like shear viscosity November 24, 203 TU Darmstadt Jochen Wambach Real Time Spectral Functions from the FRG 25