Finite Temperature Field Theory + Hard Thermal Loops

Transcription

1 Finite Temperature Field Theory + Hard Thermal Loops Michael Strickland Kent State University QCD Seminar Series I Sept 24,

2 Quantum Field Theory ParBcles are excitabons of different quantum fields QCD à quark and gluon fields When gauge- fixing is performed, an auxiliary ghost field is introduced in order to enforce gauge invariance In the weak coupling limit we can efficiently describe the propagabon and interacbon of parbcles using Feynman diagrams 2

3 The simplest theory Real- valued scalar φ 4 L = 1 µ 1 2 m2 2 g 4! 4 + L KineBc energy mass InteracBon RenormalizaBon Counterterms = i (p) (p) = 1 p 2 m 2 = ig 3

4 QCD Feynman Rules Metric tensor Gluon Propagator Quark Propagator Gauge fixing parameter (General covariant gauge) Ghost Propagator Looks like a boson but acts like a fermion (Grassmannian algebra) Strong coupling constant Quark flavor is conserved 4 x 4 Dirac Matrix SU(N c ) Special Unitary Matrix t a = ½ λ a (Gell- Mann Matrices) Fundamental representaaon: N c x N c matrix from set of N c 2-1 matrices 4

5 QCD Feynman Rules 3- gluon vertex 4- gluon vertex ghost- gluon vertex SU(N c ) structure constant [t a,t b ] = i f abc t c Adjoint RepresentaAon: (T a ) bc = - i f abc 5

6 QCD Feynman Rules One gluon exchange Something more complicated Let s focus our acenbon on a class of graphs called self- energy (or polarizabon) graphs: Π +... Σ +... Above: Leading order (one- loop) gluon polarizabon and quark self- energy graphs. 6

7 How to approach finite temperature? There are (at least) two ways to study quantum field theories at finite temperature 1. The Matsubara formalism (imaginary Bme) 2. The Schwinger- Keldysh formalism (real Bme) For systems in equilibrium, the Matsubara formalism is the most straighjorward to apply (follows naturally from thermal average) For non- equilibrium systems, one should use the Schwinger- Keldysh formalism 7

8 The Equilibrium ParAAon FuncAon Density of states and the parbbon funcbon Z =Tr[e ˆ ] = 1 T ˆ = Ĥ µ ˆN Eq. density of states In finite- temperature QFT we compute expectabon values of operators hôi = Tr[e Ĥ Ô] Tr[e Ĥ ] Thermal density of states e looks like Bme- evolubon operator in imaginary Bme Ĥ = it 8

9 Green s funcbons are Bme- ordered correlabon funcbons Expanding out explicitly Green s FuncAons G(x 1,,x n ) h Tr e i Ĥ T [ (x 1 ) (x n )] h i Tr e Ĥ G(x 1,,x n )= 1 h Tr e Ĥi X n e E n hn T [ (x 1 ) (x n )] ni In the limit T à 0 (β à + ) only the ground state (vacuum) solubon survives lim G(x 1,,x n )=h0 T [ (x 1 ) T!0 (x n )] 0i 9

10 Kubo- MarAn- Schwinger symmetry e The density of states can be seen as a Bme translabon operator with an imaginary Bme ship Ĥ e Ĥ (t i, x)e Ĥ = (t, x) Consider correlabon h G =Tr e i Ĥ T [ (t i, x i ) ] if t i is the smallest Bme h i G =Tr e Ĥ T [ ] (t i, x i ) h i G =Tr T [ ]e Ĥ (t i i, x i ) h i G =Tr e Ĥ T [ (t i i, x i ) ] Bosonic correlators are periodic with period iβ Similarly, G(t i )=G(t i i ) Fermionic correlators are ana- periodic with period iβ G(t i )= G(t i i ) Both relabons above hold for any correlabon order at any order of perturbabon theory 10

11 Matsubara Frequencies Bosonic correlators are periodic with period iβ G(t i )=G(t i i ) Fermionic correlators are ana- periodic with period iβ G(t i )= G(t i i ) Periodic funcaon in Imag. Time Fourier Transform à Discrete Fourier Transform Period given by β = 1/T Bosons have even Matsubara frequencies! n =2n T n =0, 1, 2, p 0 =!! i! n AnA- periodic funcaon in Imag. Time Fourier Transform à Discrete Fourier Transform Period given by β = 1/T Fermions have odd Matsubara frequencies! n =(2n + 1) T n =0, 1, 2, p 0 =!! i! n Z d 4 p (2 ) 4! P Z P T X P 0 =2n T Z d 3 p (2 ) 3 Z d 4 p (2 ) 4! P Z {P } T X P 0 =(2n+1) T Z d 3 p (2 ) 3 11

12 A Quick Aside Another way to understand the appearance of discrete modes is to look at the structure of a thermal distribubon funcbon in the complex plane. Let s consider a Bose distribubon first It has singularibes at complex values of p 0 e p 0 =1! p 0 = i2n T Likewise for a Fermi- Dirac distribubon Has singularibes when f B (p 0 )= f F (p 0 )= 1 e p e p 0 +1 e p 0 = 1! p 0 = i(2n + 1) T Im[p 0 ] X X X X X X X X Fermions X Bosons Re[p 0 ] 12

13 Imaginary Time (Euclidean) Formalism Consider the propagator in a simple real scalar φ 4 quantum field theory as an example 1 p 2 m 2! 1 (i! n ) 2 p 2 m 2 = 1! 2 n + p 2 + m 2 = 1 P 2 + m 2 p =(p 0, p i ) P =(! n, p i ) Minkowski metric Euclidean metric P Ring diagrams P Subleading graph Q This graph is infrared divergent! All ring diagrams must be resummed. 13

14 Hard Thermal Loops In a high temperature system we must sum a certain class of diagrams which have hard internal (loop) momentum p hard ~ T and sop external momentum p soft ~ gt Π ( ) g2 T 2 lim T (!,p)=0!!0 lim!!0 L(!,p)=m 2 D At finite temperature there are transverse and longitudinal gluons 1 T (p) = p 2 T (p) 1 L(p) = p 2 + L (p) Gluons acquire a temperature dependent mass which is proporbonal to the temperature m 2 D = 1 3 N c N f g 2 T 2 14

15 HTL The Basic Physics T (p) = L(p) = 1 p 2 T (p) 1 p 2 + L (p) + lim T (!,p)=0!!0 lim L(!,p)=m 2 D!!0 à lim!!0 lim!!0 T (p) = L(p) = 1! 2 k 2 1 k 2 + m 2 D Screening of chromoelectric interacbon with screening length r D = 1/m D SBll long range chromomagnebc interacbons in this limit (These are screened at higher order with m M ~ g 2 T à magnebc mass) V Coloumb (r) = r! V Debye (r) = r e m Dr A test charge polarizes the parbcles of the plasma and screens its charge NB: HTL also includes effect of Landau Damping (ask me later if you re interested) 15

16 HTL CollecAve Modes m 2 D = 1 3 N c N f g 2 T 2 16

17 ApplicaAon to thermodynamics QCD free energy known up to three loops since 1994 (Arnold, Zhai, and Khastening) Series in g (not g 2 ) due to plasma screening effects: Debye mass m D ~ gt Very poorly convergent Need temperatures on the order of T ~ 10 5 GeV Similar results emerge in QED and scalar theories not QCD- specific Can improve convergence by using the right DOFs from the beginning à Hard Thermal Loop perturbabon theory (HTLpt) Ideal Gas 17

18 HTLpt AcAon Can express an infinite number of HTL- dressed n- point funcbons concisely in terms of an HTL effecbve acbon, L HTL Expanding L HTL to quadrabc order in A gives dressed propagator (2- point funcbon) Expanding to cubic order in A gives the dressed gluon three- vertex Expanding to quarbc order in A gives dressed gluon four- vertex And so on... contains an infinite number of higher order verbces which all exactly sabsfy the appropriate Slavnov- Taylor idenbbes Γ 2 Γ 3 Γ 4 Γ n 18

19 3- loop CalculaAon Now simply compute all contribubons up to three loops including dressed propagators and verbces 19

20 Finite µ and T Result N. Haque, J.O. Andersen, M.G. Mustafa, MS, N. Su, hcp://arxiv.org/abs/

21 Pressure vs Temperature µ B = 0 MeV 1.0 Μ B 0MeV P P ideal T MeV NNLO HTLpt Wuppertal Budapest HotQCD 21

22 Pressure vs Temperature µ B = 400 MeV 1.0 Μ B 400 MeV 0.8 P P ideal T MeV NNLO HTLpt Wuppertal Budapest 22

23 χ 2 vs Temperature Χ2 Χ 2 f MS 344 MeV NNLO HTLpt Wuppertal Budapest RBC B, N t MILC, N t 8 TIFR, N t 8 BNL Bielefeld HISQ T MeV 23

24 χ 4 vs Temperature 1.0 MS 344 MeV Χ 4 f Χ 4 B U NNLO HTLpt BNL BI U, N t 8 p4 BNL BI S, N t 12 p4 BNL BI B, N t 8 HISQ BNL BI U, N t 8 HISQ 0.4 Wuppertal Budapest B Wuppertal Budapest U T MeV 24

25 Conclusions Very brief intro to QFT at finite temperature Hard thermal loops are a resummed class of graphs which incorporate the physics of electric screening and more Naïve applicabon of perturbabon theory gives poorly convergent series HTLpt reorganizes the calculabon around the high- temperature limit of QCD Three- loop results are in very good agreement with la{ce QCD data especially considering that there are no tunable parameters 25

26 Convergence? N. Haque, M.G. Mustafa, and MS, hcp://arxiv.org/abs/ J.O Andersen, L.E. Leganger, MS, and N. Su., arxiv: