Non-Perturbative QCD at Finite Temperature
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1 Non-Perturbative QCD at Finite Temperature Pok Man Lo University of Pittsburgh Jlab Hugs student talk, Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
2 personal information institution: University of Pittsburgh advisor: Eric Swanson Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
3 outline 1 Non-perturbative QCD at zero temperature Introduction to non-perturbative physics Tools for studying non-perturbative QCD Diagrammatics Schwinger Dyson Equation Illustration: Gap Equation Revisited 2 Finite Temperature Field Theory Survey of basic formalism of Finite Temperature of Field Theory Finite Temperature Gap Equation Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
4 Outline 1 Non-perturbative QCD at zero temperature Introduction to non-perturbative physics Tools for studying non-perturbative QCD Diagrammatics Schwinger Dyson Equation Illustration: Gap Equation Revisited 2 Finite Temperature Field Theory Survey of basic formalism of Finite Temperature of Field Theory Finite Temperature Gap Equation Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
5 Introduction The progress of theoretical physics Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
6 Introduction The progress of theoretical physics The search for exact solution: Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
7 Introduction The progress of theoretical physics The search for exact solution: Newtonian physics: 3-body problem was insoluble Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
8 Introduction The progress of theoretical physics The search for exact solution: Newtonian physics: 3-body problem was insoluble QED: 2-body and 1-body problem was insoluble Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
9 Introduction The progress of theoretical physics The search for exact solution: Newtonian physics: 3-body problem was insoluble QED: 2-body and 1-body problem was insoluble now: 0-body, namely, the vacuum is insoluble Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
10 Introduction The progress of theoretical physics The search for exact solution: Newtonian physics: 3-body problem was insoluble QED: 2-body and 1-body problem was insoluble now: 0-body, namely, the vacuum is insoluble No body is too many!! Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
11 Introduction Difficutly in calculating arbitarily complicated diagram Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
12 Introduction Difficutly in calculating arbitarily complicated diagram Impossible to sum all the diagrams Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
13 Introduction Perturbation: expansion in α Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
14 Introduction Perturbation: expansion in α is not useful if α is not small Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
15 Introduction Perturbation: expansion in α is not useful if α is not small cannot explain non-perturbative phenomena: e.g bound state formation and spontaneous symmetry breaking Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
16 Introduction Perturbation: expansion in α is not useful if α is not small cannot explain non-perturbative phenomena: e.g bound state formation and spontaneous symmetry breaking k classical mechanics example: mass attached to a spring: A sin( m t) Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
17 Outline 1 Non-perturbative QCD at zero temperature Introduction to non-perturbative physics Tools for studying non-perturbative QCD Diagrammatics Schwinger Dyson Equation Illustration: Gap Equation Revisited 2 Finite Temperature Field Theory Survey of basic formalism of Finite Temperature of Field Theory Finite Temperature Gap Equation Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
18 illustration of the failure of perturbation an innocent looking function Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
19 illustration of the failure of perturbation an innocent looking function using Taylor expansion, we write f (x) = A 0 + A 1 x + A 2 x Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
20 illustration of the failure of perturbation an innocent looking function using Taylor expansion, we write A 0, A 1, A 2,... are all strightly 0! f (x) = A 0 + A 1 x + A 2 x Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
21 illustration of the failure of perturbation differential equation... Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
22 illustration of the failure of perturbation differential equation... f (x) + x 3 f = 0 Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
23 illustration of the failure of perturbation differential equation... f (x) + x 3 f = 0 f (x) = exp 1 x 2 Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
24 illustration of the failure of perturbation differential equation... f (x) + x 3 f = 0 f (x) = exp 1 x 2 any other method will work, other than perturbation! Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
25 Different Philosophy Perturbative Vs Non-perturbative: Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
26 Different Philosophy Perturbative Vs Non-perturbative: perturbative method: sum all diagrams up to certain order in α Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
27 Different Philosophy Perturbative Vs Non-perturbative: perturbative method: sum all diagrams up to certain order in α non-perturbative method: sum a certain class of diagram to all order Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
28 Different Philosophy Perturbative Vs Non-perturbative: perturbative method: sum all diagrams up to certain order in α non-perturbative method: sum a certain class of diagram to all order in some sense, non-perturbative method includes the whole perturbation method as the latter is just a classification of diagram by the coupling constant α Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
29 Different Philosophy Perturbative Vs Non-perturbative: perturbative method: sum all diagrams up to certain order in α non-perturbative method: sum a certain class of diagram to all order in some sense, non-perturbative method includes the whole perturbation method as the latter is just a classification of diagram by the coupling constant α approximation perturbation! Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
30 Different Philosophy Perturbative Vs Non-perturbative: perturbative method: sum all diagrams up to certain order in α non-perturbative method: sum a certain class of diagram to all order in some sense, non-perturbative method includes the whole perturbation method as the latter is just a classification of diagram by the coupling constant α approximation perturbation! QFT contains more than just the S-matrix and perturbation! Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
31 Outline 1 Non-perturbative QCD at zero temperature Introduction to non-perturbative physics Tools for studying non-perturbative QCD Diagrammatics Schwinger Dyson Equation Illustration: Gap Equation Revisited 2 Finite Temperature Field Theory Survey of basic formalism of Finite Temperature of Field Theory Finite Temperature Gap Equation Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
32 motivation perturbation does not help as we want to sum diagrams to all order in α new tools other than perturbation! in some sense, we need exact relations among diagrams! two particularly useful ones: Diagrammatics Schwinger Dyson Equation Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
33 method of partial sum summing a particular class of diagrams to all order Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
34 method of partial sum summing a particular class of diagrams to all order to be explicit, let s consider the diagrams for constructing a propagator Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
35 diagrammatics of propagator in general, to construct the propagator, we need to sum... Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
36 diagrammatics of propagator in general, we cannot sum them all! Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
37 diagrammatics of propagator in general, we cannot sum them all! but let s focus on the following class of diagram: Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
38 diagrammatics of propagator in general, we cannot sum them all! but let s focus on the following class of diagram: Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
39 diagrammatics of propagator the above procedure should be compared with the summing of geometric series: 1 1 x = 1 + x + x 2 + x Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
40 diagrammatics of propagator Two comments: Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
41 diagrammatics of propagator Two comments: inspired by the series, we know we can do better: Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
42 diagrammatics of propagator Two comments: inspired by the series, we know we can do better: Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
43 diagrammatics of propagator Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
44 diagrammatics of propagator p m Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
45 diagrammatics of propagator Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
46 diagrammatics of propagator dynamical mass Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
47 Schwinger Dyson Equation an exact relations among n-point functions Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
48 Schwinger Dyson Equation an exact relations among n-point functions basic idea: dx d dx f (x) > 0 Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
49 Schwinger Dyson Equation an exact relations among n-point functions basic idea: dx d dx f (x) > 0 Z = DψDψDGe i(s+r ηψ+ψη+...) / DψDψDGe i(s) Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
50 Schwinger Dyson Equation an exact relations among n-point functions basic idea: dx d dx f (x) > 0 Z = DψDψDGe i(s+r ηψ+ψη+...) / DψDψDGe i(s) using the fact DψDψDG( δ R ηψ+ψη+...) δψ ei(s+ ) = 0 Z = e iw Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
51 Schwinger Dyson Equation as an illustration, for the Hamiltonian: H = d 3 xψ x γ0 [ i γ + m]ψ x G d 3 xd 3 yv x y ψ x T a ψ x ψ y T a ψ y Schwinger Dyson Equation (iγ x m) δ2 W + 2G δη x δη y d 4 zv γ 0 T a δ 2 W i γ 0 T a = δ xy δη x δη z δη z δη y δ2 W Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
52 Outline 1 Non-perturbative QCD at zero temperature Introduction to non-perturbative physics Tools for studying non-perturbative QCD Diagrammatics Schwinger Dyson Equation Illustration: Gap Equation Revisited 2 Finite Temperature Field Theory Survey of basic formalism of Finite Temperature of Field Theory Finite Temperature Gap Equation Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
53 Gap Equation Revisited to illustrate the main idea above, we look at the gap equation Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
54 Gap Equation Revisited to illustrate the main idea above, we look at the gap equation consider the Hamiltonian: H = d 3 xψ x γ0 [ i γ + m]ψ x G d 3 xd 3 yv x y ψ x T a ψ x ψ y T a ψ y for the contact case: V δ (3) this corresonds to the case where the quark exchange an instantaneous gluon locally Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
55 Gap Equation Revisited the gap equation tell you how the quark is dressed according to the Hamiltonian namely, the dynamical mass generated: before showing you the answer, we need to perform our final dressing... Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
56 the final dressing in fact we miss some diagram... Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
57 the final dressing Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
58 the final dressing Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
59 Gap Equation Revisited the gap equation for the general case: Generalized Gap Equation M( k) = m G Tr[TT ] N c M( k) in general is a function of k d 3 k (2π) 3 V k k [M( k ) E k it dictates how the mass is dynamically generated ˆk ˆk k M( k) E k ] k Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
60 Gap Equation Revisited for the contact case, we have G > G_critical G < G_critical Dynamical Mass Generation M current mass m Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
61 Outline 1 Non-perturbative QCD at zero temperature Introduction to non-perturbative physics Tools for studying non-perturbative QCD Diagrammatics Schwinger Dyson Equation Illustration: Gap Equation Revisited 2 Finite Temperature Field Theory Survey of basic formalism of Finite Temperature of Field Theory Finite Temperature Gap Equation Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
62 motivation FTFT is needed to study the physics of QGP, deconfinement and chiral restoration when calculating observables in QFT, we only calculate the vacuum expectation value at finite temperature, excited states start to contribute, the interesting quantity should be the thermal average of the observables we expect n E k to enter QFT partition function dictates the equilibrium Finite Temperature QFT Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
63 Outline 1 Non-perturbative QCD at zero temperature Introduction to non-perturbative physics Tools for studying non-perturbative QCD Diagrammatics Schwinger Dyson Equation Illustration: Gap Equation Revisited 2 Finite Temperature Field Theory Survey of basic formalism of Finite Temperature of Field Theory Finite Temperature Gap Equation Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
64 basics of FTFT the partition function: Z = Tr[e βh ] = dq < q e βh q > observables are given by working in imaginary time τ = it O = Tr[e βh O] =<< q O q > a corresponding path integral representation of the partition function, with Periodic/Antiperiodic boundary condition the boundary condition motivates the use of Matsubara Green s function: dk 0 1 β Σ ω n Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
65 Outline 1 Non-perturbative QCD at zero temperature Introduction to non-perturbative physics Tools for studying non-perturbative QCD Diagrammatics Schwinger Dyson Equation Illustration: Gap Equation Revisited 2 Finite Temperature Field Theory Survey of basic formalism of Finite Temperature of Field Theory Finite Temperature Gap Equation Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
66 Gap equation in Finite Temperature Generalized Gap Equation M( k) = m G Tr[TT ] N c with n E k = 1 e βe k +1 d 3 k (2π) 3 V k k [M( k ) E k ˆk ˆk k M( k) E k ](1 2n E k k ) Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
67 summary non-perturbative physics with e 1 x 2 summation of a class of diagram with 1 1 x Schwinger Dyson Equation with dx d dx f (x) Finite Temperature Field Theory with Tr[e βh ] Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
68 thank you Pok Man Lo (UPitt) Non-Perturbative QCD at finite Temperature / 39
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