Towards higher order gauge corrections to the QCD phase diagram at - PowerPoint PPT Presentation

Towards higher order gauge corrections to the QCD phase diagram at strong coupling Wolfgang Unger, Uni Bielefeld Sign 2015, Debrecen 02.10.2015 Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 1

Outline Outline Outline Outline Outline SC-LQCD phase diagram and its O ( β ) correction Review of gauge integration Young projectors and its application Characters of the symmetric group S n Higher order gauge corrections Aim of the talk: show how to compute all gauge integrals for staggered fermions in principle apply method for O ( β 2 ) Relation to other talks: MDP treatment instead of AFMC as discussed by Terukazu Ichihara Full gauge integration instead Z 3 resummation as discussed by Falk Bruckmann Attempt to do the “cumbersome” Young combinatorics mentioned by Pavel Buividovich Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 2

Motivation: Why Lattice QCD at strong coupling? Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 3

Phase Diagram of QCD: Phase Diagram of QCD: Phase Diagram of QCD: Phase Diagram of QCD: Phase Diagram of QCD: Phase diagram in the plane ( µ B , T ) with µ B baryon chemical potential : T [ MeV ] μ/ T = 1 E E ⟨ ¯ ψ ψ⟩= 0 200 a a r r l l y y U U Q uark n n T c i i v v G luon e e Crossover r r s s e CP ??? e P lasma 〈    〉≠ 0 100 Terra Incognita Hadronic Matter Nuclear Vacuum Matter μ B [ GeV ] 0 1 speculated phase diagram known phase diagram But: only little is known from lattice QCD Big question: does the critical end point of QCD exist? Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 4

Conventional vs. alternative approach to lattice QCD Conventional vs. alternative approach to lattice QCD Conventional vs. alternative approach to lattice QCD Conventional vs. alternative approach to lattice QCD Conventional vs. alternative approach to lattice QCD Sign problem is representation dependent! Conventional approach: Integrate out quarks → quark determinant det M MC simulations on colored gauge fields, requires supercomputers Direct MC simulation at µ B > 0 impossible : sign problem Alternative approach: [Rossi & Wolff, Nucl. Phys. B 258 (1984)] Integrate out gluons before the fermions! → Formulation of lattice QCD in color singlets MC on color singlets instead of colored gluons Advantages: Sign problem is almost absent , full phase diagram can be studied! Faster simulations , chiral limit is cheap! Drawback: 6 β = g 2 → 0 In the past limited to very strong coupling: i. e. coarse lattices Systematic inclusion of gauge corrections for β > 0 now possible However: Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 5

The Phase Diagram in the Strong Coupling Limit The Phase Diagram in the Strong Coupling Limit The Phase Diagram in the Strong Coupling Limit The Phase Diagram in the Strong Coupling Limit The Phase Diagram in the Strong Coupling Limit Comparison of phase boundaries ( T c , µ c ) for massless quarks [de Forcrand & U. (2011)] : T [lat. units] T [ MeV ] 1.8 〈 - 1.6 〈    〉= 0 ψ ψ 〉 = 0 200 1.4 Q uark G luon 2 nd order 2 nd order 1.2 TCP O(4) tricritical P lasma 1 point 〈    〉≠ 0 0.8 100 1 st order 1 st order 0.6 Hadronic Matter 〈 - 0.4 ψ ψ 〉 ≠ 0 Quarkyonic Nuclear Color 0.2 Matter ? Vacuum Matter Super- Neutron Stars Neutron Stars conductor? 0 0 0.5 1 1.5 2 2.5 3  B [ GeV ] 1 µ B [lat. units] measured at strong coupling speculated in continuum Similar to standard scenario of continuum QCD [Stephanov et al. PRL 81 (1998)] However, nuclear and chiral transition coincide at β = 0 Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 6

Gauge corrections to the phase diagram at strong coupling Gauge corrections to the phase diagram at strong coupling Gauge corrections to the phase diagram at strong coupling Gauge corrections to the phase diagram at strong coupling Gauge corrections to the phase diagram at strong coupling State of the art: O ( β ) corrections for SU(3) [Langelage, de Forcrand, Philipsen & U., PRL 113 (2014)] 2nd order T [lat. units] T [lat. units] T [ MeV ] tricritical 1.6 200 point 1.4 1.2 2 nd order TCP 1 O(4) 1st order 0.8 0.6 1 st 100 0 0.4 o r 0.2 d e ? 0.2 0.4 nuclear r 0.6 CEP 0 0.8 0 β 0.5 1 1 1.2 1.5 2 1.4 Vacuum Nuclear 2.5 µ B [lat. units] 3 1.6 Matter  B [ GeV ] 1 Questions we want to address: Do the nuclear and chiral transition split? Does the tricritical point move to smaller or larger µ as β is increased? Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 7

Connection Between Strong Coupling and Continuum Limit? Connection Between Strong Coupling and Continuum Limit? Connection Between Strong Coupling and Continuum Limit? Connection Between Strong Coupling and Continuum Limit? Connection Between Strong Coupling and Continuum Limit? T / m B One of several possible scenarios for the extension to the continuum: 2 nd back plane: strong coupling order phase diagram ( β = 0), N f = 1 1 st order front plane: continuum phase μ/ m B diagram ( β = ∞ , a = 0) due to fermion doubling, corresponds to N f = 4 in continuum (no rooting) β Chiral Transition Nuclear Transition Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 8

Sign problem: Are higher order gauge corrections feasible? Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 9

Severity of the Sign Problem in the SC-limit Severity of the Sign Problem in the SC-limit Severity of the Sign Problem in the SC-limit Severity of the Sign Problem in the SC-limit Severity of the Sign Problem in the SC-limit sign problem is mild across the 2nd order phase boundary , large volumes possible along 1st order boundary, sign problem gets stronger, smaller volumes suffice 0.0002 a 4 ∆ f φ =atan( µ /T) 2nd order: 0.0 0.1 0.2 0.3 0.00015 0.4 � sign � ∼ e − V T ∆ f ( µ ) open symbols 16 3 x4 0.5 filled symbols: 8 3 x4 0.6 16 3 × 4: � sign � ≈ 0 . 1 first order: 0.7 at tricritical point 0.0001 ∆ f decreases with N τ , vanishes in 5e-05 continuous time limit 0 0.25 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 ρ =a(T 2 + µ 2 ) 1/2 Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 10

Severity of the Sign Problem at O ( β ) Severity of the Sign Problem at O ( β ) Severity of the Sign Problem at O ( β ) Severity of the Sign Problem at O ( β ) Severity of the Sign Problem at O ( β ) sign problem at O ( β ) and µ = 0 becomes stronger below T c but milder above T c <sign> ∆ f 1e+00 1e-05 6 3 x4, β =0.0 6 3 x4, β =0.5 6 3 x4, β =1.0 9e-06 8 3 x4, β =0.0 8 3 x4, β =0.5 1e+00 8 3 x4, β =1.0 8e-06 1e+00 7e-06 1e+00 6e-06 5e-06 1e+00 4e-06 1e+00 3e-06 1e+00 2e-06 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 T/T c T/T c Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 11

Severity of the Sign Problem at O ( β ) Severity of the Sign Problem at O ( β ) Severity of the Sign Problem at O ( β ) Severity of the Sign Problem at O ( β ) Severity of the Sign Problem at O ( β ) T ≃ µ TCP sign problem at O ( β ) and µ T TCP remains constant below TCP and becomes milder above T c <sign> ∆ f 1e+00 7e-05 6 3 x4, β =0.0 6 3 x4, β =0.1 6 3 x4, β =0.2 8 3 x4, β =0.0 1e+00 6e-05 8 3 x4, β =0.1 8 3 x4, β =0.2 1e+00 5e-05 9e-01 4e-05 9e-01 3e-05 9e-01 2e-05 9e-01 1e-05 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 ρ / ρ c for µ /T=0.68 ρ / ρ c for µ /T=0.68 Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 12

Gauge Integrals: Review of what is known Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 13

Gauge integrals and Invariants Gauge integrals and Invariants Gauge integrals and Invariants Gauge integrals and Invariants Gauge integrals and Invariants In the strong coupling limit, link integration factorizes! One-link gauge integrals sufficient, in the strong coupling limit � dU e tr [ U M† + M U † ] J 0 ( x , y ) = SU ( N c ) N f � χ f χ f can be decomposed into linear combinations of invariants of quark fields M ij = i ( x ) ¯ j ( y ), f =1 which are products of tr � ( MM † ) t � = ( − 1) t +1 Tr � ( M xy ) t � , t = 1 . . . N c and det [ M ], det [ M † ] This talk: A similar statement is also true away from the strong coupling limit, when gauge links integrals are modified by the gauge plaquette action. Discuss recipe how to treat higher order gauge corrections! Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 14