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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

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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 Sn Higher order gauge corrections Aim of the talk: show how to compute all gauge integrals for staggered fermions in principle apply method for O(β2) Relation to other talks: MDP treatment instead of AFMC as discussed by Terukazu Ichihara Full gauge integration instead Z3 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

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Motivation: Why Lattice QCD at strong coupling?

Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 3

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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 (µB, T) with µB baryon chemical potential:

1

100 200 Vacuum

Quark Gluon Plasma

Nuclear Matter CP ??? Hadronic Matter 〈    〉≠0 ⟨ ¯ ψ ψ⟩=0 E a r l y U n i v e r s e E a r l y U n i v e r s e Crossover μ/T=1

Terra Incognita

μB [GeV] T [MeV] T c speculated phase diagram known phase diagram

But: only little is known from lattice QCD Big question: does the critical end point of QCD exist?

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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 detM MC simulations on colored gauge fields, requires supercomputers Direct MC simulation at µB > 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: In the past limited to very strong coupling: β =

6 g2 → 0

i. e. coarse lattices

However: Systematic inclusion of gauge corrections for β > 0 now possible

Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 5

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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 (Tc, µc) for massless quarks [de Forcrand & U. (2011)]:

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.5 1 1.5 2 2.5 3 tricritical point

〈- ψ ψ 〉 = 0

〈-

ψ ψ 〉 ≠ 0

1st order 2nd order µB [lat. units] T [lat. units] 1 100 200 Vacuum

Quark Gluon Plasma

Nuclear Matter TCP

 B[GeV] T [MeV]

Color Super- conductor? Quarkyonic Matter ? Hadronic Matter 〈    〉≠0 〈    〉=0 1st order Neutron Stars Neutron Stars 2nd order O(4)

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

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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)]

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 T [lat. units] 2nd order tricritical point 1st order nuclear CEP β µB [lat. units] T [lat. units] 1 100 200 Vacuum Nuclear Matter TCP

 B[GeV] T [MeV]

1st

r

d e r 2nd order O(4)

?

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

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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?

One of several possible scenarios for the extension to the continuum: back plane: strong coupling phase diagram (β = 0), Nf = 1 front plane: continuum phase diagram (β = ∞, a = 0) due to fermion doubling, corresponds to Nf = 4 in continuum (no rooting)

T /mB μ/mB

β

Chiral Transition Nuclear Transition

1st order 2nd

rder

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Sign problem: Are higher order gauge corrections feasible?

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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 sign ∼ e− V

T ∆f (µ)

163 × 4: sign ≈ 0.1 at tricritical point ∆f decreases with Nτ, vanishes in continuous time limit

5e-05 0.0001 0.00015 0.0002 0.25 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 a4∆f ρ=a(T2+µ2)1/2

pen symbols 163x4

filled symbols: 83x4 φ=atan(µ/T) 2nd order: 0.0 0.1 0.2 0.3 0.4 0.5 0.6 first order: 0.7

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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 Tc but milder above Tc

1e+00 1e+00 1e+00 1e+00 1e+00 1e+00 1e+00 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 T/Tc <sign> 2e-06 3e-06 4e-06 5e-06 6e-06 7e-06 8e-06 9e-06 1e-05 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 T/Tc ∆ f 63x4, β=0.0 63x4, β=0.5 63x4, β=1.0 83x4, β=0.0 83x4, β=0.5 83x4, β=1.0 Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 11

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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

T ≃ µTCP TTCP remains constant below TCP and becomes

milder above Tc

9e-01 9e-01 9e-01 9e-01 1e+00 1e+00 1e+00 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 ρ/ρc for µ/T=0.68 <sign> 1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 7e-05 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 ρ/ρc for µ/T=0.68 ∆ f 63x4, β=0.0 63x4, β=0.1 63x4, β=0.2 83x4, β=0.0 83x4, β=0.1 83x4, β=0.2 Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 12

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Gauge Integrals: Review of what is known

Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 13

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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 J0(x, y) =

SU(Nc)

dU etr[UM†+MU†] can be decomposed into linear combinations of invariants of quark fields Mij =

Nf

f =1

χf

i(x) ¯

χf

j(y),

which are products of tr (MM†)t = (−1)t+1Tr (Mxy)t , t = 1 . . . Nc 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

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Gauge integrals in lattice QCD without fermions (1) Gauge integrals in lattice QCD without fermions (1) Gauge integrals in lattice QCD without fermions (1) Gauge integrals in lattice QCD without fermions (1) Gauge integrals in lattice QCD without fermions (1)

Generic one-link integral for U ≡ Uµ(x) ∈ G, G =U(Nc), SU(Nc): Ir;s ≡ Ir;s

i1j1,...,ir jr ;k1l1,...,ks ls [U]

=

G

dU Ui1j1 . . . Uir jr (U†)k1l1 . . . (U†)ks ls for U(Nc): Ir;s non-zero for r = s; for SU(Nc): Ir;s non-zero for |r − s| ∈ {0, Nc, 2Nc, . . .} Well known cases: I0;0 = 1 I2;2 = 1 Nc2 − 1

δi1l1δi2l2δj1k1δj2k2 + δi1l2δi2l1δj1k2δj2k1
−

1 Nc(Nc2 − 1)

δi1l1δi2l2δj1k2δj2k1 + δi1l2δi2l1δj1k1δj2k2
I1;1 =

1 Nc δi1l1δj1k1 INc;0 = 1 Nc! ǫi1...iNc ǫj1...jNc Various methods to extend to arbiratry Ir;s [M. Creutz, J.Math Phys 19 (1978)] Here: method of Young projectors (see also [J. Myers et al. JHEP 1503 (2015) 068])

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Gauge integrals in lattice QCD without fermions (2) Gauge integrals in lattice QCD without fermions (2) Gauge integrals in lattice QCD without fermions (2) Gauge integrals in lattice QCD without fermions (2) Gauge integrals in lattice QCD without fermions (2)

For Ir,s with r = s + f Nc, we need the following identities: (a) Ui1j1 = 1 (Nc − 1)! ǫi1l2...lNc ǫj1k2...kNc (U†)k2l2 . . . (U†)kNc lNc (b) (U†)k1l1 = 1 (Nc − 1)! ǫl1i2...iNc ǫk1j2...jNc Ui2j2 . . . UiNc jNc which also imply (c) Ui2j2 . . . UiNc jNc = 1 (Nc − 1)! ǫl1i2...iNc ǫk1j2...jNc (U†)k1l1 (d) (U†)k2l2 . . . (U†)kN clNc = 1 (Nc − 1)! ǫl1i2...iNc ǫk1j2...jNc Ui1j1 Thus, for f ∈ {0, . . . , Nf} and s ≥ 0: If Nc+κ;κ s×(b) = 1 [(Nc − 1)!]s [ǫ...ǫ...]sI(s+f )Nc,0

f ×(a)

= 1 [(Nc − 1)!]f [ǫ...ǫ...]f Is+f (Nc−1);s+f (Nc−1)

f ×(c)

= 1 [(Nc − 1)!]f [ǫ...ǫ...]f Is+f ;s+f

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Gauge integrals in lattice QCD without fermions (2) Gauge integrals in lattice QCD without fermions (2) Gauge integrals in lattice QCD without fermions (2) Gauge integrals in lattice QCD without fermions (2) Gauge integrals in lattice QCD without fermions (2)

For Ir,s with r = s + f Nc, we need the following identities: (a) Ui1j1 = 1 (Nc − 1)! ǫi1l2...lNc ǫj1k2...kNc (U†)k2l2 . . . (U†)kNc lNc (b) (U†)k1l1 = 1 (Nc − 1)! ǫl1i2...iNc ǫk1j2...jNc Ui2j2 . . . UiNc jNc which also imply (c) Ui2j2 . . . UiNc jNc = 1 (Nc − 1)! ǫl1i2...iNc ǫk1j2...jNc (U†)k1l1 (d) (U†)k2l2 . . . (U†)kN clNc = 1 (Nc − 1)! ǫl1i2...iNc ǫk1j2...jNc Ui1j1 Thus, for f = 1 and s ≥ 0: INc+s;s s×(b) = 1 [(Nc − 1)!]s [ǫ...ǫ...]sI(s+1)Nc;0

(a)

= 1 (Nc − 1)! ǫ...ǫ...Is+Nc−1;s+Nc−1

(c)

= 1 (Nc − 1)! ǫ...ǫ...Is+1;s+1

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Gauge integrals of interest for O(βr+s): Gauge integrals of interest for O(βr+s): Gauge integrals of interest for O(βr+s): Gauge integrals of interest for O(βr+s): Gauge integrals of interest for O(βr+s):

Generic one-link integral for U ≡ Uµ(x) ∈ G, G =U(Nc), SU(Nc): J r;s

(i,j)1:r (k,l)1:s ≡ J r;s i1j1,...,ir jr ;k1l1,...,ks ls [U]

=

G

dU e ¯

χf i (x)Uij χf j (y)− ¯ χg i (y)(U†)ij χg j (x)

from fermion action

Ui1j1 . . . Uir jr (U†)k1l1 . . . (U†)ks ls

from gauge action

In quenched limit: fermionic action drops out, then J r;s = Ir;s In general, these integrals are not known! Brute force strategy: expand exponential, with (M†)ij =

f

¯ χf

i(x)χf j(y), Mij = g

χg

j(x) ¯

χg

i(y):

J r;s

(i,j)1:r (k,l)1:s =

κ1,κ2

Kκ1,r;κ2,s

(i,j)1:κ1+r (k,l)1:κ2+s

Kκ1,r;κ2,s

(i,j)1:κ1+r (k,l)1:κ2+s =

1 κ1!κ2!

{ia,ja,kb,lb}

κ1

a=1

(M†)iaja

κ2

b=1

Mkblb

Iκ1+r;κ2+s

(i,j)1:κ1+r (k,l)1:κ2+s

κ1 quark hoppings, κ2 anti-quark hoppings: |κ1 − κ2 + r − s| ∈ {0, Nc, 2Nc, . . .} Want to know K, not just I: color and flavor structure intimately linked!

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Revisiting the Strong Coupling Limit: Some new insights

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Results for the strong coupling limit (mesonic) Results for the strong coupling limit (mesonic) Results for the strong coupling limit (mesonic) Results for the strong coupling limit (mesonic) Results for the strong coupling limit (mesonic)

For r = s = 0, i.e. for Kκ1;κ2 ≡ Kκ1,0;κ2,0 with Mxy = MxMy an Nf × Nf matrix and (Mx)f g =

Nc

i=1

¯ χf

iχg i,

new general results: K0;0 = 1 K1;1 = 1 Nc Tr[Mxy] K2;2 = 1 2 1 Nc(Nc2 − 1)

NcTr[Mxy]2 + Tr[(Mxy)2]

K3;3 = 1 6 1 Nc(Nc2 − 1)(Nc2 − 4)

(Nc

2 − 2)Tr[Mxy]3 + 3NcTr[Mxy]Tr[(Mxy)2] + 4Tr[(Mxy)3]

Kκ;κ =

λ,τ

Cτ

λ κ

i=1

Tr[(Mxy)i]ti , with λ parameterizing Nc-dependence, τ the Nf-dependence For Nf = 1, where Tr[Mxy] = Mxy, Cλ =

τ

Cτ

λ reproduces well known result

Kκ;κ = (Nc−κ)!

Nc!κ! (Mxy)κ,

κ ≤ Nc For Nf > 1, there are contributions up to K

NcNf ;NcNf Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 19

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Results for the strong coupling limit (baryonic) Results for the strong coupling limit (baryonic) Results for the strong coupling limit (baryonic) Results for the strong coupling limit (baryonic) Results for the strong coupling limit (baryonic)

For r = s = 0, i.e. for Kκ1;κ2 ≡ Kκ1,0;κ2,0 with Mxy = MxMy an Nf × Nf matrix and (Mx)f g =

Nc

i=1

¯ χf

iχg i,

new general results: KNc;0 = 1 Nc! det[M†] KNc+1;1 = 1 Nc!(Nc − 1)! det[M†]Tr[Mxy] K2Nc;0 = 1 (Nc!)2 det[M†]2 For Nf = 1, where det[M†] = Nc!¯ BxBy, det[M] = (−1)NcNc!¯ ByBx, Bx =

1 Nc! ǫi1...iNc χi1 . . . χiNc

KNc;0 = ¯ BxBy, K0;Nc = (−1)Nc ¯ ByBx For Nf > 1, there are more contributions up to K

NcNf ;0

Reproduces the known results of SU(2), SU(3), but generalizes to arbitrary Nc, e.g. J SU(2) =

NcNf

κ1

(X + C + ¯ C)κ1 κ1!(κ1 + 1)! , X = tr[MM†], C = det[M], ¯ C = det[M†] [Eriksson, Svartholm & Skagerstam, J. Math. Phys. 22 (1981)]

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Young Tableaux Young Tableaux Young Tableaux Young Tableaux Young Tableaux

So, what are the Cτ

λ, Cτ,ρ1 λ

and for O(β2): Cτ,ρ1,ρ2

λ

, etc? answer: related to irreducible representations of the symmetric group Sn with n = κ1 + r = κ2 + s Young Tableaux: Young diagrams correspond to integer partitions λ= (λ1, . . . λk) with

i

λi = n and λi ≥ λj (i < j) Standard Young tableaux (filling a diagram with integers i ∈ {1, . . . n} such that the number are raising row- and columnwise) correspond to irreducible representations of Sn dλ =

n!

i,j

h(i,j) (Hook length formula)

used in particle physics to symmetrize/anti-symmetrize wave functions by filling in quantum numbers used to determine dimension Dλ of irreps of SU(Nc)

1 2 3 4 1 3 2 4

λ=(2,2) λ=(2,1,1)

1 2 3 4 1 3 2 4 1 4 2 3

λ=(3,1)

1 2 3 4 1 2 4 3 1 3 4 2 1 2 3 4

λ=(4) λ=(1,1,1,1)

1 2 3 4

d=1 d=3 d=2 d=3 d=1

D=(Nc+3)! (N c−1)! D= N c! (N c−4)! D=Nc(N c

2−1)(N c+2)

D=Nc(N c

2−1)(N c−2)

D=Nc

2(N c 2−1)

N c +1 +2 +3 N c +1 +2

1

N c +1

1

N c N c +1

1
2

N c

1
2
3

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Trace Structure Trace Structure Trace Structure Trace Structure Trace Structure

So, what are the Cτ

λ, Cτ,ρ1 λ

and for O(β2): Cτ,ρ1,ρ2

λ

, etc? answer: related to irreducible representations of the symmetric group Sn with n = κ1 + r = κ2 + s Trace Structure: at a given order n, the trace structure τ = (t1, . . . tk) with T = Tr[Mi

xy]ti is equivalent to a

partition of n:

i

iti = n trace structure is determined by the conjugacy classes of Sn for flavor permutations given by the cycle struture of permutations examle: π = (136)(24)(58)(7) → t1 = 1, t2 = 2, t3 = 1 Trace structure even important for Nf = 1! Cλ =

τ

Cτ

λ

1 2 3 4 1 3 2 4

τ=(2,2) τ=(2,1,1)

1 2 3 4 1 3 2 4 1 4 2 3

τ=(3,1)

1 2 3 4 1 2 4 3 1 3 4 2 1 2 3 4

τ=(4) τ=(1,1,1,1)

1 2 3 4

d=1 d=3 d=2 d=3 d=1 c=6 Tr[M

4]

Tr[M

3]Tr[M ]

Tr[M

2] 2

Tr[M

2]Tr[ M] 2

Tr[M]

4

c=8 c=3 c=6 c=1 Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 22

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Characters of the symmetric group Characters of the symmetric group Characters of the symmetric group Characters of the symmetric group Characters of the symmetric group

Character: trace of matrix representation Character table : table of all irreducible characters of a group Example: rotation 123 → 312, trace structure: τ = (3)

− 1

2 √ 3 2

−

√ 3 2

− 1

2

→

χ = − 1 2 − 1 2 = −1 λ (1,1,1) (2,1) (3) τ (1,1,1) 1 2 1 (2,1) 1

1

(3) 1

1

1 Table: Character table of S3

Can be recursively computed e.g. via Rim-hook tableaux

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Young projectors Young projectors Young projectors Young projectors Young projectors

The Young projector for a standard Young tableau λ(a) (a ∈ {1, . . . dλ}) project on one of the irreps of Sn, normalized to SU(Nc): Pλ(a)

(i,j)1:n =

1 H(λ) =

π∈λ(a)

sgn(λ(a), π)δi1π(i1) . . . δinπ(in) with H(λ) =

(i,j)∈λ

h(i, j) the hook length product. The following orthogonality relation holds:

G

dU Rλ(a)

(i,j)1:n (Rλ′(b))† (k,l)1:n = 1

dλ Pλ(a)

(i,l)1:nPλ(b) (k,j)1:nδλλ′

In;n =

dU
λ

dλ Dλ

dλ

a,b=1

Pλ(a)

(i,l)1:nPλ(b) (k,j)1:n

⇒ Kκ;κ = 1 (κ!)2

λ∈Λκ
τ∈Tκ

Cτ

λ κ

i=1

Tr[Mi]τi Cτ

λ = dλ Dλ cτχλ τ

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Table of Invariants and Characters Table of Invariants and Characters Table of Invariants and Characters Table of Invariants and Characters Table of Invariants and Characters

Traces C1,1,1,1 C2,1,1 C2,2 C3,1 C4 Sum 4000 1 9 4 9 1 24 2100 6 18

18
6

0200 3

9

12

9

3 1010 8

16

8 0001 6

18

18

6

Sum 24

Table: Table of Invariants, n = 4

Traces C1,1,1,1 C2,1,1 C2,2 C3,1 C4 Sum 4000 1 3 2 3 1 10 2100 1 1

1
1

0200 1

1

2

1

1 2 1010 1

1

1 1 0001 1

1

1

1

Sum 5 2 3 2 1

Table: Table of Characters, n = 4

Nf = 1: sum over τ is positive! No additional sign problem.

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Results for O(β), U(Nc) Results for O(β), U(Nc) Results for O(β), U(Nc) Results for O(β), U(Nc) Results for O(β), U(Nc)

For U(Nc), with Qij

g1g2 = ψg1 i (x) ¯

ψg2

j (y) and ρ1 =

ei, i ∈ {1, . . . colτ} a trace insertion vector: K0,1;1,0

i1j1

= 1 Nc Tr[Qi1j1] K1,1;2,0

i1j1

= 1 Nc(Nc2 − 1)

NcTr[Qi1j1]Tr[MxMy] + Tr[MxMyQi1j1]
K2,1;3,0

i1j1

= 1 2Nc(Nc2 − 1)(Nc2 − 4)

(Nc2 + Nc − 2)Tr[Qi1j1]Tr[Mxy]2

+ 2NcTr[MxyQi1j1]Tr[Mxy] + 4Tr[(Mxy)2Qi1j1]

Kκ−1,1;κ,0

i1j1

=

λ,τ,ρ1

Cτ,ρ1

λ κ

i=1

Tr[(Qi1j1)ρ1(MxMy)(i−ρ1)]ti ,

ρ1

Cτ,ρ1

λ

= Cτ

λ

For Nf = 1, the U(Nc) contribution is: Kκ−1,i1;κ,j0

i1j1

= (Nc − κ)! Nc!(κ − 1)! (Mxy)(κ−1)ψi1(x) ¯ ψj1(y)

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Results for O(β), SU(Nc) Results for O(β), SU(Nc) Results for O(β), SU(Nc) Results for O(β), SU(Nc) Results for O(β), SU(Nc)

For SU(Nc), with Pij

g1g2 replacing ¯

ψf1

i (x)ψf1 j (y) by 1 in the quark matrix M†:

KNc−1,1;0,0

i1j1

= 1 (Nc − 1)! det[Pi1j1

g1g2M†]

KNc,1;1,1,0

i1j1

= det[Pi1j1

g1g2M†]Tr[Mxy] − 1

Nc det[M†]Tr[Qi1j1] For Nf = 1, the additional SU(Nc) contributions are: KNc−1,1;0,0

i1j1

= 1 Nc!(Nc − 1)! ǫi1...iNc ǫk1...kNc ¯ χi2 . . . ¯ χiNc χk2 . . . ¯ χkNc KNc,1;1,1,0

i1j1

= − 1 Nc ¯ Bx ¯ Byψi1(x) ¯ ψj1(y)

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SLIDE 29

Results for O(β2), U(Nc) Results for O(β2), U(Nc) Results for O(β2), U(Nc) Results for O(β2), U(Nc) Results for O(β2), U(Nc)

For U(Nc), with Qij

g1g2 = ψg1 i (x) ¯

ψg2

j (y):

K0,2;2,0

i1j1,i2j2 =

1 Nc(Nc2 − 1)

NcTr[Qi1j1]Tr[Qi2j2] + Tr[Qi1j1Qi2j2]
K1,2,3,0

i1j1,i2j2 =

1 Nc(Nc2 − 1)(Nc2 − 4)

(Nc2 + Nc − 2)Tr[Qi1j1]Tr[Qi2j2]Tr[Mxy]

+ NcTr[Qi1j1Qi2j2]Tr[Mxy] + NcTr[MxyQi1j1]Tr[Qi2j2] + 4Tr[MxyQi1j1Qi2j2]

Kκ−2,2;κ,0

i1j1,i2j2

=

λ,τ,ρ1,ρ2

Cτ,ρ1,ρ2

λ κ

i=1

Tr[(Qi1j1)ρ1(Qi2j2)ρ2(Mxy)(i−ρ1−ρ2)]ti ,

ρ1,ρ2

Cτ,ρ1,ρ2

λ

= Cτ

λ

For Nf = 1, the U(Nc) contribution is: Iκ−2,2;κ,0

i1j1,i2j2

= (Nc − κ)! Nc!(κ − 2)! (Mxy)(κ−2)ψi1(x) ¯ ψj1(y)ψi2(x) ¯ ψj2(y)

Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 28

SLIDE 30

Monomer-Dimer-Polymer-Plaquette Ensemble (MDPP): From Gauge Integrals to Monte Carlo Simulations

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SLIDE 31

Strong Coupling Partition Function Strong Coupling Partition Function Strong Coupling Partition Function Strong Coupling Partition Function Strong Coupling Partition Function

Exact rewriting after Grassmann integration: Mapping to a MDP representation: ZF(mq, µ) =

{k,n,ℓ}
b=(x,µ)

(Nc − kb)! Nc!kb!

meson hoppings Mx My
x

Nc! nx! (2amq)nx

chiral condensate ¯

χχ

ℓ

w(ℓ, µ)

baryon hoppings ¯

Bx By

kb ∈ {0, . . . Nc}, nx ∈ {0, . . . Nc}, ℓb ∈ {0, ±1}

Grassmann constraint:

nx +

ˆ

µ=±ˆ 0,...±ˆ d

k ˆ

µ(x) + Nc

2 |ℓ ˆ

µ(x)|

= Nc

weight w(ℓ, µ) and sign σ(ℓ) ∈ {−1, +1} for oriented baryonic loop ℓ depends on loop geometry

t

finite quark mass

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SLIDE 32

Strong Coupling Partition Function Strong Coupling Partition Function Strong Coupling Partition Function Strong Coupling Partition Function Strong Coupling Partition Function

Exact rewriting after Grassmann integration: Mapping to a MDP representation: ZF(mq, µ) =

{k,n,ℓ}
b=(x,µ)

(Nc − kb)! Nc!kb!

meson hoppings Mx My

✘✘✘✘✘✘ ✘

x

Nc! nx! (2amq)nx

chiral condensate ¯

χχ

ℓ

w(ℓ, µ)

baryon hoppings ¯

Bx By

kb ∈ {0, . . . Nc}, nx ∈ {0, . . . Nc}, ℓb ∈ {0, ±1}

Grassmann constraint:

ˆ

µ=±ˆ 0,...±ˆ d

k ˆ

µ(x) + Nc

2 |ℓ ˆ

µ(x)|

= Nc

weight w(ℓ, µ) and sign σ(ℓ) ∈ {−1, +1} for oriented baryonic loop ℓ depends on loop geometry

t

chiral limit: monomers absent

Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 30

SLIDE 33

Strong Coupling Partition Function Strong Coupling Partition Function Strong Coupling Partition Function Strong Coupling Partition Function Strong Coupling Partition Function

Exact rewriting after Grassmann integration: Mapping to a MDP representation: ZF(mq, µ) =

{k,n,ℓ}
b=(x,µ)

(Nc − kb)! Nc!kb!

meson hoppings Mx My

✘✘✘✘✘✘ ✘

x

Nc! nx! (2amq)nx

chiral condensate ¯

χχ

ℓ

w(ℓ, µ)

baryon hoppings ¯

Bx By

kb ∈ {0, . . . Nc}, nx ∈ {0, . . . Nc}, ℓb ∈ {0, ±1}

Worm algorithm [Prokof’ev & Svistunov 2001]: sampling 2-monomer sector (for U(3):

[Adams & Chandrasekharan, 2003])

SU(3): Worm both in mesonic and baryonic sector

t H T

during Worm evolution

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SLIDE 34

MDPP Partition Function at O(β) MDPP Partition Function at O(β) MDPP Partition Function at O(β) MDPP Partition Function at O(β) MDPP Partition Function at O(β)

Recently derived partition function in integer variables:

[Langelage, de Forcrand, Philipsen & U., PRL 113 (2014)]

Z =

dχd ¯

χZF

P
1 +

1 g2

l∈P

zl

−1

19

s=1

Fs P + . . .

=
{k,n,ℓ,q}
b=(x,µ)

ˆ wb

x

ˆ wx

ℓ

ˆ wℓ

P

wP

ˆ wx = wxνqx

i

ˆ wb = wbρqb

Mk ,

ˆ wℓ = wℓ

Bj ∈ℓ

ρqB

Bj ,

wP = β 2Nc

−2qP

weights: ν1 = (Nc − 1)!, ν2 = Nc!, ρB1 =

1 (Nc−1)!, ρB2 = (Nc − 1)!,

modified Grassmann constraint: nx +

ˆ

µ=±ˆ 0,...±ˆ d

kˆ

µ(x) + Nc

2 |ℓˆ

µ(x)|

= Nc+qx

O(β)

Gauge Flux Baryonic Quark Flux Mesonic Quark Flux B B (MM )

3

qq q g g

(MM )

2

qg qg

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SLIDE 35

Going Beyond . . . Going Beyond . . . Going Beyond . . . Going Beyond . . . Going Beyond . . .

weights from gauge links not sufficient site weights need to be computed by integrating out Grassmann fields small number of possible vertices classified by orders κ1, κ2 and contraction of open indices (i, j) associated with every gauge flux

1 O(β) 1 1 1 O(β)' 1 1 1

1

O(β

2)

2 2 1 1 1 1 1

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SLIDE 36

Conclusions Conclusions Conclusions Conclusions Conclusions

Achievements: O(β) phase diagram seems to be valid beyond β ∼ 1 (crosschecks at µ = 0) but not conclusive concerning nuclear vs. chiral transition sign problem mild enough to tackle the next order in the strong coupling expansion new relation between one-link integrals (moments in presence of staggered fermions) and Sn group characters established all gauge integrals needed for O(β2) corrections computed Goals: complete site weight computations (Grassmann integration) run O(β2) MDPP simulations

Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 33

SLIDE 37

Backup: Example of the Murnaghan-Nakayama rule Backup: Example of the Murnaghan-Nakayama rule Backup: Example of the Murnaghan-Nakayama rule Backup: Example of the Murnaghan-Nakayama rule Backup: Example of the Murnaghan-Nakayama rule

Each rim-hook tableaux contributes a +1 or -1 depending on the rim-hook length. rims including a 2x2 block are forbidden (symmetrization + anti-symmetrization=0)

1 3 2 4 1 1 2 3 1 1 2 2 1 2 2 2 1 2 1 2 Tr[M]4 Tr[M2]Tr[M]2

+

+

+ +2 +2

1
1 2

3 4

+

1 2 1 3

+

Tr[M2]2 Tr[M3]Tr[M] 1 1 1 1

Tr[M4]

Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 34

SLIDE 38

Backup: SC-LQCD at finite temperature Backup: SC-LQCD at finite temperature Backup: SC-LQCD at finite temperature Backup: SC-LQCD at finite temperature Backup: SC-LQCD at finite temperature

How to vary the temperature? aT = 1/Nτ is discrete with Nτ even aTc ≃ 1.5, i.e. Nτ

c < 2

⇒ we cannot address the phase transition! Solution: introduce an anisotropy γ in the Dirac couplings: Z(mq, µ, γ, Nτ) =

{k,n,l}
b=(x,µ)

(3 − kb)! 3!kb! γ2kbδµ0

x

3! nx!(2amq)nx

l

w(ℓ, µ) Should we expect a/aτ = γ, as suggested at weak coupling? No: meanfield predicts a/aτ = γ2, since γ2

c = Nτ (d−1)(Nc+1)(Nc+2) 6(Nc+3)

⇒ sensible, Nτ-independent definition of the temperature: aT ≃ γ2

Nτ

Moreover, SC-LQCD partition function is a function of γ2 However: precise correspondence between a/aτ and γ2 not known

Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 35

SLIDE 39

Backup: Classification of O(β) Diagrams Backup: Classification of O(β) Diagrams Backup: Classification of O(β) Diagrams Backup: Classification of O(β) Diagrams Backup: Classification of O(β) Diagrams

Diagrams classified by external legs (monomers or external dimers)

D 2

4

(D1 D3)

2

D1 D2

3

D1 D2 D1 D3 (D1 D2)2 D1

2 D2 2

D1

3 D3

D1

3 D2

D1

4

D1 D 3 D1 B1 D1 D2 D1 B1 D1

3B1

(D1B1)

2

D1

2 B1 2

D1 B1 B2 B1 D1 B1

3

D1 B1

3

(B1 B2)

2

B1

3 B2

B1

4

mesonic sector baryonic sector

Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 36

SLIDE 40

Backup: Direct Sampling Backup: Direct Sampling Backup: Direct Sampling Backup: Direct Sampling Backup: Direct Sampling

Sampling plaquette occupation number at finite β via additional Metropolis update: qP → 1 − qP P = 2 Vd(d − 1) ∂ ∂β log(Z) = 1 β nP , nP = 2 Vd(d − 1)

P

qP saturation expected: np ≤

Nc 2d(d−1) (at most Nc adjacent plaquettes )

due to missing pure gauge sector: P → 0 for β → ∞ numerical results show indeed saturation of np direct sampling not optimal (noise, systematic errors)

0.001 0.01 0.1 1 0.001 0.01 0.1 1 10 100 <P> 1/g2 44, SU(3) to chiral limit Worm direct sampling HMC mq=0.0 reweighting: <P>=0.019937(6)

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SLIDE 41

Backup: Reweighting Backup: Reweighting Backup: Reweighting Backup: Reweighting Backup: Reweighting

Reweighting to finite β from the SC-Ensemble: P |β=0 = ZP Z with ZP the one-plaquette sector,

p qp = 1.

determine weight to update Z → ZP with detailed balance satisfied (non-trivial for anisotropic lattice) reweighting is much less noisy in O(β) truncation scheme: np ∝ β ⇒ P = const extends to various other

bservables:
gauge observables: O(β0)
fermionic observables:

O(β)

0.001 0.01 0.1 1 0.001 0.01 0.1 1 10 100 <P> 1/g2 44, SU(3) to chiral limit Worm direct sampling HMC mq=0.0 reweighting: <P>=0.019937(6)

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SLIDE 42

Backup: Gauge Observables at Finite Temperature Backup: Gauge Observables at Finite Temperature Backup: Gauge Observables at Finite Temperature Backup: Gauge Observables at Finite Temperature Backup: Gauge Observables at Finite Temperature

Polyakov loop expectation value: ratio of partition function w/o static quark Q, measured via: L =

d ¯

χdχLSCZF

d ¯

χdχZF = ZQ Z , L( x) = Tr[JNτ ,1( x)

Nτ

t=1

Jt,t+1( x)] L and P are sensitive to the chiral transition L rises, cusp is imprint of chiral transition rather than deconfinement transition

0.05 0.1 0.15 0.2 0.25 0.5 1 1.5 2 2.5 3 3.5 4 aTc aT Polyakov Loop at µ=0 43x4 63x4 83x4 123x4 163x4 aT → ∞ 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 aTc Zoom 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.5 1 1.5 2 2.5 3 3.5 4 aTc aT Spatial and Temporal Plaquette at µ=0 43x4 63x4 83x4 123x4 163x4 aT → ∞ 0.066 0.068 0.07 0.072 0.074 0.076 0.078 0.08 0.082 0.084 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 aTc Zoom into <Pt> 5 10 15 aTc aT

Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 39

SLIDE 43

Backup: Gauge Observables at Finite Density Backup: Gauge Observables at Finite Density Backup: Gauge Observables at Finite Density Backup: Gauge Observables at Finite Density Backup: Gauge Observables at Finite Density

Polyakov loop develops gap and Anti-Polyakov loop develops cusp as the transition turns 1st order large µ and/or T limit ρ → ∞ analytically computed

0.05 0.1 0.15 0.2 0.25 0.3 0.25 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 ρ → ∞ Polyakov Loop ρT µ/T on 163x4 2nd: 0.00 0.10 0.20 0.31 0.42 0.55 0.68 1st: 0.84 1.03 1.26 1.56 0.05 0.1 0.15 0.2 0.25 0.3 0.25 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 ρ → ∞ Anti-Polyakov Loop ρ=a(T2+µ2)1/2 Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 40

SLIDE 44

Backup: Gauge Observables at Finite Density Backup: Gauge Observables at Finite Density Backup: Gauge Observables at Finite Density Backup: Gauge Observables at Finite Density Backup: Gauge Observables at Finite Density

suppression of spatial plaquettes: pairs of parallel spatial dimers are rare at high T plaquette weight is nonzero only if non-trivial parallel pair of dimers/flux is present additional anisotropy in gauge couplings βs, βt in strong coupling regime: γ2 ≈ as

at ≈

βs

βt 0.001 0.01 0.1 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 aT <Ps>/<Pt>, µ/T=0 43x4 63x4 83x4 123x4 163x4 C / T4 (T>Tc) 0.001 0.01 0.1 0.8 1 1.2 1.4 1.6 1.8 2 aT <Ps>/<Pt>, µ/T=0.3 43x4 63x4 83x4 123x4 163x4 C’ / T4 (T>Tc)

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SLIDE 45

Backup: Gauge Observables at Finite Density Backup: Gauge Observables at Finite Density Backup: Gauge Observables at Finite Density Backup: Gauge Observables at Finite Density Backup: Gauge Observables at Finite Density

in strong coupling regime, anisotropy in β linked to γ: βs = βγ−2, βt = βγ2 anisotropy can be absorbed into observable: βsPs = β(γ−2Ps), βtPt = β(γ2Pt)

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.25 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 ρ → ∞ ρT temporal plaquette 2nd order: 0.00 0.10 0.20 0.31 0.42 0.55 0.68 1st order: 0.84 1.03 0.002 0.004 0.006 0.008 0.01 0.25 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 ρ → ∞ ρ=a(T2+µ2)1/2 spatial plaquette Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 40

SLIDE 46

Backup: O(β2) Corrections Backup: O(β2) Corrections Backup: O(β2) Corrections Backup: O(β2) Corrections Backup: O(β2) Corrections

At O(β2), plaquette orientations are relevant! Five types of corrections:

1

disconnected plaquettes

2

2x1 Wilson loops

3

two plaquettes sharing a site

4

two oppositely oriented plaquettes sharing a link

5

doubly occupied plaquette Link integration possible, but combinatorics difficult

1 1 1 1 1 1 1

1

2 Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 41

SLIDE 47

Backup: O(β2) Corrections Backup: O(β2) Corrections Backup: O(β2) Corrections Backup: O(β2) Corrections Backup: O(β2) Corrections

At O(β2), plaquette orientations are relevant! Five types of corrections:

1

disconnected plaquettes √

2

2x1 Wilson loops

3

two plaquettes sharing a site

4

two oppositely oriented plaquettes sharing a link

5

doubly occupied plaquette Link integration possible, but combinatorics difficult

0.5 1 1.5 2 1 2 3 4 5 6

µ=0

aT β

resummed extrapolation: aTc (Nt=6) aTc (Nt=4) aTc (Nt=2) Mean Field HMC, extrap. to mq=0 HMC aniso, extrap. to mq=0 (Nt=2) HMC aniso, extrap. to mq=0 (Nt=4) Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 41

SLIDE 48

Backup: O(β2) Corrections Backup: O(β2) Corrections Backup: O(β2) Corrections Backup: O(β2) Corrections Backup: O(β2) Corrections

At O(β2), plaquette orientations are relevant! Five types of corrections:

1

disconnected plaquettes √

2

2x1 Wilson loops √

3

two plaquettes sharing a site

4

two oppositely oriented plaquettes sharing a link

5

doubly occupied plaquette Link integration possible, but combinatorics difficult

0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 aT 2x1 Wilson loop spat 43x4 spat 63x4 spat 83x4 spat 123x4 spat 163x4 temp 43x4 temp 63x4 temp 83x4 temp 123x4 temp 163x4 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 aT 1x2 Wilson loop

Wolfgang Unger Higher order gauge corrections Debrecen, 02.10.2015 41