The QCD phase diagram:universality and continuity
Naoki Yamamoto(INT, University of Washington)
Yukawa Institute for Theoretical Physics
February 21, 2012Outline• Introduction of QCD phase diagram• Universality of phases in QCD and QCD-like theories• A no-go theorem for critical phenomena• Quark-hadron continuity/duality

Refs:

M. Hanada and NY, arXiv:1103.5480 (to appear in JHEP).
Y. Hidaka and NY, arXiv:1110.3044 (to appear in PRL).

A conjectured QCD phase diagram
RHIC/LHCT

Early universe

Quark-Gluon Plasma～200 MeV?
hadron gas
nuclearmatterquark matter?

～1 GeV

Neutron star

?

?

CFL

mB

Finite μ phase transition (T=0)
Consider T=0.
 A baryon with excitation energy mB－μB can be created for μB>mB;
Baryon (=fermion) forms a Fermi surface due to the Pauli principle.
 Phase transition at μB≈mB distinguishing between nB=0 and nB>0.

Large μ: color superconductivity (T=0)
 At high density, nuclear matter → quark matter (quark Fermi surface):
weak-coupling due to asymptotic freedom

E=|p|
μ

E

Fermi seap
Dirac sea

Large μ: color superconductivity (T=0)
 At high density, nuclear matter → quark matter (quark Fermi surface):
weak-coupling due to asymptotic freedom Attractive interaction → BCS pairing3

3×3 = 6S + 3A

q“diquark condensate”

E=|p|

μ

E

Fermi seap
Dirac seaq

A conjectured QCD phase diagram

T
Quark-Gluon Plasma?
hadron gas
nuclearmatterbaryon excitation

quark matter

?

?

?

model analyses(unsystematic)

CFL

mB
weak-coupling analysis(exact)Phase diagrams of QCD-like theories
 QCD at μI=2μ>0: chemical potentials μ & -μ for u & d quark (= π+)→ Dirac eigenvalues for u & d are complex conjugate.
→ Positive fermion determinant (no sign problem) Different gauge groups at μB>0 (instead of SU(Nc)):
・ SO(Nc) with any flavors: real(similar to QCD with adjoint fermions)・ Sp(Nc) with even degenerate flavors: pseudo-real(similar to 2-color QCD; Nc “color” version of SU(2)=Sp(2) )→ Positive fermion determinant (no sign problem)

QCD at finite isospin density
Consider T=0.

Son-Stephanov (‟01)

 At small μI:

・ π+ with excitation energy mπ－μI can be created for μI>mπ.
・ π+ (=boson) forms a Bose-Einstein condensation (BEC) at μI=mπ.

QCD at finite isospin density
Consider T=0.

Son-Stephanov (‟01)

 At small μI:

・ π+ with excitation energy mπ－μI can be created for μI>mπ.
・ π+ (=boson) forms a Bose-Einstein condensation (BEC) at μI=mπ.
 At large μI: Fermi surface of u & d quarks

1・ attractive interaction in color singlet channel: 3×3=1+8→ BCS pairingudQCD at finite isospin density
Consider T=0.

Son-Stephanov (‟01)

 At small μI:

・ π+ with excitation energy mπ－μI can be created for μI>mπ.
・ π+ (=boson) forms a Bose-Einstein condensation (BEC) at μI=mπ.
 At large μI: Fermi surface of u & d quarks

1・ attractive interaction in color singlet channel: 3×3=1+8→ BCS pairingud Same quantum numbers of condensates/ symmetry breaking pattern:
BEC-BCS crossover (continuity from hadronic to quark matter)

QCD at finite isospin density
T
Son-Stephanov (‟01)

chiral?

BEC region

BCS region

mI
BCS-BEC crossover (conjecture)pion excitation (model-independent)

weak-coupling analysis (exact)

SO & Sp gauge theories at μB>0Tchiral?

BEC region

BCS regionmB
BCS-BEC crossover (conjecture)diquark excitation (model-independent)

weak-coupling analysis (exact)

2-color QCD and adjoint QCD also have similar phase diagrams.

Universality of phase diagrams
 Question: reason for similarity? What about QCD at μB>0?

Universality of phase diagrams
 Question: reason for similarity? What about QCD at μB>0?
 Main results: Hanada-NY („11); Hidaka-NY („11)
1. Whole phase diagrams are quantitatively universal in the large-Nc
limit between QCD at μI>0 & SO/Sp gauge theories at μB>0.
2. Universality holds for QCD at μB>0 outside the BCS/BEC region.
3. No-go theorem: QCD critical point is ruled out in QCD at μB>0
outside the BCS/BEC region.

Practically this makes it possible to evade the sign problem for aclass of observables at least large Nc and hopefully Nc=3.

Universality of phase diagrams
Tchiral?
Chiral, deconfinement, BEC/BCS, ...
quantitatively universal between
QCD at μI>0, SO/Sp theories at μB>0.

BEC region

BCS regionmTchiral?
QCD at μB>0 also equivalent toabove outside BEC/BCS.

m

Large-Nc dualities/equivalences
gauge/gravity duality

orbifold equivalence

Maldacena („98), ...

Kachru-Silverstein („98), ...

strongly-coupledstrongly-coupledgauge theory

gauge theory

weakly-coupledstrongly-coupledgravity theory

gauge theory

AdS/QGP,
AdS/QCD,
AdS/CMT, ...

Cherman-Hanada-Robles-Llana;
Cherman-Tiburzi;
Hanada-NY; Hidaka-NY („11)

Large-Nc dualities/equivalences
gauge/gravity duality

orbifold equivalence

Maldacena („98), ...

Kachru-Silverstein („98), ...

strongly-coupledgauge theory

gauge theory

w/ sign problem

an example

weakly-coupledgauge theory

gravity theory

w/o sign problem

AdS/QGP,
AdS/QCD,
AdS/CMT, ...

Cherman-Hanada-Robles-Llana;
Cherman-Tiburzi;
Hanada-NY; Hidaka-NY („11)

Large-Nc orbifold equivalence[How to use]
1. Identify a discrete global symmetry of the original theory (parent).

Refs: Bershadsky-Johansen (‟98), Kovtun-Ünsal-Yaffe („03, „05, „06)

Large-Nc orbifold equivalence[How to use]
1. Identify a discrete global symmetry of the original theory (parent).
2. Eliminate all the d.o.f.s in the parent theory not invariant under thediscrete symmetry (projection).

Refs: Bershadsky-Johansen (‟98), Kovtun-Ünsal-Yaffe („03, „05, „06)

Large-Nc orbifold equivalence[How to use]
1. Identify a discrete global symmetry of the original theory (parent).
2. Eliminate all the d.o.f.s in the parent theory not invariant under thediscrete symmetry (projection).
3. This gives a new theory (daughter).

Refs: Bershadsky-Johansen (‟98), Kovtun-Ünsal-Yaffe („03, „05, „06)

Large-Nc orbifold equivalence[How to use]
1. Identify a discrete global symmetry of the original theory (parent).
2. Eliminate all the d.o.f.s in the parent theory not invariant under thediscrete symmetry (projection).
3. This gives a new theory (daughter).
4. A class of observables and correlators are shown to be identicalbetween the parent and daughter at large Nc (orbifold equivalence).

Refs: Bershadsky-Johansen (‟98), Kovtun-Ünsal-Yaffe („03, „05, „06)

Large-Nc orbifold equivalence[How to use]
1. Identify a discrete global symmetry of the original theory (parent).
2. Eliminate all the d.o.f.s in the parent theory not invariant under thediscrete symmetry (projection).
3. This gives a new theory (daughter).
4. A class of observables and correlators are shown to be identicalbetween the parent and daughter at large Nc (orbifold equivalence).

[Caution]
Valid as long as the discrete symmetry is not broken spontaneously.
Refs: Bershadsky-Johansen (‟98), Kovtun-Ünsal-Yaffe („03, „05, „06)

Example: from SO(2Nc) to SU(Nc)
Start with SO(2Nc) theory at μB>0 as a parent. Cherman et al. („11)

Example: from SO(2Nc) to SU(Nc)
Start with SO(2Nc) theory at μB>0 as a parent. Cherman et al. („11)
1. Discrete symmetry:Example: from SO(2Nc) to SU(Nc)
Start with SO(2Nc) theory at μB>0 as a parent. Cherman et al. („11)
1. Discrete symmetry:
2. Projection:

Example: from SO(2Nc) to SU(Nc)
Start with SO(2Nc) theory at μB>0 as a parent. Cherman et al. („11)
1. Discrete symmetry:
2. Projection:
3. Daughter theory: U(Nc) ≈ SU(Nc) gauge theory at μB>0 up to O(1/Nc2).

Example: from SO(2Nc) to SU(Nc)
Start with SO(2Nc) theory at μB>0 as a parent. Cherman et al. („11)
1. Discrete symmetry:
2. Projection:
3. Daughter theory: U(Nc) ≈ SU(Nc) gauge theory at μB>0 up to O(1/Nc2).

4. Orbifold equivalence:etc.

Example: from SO(2Nc) to SU(Nc)
Start with SO(2Nc) theory at μB>0 as a parent. Cherman et al. („11)
1. Discrete symmetry:
2. Projection:
3. Daughter theory: U(Nc) ≈ SU(Nc) gauge theory at μB>0 up to O(1/Nc2).

4. Orbifold equivalence:etc.

[Caution]•

Z4 ∈U(1)B broken in the BCS-BEC crossover.

•Z4 ∈SU(2)iso unbroken from SO(2Nc) to QCD at μI>0 everywhere.

“Family Tree” of QCD and QCD-like theories
sign-free theory

SO(Nc) or Sp(Nc) YM + μB
Equivalent outside the
BEC/BCS region

SU(Nc) QCD + μB
sign-full theory

Equivalent everywhere

SU(Nc) QCD + μI
Equivalent outside the sign-free theory
BEC/BCS region

Phase-quenched approximation is exact at large Nc
for a class of observables outside BEC/BCS.

“Family Tree” of QCD and QCD-like theories
sign-free theory

Nc=2

SU(2) QCD + μB

SO(Nc) or Sp(Nc) YM + μB
Equivalent outside the
BEC/BCS region

SU(Nc) QCD + μB
Nc=3sign-full theory

Equivalent everywhere

SU(Nc) QCD + μI
Equivalent outside the sign-free theory
BEC/BCS region

SU(3) QCD + μB
Phase-quenched approximation is exact at large Nc
for a class of observables outside BEC/BCS.

An intuitive derivation
 Consider 〈qq〉 at finite μI or μB for degenerate 2 flavors u & d.

An intuitive derivation
 Consider 〈qq〉 at finite μI or μB for degenerate 2 flavors u & d.
・ additional flavor loops suppressed at large Nc→ no mixing between u & d at leading order in 1/Nc.

An intuitive derivation
 Consider 〈qq〉 at finite μI or μB for degenerate 2 flavors u & d.
・ additional flavor loops suppressed at large Nc→ no mixing between u & d at leading order in 1/Nc.
・ sign of ±μ for d doesn‟t matter due to charge conjugation.
→ 〈qq〉(μB) = 〈qq〉(μI)(deconfinement argued similarly in Toublan 05)

An intuitive derivation
 Consider 〈qq〉 at finite μI or μB for degenerate 2 flavors u & d.
・ additional flavor loops suppressed at large Nc→ no mixing between u & d at leading order in 1/Nc.
・ sign of ±μ for d doesn‟t matter due to charge conjugation.
→ 〈qq〉(μB) = 〈qq〉(μI)(deconfinement argued similarly in Toublan 05)

 Not applicable to fermion-loop corrections (where u & d are mixed).
→ equivalence in the „t Hooft limit.

An intuitive derivation
 Consider 〈qq〉 at finite μI or μB for degenerate 2 flavors u & d.
・ additional flavor loops suppressed at large Nc→ no mixing between u & d at leading order in 1/Nc.
・ sign of ±μ for d doesn‟t matter due to charge conjugation.
→ 〈qq〉(μB) = 〈qq〉(μI)(deconfinement argued similarly in Toublan 05)

 Not applicable to fermion-loop corrections (where u & d are mixed).
→ equivalence in the „t Hooft limit.
 Not applicable to the quantity 〈π±〉 (where u & d are mixed).

→ equivalence for a class of observables

An intuitive derivation
 Consider 〈qq〉 at finite μI or μB for degenerate 2 flavors u & d.
・ additional flavor loops suppressed at large Nc→ no mixing between u & d at leading order in 1/Nc.
・ sign of ±μ for d doesn‟t matter due to charge conjugation.
→ 〈qq〉(μB) = 〈qq〉(μI)(deconfinement argued similarly in Toublan 05)

 Not applicable to fermion-loop corrections (where u & d are mixed).
→ equivalence in the „t Hooft limit.
 Not applicable to the quantity 〈π±〉 (where u & d are mixed).

→ equivalence for a class of observables
 Not applicable if 〈π±〉≠0 in the ground state (where u & d are mixed).
→ equivalence outside BEC/BCS region

A recent lattice result
 Critical temperature of chiral transition at small μI and μB.
Cea-Cosmai-D‟Elia-Papa-Sanfilippo
arXiv:1110.3910(staggered fermion, ma=0.05, 163×4 lattice)
Universality (or phase-quenching) looks approximately valid
up to difference ~10% in three-color QCD.

Reminder: QCD Dirac operator
 QCD Dirac operator at μ=0:・ anti-Hermiticity:・ chiral symmetry:→ γ5 Hermiticity:
This ensures the positivity(which is not true at μB >0).

 QCD at μI>0,・ property:→ positivity at any μI>0

QCD inequalities (μ=0)
 Consider correlator of

Weingarten; Witten; Nussinov („83)(∵γ5 Hermiticity)(∵ Cauchy-Schwarz ineq.)

becomes maximum when

 Correlators behave(meson is π0,±)

at large separation.

 Thus we have mΓ ≧ mπ (pion is the lightest meson).

This argument is not applicable to the flavor-singlet.
Reason: contributions of flavor-disconnected diagrams

A no-go theorem at large Nc (μ=0)
 Disconnected diagram may be suppressed phenomenologically.
e.g.) Okubo-Zweig-Iizuka rule: suppression of annihilation diagrams
 It can be justified in the large-Nc limit. Witten („79)

 Then QCD inequalities are applicable to the flavor singlet: mσ≧mπTurning on mq>0, we have mσ≧mπ > 0.
 2nd-order chiral transition implies ξ=∞ or mσ=0 at some T=Tc, which
is prohibited for any mq>0 from above constraint.

A no-go theorem at large Nc (μ>0)
 Generalization to μ>0?: not straightforward due to lack of positivity.

A no-go theorem at large Nc (μ>0)
 Generalization to μ>0?: not straightforward due to lack of positivity.
 Still it is possible in QCD-like theories at μ>0 preserving positivity.
e.g., QCD at any μI,
inequality saturated when(meson is π±)→ mσ≧mπ± > 0 for mq>0 → no 2nd-order chiral transition for mq>0

A no-go theorem at large Nc (μ>0)
 Generalization to μ>0?: not straightforward due to lack of positivity.
 Still it is possible in QCD-like theories at μ>0 preserving positivity.
e.g., QCD at any μI,
inequality saturated when(meson is π±)→ mσ≧mπ± > 0 for mq>0 → no 2nd-order chiral transition for mq>0

 Via our universality, this result can be mapped to QCD at μB>0
outside BEC/BCS region.

TμA no-go theorem at large Nc (μ>0)
 Generalization to μ>0?: not straightforward due to lack of positivity.
 Still it is possible in QCD-like theories at μ>0 preserving positivity.
e.g., QCD at any μI,
inequality saturated when(meson is π±)→ mσ≧mπ± > 0 for mq>0 → no 2nd-order chiral transition for mq>0

 Via our universality, this result can be mapped to QCD at μB>0
outside BEC/BCS region.

T Thus we obtain an exact no-go theorem:
Critical phenomena (e.g. QCD critical point)
are forbidden for mq>0 outside BEC/BCS.
Hidaka-NY („11)μ

Remarks
 In this argument, we take large Nc only to justify 2 empirical facts:
・ suppression of sea quark effects (quenched approximation)・ suppression of disconnected diagrams (OZI rule).
 In other words, the no-go theorem would be violated if either of them is
not satisfied in real QCD.

Model results Critical point & average phase factor in random matrix model
Han-Stephanov („08)

QCD at μB>0

QCD at μI>0

pion BEC phase (ends at large μ due to a model artifact)

 Similar results obtained in NJL & PNJL models
Andersen- Kyllingstad- Splittorff; Sakai-Sasaki-Kouno-Yahiro („10)

Lesson

Nature somehow hides the QCD critical point and other interesting
physics inside the region where the sign problem is maximally severe.

Tno chiral critical region

possible chiral critical region
highly severe sign problemmQuark-hadron continuity
 QCD phase diagram at μB>0 where universality breaks down?
 Still there exists a similarity in Nc=Nf=3: quark-hadron continuity
Schafer-Wilczek („99); Baym-Hatsuda-Tachibana-NY (‟06, ‟07, „08, „10);

Nf=3 QCD at finite baryon density
Consider QCD with mu=md=ms at T=0.

Nf=3 QCD at finite baryon density
Consider QCD with mu=md=ms at T=0.
 At small μB:

recent lattice QCD results by HAL QCD & NPLQCD collaborations („11)

・ H dibaryon has the smallest excit. energy per baryon since mH/2<mB.
・ For μB>mH/2, H dibaryon (= boson) forms BEC which breaks U(1)B.

Nf=3 QCD at finite baryon density
Consider QCD with mu=md=ms at T=0.
 At small μB:

recent lattice QCD results by HAL QCD & NPLQCD collaborations („11)

・ H dibaryon has the smallest excit. energy per baryon since mH/2<mB.
・ For μB>mH/2, H dibaryon (= boson) forms BEC which breaks U(1)B.
 At large μB: color flavor locking (CFL) Alford-Rajagopal-Wilczek („99)

・ Diquark condensate:
u,d,s

r,g,b

・ Gauge invariant order parameter:→ Chiral & U(1)B symmetries are broken.

Nf=3 QCD at finite baryon density
Consider QCD with mu=md=ms at T=0.
 At small μB:

recent lattice QCD results by HAL QCD & NPLQCD collaborations („11)

・ H dibaryon has the smallest excit. energy per baryon since mH/2<mB.
・ For μB>mH/2, H dibaryon (= boson) forms BEC which breaks U(1)B.
 At large μB: color flavor locking (CFL) Alford-Rajagopal-Wilczek („99)

・ Diquark condensate:
u,d,s

r,g,b

・ Gauge invariant order parameter:→ Chiral & U(1)B symmetries are broken.

 Same symmetry breaking pattern: quark-hadron continuity

QCD phase diagram (mu,d,s=0)
T
Quark-Gluon Plasmachiral transition?

chiral symmetry
brokenchiral & U(1)B broken

H BEC

mH/2

CFL

mB

Quark-hadron continuity (conjecture)

H dibaryon excitation(first principles lattice QCD)

weak-coupling analysis(exact)Color-flavor locking (contd.)
Elementary excitations• Quarks: massive by the BCS pairing (Majorana mass).
• Gluons: massive by the Higgs mechanism (SU(3)c breaking).
• Light “pions”: chiral & U(1)B symmetry breaking.

Hadron/CFL correspondence (Nc=Nf=3)
Phase

Hadron (confined)

CFL (Higgs)

symmetrybreaking

SU(3)L×SU(3)R×U(1)B

SU(3)L×SU(3)R×SU(3)C×U(1)B→ SU(3) L+R→ SU(3)L+R+C(pseudo)scalars

8 pseudo-scalars & 1 scalar

8 pseudo-scalars & 1 scalar

vectors

vector mesons

gluonsfermionsbaryons

quarkscoupling
constantstrong-couplingweak-coupling

quarksconfinedcondensed

monopoles(condensed?)confined

Existence of confined monopoles in CFL is recently shown.
Gorsky-Shifman-Yung; Eto-Nitta-NY („11)

Hadron/CFL correspondence (Nc=Nf=3)
Phase

Hadron (confined)

CFL (Higgs)

symmetrybreaking

SU(3)L×SU(3)R×U(1)B

SU(3)L×SU(3)R×SU(3)C×U(1)B→ SU(3) L+R→ SU(3)L+R+C(pseudo)scalars

8 pseudo-scalars & 1 scalar

8 pseudo-scalars & 1 scalar

vectors

vector mesons

gluonsfermionsbaryons

coupling
constantstrong-coupling

quarksconfinedmonopoles

one-to-one
correspondence

strong-weak dual?

quarksweak-coupling
condensed(condensed?) electromagnetic dual? confined
Existence of confined monopoles in CFL is recently shown.
Gorsky-Shifman-Yung; Eto-Nitta-NY („11)

Summary & Outlook•

Universality of phases in QCD and QCD-like theories at large Nc,
which seems approximately valid even at Nc=3
→ future lattice QCD applications•

Universality can be shown in all effective & holographic models(PNJL, Sakai-Sugimoto models, ...): e.g., phase-quench for Polyakov
loop up to one-meson-loop corrections.
Hanada-Matsuo-NY, to appear.

•

A no-go theorem for chiral critical phenomena.

•

The quark-hadron continuity appears in the phase diagrams withflavor symmetry.

•

Flavor symmetry breaking & finite Nc effects may be crucial for thereal dense matter.

Back up slides

What are (aren’t) equivalent?
Not all the quantities are equivalent in the orbifold equivalence.
 Projection symmetry must be unbroken.
 Observables must keep the projection symmetry (neutral).
 Symmetry breaking patterns, quantum numbers of the condensates
can be different, but their magnitudes are the same.
 Example: BCS gap (inside the BEC-BCS crossover)„t Hooft limit (large Nc, g2Nc fixed)

Perturbative proof
Bershadsky-Johansen (‟98)• Insert
for each propagator.
• Take the same „t Hooft coupling.
• Difference comes from color factors.
• Condition:

Generally,

when

for any planar diagrams.

From SO(2Nc) to SU(Nc) at finite μI
Start with SO(2Nc) gauge theory at μB>0.
1. Discrete symmetry:

2. Projection:

3. Daughter theory: U(Nc) ≈ SU(Nc) gauge theory at μI>0.

4. Orbifold equivalence:etc.

[Caution]•

Z4 ∈SU(2)iso unbroken from SO(2Nc) to QCD at μI>0 everywhere.

Approximate universality in real QCDOne can check the universality (e.g. BCS gap) at sufficiently large μ:
ratios

Universality of phase diagrams can also be shown in effectivemodels of QCD, e.g., chiral random matrix models.
・ chiral unitary matrix model

Klein-Toublan-Verbaarschot („03)

・ generalized to all the universality classes

Hanada-NY („11)

Remarks
 QCD phase diagram at μB>0 (nuclear domain) qualitatively changes as afunction of Nc inside BEC/BCS region:
・ no nuclear liquid-gas transition at large Nc

・ no color superconductivity

Torrieri-Mishustin („10).

Deryagin-Grigoriev-Rubakov („92); Shuster-Son („00).

=

non-planar

 QCD phase diagram at μB>0 is similar between Nc=3 and large Nc
outside BEC/BCS region: chiral symmetry breaking/restoration.
 The whole phase diagrams of QCD-like theories (including BEC/BCS
region) do not change qualitatively as a function of Nc.
 Universality is valid where strong Nc-dependence is absent .

Remarks
 QCD phase diagram at μB>0 (nuclear domain) qualitatively changes as afunction of Nc inside BEC/BCS region:
・ no nuclear liquid-gas transition at large Nc

・ no color superconductivity

Torrieri-Mishustin („10).

Deryagin-Grigoriev-Rubakov („92); Shuster-Son („00).

=

non-planar

 QCD phase diagram at μB>0 is similar between Nc=3 and large Nc
outside BEC/BCS region: chiral symmetry breaking/restoration.
 The whole phase diagrams of QCD-like theories (including BEC/BCS
region) do not change qualitatively as a function of Nc.
 Universality is valid where strong Nc-dependence is absent .