THE QCD PHASE DIAGRAM AND THE CRITICAL ENDPOINTApr 06, 2019 · AND THE CRITICAL ENDPOINT Fabian...

THE QCD PHASE DIAGRAMAND THE CRITICAL ENDPOINT

Fabian RenneckeBrookhaven National Laboratory

— RHIC/AGS USERS’ MEETING 2019 — BNL, 04/06/2019

THE QCD PHASE DIAGRAM

THE QCD PHASE DIAGRAM

CEP unlikely

for μB/T≤2[HotQCD, hep-lat/1701.04325]

The scale evolution of QCD

Functional RG with emergent bound states

First (preliminary) results on the phase diagram

OUTLINE

k

SCALE EVOLUTION OF QCDintegrating out fluctuations from UV to IR

energy / renormalization group scale

k

initial action: gauge fixed Euclidean QCD

SCALE EVOLUTION OF QCD

LQCD = q(�µDµ +mq)q +1

4F aµ⌫F

aµ⌫ +

1

2⇠(@µA

aµ)

2 + ca@µDabµ cb

<latexit sha1_base64="Hr/YGe7bc7sHlJ90a1fj33If7Fw=">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</latexit>

Dµ = @µ � igAaµt

a<latexit sha1_base64="0RcWZsIGnwyEp0zvVHP1XXfFrC8=">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</latexit>

F aµ⌫t

a =i

g[Dµ, D⌫ ]

<latexit sha1_base64="xYAYt/HqL3Jqcf/IHJgOSAVZ9pw=">AAAC9nichVFNSxxBEH1OTKLma9Wjl8ElEDAss0aIF0HQSC4BA1kVdtelZ+wdm53pGXp6JGbYH5E/kJvkmluu5mckvyWHvGlnAyGI3VRX9auq11VdYZ6owgbBzznv3vz9Bw8XFpcePX7y9FlreeWoyEoTyV6UJZk5CUUhE6VlzyqbyJPcSJGGiTwOJ3u1//hCmkJl+oO9zOUwFbFWYxUJS2jU2jgYVYO0HOhyeip8S9nxB2MjokpNq3ja3x/R+5KnLoejVrvbCdzybzfaaNZh1vqFAc6QIUKJFBIalnYCgYK7jy4C5MSGqIgZWsr5JaZYYm7JKMkIQXTCM+at36Ca95qzcNkRX0kohpk+nlMOHGPI6PpVSbug/k355LD41hcqx1xXeEkdknHRMb4jbnHOiLsy0yZyVsvdmXVXFmNsu24U68sdUvcZ/eXZp8cQmziPjzcuMiZH6O4X/AFN3WMF9S/PGHzX8Rm1cFo6Ft0wCvIZ6vr3WQ/HHHS2b4b6vzEb89Fmp/uqs/l+q7271Qx8AWtYxwtO9TV28RaHrCPCZ3zHNX54H70v3pX39SbUm2tyVvHP8r79AZCqoHw=</latexit>

covariant derivative field strength

asymptotic freedom: fluctuations increase strong coupling

[S. Bethke, hep-ex/0606035]

⇤ � 1GeV<latexit sha1_base64="nsE9bxKKJnSOTzluRONrXmJA0dY=">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</latexit>

k

effective four-quark interactions are generated


λT,k grows with decreasing energy scale

running controlled by IR-attractive fixed point


JHEP06(2006)024

λi

∂tλi g = 0

g ! 0

g > gcr

T > 0, g = 0

Figure 5: Sketch of a typical β function for the fermionic self-interactions λi: at zero gaugecoupling, g = 0 (upper solid curve), the Gaußian fixed point λi = 0 is IR attractive. For smallg ! 0 (middle/blue solid curve), the fixed-point positions are shifted on the order of g4. Forgauge couplings larger than the critical coupling g > gcr (lower/green solid curve), no fixed pointsremain and the self-interactions quickly grow large, signaling χSB. For increasing temperature, theparabolas become broader and higher, owing to thermal fermion masses; this is indicated by thedashed/red line.

ing larger than the regulator scale k, these functions approach zero, which reflects the

decoupling of massive modes from the flow.

Within this set of degrees of freedom, a simple picture for the chiral dynamics arises: for

vanishing gauge coupling, the flow is solved by vanishing λi’s, which defines the Gaußian

fixed point. This fixed point is IR attractive, implying that these self-interactions are

RG irrelevant for sufficiently small bare couplings, as they should be. At weak gauge

coupling, the RG flow generates quark self-interactions of order λ ∼ g4, as expected for

a perturbative 1PI scattering amplitude. The back-reaction of these self-interactions on

the total RG flow is negligible at weak coupling. If the gauge coupling in the IR remains

smaller than a critical value g < gcr, the self-interactions remain bounded, approaching

fixed points in the IR. These fixed points can simply be viewed as order-g4 shifted versions

of the Gaußian fixed point, being modified by the gauge dynamics. At these fixed points,

the fermionic subsystem remains in the chirally invariant phase which is indeed realized at

high temperature.

If the gauge coupling increases beyond the critical coupling g > gcr, the above-

mentioned IR fixed points are destabilized and the quark self-interactions become critical.

This can be visualized by the fact that ∂tλi as a function of λi is an everted parabola;

see figure 5; for g = gcr, the parabola is pushed below the λi axis, such that the (shifted)

Gaußian fixed point annihilates with the second zero of the parabola. In this case, the

gauge-fluctuation-induced λ’s have become strong enough to contribute as relevant opera-

tors to the RG flow. These couplings now increase rapidly, approaching a divergence at a

finite scale k = kχSB. In fact, this seeming Landau-pole behavior indicates χSB and, more

specifically, the formation of chiral condensates. This is because the λ’s are proportional

dimensions which were shown to influence the quantitative results for the present system only on the

percent level, if at all [50].

– 17 –

[Braun, Gies, hep-ph/0602226]

IR-attractive FP

⇠ ↵2s,k �T,k(q T q)2

beta function

k

strong coupling exceeds critical value: FP vanishes; λT,k diverges


JHEP06(2006)024

λi

∂tλi g = 0

g ! 0

g > gcr

T > 0, g = 0















high temperature.












– 17 –

[Braun, Gies, hep-ph/0602226]

resonance in condensate channel: chiral symmetry breaking

hqqi 6= 0<latexit sha1_base64="ETjkBHpZloQLdakt8j4dMedLdWQ=">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</latexit>

�S,k(qq)2 ! 1

<latexit sha1_base64="CB0syw2iDA7nH/vsrPkVMw/VonM=">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</latexit>

around the same scale: gluon mass-gap develops, confinement

0

0.5

1

1.5

2

2.5

3

0.1 1 10

glu

on p

ropagato

r dre

ssin

g 1

/ZA

p [GeV]

error estimate

mπ=140 MeV

mπ=60 MeV

mπ=285 MeV

lattice, β=5.29, mπ=150 MeV

[Cyrol, Mitter, Pawlowski, Strodthoff, hep-ph/1706.06326]

k�SB ⇡ 0.5GeV<latexit sha1_base64="+iGCH+wVegXUw18w7T0oqb3XpAs=">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</latexit>


p2GA(p2)

<latexit sha1_base64="rZr3iRqNVx4xCo3gKvjKb93kAzY=">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</latexit>

k

rewrite resonant 4-quark channels in terms of mesons (similar: baryonization from 6-quark interactions)


�T,k =h2T,k

m2�,k

Yukawa coupling

meson mass parameter

[Braun, Fister, Pawlowski, FR, hep-ph/1412.1045]

0.5 1.0 2.0 5.0 10.0 20.00

5

10

15

20

25

k @GeVD

h k

hL=5 GeV = 0.001hL=5 GeV = 30hL=10GeV = 0.001hL=10GeV = 50hL=20GeV = 0.001hL=20GeV = 50

fixed point at small g: low-energy parameters uniquely determined

k�SB ⇡ 0.5GeV<latexit sha1_base64="+iGCH+wVegXUw18w7T0oqb3XpAs=">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</latexit>


low-energy models emerge when gauge sector decouples

different tensor structures of λT,k: different bound states

U(3) generators

(qT aq)2 + (qi�5Taq)2

<latexit sha1_base64="qrgxfVLY9Gcc1XzdW+jVm3Qz0BU=">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</latexit>

⇡, K, ⌘, ⌘0, a0, , �, f0<latexit sha1_base64="qO+j021n8Hzna8M8v36rB6oDECo=">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</latexit>

successively integrate out fluctuations from UV to IR (Wilson RG)

full quantum effective action(generates 1PI correlators)

Γk is eff. action that incorporates all fluctuations down to scale k

lowering RG-scale k: zooming out / coarse graining

�⇤ �k �0 = �

FUNCTIONAL RGlow-energy QCD inherently strongly interacting: non-perturbative method

capture fundamental and emergent d.o.f.: scale dependent effective action

finite chemical potential: no sign problem

• introduce regulator to partition function to suppress momentum modes below energy scale k (Euclidean space):

k2

k2

k2

k2

Rk(p2)

12@tRk(p2)

p2

• scale dependent effective action:

� = h'iJ�k[�] = supJ

⇢Z

xJ(x)�(x)� lnZk[J ]

��Sk[J ]

Zk[J ] =

ZD' e�S[']��Sk[']+

Rx J'

�Sk['] =1

2

Zd4q

(2⇡)4'(�q)Rk(q)'(q) slow modes fast modes

DYNAMICAL HADRONIZATION

bosonize 4-quark interactions in each RG-step

[Gies, Wetterich (2002), Pawlowski (2007), Floerchinger, Wetterich (2009)][Braun, Fister, Pawlowski, FR, hep-ph/1412.1045]

0

50

100

150

200

250

0.1 1 10

(boso

niz

ed)

4-f

erm

i-in

tera

ctio

n

RG-scale k [GeV]

h2π/(2 m2

π)

λπ

[Mitter, Pawlowski, Strodthoff, hep-ph/1411.7978]

4-quark interaction encoded in Yukawa coupling

unified description of QGP and hadronic phase

[original: Wetterich 1993]

@t = kd

dk

Flow equation@t�k[�] =

1

2� � +

1

2 ⇣�(2)k [�] +R�

k

⌘�1

<latexit sha1_base64="ljc3cn7VT2dziJOOYAy1aMsMkiA=">AAAC93ichVHLThRBFD20L8AHoyzddJyYDFEnPaORdkcQxY3JaBwgmVeqm6Kp9DPdPSTDZH6CH2BH2Lpji3+B3+LCU2WPiSGG6lTfW+eee+reul4WqaJ0nKsF69btO3fvLS4t33/w8NFK7fGTnSId577s+mmU5nueKGSkEtktVRnJvSyXIvYiueuF73V890jmhUqTb+Ukk4NYBIk6UL4oCY1qL/ubKmj0t0Uci1E4nDbaa7Nev3OoBi++DrUdhZqxNpy+as1GtbrTfOe67rpjX3daTcesOqrVSWs/0cc+UvgYI4ZEgpJ+BIGCXw8tOMiIDTAlltNTJi4xwzJzx2RJMgTRkP+Ap16FJjxrzcJk+7wl4s6ZaeM590ej6JGtb5X0C9pf3McGC/57w9Qo6wontB4Vl4ziZ+IlDsm4KTOumPNabs7UXZU4gGu6UawvM4ju0/+rs8VITiw0ERsfDDOghmfOR3yBhLbLCvQrzxVs0/E+rTBWGpWkUhTUy2n167MeM2bXzNK+7szHvNNutl4321/e1DfeVgNfxFM8Q4NTXccGPqHDOnyc4AKX+GFNrFPrzDr/Q7UWqpxV/LOs778BggigDw==</latexit>

gluons ghosts quarks mesons

@tR�k

<latexit sha1_base64="d7b478y9Ubcjdu/gN7egvgUW1NQ=">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</latexit>

quarks & gluonshadronshadronic sector emerges dynamically; no model parameters

need continuous transition from quarks and gluons to hadrons

k-derivative of Γk :

APPLICATION TOTHE PHASE DIAGRAM

Task: construct appropriate Γk and solve the resulting flow equation

Difficulty: effective action can contain everything allowed by symmetries

Strategy: identify physically most relevant terms and include them: qualitative

truncation necessary

systematically add terms and look for convergence: quantitative

certainly depends on the problem at hand

Goal: initialize RG-flow in perturbative regime: microscopic QCD with strong coupling and current quark masses as parameters

low-energy sector emerges dynamically, fixed from underlying QCD dynamics

VACUUM QCDFRG results in very good agreement with the lattice (YM & Nf = 2, Landau gauge)

0

0.5

1

1.5

2

2.5

3

0.1 1 10

glu

on p

ropagato

r dre

ssin

g 1

/ZA

p [GeV]

error estimate

mπ=140 MeV

mπ=60 MeV

mπ=285 MeV


0

0.2

0.4

0.6

0.8

1

0.1 1 10

qu

ark

pro

pa

ga

tor

dre

ssin

gs

p [GeV]

error estimate

mπ=140 MeV

mπ=60 MeV

mπ=285 MeV



gluon dressing function quark propagator

p2GA(p2)

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p2Gq(p2)

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Mq(p2)

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[Cyrol, Fister, Mitter, Pawlowski, Strodthoff (2011-2018)]


VACUUM QCD

Also very expensive… (flow equations alone can take GBs in .txt files)

FRG results in very good agreement with the lattice (YM & Nf = 2, Landau gauge)[Cyrol, Fister, Mitter, Pawlowski, Strodthoff (2011-2018)]

VACUUM QCD

As it turns out: using gluon and ghost propagators as input and adding matter fluctuations works well

2

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!

@t = @t

� � + +

!

@t�1 = @t

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@t�1 = @t

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!

On the side of theories, lattice QCD simulation is anonperturbative first-principle approach, which has pro-vided us with lots of remarkable insights and understand-ing in recent years, such as the determination of freeze-our parameters through the confrontation of lattice re-sults with experiments [? ? ? ], the QCD equationof state and fluctuations of conserved charges at finitechemical potentials using Taylor expansions or analyti-cal continuation [? ? ? ? ], etc. However, because ofthe sign problem, lattice calculations are usually limitedto some specific region in the QCD phase diagram, sayµB/T 2, which corresponds to center-of-mass energies& 12 GeV, and in this regime the existence of a CEP isdisfavored [? ]. Functional continuum field approaches,e.g., the Dyson-Schwinger equation [? ? ? ? ], and thefunctional renormalization group (FRG) [? ? ? ? ? ? ],etc., which are complementary to the lattice QCD, havealso seen significant progresses on the studies of the non-perturbative QCD and QCD thermodynamics in recentyears.

The FRG approach is a nonperturbative continuumfield theory [? ], which encodes successively quantumfluctuations of di↵erent scales with the evolution of therenormalization group (RG) scale. And thus a full quan-tum e↵ective action is obtained from a classical one, af-ter the RG scale evolves from the ultraviolet to infraredregion. For more discussions about the FRG, see, e.g.,QCD related reviews in [? ? ? ? ? ] and recent devel-

opments in [? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?].

In the past several yeas a remarkable progress on thestudies of FRG is the first-principle calculations in theYang-Mills theory and QCD, e.g. the quenched [? ] andunquenched [? ] QCD in the vacuum, the Yang-Millsgauge theory in the vacuum [? ] and finite tempera-ture [? ], the unquenched QCD in the vacuum with asimplified truncation [? ? ]. In this work, we wouldlike to perform the first-principle QCD calculations atfinite temperature and baryon chemical potential withinthe FRG approach. QCD phase transitions including thechiral phase transition and the color deconfinement phasetransition will be investigated. We will also study theQCD correlation functions and their dependence on theexternal parameters. Moreover, a T �µB phase diagramwill be presented based on our computation.

This paper is organized as follows. In Sec. ?? we in-troduce the functional renormalisation group approach toQCD. In Sec. ?? correlation functions including propaga-tors, the strong couplings, and the dynamical hadroniza-tion are discussed. In Sec. ?? numerical results and re-lated discussions are presented, and a summary and out-look is given in Sec. ??. Details about our calculations,such as the flow equations for the e↵ective potential andcouplings, anomalous dimensions, the glue potential, nu-merical setup, and the threshold functions, are presented

propagators as input quark back-coupling

0 1 2 3 4 5 6 70

1

2

3

4

p @GeVD

1êZA,k=p

0 1 2 3 4 5 6 70

1

2

3

4

p @GeVD

1êZA,k=p

0 1 2 3 4 5 6 70

1

2

3

4

p @GeVD

1êZA,k=p

Nf = 0 , Sternbeck et al (2005)

Nf = 2 , Sternbeck (2015)

Nf = 0 , input

Nf = 2

Nf = 2 , (vacuum polarization only)

[Braun, Fister, Pawlowski, FR, hep-ph/1412.1045]

Also very expensive…

FRG results in very good agreement with the lattice (YM & Nf = 2, Landau gauge)[Cyrol, Fister, Mitter, Pawlowski, Strodthoff (2011-2018)]

(flow equations alone can take GBs in .txt files)

TRUNCATIONdominant 4-quark channel at low and moderate μ: �S-P,k

⇥(qq)2 + (qi�5~⌧q)

2⇤

<latexit sha1_base64="LCC/8KqeAS4KTtWkOVWekVKa2rI=">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</latexit>

if resonant: chiral condensate, σ/f0(500) and pions emerge

spontaneous chiral symmetry breaking!

QCD phase structure at finite temperature and density

Wei-jie Fu,1, 2, ⇤ Jan M. Pawlowski,2, 3, † and Fabian Rennecke4, ‡

1School of Physics, Dalian University of Technology, Dalian, 116024, P.R. China

2Institut fur Theoretische Physik, Universitat Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

3ExtreMe Matter Institute EMMI, GSI, Planckstr. 1, 64291 Darmstadt, Germany

4Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA

We discuss the QCD phase structure at finite temperature and chemical potential within thefunctional renormalisation group approach. Results are presented for Nf = 2 and Nf = 2 + 1flavours. These results are compared with lattice results for low densities and with other functionalresults for the whole density regime. The curvature of the phase boundary for small chemicalpotential, = 0.0159±0.0020, is consistent with recent lattice results. For Nf = 2+1 a critical endpoint is found at (TCEP , µBCEP

) = (117± 5MeV, 645± 40MeV). This density regime is beyond thevalidity regime of the current approximation, and we discuss the necessary systematic enhancements.

PACS numbers: 11.30.Rd, 11.10.Wx, 05.10.Cc, 12.38.Mh

I. INTRODUCTION

QCD phase structure is a highly active scientific re-search topic in recent years. A plethora of relevant impor-tant results has been obtained from both experimentalmeasurements and theoretical studies, see, e.g., reviewsin [? ? ? ? ? ? ? ? ] and references therein. How-ever, our knowledge concerning the QCD phase diagramat high baryon chemical potential remains rare and elu-sive, and in particular, the key questions whether thereis a critical end point (CEP) in the phase diagram, andif it exits where it is, are still open to be answered. For-

tunately, lots of e↵orts are currently being performed inorder to unravel the mysterious veil. Experimentally, theBeam Energy Scan (BES) Program at Relativistic Heavy-Ion Collider (RHIC) has presented significant measure-ments [? ? ? ? ], and BES phase II is under wayfor the moment; searching for CEP and studies of QCDphase structure are also planned at other facilities allover the world with di↵erent collision energies and lumi-nosities, such as the FAIR/CBM in Germany [? ] , theNICA/MPD in Russia [? ], the J-PARC-HI in Japan [?], and the HIAF in China [? ].

@t = @t

� � �

� � �

� � �!

⇤ E-mail: [email protected] † E-mail: [email protected]‡ E-mail: [email protected]

[Braun, Leonhardt, Pospiech, hep-ph/1801.08338]

2

@t = @t

� � �

!

@t = @t

� � + +

!



opments in [? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?].In the past several yeas a remarkable progress on the

studies of FRG is the first-principle calculations in theYang-Mills theory and QCD, e.g. the quenched [? ] andunquenched [? ] QCD in the vacuum, the Yang-Millsgauge theory in the vacuum [? ] and finite tempera-ture [? ], the unquenched QCD in the vacuum with asimplified truncation [? ? ]. In this work, we wouldlike to perform the first-principle QCD calculations atfinite temperature and baryon chemical potential withinthe FRG approach. QCD phase transitions including thechiral phase transition and the color deconfinement phasetransition will be investigated. We will also study theQCD correlation functions and their dependence on theexternal parameters. Moreover, a T �µB phase diagramwill be presented based on our computation.This paper is organized as follows. In Sec. ?? we in-

troduce the functional renormalisation group approach toQCD. In Sec. ?? correlation functions including propaga-tors, the strong couplings, and the dynamical hadroniza-tion are discussed. In Sec. ?? numerical results and re-lated discussions are presented, and a summary and out-look is given in Sec. ??. Details about our calculations,such as the flow equations for the e↵ective potential andcouplings, anomalous dimensions, the glue potential, nu-merical setup, and the threshold functions, are presentedin appendices.

2

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!

@t = @t

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!

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✓+

◆

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� 1

2� �

!

@t�1 = @t

� + � 1

2

!



opments in [? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?].

In the past several yeas a remarkable progress on thestudies of FRG is the first-principle calculations in theYang-Mills theory and QCD, e.g. the quenched [? ] andunquenched [? ] QCD in the vacuum, the Yang-Millsgauge theory in the vacuum [? ] and finite tempera-ture [? ], the unquenched QCD in the vacuum with asimplified truncation [? ? ]. In this work, we wouldlike to perform the first-principle QCD calculations atfinite temperature and baryon chemical potential withinthe FRG approach. QCD phase transitions including thechiral phase transition and the color deconfinement phasetransition will be investigated. We will also study theQCD correlation functions and their dependence on theexternal parameters. Moreover, a T �µB phase diagramwill be presented based on our computation.

This paper is organized as follows. In Sec. ?? we in-troduce the functional renormalisation group approach toQCD. In Sec. ?? correlation functions including propaga-tors, the strong couplings, and the dynamical hadroniza-tion are discussed. In Sec. ?? numerical results and re-lated discussions are presented, and a summary and out-look is given in Sec. ??. Details about our calculations,such as the flow equations for the e↵ective potential andcouplings, anomalous dimensions, the glue potential, nu-merical setup, and the threshold functions, are presented

quark, gluon, meson (& ghost) propagators gauge couplings

dynamical hadronization of this channel: dynamical π,σ + Yukawa interactions

external input:

effective meson potential: arbitrary orders of meson self-interactions

not one-loop!flow equations: all propagators and vertices are coupled solving them: resummation of loops of infinite order

vacuum QCD & finite-T YM

[Fu, Pawlowski, FR (in preparation)]

PRELIMINARY RESULTS[Fu, Pawlowski, FR (in preparation)]

Strong couplingsJHEP06(2006)024

λi

∂tλi g = 0

g ! 0

g > gcr

T > 0, g = 0















high temperature.












– 17 –


gluon propagator quark & meson masses


The 2+1 flavor phase diagram

(TCEP, µBCEP) = (110+5�5, 590

+30�20)MeV

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psCEP ⇡ 3.8� 4.5GeV

<latexit sha1_base64="+n6fpXnSH3WCc4dzcdW+o8isW4I=">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</latexit>

BES Fixed Target!

OUTLOOK

Compute gluon background potential directly

• so far: external input from the lattice (introduces a pheno parameter)

• Polyakov loop potential at finite density

Short- to mid-range goals

Include additional 4-quark channels (Fierz-completeness)

• so far: most dominant channel including χSB

• ω0 vector channel / diquark channel become relevant at large μ (and `small´ T)

• access to color-superconducting phases, liquid-gas transition, etc.

[Braun, Gies, Pawlowski, hep-th/0708.2413][Fischer, Fister, Luecker, Pawlowski, hep-ph/1306.6022]

[Braun, Leonhardt, Pospiech, hep-ph/1801.08338][Mitter, Pawlowski, Strodthoff, hep-ph/1411.7978]

0

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0.2

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0.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

T/k 0

µ /k0

Fierz completetwo channelsone channel

scalar-pseudoscalar

scalar-pseudoscalar+ diquark

all channels

Include additional quark-gluon channels (non-classical)

• crucial for χSB!

• so far: only classical channel (need additional pheno parameter to compensate for missing channels)


SUMMARY

First steps towards a systematic computation of the QCD phase diagram from first principles

indications that CEP is within BES Fixed Target region

THE QCD PHASE DIAGRAM AND THE CRITICAL ENDPOINTApr 06, 2019 · AND THE CRITICAL ENDPOINT Fabian...

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