Impact of confining dynamics on chiral symmetry breakingdietrich/SLIDES3/Leonhardt.pdf · Bad...

Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt

Impact of confining dynamics on chiral symmetry breaking

Marc Leonhardt Institut für Kernphysik, Technische Universität Darmstadt

1

with Jens Braun1,2, Stefan Rechenberger3 and Paul Springer4 1TU Darmstadt, 2ExtreMe Matter Institute EMMI, GSI,

3Goethe-Universität Frankfurt, 4TU München

Understanding the LHC 637. Wilhelm und Else Heraeus-Seminar

Physikzentrum Bad Honnef February 2017

Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 2

• Motivation

• FRG and the low energy model of QCD

• Results

• Fixed-point analysis

• Phase diagram

• Conclusions and outlook

Outline

QCD phase diagram


K

Universe (<10-6 s)Primordial

"Solid" state

Quark-Gluon Plasma

Neutron Stars

"Ordinary" stateNet baryonic density

(normalised, d/d0)

Hadronic gas

0 1 5 8

50

150

100

250

200

Transition

PhasepTe

mpe

ratu

re (M

eV)

[ALI

CE@

CER

N]

Hadronic Phase

Quark-Gluon Plasma

QCD phase diagram


[ALI

CE@

CER

N]

Deconfinement Restoration of chiral symmetry

Free color charges possible

Quark masses remain small

Deconf. Phase Transition

Chiral Phase Transition

ConfinementChiral symmetry breaking (χSB)

Quarks confined in bound states

forming color singlets

Dynamical generation of (constituent) quark masses

Hadronic PhaseK


"Solid" state

Quark-Gluon Plasma

Neutron Stars


(normalised, d/d0)

Hadronic gas

0 1 5 8

50

150

100

250

200

Transition

PhasepTe

mpe

ratu

re (M

eV)

Quark-Gluon Plasma

QCD phase diagram


[ALI

CE@

CER

N]

Deconfinement Restoration of chiral symmetry

Free color charges possible

Quark masses remain small

Deconf. Phase Transition

Chiral Phase Transition

ConfinementChiral symmetry breaking (χSB)

Quarks confined in bound states

forming color singlets

Dynamical generation of (constituent) quark masses

Closely linked?

Hadronic PhaseK


"Solid" state

Quark-Gluon Plasma

Neutron Stars


(normalised, d/d0)

Hadronic gas

0 1 5 8

50

150

100

250

200

Transition

PhasepTe

mpe

ratu

re (M

eV)

Quark-Gluon Plasma

Functional renormalization group (FRG)


�kk!0��! �

�kk!⇤��! S

Theory space

S

��k

Partition function/ generating functional

Effective action �

Effective average action Rk �k

[C.Wetterich, Phys. Lett. B, 301, 1993]

Flow equation t = ln(k/⇤)@t�k =

1

2STr

⇢h�(2)k +Rk

i�1· (@tRk)

�

5

Z = trhe��H

i=

ZD' e�S[']

UV:

IR:

Low energy model of QCD


QCD Lagrangian in the chiral limit ( ):mq �! 0

SUV (Nf )⌦ SUA(Nf )⌦UV (1)⌦UA(1)





SUV (Nf )⌦ SUA(Nf )⌦UV (1)

�k

[ , ] =

Z

x

⇢Z

�i/@ + i�0µq

� +

�

2

⇥( )2 � ( ~⌧�5 )

2⇤�

NJL-type model of low-energy QCD, , Nf = 2 Nc

µq : The quark chemical potential introduces an imbalance between quarks and antiquarks

µq > 0qq

qq

q q qqq

qqq



gs gs

undefined

1

undefined

1

gs gs

�



SUV (Nf )⌦ SUA(Nf )⌦UV (1)

�k

[ , ] =

Z

x

⇢Z

�i/@ + i�0µq

� +

�

2

⇥( )2 � ( ~⌧�5 )

2⇤�

NJL-type model of low-energy QCD, , Nf = 2 Nc

µq : The quark chemical potential introduces an imbalance between quarks and antiquarks

µq > 0qq

qq

q q qqq

qqq

Incorporating confining dynamics


Order parameter confinement-deconfinement phase transition:Polyakov Loop: hP [A0]i




mq �! 1QCD with : hP [A0]i / e��Fq

Free energy of a single static color source






T < Td : () �Fq �! 1 () hP [A0]i = 0T � Td : () �Fq () hP [A0]i 6= 0

ConfinedDeconf. finite






T < Td : () �Fq �! 1 () hP [A0]i = 0T � Td : () �Fq () hP [A0]i 6= 0

ConfinedDeconf. finite

T/Td

P [hA0i]1.0

0.5

0.00.9 1.0 1.1

Confined Deconfined

in Polyakov gaugeP [hA0i][J. Braun and T.K. Herbst, 2012][F. Marhauser and J. M. Pawlowski, 2008]

[J. Braun, H. Gies, and J. M. Pawlowski, 2010]: Computation of in YM theory for hA0i Nc = 3



�k

[ , , hA0i] =Z

x

⇢Z

�i/@ + g

s

�0hA0i+ i�0µq

� +

�

2

⇥( )2 � ( ~⌧�5 )

2⇤�

Ansatz for the effective average action

T/Td

P [hA0i]1.0

0.5

0.00.9 1.0 1.1

Confined Deconfined

Background field as external inputTdCritical temperature




�k

[ , , hA0i] =Z

x

⇢Z

�i/@ + g

s

�0hA0i+ i�0µq

� +

�

2

⇥( )2 � ( ~⌧�5 )

2⇤�


T/Td

P [hA0i]1.0

0.5

0.00.9 1.0 1.1

Confined Deconfined

Background field as external input

@t�k =1

2STr

(@tRk

�(2)k +Rk

)

Wetterich eq.

@t� ⌘ ��

RG flow eq.

TdCritical temperature




�k

[ , , hA0i] =Z

x

⇢Z

�i/@ + g

s

�0hA0i+ i�0µq

� +

�

2

⇥( )2 � ( ~⌧�5 )

2⇤�


T/Td

P [hA0i]1.0

0.5

0.00.9 1.0 1.1

Confined Deconfined

Background field as external input

@t�k =1

2STr

(@tRk

�(2)k +Rk

)

Wetterich eq.

@t� ⌘ ��

RG flow eq.

, 1

� ! 0Onset

of χSB

TdCritical temperature



� � �� ⌘ @t� ⇠ Fixed point: �(�⇤ ) = 0

Fixed-point analysis of the RG flow equation

� �⇤

@t�

T = 0, µq = 0, hA0i = 0


� � �� ⌘ @t� ⇠ Fixed point: �(�⇤ ) = 0


χSBχSym

�UV

� �⇤

@t�

T = 0, µq = 0, hA0i = 0



� � �� ⌘ @t� ⇠

T/k

�⇤ (⌧, hA0i, µq)

Pseudo fixed-point

� �⇤

@t�

�UV

µq > 0

T > 0, µq > 0, hA0i = 0

T = 0, µq = 0, hA0i = 0T > 0


� � �� ⌘ @t� ⇠

� �⇤

@t�

�UV

µq > 0

T/k

�⇤ (⌧, hA0i, µq)

Pseudo fixed-point

T > 0, µq > 0, hA0i = 0

T = 0, µq = 0, hA0i = 0T > 0

hA0i > 0

^Nc ! 1

T > 0, µq > 0, hA0i > 0


: 1/� T�

Locking in the - plane


(T,�UV )

for k �! 0Chiral temperature

: 1/� T�



(T,�UV )


hA0i = 0

1.1

µq = 0 MeV

µq = 175 MeV

1.0

0.5

0.01.0 1.2 1.3 1.4

T�/T

d

�UV /�⇤

1.1

: 1/� T�



(T,�UV )


µq = 0 MeV

µq = 175 MeV

hA0i

T� ⇡ TdLocking:1.0

0.5

0.01.0 1.2 1.3 1.4

T�/T

d

�UV /�⇤

: 1/� T�



(T,�UV )


µq = 0 MeV

µq = 175 MeV

hA0i


0.5

0.01.0 1.1 1.2 1.3 1.4

T�/T

d

�UV /�⇤

Td

0 50 100 150µq [MeV]

: 1/� T�



(T,�UV )


µq = 0 MeV

µq = 175 MeV

hA0i


0.5

0.01.0 1.1 1.2 1.3 1.4

T�/T

d

�UV /�⇤

Td

0 50 100 150µq [MeV]

T�(µ2q) ⇡ T�

1� ·

µ2q

T 2�

!

�UV = 1.15 : ⇡ 0

�UV = 1.1 : ⇡ 1.368

⇡ 0.0032(1) [Fodor et al., 2004]Lattice QCD (2+1 flavors):

Partial bosonization and finite current quark masses


� h

� ⇠

~⇡ ⇠ ~⌧�5 h



� h

� ⇠

~⇡ ⇠ ~⌧�5 h

U�(�2)

��0

i mc ⌘ c�



� h

� ⇠

~⇡ ⇠ ~⌧�5 h

f⇡ = 93 MeV

m = 300 MeV

m⇡ = 139 MeV

Low- energy observables:

U�(�2)

��0

i mc ⌘ c�



� h

� ⇠

~⇡ ⇠ ~⌧�5 h

f⇡ = 93 MeV

m = 300 MeV

m⇡ = 139 MeV

Low- energy observables:

U�(�2)

��0

i mc ⌘ c�

1.0

0.5

0.00 50 100 150200 250

With hA0i

hA0i = 0

T�/T

d

µq [MeV]

Td

Conclusions and outlook


• Dynamically locking of the chiral phase transition to the deconfinement phase transition, even at small quark chemical potential

• The phase diagram and the curvature of the phase boundary is very sensitive to the initial UV value

• Adjusted to low-energy observables and at finite current quark masses the physical point is located inside the locking window, suggesting a dominance of the confining dynamics

• Outlook: Inclusion of dynamical gauge fields and back reaction of the matter sector on the gauge sector

(T, µq) �UV

Backup


Backup: Adjoint matter


10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 1.2 1.4 1.6 1.8 2 2.2 2.4

N = 2 (fund.)

N = 3 (fund.)

T�/T

d

�UV /�⇤

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 1.2 1.4 1.6 1.8 2 2.2 2.4

N = 2 (adj.)

N = 2 (fund.)

T�/T

d

�UV /�⇤

Figure 2. Left panel: Phase diagram in the plane spanned by the temperature and the rescaled coupling �UV /�⇤

for Nf = 2massless quark flavors in the fundamental representation and N = 2 colors (red/solid line) as well as for N = 3 colors(blue/dashed line), see also Ref. [28]. Note that there is no splitting of the phase boundary (i. e. T� ' Td) for small �UV

in the

large-N limit, see Eq. (34) and discussion thereof. Right panel: T�/Td as a function of �UV /�⇤

for Nf = 2 massless quarks inthe fundamental representation (N = 2) (red/solid line) as well as for quarks in the adjoint representation (blue/dashed line).

can be traced back to the deformation of the fermionicfixed-point structure in the presence of gauge dynamics.

To obtain the numerical results in Fig. 2, we have em-ployed data for hA0i(T ) as obtained from an RG studyof the associated order parameter potential for SU(2)and SU(3) Yang-Mills theory [33, 70]. However, wedid not take into account the back-reaction of the mat-ter fields on the order parameter potential associatedwith hA0i. In the case of fundamental matter, we expectthat this back-reaction will shrink the size of the lockingwindow since it further increases the quantity PF(T ) atlow temperatures. For adjoint quarks, the back-reactionwill also increase PA(T ). Nevertheless, it may remainnegative over a wide range of temperatures. Thereforewe may still have T

�

> Td for all values of �UV

/�

⇤

> 1,at least for N = 2.

Let us add a word of caution on the treatment of thequantity trRLR[hA0i] in standard PNJL/PQM model ap-proaches. In these studies, one relies on the assumptionthat trRLR[hA0i] = htrRLR[A0]i. For htrRLR[A0]i, onethen uses lattice data as input. Whereas such an ap-proach would lead to similar conclusions for fundamentalquarks (htrFLF[A0]i � 0 and trFLF[hA0i] � 0), the sit-uation is di↵erent for adjoint quarks. In the latter case,we have htrALA[A0]i > 0 but trALA[hA0i] can assumeboth positive and negative values as discussed above.

Before we now enter the discussion of the RG flows ofthe partially bosonized formulation of the matter sector,we would like to comment on the number of parametersin our study. Up to this point, our discussion suggeststhat our study only relies on a single parameter in thematter sector apart from the UV cuto↵ ⇤, namely on theinitial value �

UV

. Strictly speaking, however, the non-trivial fixed-point of the four-fermion interaction is anartifact of our point-like approximation. With the aid

of the partially bosonized formulation, we will resolvepart of the momentum dependence of the four-fermioninteraction. We will then find that the matter sectordepends on three parameters: the Yukawa coupling h,the bosonic mass parameter m and the UV cuto↵ ⇤,see Eq. (23). This is a substantial di↵erence to, e. g.,fermion models in d < 4 space-time dimensions, wherewe only have a single parameter in both formulations, seee. g. Ref. [105]. There, the non-trivial fixed-point of thefour-fermion coupling can be mapped onto a correspond-ing non-trivial fixed-point in the plane spanned by therenormalized Yukawa coupling h and the dimensionlessrenormalized bosonic mass parameter m. In our case, therole of the non-trivial fixed-point in the purely fermionicformulation is taken over by a separatrix in the (h2

,m

2)-plane in the partially bosonized formulation. The shiftof the non-trivial fixed-point of the four-fermion couplingdue to the gauge dynamics then turns into a correspond-ing shift of this separatrix. The mapping between the twoformulations is discussed in detail in the subsequent sec-tion. Being aware of this subtlety, the discussion of thefermionic fixed-point structure is still useful and nicelyillustrates the mechanism underlying the interplay of thechiral and the deconfinement phase transition.

IV. PARTIAL BOSONIZATION AND THELARGE-d(R) EXPANSION

A. Gap Equation

In this subsection, we briefly discuss how our studyof fermionic RG flows is related to the gap equation forthe fermion mass in the large-d(R) limit. For related

14

0

0.5

1

1.5

2

0 50 100 150 200 250

N = 3 (Large N)

N = 3 (Beyond Large N)

T�/T

d

f⇡ [MeV]

fundamental matter

0

0.5

1

1.5

2

0 50 100 150 200 250

N = 2 (Large d(A))

N = 2 (Beyond Large d(A))

T�/T

d

f⇡ [MeV]

adjoint matter

Figure 5. In the left panel, we show the phase diagram for two massless fundamental quarks and N = 3 in the plane spannedby the rescaled temperature T�/Td and the value of the pion decay constant f⇡ at T = 0. In the right panel, the correspondingphase diagram for two massless quark flavors in the adjoint representation and N = 2 is shown. In both panels, the resultsfrom the large-d(R) approximation are given by the red (solid) line, whereas the blue (dashed) line depicts the results from ourstudy including corrections beyond the large-d(R) limit.

in Sect. II B, the mass parameter m

2 assumes negativevalues in the regime with broken chiral symmetry in theground state and the vacuum expectation value h�i ⌘ �0

becomes finite. It is therefore convenient to study the RGflow of �0 and �� rather than that of m2 and ��. Theflow equation of �0 can be obtained from the stationarycondition:

d

dt

@

@�2

✓

1

2m

2�2 +1

8��

4

◆�

�0

!= 0 . (49)

To be specific, we find the following RG flow equationsfor the regime with broken chiral symmetry in the groundstate:

⌘� =2

3⇡2

d(R)X

l=1

M(F)4,?(⌧,m

2q, ⌫l|�|)h2

, (50)

⌘

= 0 , (51)

@

t

h

2 = (2⌘

+ ⌘�)h2, (52)

@

t

�20 = �(⌘�+2)�2

0

� 8

⇡

2

d(R)X

l=1

l

(F)1 (⌧,m2

q, ⌫l|�|)h

2

��, (53)

@

t

�� = 2⌘�� 8

⇡

2

d(R)X

l=1

l

(F)2 (⌧,m2

q, ⌫l|�|)h4, (54)

where �20 = k

�2Z��2

0 and the (dimensionless) renormal-ized constituent quark mass reads

m

2q = h

2�20 .

In the following we will identify the pion decay con-

stant f⇡

with Z

1/2� �0. The (dimensionless) renormalized

meson masses are given by

m

2⇡

= 0 and m

2�

= ��20 .

Since we are working in the large-d(R) limit in this sec-tion, the latter do not appear explicitly on the right sideof the flow equations.Recall that the scale for mq and m

�

is set by thesymmetry breaking scale kSB which is set by our choicefor h2

⇤/m2⇤. The role of the Yukawa coupling (as an ad-

ditional parameter) becomes now apparent from the re-lation

m

2�

= ��20 ⇠ h

4�20 ⇠ h

2m

2q ,

which follows from the flow equations of the couplings.Since the flow of the Yukawa coupling is not governedby the presence of a non-trivial IR attractive fixed-point,its value depends on kSB and the initial value h⇤, asdiscussed above. Therefore the ratio m

2�

/m

2q depends on

our choice for h⇤. On the other hand, the initial valueof the coupling �� does not represent a free parameterof the theory. It is set to zero at k = ⇤ and thereforegenerated dynamically in the RG flow, see also Eq. (23).Using the flow equations (40)-(44) and (50)-(54), we

can now proceed and compute the phase diagram in theplane spanned by the temperature and the value of thepion decay constant at T = 0. In Fig. 5 (left panel) weshow our results for quarks in the fundamental represen-tation and N = 3. For adjoint matter and N = 2, ourresults can be found in the right panel of Fig. 5. To ob-tain these results, we have used ⇤ = 1GeV. Moreover,we have again employed the data for the ground-statevalues of hA0i as obtained from a RG study of SU(N)Yang-Mills theories [33, 70].In the case of fundamental matter and N = 3, we ob-

serve that the upper end of the locking window (Td ⇡ T

�

)roughly coincides with the physical value of the piondecay constant, provided that we fix the initial condi-tion of the Yukawa coupling such that mq ⇡ 300MeV

[Jens Braun, Tina K. Herbst, 2012]arXiv:1205.0779


Deconfinement Phase Transition and the Polyakov Loop

[FA

IR@

GSI

]

Confinement

qq

�Fqq(r) / �r



[FA

IR@

GSI

]

Confinement

qq

mq �! 1Quenched QCD

�Fqq(r) / �r



[FA

IR@

GSI

]

Confinement

qq


�Fqq(r) / �r

Quark-Antiquark Pair (static), for r �! 1

() �Fqq �! 1 () e��Fqq = 0() �Fqq finite () e��Fqq > 0Deconf.

Conf.finite



[FA

IR@

GSI

]

Confinement

qq


�Fqq(r) / �r



Conf.finite

Single Quark (static)

Deconf.Conf. () �Fq �! 1 () e��Fq = 0

() �Fq finite () e��Fq > 0finite



[FA

IR@

GSI

]

Confinement

qq


�Fqq(r) / �r



Conf.finite

Single Quark (static)

Deconf.Conf. () �Fq �! 1 () e��Fq = 0

() �Fq finite () e��Fq > 0finite

e��Fq / hP [A0]iPolyakov

Loop

forT� :1

� �! 0 k �! 0Chiral

Temperature

�UV /�⇤

T�/T

d

µq/I = 0 MeV

hA0i > 0

1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

0.8

1.0T� ⇡ Td

Phase Diagram�T,�UV

�


Phase Diagram�T,�UV

�

forT� :1

� �! 0 k �! 0Chiral

Temperature

�UV /�⇤

T�/T

d

1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

0.8

1.0

µq/I = 0 MeV

µq/I = 100 MeV

µq/I = 200 MeV

µq/I = 300 MeV

Nc �! 1



Partial Bosonization (Hubbard-Stratonovich Transformation)

�T =��,~⇡T

�

�k

[ , ] =

Z

x

⇢Z

i/@ +�

2

⇥( )2 � ( ~⌧�5 )

2⇤�

� ⇠

~⇡ ⇠ ~⌧�5



�k

[ , , �] =

Z

x

⇢Z

i/@ + ih (� + i~⌧~⇡�5)

+1

2Z�(@µ�)

2 +1

2m2�2 +

1

8��

4

�

�T =��,~⇡T

�

�k

[ , ] =

Z

x

⇢Z

i/@ +�

2

⇥( )2 � ( ~⌧�5 )

2⇤�

� ⇠

~⇡ ⇠ ~⌧�5



�k

[ , , �] =

Z

x

⇢Z

i/@ + ih (� + i~⌧~⇡�5)

+1

2Z�(@µ�)

2 +1

2m2�2 +

1

8��

4

�

�

hh

� =h2

m2�T =

��,~⇡T

�

�k

[ , ] =

Z

x

⇢Z

i/@ +�

2

⇥( )2 � ( ~⌧�5 )

2⇤�

� ⇠

~⇡ ⇠ ~⌧�5



�k

[ , , �] =

Z

x

⇢Z

i/@ + ih (� + i~⌧~⇡�5)

+1

2Z�(@µ�)

2 +1

2m2�2 +

1

8��

4

�

�

hh

� =h2

m2�T =

��,~⇡T

�

�k

[ , ] =

Z

x

⇢Z

i/@ +�

2

⇥( )2 � ( ~⌧�5 )

2⇤�

� ⇠

~⇡ ⇠ ~⌧�5

limk!⇤

m2 > 0, limk!⇤

�� = 0, limk!⇤

Z� = 0.limk!⇤

Z = 1.Boundary Conditions at UV Cutoff


�k

[ , , �] =

Z

x

⇢Z

i/@ + ih (� + i~⌧~⇡�5)

+1

2Z�(@µ�)

2 +1

2m2�2 +

1

8��

4

�

�T =��,~⇡T

�

�k

[ , ] =

Z

x

⇢Z

i/@ +�

2

⇥( )2 � ( ~⌧�5 )

2⇤�

hh

�

� =h2

m2

� ⇠

~⇡ ⇠ ~⌧�5

U�(�2)



�k

[ , , �] =

Z

x

⇢Z

i/@ + ih (� + i~⌧~⇡�5)

+1

2Z�(@µ�)

2 +1

2m2�2 +

1

8��

4

�

�T =��,~⇡T

�

�k

[ , ] =

Z

x

⇢Z

i/@ +�

2

⇥( )2 � ( ~⌧�5 )

2⇤�

hh

�

� =h2

m2

� ⇠

~⇡ ⇠ ~⌧�5

U�(�2)

m2 < 0 �0 = h i > 0



�k

[ , , �] =

Z

x

⇢Z

i/@ + ih (� + i~⌧~⇡�5)

+1

2Z�(@µ�)

2 +1

2m2�2 +

1

8��

4

�

�T =��,~⇡T

�

�k

[ , ] =

Z

x

⇢Z

i/@ +�

2

⇥( )2 � ( ~⌧�5 )

2⇤�

Onset of SSB� , m2 ! 0 , 1

� ! 0

hh

�

� =h2

m2

� ⇠

~⇡ ⇠ ~⌧�5

U�(�2)

m2 < 0 �0 = h i > 0




�k

[ , , �, hA0i] =Z

x

⇢Z

�i/@ + g

s

�0hA0i+ i�0µq

� +

1

2Z�(@µ�)

2

+ ih (� + i~⌧~⇡�5) + U�(�2)

�

U�(�2) =

1

2m2�2 +

1

8��

4

Chirally Symmetric Regime

24



�k

[ , , �, hA0i] =Z

x

⇢Z

�i/@ + g

s

�0hA0i+ i�0µq

� +

1

2Z�(@µ�)

2

+ ih (� + i~⌧~⇡�5) + U�(�2)

�

U�(�2) =

1

2m2�2 +

1

8��

4


Z = Z� = 1

Scale Dependent Variables

h, m2, ��

24



�k

[ , , �, hA0i] =Z

x

⇢Z

�i/@ + g

s

�0hA0i+ i�0µq

� +

1

2Z�(@µ�)

2

+ ih (� + i~⌧~⇡�5) + U�(�2)

�

U�(�2) =

1

2m2�2 +

1

8��

4


Z = Z� = 1


h, m2, ��

Regime of Broken Chiral Symmetry

U�(�2) =

1

8�� (⇢� ⇢0)

2

⇢ := �2with

24



�k

[ , , �, hA0i] =Z

x

⇢Z

�i/@ + g

s

�0hA0i+ i�0µq

� +

1

2Z�(@µ�)

2

+ ih (� + i~⌧~⇡�5) + U�(�2)

�

U�(�2) =

1

2m2�2 +

1

8��

4


Z = Z� = 1


h, m2, ��

Z = Z� = 1


h, ⇢0, ��

Regime of Broken Chiral Symmetry

U�(�2) =

1

8�� (⇢� ⇢0)

2

⇢ := �2with

24

Free Parameters and Low-Energy Observables


Purely Fermionic

�UV

25



Purely Fermionic

�UV

Partially Bosonized

�UV =

h2

m2

��UV

, �UV� = 0

25



Purely Fermionic

�UV , h2

UV

Partially Bosonized

�UV =

h2

m2

��UV

, �UV� = 0

25



Purely Fermionic

�UV , h2

UV

Partially Bosonized

�UV =

h2

m2

��UV

, �UV� = 0

Adjustment to Physical Values of Low-Energy Observables at :T = µq = 0

25



Purely Fermionic

�UV , h2

UV

Partially Bosonized

�UV =

h2

m2

��UV

, �UV� = 0

f⇡ = Z1/2� �0

Pion Decay Constant

m = h�0

Constituent Quark Mass


25



Purely Fermionic

�UV , h2

UV

Partially Bosonized

�UV =

h2

m2

��UV

, �UV� = 0

m2UV = 0.445

h2UV = 5.889

f⇡ = 87 MeV

m = 280 MeV

Initial UV Values(⇤ = 1 GeV)

Values of Low-Energy Observablesf⇡ = Z1/2

� �0

Pion Decay Constant

m = h�0

Constituent Quark Mass


25

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Longitudinal Susceptibility


T/Td

T� ⇡ 162 MeV

T� = Td

��/��(T

=0) With hA0i

hA0i = 0

�� ⇠ 1

m2�

26

Impact of confining dynamics on chiral symmetry breakingdietrich/SLIDES3/Leonhardt.pdf · Bad...

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