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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
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 3
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
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 4
[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
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)
Quark-Gluon Plasma
QCD phase diagram
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 4
[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
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)
Quark-Gluon Plasma
Functional renormalization group (FRG)
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt
�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
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 6
QCD Lagrangian in the chiral limit ( ):mq �! 0
SUV (Nf )⌦ SUA(Nf )⌦UV (1)⌦UA(1)
Low energy model of QCD
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 6
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
q q qqq
qqq
Low energy model of QCD
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 6
gs gs
undefined
1
undefined
1
gs gs
�
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
q q qqq
qqq
Incorporating confining dynamics
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 7
Order parameter confinement-deconfinement phase transition:Polyakov Loop: hP [A0]i
Incorporating confining dynamics
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 7
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
Incorporating confining dynamics
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 7
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
Incorporating confining dynamics
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 7
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
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
Incorporating confining dynamics
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 8
�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
[J. Braun, H. Gies, and J. M. Pawlowski, 2010]: Computation of in YM theory for hA0i Nc = 3
Incorporating confining dynamics
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 8
�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 input
@t�k =1
2STr
(@tRk
�(2)k +Rk
)
Wetterich eq.
@t� ⌘ ��
RG flow eq.
TdCritical temperature
[J. Braun, H. Gies, and J. M. Pawlowski, 2010]: Computation of in YM theory for hA0i Nc = 3
Incorporating confining dynamics
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 8
�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 input
@t�k =1
2STr
(@tRk
�(2)k +Rk
)
Wetterich eq.
@t� ⌘ ��
RG flow eq.
, 1
� ! 0Onset
of χSB
TdCritical temperature
[J. Braun, H. Gies, and J. M. Pawlowski, 2010]: Computation of in YM theory for hA0i Nc = 3
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 9
� � �� ⌘ @t� ⇠ Fixed point: �(�⇤ ) = 0
Fixed-point analysis of the RG flow equation
� �⇤
@t�
T = 0, µq = 0, hA0i = 0
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 9
� � �� ⌘ @t� ⇠ Fixed point: �(�⇤ ) = 0
Fixed-point analysis of the RG flow equation
χSBχSym
�UV
� �⇤
@t�
T = 0, µq = 0, hA0i = 0
Fixed-point analysis of the RG flow equation
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 10
� � �� ⌘ @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
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 11
� � �� ⌘ @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
Fixed-point analysis of the RG flow equation
: 1/� T�
Locking in the - plane
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 12
(T,�UV )
for k �! 0Chiral temperature
: 1/� T�
Locking in the - plane
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 12
(T,�UV )
for k �! 0Chiral temperature
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�
Locking in the - plane
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 13
(T,�UV )
for k �! 0Chiral temperature
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�
Locking in the - plane
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 14
(T,�UV )
for k �! 0Chiral temperature
µq = 0 MeV
µq = 175 MeV
hA0i
T� ⇡ TdLocking:1.0
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�
Locking in the - plane
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 14
(T,�UV )
for k �! 0Chiral temperature
µq = 0 MeV
µq = 175 MeV
hA0i
T� ⇡ TdLocking:1.0
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
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 15
� h
� ⇠
~⇡ ⇠ ~⌧�5 h
Partial bosonization and finite current quark masses
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 15
� h
� ⇠
~⇡ ⇠ ~⌧�5 h
U�(�2)
��0
i mc ⌘ c�
Partial bosonization and finite current quark masses
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 15
� h
� ⇠
~⇡ ⇠ ~⌧�5 h
f⇡ = 93 MeV
m = 300 MeV
m⇡ = 139 MeV
Low- energy observables:
U�(�2)
��0
i mc ⌘ c�
Partial bosonization and finite current quark masses
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 15
� 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
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 16
• 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
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Backup: Adjoint matter
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 18
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
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 19
Deconfinement Phase Transition and the Polyakov Loop
[FA
IR@
GSI
]
Confinement
�Fqq(r) / �r
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 19
Deconfinement Phase Transition and the Polyakov Loop
[FA
IR@
GSI
]
Confinement
mq �! 1Quenched QCD
�Fqq(r) / �r
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 19
Deconfinement Phase Transition and the Polyakov Loop
[FA
IR@
GSI
]
Confinement
mq �! 1Quenched QCD
�Fqq(r) / �r
Quark-Antiquark Pair (static), for r �! 1
() �Fqq �! 1 () e���Fqq = 0() �Fqq finite () e���Fqq > 0Deconf.
Conf.finite
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 19
Deconfinement Phase Transition and the Polyakov Loop
[FA
IR@
GSI
]
Confinement
mq �! 1Quenched QCD
�Fqq(r) / �r
Quark-Antiquark Pair (static), for r �! 1
() �Fqq �! 1 () e���Fqq = 0() �Fqq finite () e���Fqq > 0Deconf.
Conf.finite
Single Quark (static)
Deconf.Conf. () �Fq �! 1 () e���Fq = 0
() �Fq finite () e���Fq > 0finite
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 19
Deconfinement Phase Transition and the Polyakov Loop
[FA
IR@
GSI
]
Confinement
mq �! 1Quenched QCD
�Fqq(r) / �r
Quark-Antiquark Pair (static), for r �! 1
() �Fqq �! 1 () e���Fqq = 0() �Fqq finite () e���Fqq > 0Deconf.
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
�
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 20
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
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 21
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 22
Partial Bosonization (Hubbard-Stratonovich Transformation)
�T =��,~⇡T
�
�k
[ , ] =
Z
x
⇢Z
i/@ +�
2
⇥( )2 � ( ~⌧�5 )
2⇤�
� ⇠
~⇡ ⇠ ~⌧�5
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 22
Partial Bosonization (Hubbard-Stratonovich Transformation)
�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
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 22
Partial Bosonization (Hubbard-Stratonovich Transformation)
�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
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 22
Partial Bosonization (Hubbard-Stratonovich Transformation)
�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
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 23
�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)
Partial Bosonization (Hubbard-Stratonovich Transformation)
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 23
�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
Partial Bosonization (Hubbard-Stratonovich Transformation)
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt 23
�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
Partial Bosonization (Hubbard-Stratonovich Transformation)
Partial Bosonization (Hubbard-Stratonovich Transformation)
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt
�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
Partial Bosonization (Hubbard-Stratonovich Transformation)
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt
�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
Z = Z� = 1
Scale Dependent Variables
h, m2, ��
24
Partial Bosonization (Hubbard-Stratonovich Transformation)
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt
�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
Z = Z� = 1
Scale Dependent Variables
h, m2, ��
Regime of Broken Chiral Symmetry
U�(�2) =
1
8�� (⇢� ⇢0)
2
⇢ := �2with
24
Partial Bosonization (Hubbard-Stratonovich Transformation)
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt
�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
Z = Z� = 1
Scale Dependent Variables
h, m2, ��
Z = Z� = 1
Scale Dependent Variables
h, ⇢0, ��
Regime of Broken Chiral Symmetry
U�(�2) =
1
8�� (⇢� ⇢0)
2
⇢ := �2with
24
Free Parameters and Low-Energy Observables
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt
Purely Fermionic
�UV
25
Free Parameters and Low-Energy Observables
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt
Purely Fermionic
�UV
Partially Bosonized
�UV =
h2
m2
����UV
, �UV� = 0
25
Free Parameters and Low-Energy Observables
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt
Purely Fermionic
�UV , h2
UV
Partially Bosonized
�UV =
h2
m2
����UV
, �UV� = 0
25
Free Parameters and Low-Energy Observables
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt
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
Free Parameters and Low-Energy Observables
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt
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
Adjustment to Physical Values of Low-Energy Observables at :T = µq = 0
25
Free Parameters and Low-Energy Observables
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt
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
Adjustment to Physical Values of Low-Energy Observables at :T = µq = 0
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
Bad Honnef, February 2017 | Understanding the LHC | Marc Leonhardt
T/Td
T� ⇡ 162 MeV
T� = Td
��/��(T
=0) With hA0i
hA0i = 0
�� ⇠ 1
m2�
26