Table Of ContentJanuary 5, 2016 BI-TP 2014/26v2
PoS(CPOD2014)070
arXiv: 1501.01127v3[hep-ph]
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Confining forces *
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Dirk Rollmann1,a and David E. Miller1,2,b
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p 1 FacultyofPhysics,UniversityofBielefeld, D-33501Bielefeld, Germany
e 2 Department ofPhysics,PennsylvaniaStateUniversity,Hazleton Campus,
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[ Hazleton, PA 18202,USA
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1 Abstract
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We discuss the forces on the internal constituents of the hadrons based on the bag model. The
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1 groundstateofthehadronsformsacolorsingletsothattheeffectsofthecoloredinternalstates
:
v areneutralized. Fromthebreakingofthedilatationandconformalsymmetriesunderthestrong
i interactionsthecorrespondingcurrentsarenotconserved. Thesecurrentsgiverisetotheforces
X
changingthemotionoftheinternal particleswhichcauses confinement.
r
a
*Poster presentation at the 9th International Workshop on Critical Point and Onset of Deconfinement -
CPOD2014, 17-21 November 2014, ZiF (Center for Interdisciplinary Research), University of Bielefeld, Ger-
many.
FirstpublishedinProceedingsofScience,CPOD2014[PoS(CPOD2014)070].
aE-mail:[email protected]
bE-mail:[email protected]@psu.edu
1 Dilatation Current and Conformal Currents
The starting points of our discussion are the dilatation and the conformal currents [1]. In four-
dimensionalMinkowskispaceM4 thedilatationcurrent isgivenby
m mn
D (x)=xn T (T) (1.1)
and thefourconformalcurrents can bewritten as
Kma (x)=(2xa xn gna x2)Tmn (T). (1.2)
−
Withthecovariantpositionfour-vectorxn andtheinnerproductofthemetrictensorwithpositiv
timemetricgan =gnr gra =d an =1na . Thefourconformalcurrentsarelabeledwitha =0,1,2,3.
To investigatetheconservationofthesecurrents under stronginteractionswe lookat thediver-
genceofthesecurrents
¶ m Dm (x) = Tmm (T) (1.3)
¶ m Kma (x) = 2xa Tmm (T). (1.4)
m ma
ThusthedivergencerelationsofthesecurrentsD (x)andK (x)relatedirectlytothethermally
m
averaged traceoftheenergy-momentumtensorTm (T).
2 Breaking of Scale and Conformal Symmetries
It is well known in quantumchromodynamics (QCD) that the scale and conformal symmetries
must be broken to avoid physically absurd mass spectra [1]. With exact scale and conformal
symmetries all particles have to be massless or their mass spectra continuous [2]. Thus the
related currents can not be conserved. From the divergence relations in Eqs. (1.3) and (1.4)
onecansee,thatthebreakingofscaleandconformalsymmetriesarisefromafinitetraceofthe
energy-momentumtensor(traceanomaly)[3]
m
Tm (T)=e (T) 3p(T). (2.1)
−
m
Thus, Tm (T) acts as an order parameter and gives the magnitude of scale and conformal sym-
metrybreaking.
3 Lattice Gauge Theorie and Equation of State
Lattice gauge simulations of QCD compute not directly the trace of the thermally averaged
m
energy-momentumtensorTm (T), butthedimensionlessinteractionmeasureD (T)
D (T)=(e (T) 3p(T))/T4. (3.1)
−
2
Here are the differences between the energy density e (T) and three times the pressure 3p(T)
strongly suppressed by the division of T4. To obtain the equation of state e (T) 3p(T) as an
−
actual physicalquantityonehas tomultiplytheinteractionmeasurebyT4
D (T) T4 =e (T) 3p(T). (3.2)
· −
In Ref. [4] thenumerical values for the interaction measurewere calculated from lattice gauge
simulations(lattice QCD) for pure SU(3) gauge theory for different lattice sizes. The obtained
graphs of theequation of stateare shownin Fig. 2 in Ref. [3]. For all latticesizes theequation
of state shows a rapid growth at the deconfinement temperature T and a transition to a slower
d
m
but continual growth in the range where data exists. There is no obvious sign that Tm (T) =
e (T) 3p(T)decreasestozeroathightemperatures. Thisisalsotrueforthelatticesimulations
−
with massive dynamical quarks even if the growth of the equation of state is much slower, see
Fig. 9in Ref. [3]. Thatis, theQuark gluonplasma(QGP) isnot an idealultrarelativisticgasas
it would be for e (T) =3p(T). The thermally averaged trace of the energy-momentum tensor
m
Tm (T)also enters thegluoncondensate[5]
m
G2 T = G2 0 Tm T. (3.3)
h i h i −h i
m
WithincreasingTm (T)theexpectationvalueofthevacuumgluoncondensate G2 0 isprogres-
h i
sively reduced and the gluon condensate G2 becomes negative and continues to fall. This
T
alsoholdsifEq. (3.3)isgeneralized to mahssiviequark fieldsm y y [6]
q q q
h i
m
hG2iT =hG2i0+mqhy qy qi0−mqhy qy qiT −hTm mqiT. (3.4)
For further details see Fig. 10-15 in Ref. [3]. Since the gluon condensate, as a correlation
mn
functionofthegluonfield strengthtensorG [5]
a
b (g)
G2 ≡ −2g3 Gamn Gamn , (3.5)
doesnotvanishforallcomputedtemperaturesaboveT ,theQGPremainsastronglyinteracting
d
gas. Where a denote thecolor index for SU(N ), b (g) therenormalization group beta function
c
andgthecoupling. However,withanonvanishingtraceoftheenergy-momentumtensor,scale
and conformal symmetries under the strong interactions remain broken even with vanishing
massesand at high temperatures. Thissituationis different to thebreaking ofchiral symmetry,
whichisrestoredinthechirallimit y y aswellasinthehightemperaturelimitT ¥ .1
h q qimq→0 →
4 Confining Forces
The physical insight of the divergence relations in Eqs. (1.3) and (1.4) is breaking of scale
and conformal symmetries caused by the trace anomaly. Now we look [3, 6] at the physical
1For a recent work on chiral symmetry breaking in confined quarkyonic matter see the talk Inhomoge-
neous and Quarkyonic phases of High Density QCD given by L. McLerran at this CPOD 2014 conference
[PoS(CPOD2014)046].
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dimensionofthecorrespondingcurrents in Eqs. (1.1)and (1.2). Theenergy-momentumtensor
asanenergy densityisofthedimensionofenergy perunitvolume. Thiscorrespondstoaforce
perarea. Multipliedby alengthtoasquarearises aforce. Thus
ma
K (x) isa force.
m
The dilatation current D (x) is of the dimension energy per area or force per length. This mo-
m
tivatedtheprojectionofthedilatationcurrent on thecoordinateaxes: D (x)xm . Then likewise
m
D (x)xm isa force.
The character of these forces arise from the type of symmetry breaking. The breaking of con-
formal symmetry give rise to an angular change of the world line of a parton. In a numerical
ma
evaluationintwodimensionswewillseethatK (x)causesastrongsteeringeffectofthemo-
ma
tionofapartoninaconfinedregion. Accordingto[3]wecallK (x)fourspan. Thebreakingof
m
dilatationsymmetrycausesastretchingforcealongtheworldlineofaparton. WecallD (x)xm
dyxle [3]. The relationship between the breaking of dilatation and conformal symmetries and
theresultingforcesissimilartotherelationshipbetweenahomogeneousspaceandmomentum
conservation. A violation of homogeneity would cause an extra momentum, according to a
force. Onlytheeffects offourspanand dyxleare different,namelya changeofdirection.
5 Ground State and Bag-Model
mn
Thethermallyaveraged energy-momentumtensorT (T)can beseparated intoavacuumpart
q mn ,which isnottemperaturedependent,and afinitetemperaturecontributionq mn (T) [3]
0
Tmn (T)=q mn +q mn (T). (5.1)
0
The thermal part q mn (T) is subject to lattice-QCD simulations. For the sake of simplicity we
carry out our numerical evaluations in the ground state. To step around the standard problems
mn
with infinities of any ground state we use a bag type of model [7] for the vacuum part q .
0
In this model the energy density e in the ground state is given by the bag-energy B and the
pressure p by B. The bag energy raises the hadron ground state above the QCD vacuum
−
and the negative bag pressure balances the parton pressure to ensure their confinement in a
hadron. Then thetraceoftheenergy-momentumtensorinthegroundstateinfour-dimensional
Minkowskispacebecomes
m
q =e 3p=4B. (5.2)
m 0 −
Andin two-dimensionalMinkowskispaceM2 weyield
m
q =2B. (5.3)
m 0
Then the two conformal breaking forces (now denoted as twospan) and the dyxle in two-
dimensionalMinkowskispace, m ,a =0,1,can bewrittendownas
ma (x0)2+(x1)2 2x1x0
K (x)=2B (5.4)
(cid:18) 2x0x1 (x0)2+(x1)2(cid:19)
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and
Dm (x)xm =B[(x0)2 (x1)2]=Bt 2. (5.5)
−
With the time coordinate x0 and the spatial coordinate x1. The dyxle Dm (x)xm depends on the
propertimesquaredt 2 correspondingtoourinterpretationasastretchingforcealongtheworld
lineofaparton.
6 Transformationon two-dimensionalLight-Cone Coordinates
In two dimensions the conformal matrix Eq. (5.4) is symmetrical and hence only one con-
formal current is independent. Furthermore the two conformal currents are mirrored on the
positivelight-conecoordinate. This motivateda transformation on two-dimensionallight-cone
coordinates
1
xˆ = (xˆ0 xˆ1). (6.1)
±
√2 ±
Inadditionweprovidetheforceswithaminussigncorrespondingtotheconfiningcharacterof
theseforces. Thenweobtainthetwoforces ofthetwospan
K+(x+)=Km 0(x)+Km 1(x)=√2q m (x+)2( xˆ+) (6.2)
m 0 −
K (x )=Km 0(x) Km 1(x)=√2q m (x )2( xˆ ). (6.3)
− − − m 0 − − −
AfterthistransformationtheK+(x+)andK (x )uncoupleanddependonlyonthepositivex+
− −
and negativex light-conecoordinate,respectively. Withthetransformation
−
1 1 1
x+x = (x0+x1) (x0 x1)= ((x0)2 (x1)2) (6.4)
−
√2 √2 − 2 −
weyieldforthedyxle
Dm (x)xm = 2Bx+x−. (6.5)
−
7 Numerical Evaluationsin 1+1 dimensional Minkowski Space
In numerical evaluations [8] the trajectories of a parton in a bag or hadron which underly
twospan and dyxle were calculated in two-dimensional Minkowski space M2 in the ground
state. Since the ground state of the hadrons forms a color singlet so that the effects of the col-
ored internal states are neutralized we regard a colorless parton model [9]. We use the law of
motion~F =m~a and~v=~at with the proper time t = (x0)2 (x1)2, since the dyxleitself de-
−
pendsonthepropertime. Ineveryloopthetimet,reppresentingaprogressivetime,isincreased
by a time step D t. The light-cone coordinates (blue axes) form a natural limit of the bag. The
valueforthemassischosenasm=0.1justaboutthemassofastrangequarkm =0.095GeV.
s
Fromthevacuumgluoncondensate G2 =0.012GeV4 [10]wechosethebagconstantin the
0
h i
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x0 x0
0.30 0.30
0.25 0.25
0.20 0.20
0.15 0.15
0.10 0.10
0.05 0.05
-0.3 -0.2 -0.1 0.1 0.2 0.3x1 -0.15 -0.10 -0.05 0.05 0.10 0.15 x1
Figure 1: (color) Trajectory by the dyxle (left) with initial values x0 =2.0, x1 =1.0, D t =0.1,
2000loopsand thetwospan(right)withinitialvaluesx0 =0.2,x1 =0.1, D t =0.01,400loops.
ground state as B=0.01. The trajectory of a parton by the dyxleis shown in Fig. 1 (left). The
parton moves from the outer light-cone to the origin. Thereby the minus sign in the dyxle, Eq.
(5.5), arising from Minkowskimetric, causes a subtle directing effect: The parton trajectory is
directed to the time axis x0. In the further course the parton moves toward the origin along the
timeaxis. Thechangesinthepositionofthepartonarelesswithdecreasingvaluesofx+x and
−
thepropertimet . ThetrajectoryofapartonbythetwospanisshownontherightsideofFig. 1.
K+(x+) and K (x ) act along the positive xˆ+ and negative xˆ light cone, respectively, in
− − −
− −
thedirectionoftheorigin. K+(x+)vanishesonthenegativelightconeandK (x )vanisheson
− −
the positivelight cone. At the positivelight cone K+(x+) takes over and pulls the parton back
to the "center" of the bag. In combination of both these forces result a steering effect which
pulls the parton from one light cone to the other, approaching the origin. The trajectories of a
parton by the dyxle and twospan together are shown in Fig. (2). These are the physically rel-
evant evaluations since the trace anomaly breaks both symmetries simultaneously. The parton
makes only one movement to the negativelight cone and then it moves to the time axis due to
the dyxle. Again, the parton first moves from the outer light-cone to the origin and near the
originthechanges in thepositionoftheparton are less withdecreasing valuesof x+x and the
−
propertimet . Theoriginacts as fixed pointtype. In both evaluationsin Fig. 2 werecalculated
10000loopsofpartondisplacements. ThecrucialpointofthesimulationsarethetimestepsD t.
x0 x0
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
-1.0 -0.5 0.5 1.0 x1 -1.0 -0.5 0.5 1.0 x1
Figure2: (color)Trajectoriesbytwospananddyxlecombined. Initialvaluesx0=1.0,x1=0.2;
D t = 0.1, 10000 loops (left) and with the same parameters, but smaller time steps D t = 0.01
(right).
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Fig. 2 shows the trajectories for two different time steps D t =0.1 (left) and D t =0.01 (right).
Smallertimestepscouldprovidemorerealisticsimulations.
8 Summary and Outlook
In this numerical evaluation in two-dimensional Minkowski space, twospan and dyxle, the
forceswhichresultfromthebreakingofconformalandscalesymmetriesinQCD, pullthepar-
ton from the boundary to the center of the bag. The forces are particularly strong on the outer
light-cone coordinates, which we regard as the limit of a bag or a hadron. We consider this as
confinement of a parton in a hadron or more generally as quark confinement. At the origin the
forces become particularly weak corresponding to asymptotic freedom of quarks. Concerning
the direction of the time we notice that the time beyond the horizon, which is formed by the
bag limit, does not have the same meaning like the "normal" physical time.2 But this is still a
subject to be investigated. For further statements simulations of the fourspan and the dyxle in
four-dimensionalMinkowskispaceare required as well as arefinement ofthenumerical meth-
ods. But we expect that the confining character of the fourspan and the dyxle also will hold
in four dimensions. Since the parton couples to the energy, the fourspan and the dyxle could
also be significant for deconfined matter and high-energy particle collisions. For instance, the
ma
off-diagonal entries of K (x) correspond to the shearing forces, which also occur in particle
collisions.
For one of us (DR) the participation at the CPOD 2014 conference was supported by the
ExtreMeMatterInstitute(EMMI)byHelmholtzAssociation.
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