Trouble Finding the Optimal AdS/QCD

0.16 MB·English

by  K. Veschgini

Trouble Finding the Optimal AdS/QCD
K. Veschgini,∗ E. Meg´ıas,† and H.J. Pirner‡
Institut fu¨r Theoretische Physik der Universita¨t Heidelberg, Heidelberg, Germany
In the bottom-up approach to AdS/QCD based on a five-dimensional gravity dilaton action the
exponential of the dilaton field is usually identified as the strong or ’t Hooft coupling. There is
currently no model known which fits the measurements of the running coupling and lattice results
forpressureatthesametime. Withaoneparametrictoymodelwedemonstratetheeffectoffitting
thepressure on the coupling and vice versa.
PACSnumbers: 11.10.Wx11.15.-q11.10.Jj12.38.Lg
Keywords: ads;qcd;holographicprinciple;finitetemperature;
1
1 I. INTRODUCTION namely1
0
2 In the bottom-up approach to AdS/QCD we modify 12 8 α β(α˜) β(α) 2
Jan MthiellsMtahledoarcyenaandduTaylpiteyI[I1B] sbtertiwngeetnheNory=of4AsduSp5er×-YSa5ntgo- V(α)=−ℓ2 exp−9Z0 α˜2 dα˜ 1−(cid:18) 3α (cid:19) ! .
findaholographicdualforQCD.Webreaktheconformal   (1.7)
1
invariancebyaddinganon-trivialdilatonpotentialV(φ) W is defined by eq. (1.5) to reduce Einstein equations
1 0
to the bulk action [2–4] to first oder. Thus, up to the constant factor 12/ℓ2 the

dilaton potential is fixed by the β-function.
]
p-th Sbulk = 16−π1G5 Z √G(cid:18)R− 34(cid:0)∂µφ(cid:1)2−V(φ)(cid:19)d5(x1..1) pthheAysgirchasovliaotsyg,rdaaepcshccoircripdtmiinoogndtoeolfttihsheemsuAepdaenSrt/-YCtaFonTgc-aMcpoitrlulrserestpheoinonfrdryaenarepcde-
e Thefive-sphereS isofnoimportanceforthe purposeof plies to the large ’t Hooft coupling limit λ = g2 N
h 5 t YM c →
this paper. The integration is performed over Euclidean . The ultraviolet behavior physics computed from a
[
∞
space-time with periodic time axis and the bulk coordi- gravity dual is known to show often a wrong behavior.

2 nate z. The main challenge is to find the correct poten- For example, consider a holographic model with the β-
v
tial. One approach would be to make directly an ansatz function
9
for V. Instead we will use b (z), given by the zero tem-
3 0 β (α)= β α2 β α3, (1.8)
6 perature solution of the Einstein equations pert − 0 − 1
4 intheultraviolet. Theasymptoticbehaviorofthespatial
09. ds2 =b20(z)(cid:16)dτ2+d~x·d~x+dz2(cid:17) , (1.2) sTtr→ing∞tenissiσosn∝comT2pαu4t/e3d[f6ro,m7]tihnestgeraadviotyf σdsua∝liTn2tαhe2laims iitt
0 to define an energy scale. The β-function is then given followsfromdimensionalreductionargumentsandlattice
1 by simulations[8,9]. Similarlyifwecomparetheasymptotic
:
v behavior of the pressure [7, 10]
i dα
X β(α)=b0 db . (1.3) p= π3ℓ3 T4 1 4β α + 2 4β2 3β α2
r 0 16G − 3 0 h 9 0 − 1 h
a 5 (cid:18) (cid:16) (cid:17) (cid:19) (1.9)
The running coupling α on the gauge side of the duality =p 1 2.33α +1.86α2 ,
correspondstotheexponentialofthedilatonfieldexp(φ). SB − h h
Theβ-functionthenfixesthepotentialwhichisobtained (cid:16) (cid:17)
with the perturbative result from QCD [11]
from the zero temperature Einstein equations [2, 3, 5]
8π2 15α α 3/2
16 3 p= T4 1 +30
∂zW0 = 9 b0W02+ 4b0V(α0), (1.4) 45 − 4 π (cid:18)π(cid:19) ! (1.10)
∂zb0 =−94b20W0, (1.5) =pSB 1−1.19α+5.4α3/2 ,
(cid:16) (cid:17)
∂ α =α b ∂ W . (1.6) weseethatnotonlythecoefficientsaredifferentbutalso
z 0 0 0 z 0
the power of α in the next to leading order. Note, per-
p tubation theory gives the coupling at the black horizon
∗ [e-mail]
† [e-mail] 1 Theminussignsinthedilatonpotential andinfrontofV(φ)in
‡ [e-mail] eq.(1.1)areamatter ofconvention.
2
α = α(z ) equal to α(πT) in QCD in lowest order. In
h h
many cases, like the spatial tension, the ultraviolet limit
does not prevent the model from capturing the infrared
physics, but in the case of the pressure the situation is
different. The gravity model suggests a smaller pressure
thanperturbativeQCDatveryhightemperatures,aswe
can see from a comparison of the coefficients at (α) in
O
eq. (1.9) and eq. (1.10). If the pressure is already much
toosmallat103T ,ithasatendencytobealsomuchtoo
c
small at lower temperatures.
We will consider a simple β-function and demonstrate
that as we switch over at smaller values of αˆ from β
pert
toanasymptoticlinearbehaviorthepressuregetslarger.
Fitting lattice results requires αˆ .0.04.
II. THE MODEL
FIG. 1. Pressure normalized to the Stefan-Boltzmann pres-
Weassumethefollowingtoyβ-functiontodemonstrate sure as a function of T/T compared with lattice data for
c
our case: N =3 taken from ref. [8].
c
β (α) βˆ α4 if α αˆ
pert 2
β(α)= − ≤ ,
(βpert(αˆ) βˆ2αˆ4 3(α αˆ) if α>αˆ The zero temperature solution serves as background.
− − − Any finite temperature solution shares the same ultra-
(2.1)
violet behavior as the zero temperature solution up to
where
(z4). The physicalaction is givenby the difference be-
3 2β αˆ 3β αˆ2 Otween the action of the black hole solutions and the zero
βˆ = − 0 − 1 , (2.2)
2 4αˆ3 temperature solution
is chosen such that ∂αβ(α) is continuous. There is only Sphys. =ST −S0, (3.3)
one parameter αˆ which controls the transition point.
where and are actions given by the bulk term
Notethatβˆ2 willbecomeverylargeforsmallαˆ suchthat eq.(1.1S)TplustheS0Gibbons-Hawking-Yorkboundaryterm
β(α) deviates from β (α) already at α much smaller
pert [13, 14]
than αˆ. The slope of the linear term is chosen to be
3. With this choice, according to eq. (1.7) we obtain 1
a−monotonic potential with the following asymptotic be- Sbound. = 8πG dx4√gK. (3.4)
5
haviors: Z
The induced metric on the boundary is g and K is the
V(α) 12/ℓ2;α 0, (2.3) trace of the second fundamental form of the boundary.
→− → For temperatures T larger than some T we find three
V(α) α5/3 ;α . (2.4) min
∼− →∞ solutions,thezerotemperaturesolution(temperaturein-
dependent) and two black hole solutions. For T < T
min
the black hole solutions are not present. The solution
III. THE PRESSURE
with the smaller physical action defines the stable glu-
onic matter. The phase of confined gluons is given from
The deconfinedphaseofthe gluonplasmais described
T = 0 up to T = T by the zero temperature solution.
c
inthe holographicpicture bya blackhole geometrywith
Black hole solutions exist for temperatures higher than
the metric
someT ,thebigblackholeshaveasmalleractionthan
min
the small black holes and for T >T a negative physical
dz2 c
ds2 =b2(z) f(z)dτ2+d~x d~x+ . (3.1) action (relative to the T = 0 solution). The black hole
· f(z)! geometry corresponds to the deconfined phase. Details
of the computation can be found in ref. [7]. The free en-
The horizon lies in the bulk at z where f(z ) = 0. We
h h ergy can be computed directly from the physical action
normalizef(0)to1. Inorderto avoida conicalsingular-
as =T or by integrating the entropy
phys.
ity the periodicity of the τ axes β = T−1 must be fixed F S
to [12, 13]
= SdT . (3.5)
F
Z
4π
β = . (3.2) The entropy of a classical black hole is a well defined
−∂zf(z)(cid:12) quantity given by the Bekenstein-Hawking formula [15,
(cid:12)z=zh
(cid:12)
(cid:12)
(cid:12)
3
16]
A Vol(3)b3(z )
h
S = = , (3.6)
4G 4G
5 5
where A is the area of the black hole horizon. Both
methodsproducethesameresultbutthelatterapproach
is numerically favorable [7]. Thermodynamic quantities
can then be computed from the free energy. The five-
dimensional gravitational constant G is fixed by nor-
5
malizing the pressure given in eq. (1.9) to the Stefan-
Boltzmann pressure in the limit T :
→∞
45πℓ3
G = . (3.7)
5 16(N2 1)
c −
This value differs from the conformal case because we
have fewer degrees of freedom in QCD than in = 4
N
super Yang-Mills theory [17]. We have computed the FIG. 2. The running coupling α as a function energy E =
pressure using Bekenstein-Hawking entropy for different ΛEb0. Data points are taken from ref. [18]. The curve
choices of αˆ. The results normalized to the ideal gas corresponding to αˆ = 0.02 is shown for two different choices
limit are shown in fig. 1. Lattice data from Tc up to 2Tc of ΛE.
are fitted very well for αˆ = 0.04. The choice αˆ = 0.02
gives the best fit from T up to 3T . Some other curves
c c
whichis a muchbetter fit to the MS value inthe plotted
are also shown for comparison. Data above 3T are not
c
range.
fitted precisely by any choice of αˆ.
The chosen β-function for αˆ = 0.5 can reproduce the
running coupling in vacuum quite well. The coupling
IV. THE RUNNING COUPLING AND at finite temperatures defined as αh(T) = α(zh) follows
SCREENING MASS the vacuum coupling constant in the ultraviolet [4, 7].
Thusα isalsoverysmallathightemperatures. Inorder
h
totestthemodelatfinitetemperaturesT,itisimportant
It is interesting to see how the running coupling α is
tomonitorthe behaviorofanobservablewhichisclosely
affectedbythechoiceofαˆ. Theweakcouplingexpansion
relatedtothecouplingatfinitetemperatures. Tothisend
eq.(1.9)alreadysuggeststhatthecouplingmustbe very
we have computed the Debye mass in the gluon plasma.
small when we want to fit the pressure. First we discuss
We usethe methodpresentedinref.[19]. The resultsfor
the zero temperature running coupling as a function of
αˆ = 0.02 and αˆ = 0.5 are plotted in fig. 3 together with
energy E, which we denote by α . Since we define the
E latticedata [20,21]. The modelwithαˆ =0.02predicts a
β-function with b as given in eq. (1.3), the energy is
0 valuefortheDebyemasswhichisafactor4smallerthan
proportional to b (z). The proportionality constant Λ
0 E lattice data. This is a consequence of the fact that the
is not fixed by the model. Different values of Λ corre-
E running coupling α is very small at high temperature
spond to different initial conditions for the renormaliza- h
for this choice of αˆ. For the large αˆ = 0.5 the Debye
tion groupequation(1.3). Fig. 2 shows α as a function
E mass is quite close to the lattice data as one can see in
of energy E in logarithmic scale. Different values of Λ
E fig. 3.
shiftthecurvestotheleftorrightwithoutaffectingtheir
shape since log(E = Λ b ) = log(Λ )+log(b ). This is
E 0 E 0
demonstrated for the case αˆ = 0.02, where we fix Λ
E V. SUMMARY AND CONCLUSION
either by
The β-function, on the one hand defines the running
α (1.78GeV)=0.33 (4.1)
E
of the coupling and on the other hand determines the
or by dilaton potential and hence the full thermodynamics in-
cluding the pressure and the screening mass. Follow-
α (M =91.2GeV)=0.118 (4.2) ing the idea of AdS/CFT we identify the exponential of
E Z
the dilaton field as the running coupling of QCD. But,
accordingtoref.[18]. Asmallvalueofαˆ resultsinavery when we fit the pressure, the running coupling α and
smallcouplingathighenergies. Asasideeffectthecurve the screening mass in the plasma m deviate strongly
D
becomesalsoveryflat,thereisalmostnorunningathigh from phenomenology. On the other side, if we choose
energies. Onthe otherhandthecouplingrisesextremely to fit α or m , we obtain a pressure that is about 40%
D
fast in the infrared. This differs from the case αˆ = 0.5 too small. This outcome is not just a property of the
4
3.0 m (cid:144)T,323´4 has been achieved, however at the expense of not be-
D
ing able to relate the dilaton potential with the running
m (cid:144)T,323´8
- D α(E) in the MS scheme. Note, one could argue, that
2.5
mD(cid:144)T in the scheme used here with a large coefficient βˆ α4
2
-- H4ΠΑ L1(cid:144)2 in the β-function, the MS-value of α (E) = 0.33 at
2.0 h MS
E =1.78 GeVisnotrealistic. ButasshowninsectionIV
T
(cid:144) anyotherchoiceleadstothesameresult. Alsothesecond
mD 1.5 Αï=0.5 thermodynamic observable, the Debye mass, points to a
real deficit for this choice of small αˆ. For our toy model
1.0 as well as for the model of ref. [22] there is currently
no known mapping between the assumedβ-function and
0.5 Αï=0.02 known MS values of α [24]. This would be very much
neededifonewantstoapplythismodeltohadronization.
0.0 In ref. [10] an extrapolation of the β-function was used
0 2 4 6 8 10 12 which agrees well the perturbative running of α [5].
MS
The calculations of the thermodynamics however have
T(cid:144)T
c
the default shown by the toy-model for αˆ 0.5. It pro-
≈
duces a pressure which is at all temperatures too small,
FIG. 3. Debyemass over temperature as a function of T/T .
c whereas the calculation of the Debye mass comes out in
Thefull (blue)curvescorrespond tocomputations usingαˆ=
agreement with the lattice data. At the moment we do
0.02 and αˆ = 0.5. Lattice data are taken for gluodynamics
SU(3)withN3×N =323×4and323×8fromrefs.[20,21]. not know any solution of this dilemma.
σ τ
Thedotdashed (red)curvesrepresent thefinitetemperature
runningcoupling (4πα(z ))1/2 for both cases.
h ACKNOWLEDGMENTS
E.M.wouldliketothanktheHumboldtFoundationfor
their stipend. This work was also supported in part by
model introduced here cf. [10, 22, 23]. In the litera- the ExtreMe Matter Institute EMMI in the framework
ture refs.[10, 22]both approacheshavebeen covered. In ofthe Helmholtz Alliance Programof the Helmholtz As-
ref. [22] a very good description of the thermodynamics sociation.
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