su(n) gauge theories on the lattice ii -- results and...

SU(N) gauge theories on the LatticeII – Results and Interpretation

Biagio Lucini

4th Odense Winter School on Geometry and PhysicsOdense, Denmark, 18th-19th December 2011

B. Lucini SU(N) gauge theories on the Lattice

Summary

Yesterday, we have seen thatthe large N limit is an useful tool to understand qualitativefeatures of QCDMonte Carlo calculations in the lattice formulation of SU(N)gauge theories allow us to compute observables at finite N

Today we will cover the extrapolation of numerical results toN = ∞ for a significant set of observable and see what we canlearn for large-N QCD and analytical calculations based on thegauge-gravity correspondence.


The challenge

Large N theory

String background

Gauge-‐string duality

QCD

La;ce


Outline

1 The large N limit on the lattice

2 Confinement and asymptotic freedom

3 Glueballs

4 Mesons

5 Finite Temperature

6 Conclusions


Other topics

I deliberately will not discussTopologyk -stringsVolume reductionEguchi-KawaiOther large N limits (Veneziano, Orientifold PlanarEquivalence). . .


Generalities

Part I

The large N limit on the lattice


Generalities

Large N limit on the lattice

The lattice approach allows us to go beyond perturbative anddiagrammatic arguments. For a given observable

1 Continuum extrapolationDetermine its value at fixed a and NExtrapolate to the continuum limitExtrapolate to N →∞ using a power series in 1/N2

2 Fixed lattice spacingChoose a in such a way that its value in physical units iscommon to the various NDetermine the value of the observable for that a at any NExtrapolate to N →∞ using a power series in 1/N2

Study performed for various observables both at zero and finitetemperature for 2 ≤ N ≤ 8


Generalities

Analysis of numerical data

Monte Carlo methods are importance sample methods ⇒expectation values of observables are given by simpleaverages of the numerical results

Due to the finiteness of the sampling, each observable carriesa statistical error

Monte Carlo dynamics produces data that are correlated ⇒Gaussian statistics does not apply straightforwardly

Correct data analysis keeps into account correlations in bothaverages (through binning) and fits (using correlated fits)

Bias in observables are dealt with methods such as bootstrapor jack-knife


Generalities

Bulk phase transition ⇒ avoided if β is large enough’t Hooft’s coupling

λ = g20N ⇒ β(N)

β(N ′)=

N2

N ′2

Tadpole improvement

g2I =

g20

〈UP〉⇒ βI(N)

βI(N ′)=

N2

N ′2


Confinement

Part II

Confinement and asymptotic freedom


Confinement

(k )-string tensions (closed string channel)

Confining potential: V (R) = σR

V (R) can be extracted from Wilson loops (open string channel)or from Polyakov loops (closed string channel)

In the closed string channel

CRR(t) =⟨(

PR(0))†

PR(t)⟩

=∑

j

|cRRj |2e−amRj t →t→∞

|cRRl |2e−amRl t

PR(t) =1

dR

∑k ,~n⊥

TrRl∏

j=0

Uk (~n + j k , t) , amRl 'a2σRl − cR

π(D − 2)

6l


Confinement

Confining flux tubes

14 MTeper printed on December 17, 2009

l!

!

am(l)

654321

1.5

1

0.5

0

Fig. 5. Ground state energy of a flux loop winding around a spatial torus of lengthl, in SU(6) and D = 3 + 1.

apparent from the plot that we do indeed have the (approximately) linearincrease with l that indicates linear confinement. So that you can judgewhat is the length l in physical units, I have used the value of a

!! from

our fits to translate the lattice size l = aL into physical units using l!

! =aL!

! = L" a!

!. Since we expect the intrinsic width of a flux tube to beO(1/

!!) we can see that our largest values of l are indeed large compared

to the flux tube width and it is reasonable to infer that what we are seeingis the onset of an asymptotic linear behaviour.

The dashed line shown on the plot represents a linear piece modified bythe Luscher correction term

m(l) = !l # "

3l. (20)

This O(1/l) correction is universal and the value used here corresponds tothe universality class of a simple bosonic string where the only masslessmodes are those of the transverse oscillations. We can see from Fig. 5 thatthis correction captures the bulk of the observed deviation from linearity.(One of course expects further corrections that are higher powers of 1/l.)So we have good evidence not only that linear confinement persists at largeN , but that it remains in the same universality class as has been establishedby previous work for SU(2) and SU(3).

[H. Meyer and M. Teper, JHEP 0412 (2004) 031]


Confinement

Universality of the cut-off effects

[B. Lucini and M. Teper, JHEP 01106 (2001) 050]


Confinement

In three dimensions

g20 has dimensions of mass

β =2Nag2

0

For the string tension

βIa√σ = c0 +

c1

βI, c0 =

2N√σ

g20


Confinement

√σ/g2

0 vs. N in D=2+1

A fit gives√σ

g20N

= 0.19755(35)− 0.120(3)

N2


Confinement

The running coupling in SU(4) gauge theoryRunning coupling in pure SU(4) Yang-Mills theory Gregory Moraitis

1 10E/GeV

2

4

6

8

10g2N

SU(2)SU(3)SU(4)

Figure 3: Running of the t’Hooft coupling g2N for N = 2,3,4.

running coupling from 7GeV down to the scale of√

σ , effectively linking the perturbative to thenon-perturbative regime. The result is similar to the previous SU(2) and SU(3) calculations alreadyperformed in that the data is well approximated by perturbation theory down to energies of the orderof the string tension. Of course, this should not be taken to mean that perturbation theory correctlyaccounts for all phenomena down to this energy, as the coupling defined through the Schrödingerfunctional may simply be an exceptional case.

Similarly, the t’Hooft coupling g2N is seen to be a universal function of E, only weakly de-pendent on N even down to N = 2, and we have extracted the N dependence of ΛSF to leadingorder in 1/N2. The large-N expectation of universality thus holds true down to energies of order√

σ , though we are unable to make any certain claims as to whether universality really extends tothe non-perturbative level, since perturbation theory itself proved adequate in the ranges of energystudied.

References

[1] M. Lüscher, R. Sommer, U. Wolff and P. Weisz, Nucl. Phys. B 389 (1993) 247 [arXiv:hep-lat/9207010].

[2] M. Lüscher, R. Sommer, P. Weisz and U. Wolff, Nucl. Phys. B 413 (1994) 481 [arXiv:hep-lat/9309005].

[3] M. Lüscher, R. Narayanan, P. Weisz and U. Wolff, Nucl. Phys. B 384 (1992) 168[arXiv:hep-lat/9207009].

[4] M. Lüscher, P. Weisz and U. Wolff, Nucl. Phys. B 359 (1991) 221.

[5] B. Lucini, M. Teper and U. Wenger, JHEP 0406 (2004) 012 [arXiv:hep-lat/0404008].

[6] B. Lucini, M. Teper and U. Wenger, JHEP 0502 (2005) 033 [arXiv:hep-lat/0502003].

[7] B. Lucini and M. Teper, JHEP 0106 (2001) 050 [arXiv:hep-lat/0103027].

7

[B. Lucini and G. Moraitis, PoSLAT2007:058 (2007)]


Confinement

The running of the improved bare coupling


µ = 1a!

!

g2I (µ)N

121086420

6.5

5.5

4.5

3.5

Fig. 9. The running of the (improved) lattice ’t Hooft coupling for various N , fromSU(2) (green open circles) to SU(8) (red solid points).

[29] which I display in Fig. 10. For comparison I show in Fig. 11 a quiterecent compilation of experimental determinations of the running couplingthat I have borrowed from [30]. As you can see, the lattice calculation(which includes an extrapolation to the continuum limit) is more accuratethan the experimental one, and extends over a range of scales that is atleast as large. We see a very impressive comparison with 3-loop continuumrunning, beginning at very high energies where we can have confidence inthe applicability of perturbation theory.

However my purpose here is not to dwell upon these calculations in anydetail, but to point out that there have recently [31] been calculations of thiskind in SU(4). I show the corresponding plot, borrowed from [31], in Fig. 12.The range of energies is less impressive but is still non-trivial. Extractingthe ! parameter from the fits, and converting to the standard MS scheme,and using the results of earlier calculations for N = 2 and N = 3, one finds[31]

!MS!!

= 0.528(40) +0.18(36)

N2: N " 3 (31)

Since this is a straight-line fit to just 2 points, N = 3 and N = 4, it willrequire further confirmation, but it is reassuring that it is consistent withthe result in eqn(30), obtained from the quite di"erent approach of fitting

[C. Allton, M. Teper and A. Trivini, JHEP 0807 (2007) 021]


Confinement

The Λ parameter

0 0.05 0.1 0.15 0.2 0.25

1/N20

0.2

0.4

0.6

0.8

1

!MS/"

1/2

Figure 4: The ratio !MS/!

! determined in this work as a function of 1/N2. The continuousline is the extrapolation to N = " from Ref. [22], while the dot-dashed lines delimit theregion at one sigma of confidence level (only the statistical errors are shown).

computed with two independent methods shows that at large N the systematic shouldbe under control in both cases.

5 Systematic errors

As we have seen in the previous section, albeit with larger errors, our results agreewith those reported in [22]. In order to better assess the scope of our findings, in thissection we shall discuss in detail the systematic errors of our calculation.

The main sources of systematic errors are the following:

1. Extrapolation of LMAX!

! to a = 0. We have already mentioned that this is thebiggest source of error in determining the !-parameter. In order to extrapolateLMAX

!! to a = 0, the string tension is measured for a number of lattice spacings

and then the a # 0 limit is taken according to Eq. (20). The di"culty is thelargeness of both coe"cients c1 and c2. We also have no information about therest of the series. By changing the extremes of the fitting interval and compar-ing with fits including also cubic terms (where there are su"cient values of thestring tension), we can measure the spread of those results to obtain a handle onthe systematic error connected with the extrapolation. These rather large sys-

12

[B. Lucini and G. Moraitis, Phys. Lett. B668 (2008) 226]


Glueballs

Part III

Glueballs


Glueballs

Extrapolation to the continuum limit

ExampleGlueball masses in SU(4)


Glueballs

Glueball masses at large N

[B. Lucini, M. Teper and U. Wenger, JHEP 0406 (2004) 012]


Glueballs

Masses at N = ∞

0++ m√σ

= 3.28(8) +2.1(1.1)

N2

0++∗ m√σ

= 5.93(17)− 2.7(2.0)

N2

2++ m√σ

= 4.78(14) +0.3(1.7)

N2

Accurate N = ∞ value, small O(1/N2) correction


Glueballs

0++ excitations

0 0.05 0.11/N2

0

0.5

1

1.5

2

2.5

am

A1++

A1++*

A1++**

Figure 10: Extrapolation to N !" of the states in the A++1 channel.

0 0.05 0.11/N2

0

0.5

1

1.5

2

2.5

am

A1-+

Figure 11: Extrapolation to N !" of the states in the A!+1 channel. We denote the less reliable states

with open symbols.

– 31 –

Lattice spacing fixed by requiring aTc = 1/6


Glueballs

Spectrum at aTc = 1/6

0

0.5

1

1.5

2

2.5

3

am

Ground StatesExcitations[Lucini,Teper,Wenger 2004]

A1++ A1

-+ A2+- E++ E+- E-+ T1

+- T1-+ T2

++ T2--T2

-+T1++E-- T2

+-

Figure 20: The spectrum at N = !. The yellow boxes represent the large N extrapolation of masses

obtained in ref. [38].

0 2 4 6 8 10 12 14 16 18

!!M2

2"#

0

1

2

3

J

PC = + +PC = + -PC = - +PC = - -

Figure 21: Chew-Frautschi plot of the glueball spectrum

– 36 –

[B. Lucini, A. Rago and E. Rinaldi, JHEP 1008 (2010) 119]


Glueballs

In three dimensions


Glueballs

Extrapolation to N = ∞

Expected large N behaviourmG

g20N

= d0 +d1

N2

Fit resultsstate d0 d1

0++ 0.8116(36) -0.090(28)0++∗ 1.227(9) -0.343(82)0++∗∗ † 1.65(4) -2.2(7)0−− 1.176(14) 0.14(20)0−−∗ 1.535(28) -0.35(35)2++ † 1.359(12) -0.22(8)2++∗ 1.822(62) -3.9(1.3)2−− 1.615(33) -0.10(42)


Mesons

Part IV

Mesons


Mesons

Fermionic operators

Meson masses extracted from large time behaviour of mesoniccorrelators

Particle Bilinear JPC

σ or f0 I, ψψ 0++

π ψγ5ψ, ψγ0γ5ψ 0−+

ρ ψγiψ, ψγ0γiψ 1−−

a1 ψγ5γiψ 1++

b1 ψγiγjψ 1+−

Focus on π (0−+) and ρ (1−−)


Mesons

Fixing the bare parameter

β fixed by imposing that aTc = 1/5

Another measured quantity (e.g. σ) could be used ⇒differences are O(1/N2)

Bare quark mass fixed (a posteriori!) by κ

StrategyStudy masses at fixed lattice spacing and various κ and fit tothe expected behaviour to compare various N


Mesons

A typical correlator

Operator: γ5 (describes the π)

0 10 20 30t

1e-06

0.0001

0.01

1C

Γ,Γ(t)

k=0.161k=0.160k=0.159k=0.1575k=0.156

Expected behaviour: C(T ) = A cosh(m(T − Nt/2))


Mesons

Effective mass

0 5 10 15 20 25 30t

0

0.2

0.4

0.6

0.8

1

am0(t)

k=0.161k=0.160k=0.159k=0.1575k=0.156

meff (T ) = acosh(

C(T − 1) + C(T + 1)

2C(T + 1)

)B. Lucini SU(N) gauge theories on the Lattice

Mesons

mρ vs. m2π

0 0.1 0.2 0.3 0.4 0.5 0.6a2mπ

2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

amρ

SU(2)SU(3)SU(4)SU(6)N = inf.

At small mπ

mρ = Am2π + B


Mesons

A vs. 1/N2

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0 0.05 0.1 0.15 0.2 0.25

Ang. Coeff.

1/N2

Fit with an O(1/N2) correction only


Mesons

B vs. 1/N2

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0 0.05 0.1 0.15 0.2 0.25

Intercept

1/N2

Fit with an O(1/N2) correction only


Mesons

mρ vs. m2π at N = ∞

0 0.1 0.2 0.3 0.4 0.5 0.6a2mπ

2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

amρ


[L. Del Debbio, B. Lucini, A. Patella and C. Pica, JHEP 0803(2008) 062]


Mesons

Chiral extrapolation of mπ - fixing σ

statistical errors. So for the ! we simply use the results from our largest volumes. Simi-

larly, at larger masses it is not necessary to include finite-volume e!ects since they will be

smaller than our statistical errors, for both the " and the !.

We note that our chiral extrapolations, described below, are not very sensitive to the

details of our finite-volume corrections. This is because the chiral fits are mostly controlled

by the data at higher mass values, which have significantly smaller errors, and are much

less sensitive to potential finite-volume corrections.

3.3 Determination of the critical hopping parameter

3.3.1 #c at finite N

0

0.1

0.2

0.3

6.3 6.4 6.5 6.6 6.7 6.8 6.9

(a m

!)2

"-1

SU(6)SU(4)SU(3)SU(2)

Figure 3: (am!)2 as a function of 1/#, eq. (3.3).

The Wilson action quark mass undergoes an additive renormalization (2a#c)!1, see

eq. (2.3). Thus we expect that the pion mass will be related to # (up to quenched chiral

logs) by,

(am!)2 = A

!

1

#!

1

#c

"

. (3.3)

By fitting to this equation we can extract #c for each N . We obtain good fits for each N ,

which we display in figure 3. Note also that the pion masses are horizontally aligned across

the di!erent N -values, indicating that we have succeeded to approximately match the #-

values to lines of constant physics, thus eliminating another possible source of systematic

bias. The values of #c that we obtain are shown in table 6. We note that while the lattice

– 8 –

[G. Bali and F. Bursa, JHEP 0809 (2008) 110]


Mesons

mπ vs. mρ - fixing σ

table 6 to the form,

!c = !c(N =!) +c

N2. (3.5)

After including the systematic uncertainties from the chiral extrapolation we obtain values

!c(!) = 0.1596(2) and c = "0.028(3). Some of the systematics will be correlated and we

obtain a rather small "2/n # 0.27. We display the 1/N2 extrapolation of !c in figure 4. We

also include the 1/N2 extrapolation of the lattice ’t Hooft coupling at fixed string tension.

Note that this extrapolates to the value #(N =!) = 2.780(4).

3.4 The $ meson mass

3.4.1 m! at finite N

1.6

1.8

2

2.2

2.4

2.6

2.8

3

0 1 2 3 4 5 6 7

m!/"

1/2

m#2/"

SU(2)SU(3)SU(4)SU(6)

Figure 5: Fits of m!/$

% as functions of m2"/%, eq. (3.6), at di!erent N .

In chiral perturbation theory, as well as inN m!(0)/

$% B "2/n

2 1.60(5) 0.182(10) 0.2

3 1.63(4) 0.184( 9) 1.0

4 1.70(3) 0.174( 6) 0.5

6 1.64(3) 0.185( 4) 0.6

Table 7: The fit parameters of eq. (3.6)for each N , with the reduced "2-values.

the heavy quark limit, m! depends linearly on the

quark mass mq. Within our range of pion masses

m"/$

% # 1.3 . . . 2.6 we find m2" to linearly de-

pend on !!1 and therefore to be proportional to

the quark mass. Hence, we can fit our $ masses to

the parametrization,

m!$%

=m!(0)$

%+ B

m2"

%, (3.6)

– 10 –

[G. Bali and F. Bursa, JHEP 0809 (2008) 110]


Finite Temperature

Part V

Finite Temperature


Finite Temperature

Deconfinement phase transition

Temperature T introduced by taking a lattice Ns × Nt ,Ns � Nt [T = (Nta(β))−1]A critical temperature Tc exists above which quarks andgluons are deconfinedThe phase transition can be seen as a change ofsymmetry in the ground stateThe relevant symmetry is Z(N), under which the groundstate is not invariant above Tc

The order parameter is the Polyakov loop

L =1N

∑~n

TrNt−1∏j=0

U4(~n, j)


Finite Temperature

SU(5) near Tc

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2Re L

-0.2

-0.1

0

0.1

0.2

Im L

β = 16.8785β = 16.7250

SU(5) on 83x5

degenerate vacuain the broken Z5 phase

symmetric phase

[BL, A. Rago, E. Rinaldi, in preparation]


Finite Temperature

Order parameter near Tc4 Ns = 14

Figure 10: Observables measured on 600000 configurations obtained after thermalisation and everycompound sweep (each compound is 1 heatbath update and 4 overrelaxation steps). Analysis usedjackknife blocks 3 times bigger than the integrated autocorrelation time. The temporal polyakov historiesshow O(10) times tunnelling between the symmetric and the broken phase.

8

SU(7) theory, Nt = 7, Ns = 14


Finite Temperature

History and histogram of L near Tc

Figure 11: (Upper panel): histories of the temporal polyakov loop absolute value. The � values ofthese histories are near the critical one. The high number of tunnelling is clearly visible. (Lower panel):histograms of the above histories. The distributions are the one choosen for the reweighting procedure.They are both in the broken and in the symmetric phase.

Figure 12: Multi–histogram reweight of the temporal polyakov loop susceptibility. 6 di↵erent � pointswere used.

9



Finite Temperature

Susceptibility of L near Tc

Figure 11: (Upper panel): histories of the temporal polyakov loop absolute value. The � values ofthese histories are near the critical one. The high number of tunnelling is clearly visible. (Lower panel):histograms of the above histories. The distributions are the one choosen for the reweighting procedure.They are both in the broken and in the symmetric phase.

Figure 12: Multi–histogram reweight of the temporal polyakov loop susceptibility. 6 di↵erent � pointswere used.

9



Finite Temperature

Error analysis of χ

Figure 13: Maximum of the temporal polyakov loop susceptibility extracted from each reweighted boot-strap sample. The � value of the maximum is determined just by looking at each point in the boostrapsample (see Fig. 12). To improve this result we could fit each boostrap sample separately and thenestimate the error on the location of the maximum.

10



Finite Temperature

Infinite volume extrapolation of βc

5 Infinite volume limit

We did a finite size scaling study to determine the value of the critical value �c(1) in the thermodyamiclimit, starting from �c(Ls) for di↵erent values of Ls. The results are summarized in Tab. 1.The location of the finite volume critical coupling (actually it’s the value of the coupling � correspondingto the maximum in the temporal Polyakov loop susceptibility) shifts with the physical volume of thelattice according to

�c(Ls) = �c(1)� hL3

t

L3s

, (1)

depending linearly by the inverse of the volume.Fitting the data of Tab. 1 using Eq. (1) and discarding the smallest volume Ls = 7, we obtain

�c(1) = 43.9793(16) (2)

andh = 0.150(10) (3)

with a very good �2 = 0.78. The result is compatible with the old one from arXiv:hep-lat/0502003v1and it is shown in Fig. 17. Also a compatible �c(1) is obtained by including the smallest volume, evenif the goodness of the fit is lower. A quadratic fit including all the points works very well but give anoverestimated �c(1) = 43.9848(16) which is not compatible with Eq. (2).

Ls 7 8 10 11�c 43.9318(17) 43.9429(14) 43.9593(16) 43.9656(10)

Table 1: Location of the maximum of the temporal Polyakov loop susceptibility for di↵erent lattice sizesLs.

Figure 17: Fits of the whole set of data and with the exclusion of the point at the smallest volume are

shown. A quadratic fit is shown in red and the infinite volume critical couplings are plotted at L3t

L3s

= 0

12

SU(8) theory, Nt = 5, various Ns


Finite Temperature

Deconfinement temperature


such relation – but, on the other hand, this opens the door to AdS/CFTcalculations.

In Fig. 14 I show lattice calculations of Tc in units of the string tensionfor N ! [2, 8] in D = 3 + 1. All the values shown are after extrapolation tothe continuum limit [22]. If we fit with an O(1/N2) correction we obtain

Tc"!

N!2= 0.597(4) +

0.45(3)

N2: D = 3 + 1 (33)

which thus provides a prediction #N . It is perhaps surprising that this sim-ple analytic form should fit all the way down to N = 2 where the transitionhas changed from first to second order. Especially so, given that the errorsfor SU(2) and SU(3) are very small, about 0.5%. We also note that thecoe!cient of the correction is O(1) in natural units.

1/N2

Tc"!

0.30.250.20.150.10.050

1

0.8

0.6

0.4

0.2

0

Fig. 14. Deconfining temperature in units of the string tension, for continuumSU(N) gauge theories in D = 3 + 1; with O(1/N2) extrapolation to N = $.

It is interesting to see what happens in D = 2 + 1. The correspondingresults for Tc [34] are shown in Fig. 15. Now both N = 2 and N = 3 aresecond order, but a fit with just the leading correction still works for all N ,giving

Tc"!

N!2= 0.903(3) +

0.88(5)

N2: D = 2 + 1 (34)

[B. Lucini, M. Teper and U. Wenger, JHEP 0401 (2004) 061]


Conclusions

Part VI

Conclusions


Conclusions

Gluonic observables from guage-string duality

HQCD lattice Nc = 3 lattice Nc !" Parameter

[p/(N2c T 4)]T=2Tc

1.2 1.2 - V 1 = 14

Lh/(N2c T 4

c ) 0.31 0.28 [25] 0.31 [27] V 3 = 170

[p/(N2c T 4)]T!+" !2/45 !2/45 !2/45 Mp" = [45!2]#1/3

m0++/#

# 3.37 3.56 [28] 3.37 [29] "s/" = 0.15

m0!+/m0++ 1.49 1.49 [28] - ca = 0.26

$ (191MeV )4 (191MeV )4 [31] - Z0 = 133

Tc/m0++ 0.167 - 0.177(7)

m0"++/m0++ 1.61 1.56(11) 1.90(17)

m2++/m0++ 1.36 1.40(4) 1.46(11)

m0"!+/m0++ 2.10 2.12(10) -

Table 2: Collected in this table is the complete set of physical quantities that we computed

in our model and compared with data. The upper half of the table contains the quantities

that we used as input (shown in boldface) for the holographic QCD model (HQCD). Each

quantity can be roughly associated to one parameter of the model (last column). The lower

half of the table contains our “postdictions” (i.e. quantities that we computed after all the

parameters were fixed) and the comparison with the corresponding lattice results. The

value we find for the critical temperature corresponds to Tc = 247MeV .

graphic models we discuss generically provides a qualitatively accurate description ofthe strong Yang-Mills dynamics. This is a highly non-trivial fact, that strongly indi-cates that a realistic holographic description of real-world QCD might be ultimately

possible.

Although the asymptotics of our potential is dictated by general principles, webased our choice of parameters by comparing with the lattice results for the ther-

– 27 –

U. Gursoy et al., Nucl. Phys. B820 (2009) 148-177[arXiv:0903.2859]


Conclusions

Meson masses from guage-string duality

determined asM2

⇡

⇤2b

= 2.75

r⇡

gsN

mq

⇤b

. (6.19)

6.2.3 Vector mesons

The vector mesons in the model are, as in the basic D7 brane embeddings, described by the

gauge fields in the DBI action describing the D7 branes. Again solutions of the form Aµ =

g(⇢) sin(k · x)✏µ provide the masses of the ⇢ and its radially excited states [148]. The n = 0,

unexcited, state has massM2

⇢

⇤2b

= 2.16

r⇡

gsN. (6.20)

As expected it is massive, reflecting the dynamical generation of a quark mass.

In figure 6.5 we plot the dependence of the rho meson mass on the pion mass squared in this

model, in dimensionless units fixed by the choice of the supergravity scale b. The rho mass as

function of the pion mass squared has recently also been computed for large N within lattice

gauge theory [19, 20], and a direct comparison of gauge/gravity and lattice results is possible.

In the lattice computations, the scale is set by the lattice spacing a. We choose our units such

as to be able to compare directly with the lattice results of [20], which are shown on the right

hand side of figure 6.5. In units such that the o↵set at m⇡ = 0 coincides with [20], we find a

linear dependence of the rho mass on the pion mass squared, with slope 0.57.

0.0 0.1 0.2 0.3 0.4 0.5m!20.3

0.4

0.5

0.6

0.7

0.8

0.9m"

0 0.1 0.2 0.3 0.4 0.5 0.6mπ2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

mρ


Figure 6.5: A plot of m⇢ vs m2⇡ in the Constable-Myers background on the left (we thank

Andrew Tedder for generating this plot). Lattice data [20] (preliminary, quenched and at finitespacing) for the same quantity is also shown on the right.

For the lattice results of [20], the simulations are performed in the quenched approximation.

This is appropriate for the large N limit, if not for smaller N . The lattice data of [20] is

preliminary and at a fixed, finite lattice spacing. Nevertheless it is striking that not only does

the lattice data display the same linearity as the gauge/gravity model, but also the slope in

the large N limit is 0.52 and therefore is very close to the gauge/gravity dual result. The fact

67

Erdmenger et al., Eur.Phys.J. A35 (2008) 81-133[arXiv:0711.4467]


Conclusions

Conclusions

The lattice is a useful tool a non-perturbative investigationsof the large N limitThese results can be used to check (or inspire?)constructions of string backgroundSeveral results for various observables available by knowMany (all?) observables are well described by theexpected leading correction for 2 ≤ N ≤ 8 ⇒ SU(3) differsfrom SU(∞) by small O(1/N2) correctionsFuture work includes

Excited statesImproved meson spectrumEffects of fermionsOther large N generalisations (e.g. Veneziano, OrientifoldPlanar Equivalence)


su(n) gauge theories on the lattice ii -- results and...

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