Mesons at large N

Outline SU(N) Gauge theories at large N The meson spectrum Conclusions

Mesons at large N

Biagio Lucini

with L. Del Debbio, A. Patella and C. Pica

DEWBS, Odense, September 2008

Biagio Lucini Mesons at large N


References

L. Del Debbio, B. Lucini, A. Patella and C. Pica,Quenched mesonic spectrum at large N,JHEP 0803, 062 (2008) [arXiv:0712.3036].

G. S. Bali and F. Bursa,Mesons at large Nc from lattice QCD,arXiv:0806.2278 [hep-lat].



Outline

1 SU(N) Gauge theories at large NThe large N limitSU(N) Lattice Gauge theory

2 The meson spectrumMasses from correlatorsMeson masses

3 Conclusions



Motivations

The lattice is an invaluable tool for studying QCD from firstprinciples

Analytical approaches are needed to complement/interpretlattice results

The large N limit is a promising analytical approachrecently revived by the AdS/CFT correspondence

The lattice allows to extract non-perturbative physics in thelarge N limit⇒ comparison with other approaches



The large N limit

SU(N) gauge theory (possibly enlarged with fermionsand/or scalars in the fundamental or adjoint representation)

In the limit g → 0, N →∞ at fixed λ = g2N only thosediagrams survive that can be drawn in a plane withoutcrossing of the lines (’t Hooft)

An SU(∞) gauge theory can be defined rigorously byresumming the perturbative series provided that g issufficiently small and all masses are sufficiently large

Corrections are O(1/N) for the full theory and O(1/N2) forthe pure gauge theory

The lattice approach allows us to go beyond perturbativeand diagrammatic arguments



On the lattice

For a given observable1 Continuum extrapolation

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

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/N

Studies performed in pure gauge for various observables bothat zero and finite temperature for 2 ≤ N ≤ 8



Example: glueball masses at N =∞

(from B. Lucini, M. Teper, U. Wenger, hep-lat/0404008)

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



Lattice action

Path integral

Z =

∫

(DUµ(i)) (det M(Uµ))Nf e−Sg(Uµν (i))

with

Uµ(i) = Pexp

(

ig∫ i+aµ̂

iAµ(x)dx

)

and

Uµν(i) = Uµ(i)Uν(i + µ̂)U†µ(i + ν̂)U†

ν(i)

Gauge part

Sg = β∑

i ,µ

(

1− 1NRe Tr(Uµν(i))

)

, with β = 2N/g2



Fermions on the lattice

Naive discretisation of the fermionic field gives rise to fermiondoubling (16 species of fermions in (3 + 1) dimensions)

No-go theorem (Nielsen-Ninomiya): no lattice formulation offermions can be at the same time ultra-local, chirally symmetricand avoid fermion doubling

Solutions:1 Wilson fermions← give up chirality2 Staggered fermions← put up with doubling3 Domain wall and overlap fermions← couple all sites

In our calculation we opted for Wilson fermions



Why Wilson fermions?

Different fermion formulations must give the same results, but agiven discretisation can provide some advantages at finite a

The Wilson formulation has been chosen because1 Unlike non-ultralocal chiral fermions, Wilson fermions are

cheap to simulate2 Unlike the staggered fermions, a generic number of

flavours can be simulated3 Chiral symmetry can be recovered by tuning the hopping

parameter4 New technical breakthroughs allow to go close enough to

the chiral limit



Wilson fermions

Take the naive Dirac fermions and add an irrelevant term thatgoes like the Laplacian

Mαβ(ij) = (m + 4r)δijδαβ

− 12

[

(r − γµ)αβ Uµ(i)δi ,j+µ + (r + γµ)αβ U†µ(j)δi ,i−µ

]

This formulation breaks explicitly chiral symmetry

Define the hopping parameter

κ =1

2(m + 4r)

Chiral symmetry recovered in the limit κ→ κc (κc to bedetermined numerically)



Quenched approximation

For an observable O

〈O〉 =

∫

(DUµ(i)) (det M(Uµ))Nf f (M)e−Sg(Uµν (i))∫

(DUµ(i)) (det M(Uµ))Nf e−Sg(Uµν (i))

Assume det M(Uµ) ≃ 1 i.e. fermions loops are removed fromthe action

The approximation is exact in the m →∞ and N →∞ limit(g2N is fixed)→֒ the large N spectrum is quenched for m 6= 0

As N increases, unquenching effects are expected for smallerquark masses



Correlation functions from propagators

Generic fermion operator O(~x , t) = ψ̄1(~x , t)Γψ2(~x , t), Γ productof Dirac matrices (G = M−1(~x , t + T ; ~y , t))

C(~x − ~y ,T ) = 〈O†(~x , t + T )O(~y , t)〉= −〈Tr

(

γ0Γ†γ0GΓγ5G

†γ5

)

〉

Flavour singlet⇒ also disconnected diagram

If O is flavour non-singlet, it is enough to know the propagatorfrom one point to all the others (one column of the M−1)

Zero momentum propagator: C(t) =∑

~x C(~x − ~y ,T )∝ e−mT

when T →∞



Fermionic operators

Particle Bilinear JPC

σ or f0 I, ψ̄ψ 0++

π ψ̄γ5ψ, ψ̄γ0γ5ψ 0−+

ρ ψ̄γiψ, ψ̄γ0γiψ 1−−

a1 ψ̄γ5γiψ 1++

b1 ψ̄γiγjψ 1+−

Focus on π (0−+) and ρ (1−−)



Fixing the bare parameter

β fixed by imposing that aTc = 1/5

Another observable (e.g. σ) could be used(e.g.: Bali-Bursa: a

√σ = 0.2093 for all N)

⇒ differences are O(1/N2)

Bare quark mass fixed (a posteriori!) by κ

Strategy

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



A typical correlator

Group: SU(4); Operator: γ5 (describes the π)

0 10 20 30t

1e-06

0.0001

0.01

1

CΓ,

Γ(t)

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

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



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)

)



Effective masses vs. N at fixed κ

0 5 10 15 20 25 30t

0

0.5

1

1.5

2

am0(t

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



mπ vs. κ

From χPT

mπ ∝√

mq ⇒ ansatz mπ = A√

1/κ− 1/κc

At fixed quark mass the N dependence is in A and κc

From lattice perturbation theory

κc(N) = κc(N =∞) + k/N2

Similarly we use a large N-inspired ansatz for A(N):

A(N) = A(N =∞) + α/N2



κc vs. 1/N2

0 0.05 0.1 0.15 0.2 0.25

1/N2

0.15

0.155

0.16

0.165

0.17

kc

No 1/N term, in agreement with perturbation theory



Numerical results for mπ vs. κ

5.9 6 6.1 6.2 6.3 6.4 6.5 1/k

0

0.2

0.4

0.6

0.8

1a2 m

π2

SU(2)SU(3)SU(4)SU(6)SU(∞)



Quark mass from PCAC

From the PCAC relation

∂µAµa(x) = 2mq ja(x) with

Aµa(x) = ψ̄(x)γµγ5T aψ(x)ja = ψ̄(x)γ5T aψ(x)mq = Zm̃q

we get

mq =12〈∑~x (∂0A0a(x))ja(y))〉〈∑~x ja(x)ja(y)〉

On the lattice

mq(T ) =14〈A0a(0)ja(T + 1)〉 − 〈A0a(0)ja(T − 1)〉

〈ja(0)ja(T )〉



mq from correlators (κ = 0.156)

0 5 10 15 20t

0

0.1

0.2

0.3

0.4am

PCA

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



m2π vs. mq for all N

0 0.05 0.1 0.15 0.2am

PCAC

0

0.1

0.2

0.3

0.4

0.5

0.6

a2 mπ2

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

Fit:m2

π = amq + b (b 6= 0 due to lattice discretisation errors)



mρ vs. m2π

At small mπ

mρ(N,mπ) = A(N)m2π + B(N)

Large N arguments suggest

A(N) = a1 + a2/N2

B(N) = b1 + b2/N2

Then at N =∞

mρ(mπ) = a1m2π + b1



Numerical results for mρ vs. m2π

0 0.1 0.2 0.3 0.4 0.5 0.6

a2mπ2

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.



Extrapolation to N =∞

Summary of the results (aTc = 1/5)

mπ(N) =

(

1.262(8)− 0.82(6)

N2

)(

1κ− 5.945(4) +

0.0398(6)

N2

)1/2

mρ(N) =

(

0.539(3)− 0.62(3)

N2

)

+

(

0.5224(8) +1.10(1)

N2

)

m2π(N)



Extrapolation to the continuum - I

Infinite N results

Del Debbio et al. (a√σ = 0.3350)

mρ√σ

= 1.609(9) + 0.1750(3)m2

π

σ

Bali and Bursa (a√σ = 0.2093)

mρ√σ

= 1.670(24) + 0.182(5)m2

π

σ



Extrapolation to the continuum - II

1.6

1.8

2

2.2

2.4

2.6

2.8

3

0 1 2 3 4 5 6 7

mρ/

σ1/2

mπ2/σ

a=0a=0.209σ-1/2

a=0.335σ-1/2

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



Systematic errors

Finite size effects negligible (Bali and Bursa)

The distribution of the smallest eigenvalues is more stableas N increases

Chiral log effects seem numerically negligible

Small O(a) corrections



Comparison with AdS/CFT

Continuum extrapolation for mρ vs. mπ

mρ(mπ)

mρ(0)= 1 + 0.341(4)

(

mπ

mρ(0)

)2

A calculation in the Constable-Myers background (seeErdmenger et al., Phys. Rev. D69 (2004) [hep-th/0306018])

mρ(mπ)

mρ(0)= 1 + 0.307

(

mπ

mρ(0)

)2

remarkably close to large N QCD



Conclusions

Lattice calculation of the lowest-lying states in the large Nmeson spectrum

Determination of mπ vs. mq and mπ vs mρ at N =∞Verification of χPT at large N

mπ and mρ are described by the perturbative leading orderin 1/N

No evidence for deviation from leading order large N PThas been observed

AdS/CFT predictions close to the lattice results



Future developments

Unquench the theory and go closer to the chiral limit

Extend the calculation to higher-mass states and to flavoursinglet statesChange the representation of the fermions

antisymmetric and adjoint⇒ Orientifold planar equivalencesymmetric⇒ strongly interacting BSM dynamics

(Softly broken) N = 1 Supersymmetry?


Mesons at large N

Documents

Transcript of Mesons at large N