Large N: lattice results and perspectives

Large N

B. Lucini

The ’t Hooftlarge-N limit

Quenchedorientifoldplanarequivalence

Reducedmodels

ConclusionsandPerspectives

Large N:Lattice Results and Perspectives

Biagio Lucini

SFB TR 55 Meeting, Regensburg, 1st August 2014

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Outline

1 The ’t Hooft large-N limit

2 Quenched orientifold planar equivalence

3 Reduced models

4 Conclusions and Perspectives

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Outline



3 Reduced models


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Why a lattice study?

In QCD some phenomena (confinement, chiral symmetry breaking) arenon-perturbative

Lattice calculations are successful in this regime, but more effective if someanalytic guidance is available

If we embed QCD in a larger context (SU(N) gauge theory with Nf quarkflavours), the theory simplifies in the limit N →∞, Nf and g2N = λ fixed.Yet, it is non-trivial. Perhaps QCD is physically close to this limit?

However large-N QCD is still complicated enough that an analytic solutionhas not been found

Lattice calculations can shed light on the existence of the limiting theory andon the proximity of QCD to it

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Meson masses from gauge-string duality

0 0.25 0.5 0.75 1

(mπ / mρ0)2

1

1.2

1.4

mρ /

mρ0

Lattice extrapolationAdS/CFT computationN= 3N= 4N= 5N= 6N= 7N=17

Ads/CFT data from Erdmenger et al., Eur.Phys.J. A35 (2008) 81-133

[arXiv:0711.4467]

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The spectrum from the topological string0 1 2 3 4 5 60

1

2

3

4

5

m 2� � QCD

2

korn

s �0

s �2

s �4

s � 1

s � 3

s � 5

s �0

s �1

s �2

F i g u r e 1 . T h e g l u e b a l l a n d m e s o n s p e c t r u m o f l a r g e - N m a s s l e s s Q C D : T h e p o i n t s i n b l a c k ,a n d t h e s t r a i g h t t r a j e c t o r i e s l a b e l l e d b y t h e s p i n s , r e p r e s e n t t h e s p e c t r u m i m p l i e d b y t h e l a w sE q s . ( 1 . 1 ) . T h e r e d a n d y e l l o w p o i n t s r e p r e s e n t r e s p e c t i v e l y g l u e b a l l s a n d m e s o n s a c t u a l l y f o u n di n t h e l a t t i c e c o m p u t a t i o n s [ 1 – 4 ] .

0 1 2 3 4 5 60

1

2

3

4

5

m 2� � QCD

2

s

k �1

k �2

k �3

k �4

k �1

k �2

n�0

n�1

n�2

n�3

F i g u r e 2 . T h e g l u e b a l l a n d m e s o n s p e c t r u m o f l a r g e - N m a s s l e s s Q C D : T h e p o i n t s i n b l a c k , a n dt h e s t r a i g h t R e g g e t r a j e c t o r i e s l a b e l l e d b y t h e i n t e r n a l q u a n t u m n u m b e r s , k f o r g l u e b a l l s a n d n f o rm e s o n s , r e p r e s e n t t h e s p e c t r u m i m p l i e d b y t h e l a w s E q s . ( 1 . 1 ) .

– 2 –[M. Bochicchio, Glueball and meson spectrum in large-N massless QCD,

arXiv:1308.2925 [hep-th]]

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Glueball 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]

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The meson spectrum at large N

[G. Bali, F. Bursa, L. Castagnini, S. Collins, L. Del Debbio, B. Lucini and M.

Panero, JHEP 1306 (2013) 071]

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Comparison with QCD

0

1

2

3

4

5

m /

√σ

Ground statesExcited states

Experiment, √σ = 395 MeV

Experiment, √σ = 444 MeV

mq = m

ud

^Fπ

^fρ

π ρ a0

a1

b1

0

4.6

9.2

13.8

18.4

23.0

m /

^ F

∞

√σ fixed from the condition F̂∞ = 85.9 MeV, mud from mπ = 138 MeV

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Finite a corrections – SU(7)

0

0.5

1

1.5

2

2.5

3

0 0.05 0.1 0.15 0.2 0.25 0.3

X/√σ

a√σ

11%

3.3%

16%

4.2%

11%

5.5%

a1b1a0ρ

f̂ρF̂π

[G. Bali, L. Castagnini, BL and M. Panero, in preparation]

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Monte Carlo history of Q – SU(3)

V=164, β = 6.0

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Monte Carlo history of Q – SU(5)

V=164, β = 17.45

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A first exploration of open boundary conditions

Gauge group: SU(7)

Lattice spacing: a√σ ∼ 0.21

Lattice sizes: 163 × Nt, Nt = 32, 48, 64

Statistics: ∼ 500 configurations, separated by 200 composite sweeps (1composite sweep is 1 hb + 4 overrelax)

Purpose: comparing results with PBC and OBC at fixed lattice parametersin a case where there is a severe ergodicity problem

Observables: Q, gluonic correlators in the 0++ and 0−+ channels andinstantons

Both cooling and Wilson Flow used to filter UV modes

[A. Amato, G. Bali and BL, in progress]

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Monte Carlo history of Q

0 50 100 150 200 250 300 350 400 450

CFG

0

1

2

3

4

5

6

7

His

tory

ofQ

PERIODIC

0 100 200 300 400 500 600

CFG

−10

−5

0

5

10

His

tory

ofQ

OPEN

Significantly better decorrelation for OBC

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Distribution of Q

PBC OBC

OBC seem to cure the problem of ergodicity

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Outline



3 Reduced models


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Orientifold planar equivalence

The antisymmetric and the antifundamental representations coincide forSU(3) (but not in general for SU(N))⇒ different SU(N) generalizations ofQCD.

In the planar limit, the (anti)symmetric representation is equivalent toanother gauge theory with the same number of Majorana fermions in theadjoint representation (in a common sector). In particular, QCD with onemassless fermion in the antisymmetric representation is equivalent toN = 1SYM in the planar limit⇒ copy analytical predictions from SUSY to QCD.

The orientifold planar equivalence holds if and only if the C-symmetry is notspontaneously broken in both theories⇒ a calculation from first principlesis mandatory.

Assuming that planar equivalence works, how large are the 1/Ncorrections?

A. Armoni, M. Shifman and G. Veneziano. SUSY relics in one-flavor QCD from anew 1/N expansion. Phys. Rev. Lett. 91, 191601, 2003.

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A. Armoni, M. Shifman and G. Veneziano. SUSY relics in one-flavor QCD from anew 1/N expansion. Phys. Rev. Lett. 91, 191601, 2003.

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M. Unsal and L. G. Yaffe. (In)validity of large N orientifold equivalence. Phys. Rev.D74:105019, 2006.

A. Armoni, M. Shifman and G. Veneziano. A note on C-parity conservation and thevalidity of orientifold planar equivalence. Phys.Lett.B647:515-518,2007.

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Dynamical fermions difficult to simulate⇒ start with the quenched theory

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Condensates on the lattice

Aim

To measure the bare quark condensate with staggered fermions in the two-indexrepresentations of the gauge group, in the quenched lattice theory.

Wilson action.

Staggered Dirac operator. D = m− K.

The two-index representations.

The bare condensate.

A. Armoni, B. Lucini, A. Patella and C. Pica, Phys. Rev. D78 (2008) 045019

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Aim


Wilson action.




SYM = −2Nλ

∑p

<e tr U(p)

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Aim


Wilson action.




Dxy = mδxy − Kxy =

= mδxy +12

∑µ

ηµ(x){

R[Uµ(x)]δx+µ̂,y − R[Uµ(x− µ̂)]†δx−µ̂,y

}

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Aim


Wilson action.




tr Adj[U] = | tr U|2 − 1

tr S/AS[U] =(tr U)2 ± tr(U2)

2

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Aim


Wilson action.




For S/AS representations:

〈ψ̄ψ〉q =1V〈Tr(m− K)−1〉YM

For the adjoint representation:

〈λλ〉q =1

2V〈Tr(m− K)−1〉YM

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Proof of the “quenched” equivalence

Equivalence

limN→∞

1VN2〈Tr(m− KS/AS)−1〉 = lim

N→∞

12VN2

〈Tr(m− KAdj)−1〉

Expand in m−1.

Replace the two-index representations.

Take the large-N limit.

Mathematical details. The condensate is an analytical function of each realmass. The large-N limit can be exchanged with the series.

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Equivalence

limN→∞


N→∞

12VN2


Expand in m−1.




1VN2〈Tr(m− K)−1〉 =

1VN2

∞∑n=0

1mn+1

〈Tr Kn〉 =

=1

VN2

∑ω∈C

c(ω)

mL(ω)+1〈tr R[U(ω)]〉

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Equivalence

limN→∞


N→∞

12VN2


Expand in m−1.




1VN2〈Tr(m− KS/AS)−1〉 =

12V

∑ω∈C

c(ω)

mL(ω)+1

〈[tr U(ω)]2〉 ± 〈tr[U(ω)2]〉N2

12VN2

〈Tr(m− KAdj)−1〉 =

12V

∑ω∈C

c(ω)

mL(ω)+1

〈| tr U(ω)|2〉 − 1N2

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Equivalence

limN→∞


N→∞

12VN2


Expand in m−1.





12V

∑ω∈C

c(ω)

mL(ω)+1

〈[tr U(ω)]2〉 ± 〈tr[U(ω)2]〉N2

12VN2


12V

∑ω∈C

c(ω)

mL(ω)+1

〈| tr U(ω)|2〉 − 1N2

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Equivalence

limN→∞


N→∞

12VN2


Expand in m−1.





12V

∑ω∈C

c(ω)

mL(ω)+1

〈tr U(ω)〉〈tr U(ω)〉N2

12VN2


12V

∑ω∈C

c(ω)

mL(ω)+1

〈tr U(ω)〉〈tr U(ω)†〉N2

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Equivalence

limN→∞


N→∞

12VN2


Expand in m−1.




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A convenient parameterization

1N2〈ψ̄ψ〉S/AS =

12V

∑ω∈C

c(ω)

mL(ω)+1

〈[tr U(ω)]2〉 ± 〈tr[U(ω)2]〉N2

1N2〈λλ〉Adj =

12V

∑ω∈C

c(ω)

mL(ω)+1

〈| tr U(ω)|2〉 − 1N2

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A convenient parameterization

1N2〈ψ̄ψ〉S/AS = f

(m,

1N2

)±

1N

g(

m,1

N2

)1

N2〈λλ〉Adj = f̃

(m,

1N2

)−

12N2〈ψ̄ψ〉free

Planar equivalence: f (m, 0) = f̃ (m, 0).

Strategy

1 Simulate the condensates at various values of the mass.2 Extract the functions f , g, f̃ .3 Fit at fixed mass:

f̃ = a0 +b0

N2g = a1 +

b1

N2f − f̃ =

a2

N2+

b2

N4

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Simulation details

N = 2, 3, 4, 6, 8β(N) chosen in such a way that (aTc)−1 = 5 (a ' 0.145 fm)144 lattice, which corresponds to L ' 2.0 fm22 values of the bare mass in the range 0.012 · · · 8.0

0 2 4 6 80

0.2

0.4

0.6

0.8

<ψ

ψ>

region Iregion IIregion III

0 0.005 0.01 0.015 0.02m

0

0.04

0.08

0.12

<ψψ

>

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Function f̃

N=2

N=3

N=4

N=6

N=8

Infinite N

0 0.2 0.4 0.6 0.8 1m

0.15

0.2

0.25f~

For m ≤ 0.2 we get χ2/dof ≤ 0.53 (we use N = 4, 6, 8).

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Function f

N=2

N=3

N=4

N=6

N=8

Infinite N

0 0.2 0.4 0.6 0.8 1m

0.15

0.2

0.25f

For m ≤ 0.2 we get χ2/dof ≤ 0.37 (we are fitting here f − f̃ ; we use N = 4, 6, 8).

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Function g

N=2N=3N=4N=6N=8Infinite N

0 0.2 0.4 0.6 0.8 1m

0.2

0.3

0.4

0.5

g

For m ≤ 0.2 we get χ2/dof ≤ 0.17 (we use N = 4, 6, 8).

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Reducedmodels


Condensate in the adjoint representation

0 0.1 0.2 0.3 0.4 0.51/N

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

cond

ensa

te/N

^2

〈λλ〉Adj(m = 0.012)

N2= 0.23050(22)−

0.3134(72)

N2

At N = 3, relative error ' 0.8%.

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Condensate in the antisymmetricrepresentation

0 0.1 0.2 0.3 0.4 0.51/N

0

0.05

0.1

0.15

0.2

cond

ensa

ta/N

^2

〈ψ̄ψ〉AS(m = 0.012)

N2= 0.23050(22)−

0.4242(11)

N−

0.612(43)

N2−

0.811(25)

N3

At N = 3, condensate < 0!

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Condensate in the symmetric representation

0 0.1 0.2 0.3 0.4 0.51/N

0.22

0.24

0.26

0.28

0.3

0.32

0.34

cond

ensa

te/N

^2

〈ψ̄ψ〉S(m = 0.012)

N2= 0.23050(22) +

0.4242(11)

N−

0.612(43)

N2+

0.811(25)

N3

At N = 3, relative error ' 4%.

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Mesonic two-point functions on the lattice

Aim

To measure the mesonic two-point functions with Wilson fermions in the two-indexrepresentations of the gauge group, in the quenched lattice theory.

Wilson action.

Wilson Dirac operator.


The mesonic two-point correlation functions.

B. Lucini, G. Moraitis, A. Patella and A. Rago, Phys. Rev. D82 (2010) 114510

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Aim


Wilson action.




SYM = −2Nλ

∑p

<e tr U(p)

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Aim


Wilson action.




Dxy;αβ = (m + 4r)δxyδαβ − Kxy;αβ

Kxy;αβ = −12

[(r − γµ)αβ R

[Uµ(x)

]δy,x+µ̂ + (r + γµ)αβ R

[U†µ(y)

]δy,x−µ̂

]

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Aim


Wilson action.




tr Adj[U] = | tr U|2 − 1

tr S/AS[U] =(tr U)2 ± tr(U2)

2

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Aim


Wilson action.




CRΓ1Γ2

(x, y) = rR

⟨ψ̄R

a (x)Γ†1ψRb (x)ψ̄R

b (y)Γ2ψRa (y)

⟩YM

= rR

⟨trR

(D−1

yx;αβΓγβ?1 D−1xy;γδΓ

δα2

)⟩YM{

rR = 1 R = S/ASrR = 1/2 R = Adj

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Heuristic proof of the quenched equivalence

Equivalence

limN→∞

1N2

CS/ASΓ1Γ2

(x, y) = limN→∞

1N2

CAdjΓ1Γ2

(x, y)

Expand in Wilson loops.



Use invariance under charge conjugation.

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Equivalence

limN→∞

1N2

CS/ASΓ1Γ2

(x, y) = limN→∞

1N2

CAdjΓ1Γ2

(x, y)





1N2

CRΓ1Γ2

(x, y) =rR

N2

∑C⊃(x,y)

αC 〈 trR WC 〉

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Equivalence

limN→∞

1N2

CS/ASΓ1Γ2

(x, y) = limN→∞

1N2

CAdjΓ1Γ2

(x, y)





1N2

CS/ASΓ1Γ2

(x, y) =12

∑C⊃(x,y)

αC〈[tr WC ]2〉 ± 〈tr[W2

C ]〉N2

1N2

CAdjΓ1Γ2

(x, y) =12

∑C⊃(x,y)

αC〈| tr WC |2〉 − 1

N2

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Equivalence

limN→∞

1N2

CS/ASΓ1Γ2

(x, y) = limN→∞

1N2

CAdjΓ1Γ2

(x, y)





1N2

CS/ASΓ1Γ2

(x, y) =12

∑C⊃(x,y)

αC〈[tr WC ]2〉 ± 〈tr[W2

C ]〉N2

1N2

CAdjΓ1Γ2

(x, y) =12

∑C⊃(x,y)

αC〈| tr WC |2〉 − 1

N2

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Equivalence

limN→∞

1N2

CS/ASΓ1Γ2

(x, y) = limN→∞

1N2

CAdjΓ1Γ2

(x, y)





1N2

CS/ASΓ1Γ2

(x, y) =12

∑C⊃(x,y)

αC〈tr WC〉〈tr WC〉

N2

1N2

CAdjΓ1Γ2

(x, y) =12

∑C⊃(x,y)

αC〈tr WC〉〈tr W†C〉

N2

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Equivalence

limN→∞

1N2

CS/ASΓ1Γ2

(x, y) = limN→∞

1N2

CAdjΓ1Γ2

(x, y)





〈tr W†C〉 = 〈tr WC〉 ⇒ limN→∞

1N2

CS/ASΓ1Γ2

(x, y) = limN→∞

1N2

CAdjΓ1Γ2

(x, y)

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Equivalence

limN→∞

1N2

CS/ASΓ1Γ2

(x, y) = limN→∞

1N2

CAdjΓ1Γ2

(x, y)





A more formal proof of the equivalence exists which does not use the expansion inWilson loops, but is much more involved

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Simulation strategy

Simulations performed for N = 2, 3, 4, 6

β(N) chosen in such a way that (aTc)−1 = 5 (a ' 0.145 fm)

Calculations on a 32× 163 lattice, which corresponds to L ' 2.3 fm

CRΓ1Γ2

determined for Γ1 = Γ2 = γ5 (π channel) and Γ1 = Γ2 = γi (ρchannel)

Mass extracted from the ansatz CRΓ1Γ2

(t) = A cosh (m(t − T/2))

Chiral extrapolation of mρ using mρ(mπ) = cm2π + mρ(mπ = 0)

Extrapolation to large N

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Simulation strategy




CRΓ1Γ2



(t) = A cosh (m(t − T/2))



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Simulation strategy




CRΓ1Γ2



(t) = A cosh (m(t − T/2))



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Reducedmodels


Simulation strategy




CRΓ1Γ2



(t) = A cosh (m(t − T/2))



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Reducedmodels


Simulation strategy




CRΓ1Γ2



(t) = A cosh (m(t − T/2))



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Reducedmodels


Simulation strategy




CRΓ1Γ2



(t) = A cosh (m(t − T/2))



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Reducedmodels


Simulation strategy




CRΓ1Γ2



(t) = A cosh (m(t − T/2))



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Reducedmodels


mρ vs. mπ in SU(3)

0 0.1 0.2 0.3 0.4 0.5

mπ2

0

0.2

0.4

0.6

0.8

1

mρ

AdjSAS

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mρ vs. mπ in SU(6)

0 0.1 0.2 0.3 0.4 0.5

mπ2

0

0.2

0.4

0.6

0.8

1

mρ

AdjSAS

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Reducedmodels


mρ vs. mπ (Antisymmetric)

0 0.1 0.2 0.3 0.4 0.5

mπ2

0

0.2

0.4

0.6

0.8

1

mρ

N=3N=4N=6

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Large N

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Reducedmodels


mρ vs. mπ (Symmetric)

0 0.1 0.2 0.3 0.4 0.5

mπ2

0

0.2

0.4

0.6

0.8

1

mρ

N=3N=4N=6

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Large N

B. Lucini



Reducedmodels


mρ vs. mπ (Adjoint)

0 0.1 0.2 0.3 0.4 0.5

mπ2

0

0.2

0.4

0.6

0.8

1

mρ

N=2N=3N=4N=6

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Large N

B. Lucini



Reducedmodels


Chiral extrapolation of mρ

0 0.05 0.1 0.15 0.2 0.25

1/N2

0

0.2

0.4

0.6

0.8

1

mρ

AdjSA

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Large N

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Reducedmodels


Order of corrections

The correlator in the adjoint representation decays with a mass mAdjρ that can be

expressed as a power series in 1/N2, while mASρ and mS

ρ have 1/N corrections thatare related:

mAdjρ (N) = F

(1

N2

);

mSρ(N) = M

(1

N2

)+

1Nµ

(1

N2

);

mASρ (N) = M

(1

N2

)−

1Nµ

(1

N2

).

M =(mSρ + mAS

ρ

)/2 and µ = N

(mSρ − mAS

ρ

)/2 can be expressed as a power

series in 1/N2

Orientifold planar equivalence is the statement F(N =∞) = M(N =∞)

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Large N

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Reducedmodels






mAdjρ (N) = F

(1

N2

);

mSρ(N) = M

(1

N2

)+

1Nµ

(1

N2

);

mASρ (N) = M

(1

N2

)−

1Nµ

(1

N2

).

M =(mSρ + mAS

ρ

)/2 and µ = N

(mSρ − mAS

ρ


series in 1/N2


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Large N

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Reducedmodels






mAdjρ (N) = F

(1

N2

);

mSρ(N) = M

(1

N2

)+

1Nµ

(1

N2

);

mASρ (N) = M

(1

N2

)−

1Nµ

(1

N2

).

M =(mSρ + mAS

ρ

)/2 and µ = N

(mSρ − mAS

ρ


series in 1/N2


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Large N

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Reducedmodels



0 0.05 0.1 0.15 0.2 0.25

1/N2

0

0.2

0.4

0.6

0.8

1

mρ

AdjSAMµ

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Large N

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Reducedmodels



0 0.05 0.1 0.15 0.2 0.25

1/N2

0

0.2

0.4

0.6

0.8

1

mρ

AdjMµ

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Large N

B. Lucini



Reducedmodels


Large-N fits

-0.05 0 0.05 0.1 0.15 0.2 0.25

1/N2

0

0.2

0.4

0.6

0.8

1

mρ

AdjMµ

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Large N

B. Lucini



Reducedmodels


Large-N fits

-0.05 0 0.05 0.1 0.15 0.2 0.25

1/N2

0

0.2

0.4

0.6

0.8

1

mρ

AdjSA

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Large N

B. Lucini



Reducedmodels


Fit results

mAdjρ = 0.6819(51)−

0.202(67)

N2;

mSρ = 0.701(25) +

0.28(12)

N−

0.85(24)

N2+

1.4(1.0)

N3;

mASρ = 0.701(25)−

0.28(12)

N−

0.85(24)

N2−

1.4(1.0)

N3.

Orientifold planar equivalence verified within 3.5%

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Large N

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Reducedmodels


Fit results

mAdjρ = 0.6819(51)−

0.202(67)

N2;

mSρ = 0.701(25) +

0.28(12)

N−

0.85(24)

N2+

1.4(1.0)

N3;

mASρ = 0.701(25)−

0.28(12)

N−

0.85(24)

N2−

1.4(1.0)

N3.

Orientifold planar equivalence verified within 3.5%

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Large N

B. Lucini



Reducedmodels


Towards two-index fermion unquenchedsimulations

In the context of strongly interacting dynamics beyond the Standard Model,unquenched lattice studies exist for

SU(2) with Nf = 1, 2 adjoint (symmetric) Dirac fermions

SU(3) with Nf = 2 symmetric Dirac fermions

SU(3) with Nf = 2 adjoint Dirac fermions

SU(4) with Nf = 6 antisymmetric Dirac fermions

. . .

The lesson we learn is that at small N the equivalence can have strong deviations

due to the dependence on N (and on the representation) of the lower edge of the

conformal window.

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Large N

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Reducedmodels


Outline



3 Reduced models


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Large N

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Reducedmodels


Large-N reduction

At N =∞ the SU(N) gauge theory can be formulated on a single latticepoint (Eguchi-Kawai model)

Non-perturbatively, centre symmetry needs to be preserved

Earlier proposals (Bhanot, Heller and Neuberger; González-Arroyo andOkawa) disproved by recent simulations (Bringoltz and Sharpe; Teper andVairinhos)

New proposal by González-Arroyo and Okawa currently being tested, withencouraging results

Another possibility is partial volume reduction (Narayanan and Neuberger)

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Large N

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Reducedmodels


Test of TEK (González-Arroyo and Okawa)

The expected large-N result seems to be obtained

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Large N

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Reducedmodels


Centre stabilised Yang-Mills theories(Unsal and Yaffe)

The centre can be stabilised by adding a deformation to the action thatcounteracts the centre-breaking effective Polyakov loop potential

SW → SW +1L3

[N/2]∑n=1

an |Tr (Pn)|2

for appropriate values of the an

The deformed theory (which is large-N equivalent to the original Yang-Millstheory) can then be shrunk to a single point

Preliminary lattice studies in progress (Vairinhos)

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Large N

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Reducedmodels


Reduction with adjoint fermions

The action with adjoint fermions is centre-symmetric

Adjoint fermions with periodic boundary conditions helps in stabilising thecentre symmetry (Kovtun, Unsal and Yaffe)

This can be used to to establish a chain of large-N equivalences from QCDto a reduced adjoint model

Non-perturbative lattice tests exist, but their are inconclusive

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Large N

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Reducedmodels


Test of adjoint reduction (Bringoltz and Sharpe)

A region of preserved centre symmetry seems to exist in the continuum limit

However, a study by Hietanen and Narayanan reaches the opposite conclusion

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Large N

B. Lucini



Reducedmodels


Phases with a compactified direction

The phase structure is non-trivial at finite mass (Cossu and D’Elia)

(Also confirmed analytically by Myers and Ogilvie and Unsal and Yaffe)

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Large N

B. Lucini



Reducedmodels


Outline



3 Reduced models


B. Lucini Large N

Large N

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Reducedmodels


Where from here?

1 The large-N limit à la ’t Hooft is a mature field of investigation, but moreshould be done:

Better control over continuum limitUnquenching effectsScattering and decaysVeneziano limit?

2 Orientifold/Orbifold equivalences are an almost unexplored territoryUnquenched simulationsLink with supersymmetriesLessons from lattice studies of near-conformal gauge theories?

3 There have been several studies of reduced models (with partial or totalreduction)

Important questions about viability remainNeed of efficient algorithms

B. Lucini Large N

Science

Large N: lattice results and perspectives