Studies of large-N reduction with adjoint fermions
Barak Bringoltz
University of Washington
“Non-perturbative volume-reduction of large-N QCD with adjoint fermions”
BB and S.R.Sharpe, 0906.3538
“Large-N volume reduction of lattice QCD with adjoint Wilson fermions at weak coupling”
BB, JHEP, 0905.2406
Main motivation
Study a large-N limit of QCD where quarks are back-reacting on gauge fields
Natural to put quarks in two-antisymmetric :
QCD(AS)
Corrigan-Ramond,
Armoni-Shifman-Veneziano,
Sannino et al.
Armoni-Veneziano-Shifman:
Orientifold
QCD(AS)
infinite volume
QCD(Adj)
infinite volume
Kovtun-Unsal-Yaffe:
QCD(Adj)
infinite volume
Orbifold
QCD(Adj)
“zero” volume
“Eguchi-Kawai”
volume reduction
works if have light
adjoint quarks around
Study QCD(Adj) on a single-site gives physics of QCD(AS) at infinite volume
More motivations
QCD + adjoints is an interesting theory:
•
Nf=1/2 is softly broken N=1 SUSY
•
•
Nf=2 is in (or close by to) the conformal window.
Can study large-N limit
of all these with method I
will describe.
Any value of Nf=2 and heavy enough quarks is YM.
Does it really provide a workable Eguchi-Kawai ? (finally, after 27 years...)
•
•
Original Eguchi-Kawai.
Jan `82.
Quenched Eguchi-Kawai.
Bhanot
Heller , Feb
Neuberger
•
Twisted Eguchi-Kawai.
•
Adjoint Eguchi-Kawai.
•
Deformed Eguchi-Kawai.
But Bhanot-Heller-Neuberger Feb `82.
`82
Gonzalez-Arroyo ,
Okawa
Kovtun- ,
UnsalYaffe
UnsalYaffe
July `82
2007
, 2008
But BB-Sharpe, 2008.
But,
Teper-Vairinhos,
Hanada-et-al, 2006.
Bietenholtz et al.
This talk.
No “But”’s ?
BB, Vairinhos, 2008
I.
What is the Eguchi-Kawai equivalence.
Given SU(N) gauge theory on an L1xL2xL3xL4 lattice, defined by g 2 N, am, aµ, . . .
Then if:
•
Translation symmetry is intact.
•
ZN center symmetry is intact.
•
large-N factorization holds.
at N=oo
Wilson loops, Hadron spectra,
condensates, etc.
are independent of L1,2,3,4.
I.
What is the Eguchi-Kawai equivalence.
Given SU(N) gauge theory on an L1xL2xL3xL4 lattice, defined by g 2 N, am, aµ, . . .
Then if:
•
Translation symmetry is intact.
•
ZN center symmetry is intact.
•
large-N factorization holds.
at N=oo
Wilson loops, Hadron spectra,
condensates, etc.
are independent of L1,2,3,4.
This talk: fate of ZN. First at weak coupling, then non-perturbatively
I.
What is the Eguchi-Kawai equivalence.
Given SU(N) gauge theory on an L1xL2xL3xL4 lattice, defined by g 2 N, am, aµ, . . .
Not “academic”
requirements:
Then if:
•
Translation symmetry is intact.
breakdown of EK equivalence by
formation of a baryon crystal
BB, PRD,0811.4141, 0901.4035
•
ZN center symmetry is intact.
•
large-N factorization holds.
at N=oo
QEK model
BB+Sharpe 2008
Wilson loops, Hadron spectra,
condensates, etc.
are independent of L1,2,3,4.
This talk: fate of ZN. First at weak coupling, then non-perturbatively
II.
Weak coupling analysis: L2,3,4 = oo, L1=1.
Fields:
BB, JHEP, 0905.2406
Uµ=0 (!x, x0 = 1) ≡ Ω!x
Uµ=1,2,3 (!x, x0 = 1) ≡ Uµ=1,2,3 (!x)
ψ("x, x0 = 1) ≡ ψ!x
Action:
Gauge: Wilson, b = 1/g 2 N
1
Fermions: Wilson, κ =
8 + 2am0
Weak coupling expansion:
g N →0:
2
ab
Ω!x
=δ
kappa=1/8 : chiral quarks
kappa=0 : infinitely massive quarks
ab iθ a
e
+ δΩ ("x) + . . .
ab
U!x,µ = 1 + iAµ (!x) + . . .
II.
Weak coupling analysis: L2,3,4 = oo, L1=1.
BB, JHEP, 0905.2406
Get familiar form
# a
$'
(
+,
b
! " # dp $3 % &
)
*
θ
−
θ
log p̂2 + 4 sin2
− 2Nf log p̂ˆ2 + sin2 θa − θb + m2W (θ, p)
V (θ) =
2π
2
a!=b
p̂ = 4
2
3
!
i=1
a→0
sin pi /2 −→ p!
2
2
p̂ˆ2 =
3
!
i=1
Calculate V (θ) potential for different
Chiral
quarks
κ = 1/8
κ=0
YM
model
(original EK)
a→0
sin pi −→ p!
2
2
mW
! 2
"
2
a
b
= am0 + 2 sin (p/2) + sin ((θ − θ )/2)
θa corresponding to ZN , ZN → Ø , ZN → Z2 .
These are
good news:
Reduction
works !
II.
Weak coupling analysis: L2,3,4 = oo, L1=1.
BB, JHEP, 0905.2406
Aside : Comparing some weak coupling calculations.
Kovtun et al. ‘07
4D YM +
adjoints continuum
ZN
Lattice calculation
Bedaque et al. ‘08
4D YM+Wilson
adjoints and L1=1
Think about L1=1 as
3D theory defined
on spatial continuum
ZN
ZN → Z2
Useful to understand the reason for the difference
II.
Weak coupling analysis: L2,3,4 = oo, L1=1.
BB, JHEP, 0905.2406
Bedaque et al. (focus on gauge dynamics first)
Sgauge
"
!
2N
†
†
=
Re
Tr Ux,i Ux+i,j Ux+j,i
Ux,j
λ
x
i<j
#
#
! "
2N
†
†
+
Re
Tr Ux,i Ωx+i Ux,i Ωx
λ
x
i
S
“as → 0”
one−site
=
!

1
d x  2 Tr
g
3
i<j∈[1,3]
Di Ω(x) = ∂i Ω(x) + i[Ai , Ω]
Ω ∈ SU (N )
$
Fij2
+ f Tr
2
$
i

2
|Di Ω| 
II.
Weak coupling analysis: L2,3,4 = oo, L1=1.
BB, JHEP, 0905.2406
Bedaque et al. (focus on gauge dynamics first)
Sgauge
"
!
2N
†
†
=
Re
Tr Ux,i Ux+i,j Ux+j,i
Ux,j
λ
x
i<j
#
#
! "
2N
†
†
+
Re
Tr Ux,i Ωx+i Ux,i Ωx
λ
x
i
S
“as → 0”
one−site
=
!

1
d x  2 Tr
g
3
$
i<j∈[1,3]
Di Ω(x) = ∂i Ω(x) + i[Ai , Ω]
Fij2
+ f Tr
2
$
i
2
|Di Ω| 
Ω ∈ SU (N )
%
# a
$&
%
# a
$&
" # $3
b
b
!
! " # dp $3
θ −θ
dp
1
θ −θ
2
2
2 2
3
log at p + sin
=Λ +
log 1 + 2 2 sin
V (θ) =
2π
2
2π
at p
2
a!=b
a!=b

II.
Weak coupling analysis: L2,3,4 = oo, L1=1.
BB, JHEP, 0905.2406
Bedaque et al. (focus on gauge dynamics first)
Sgauge
"
!
2N
†
†
=
Re
Tr Ux,i Ux+i,j Ux+j,i
Ux,j
λ
x
i<j
#
S
“as → 0”
one−site
=
!

1
d x  2 Tr
g
3
$
Fij2
+ f Tr
2
$
i
i<j∈[1,3]
Di Ω(x) = ∂i Ω(x) + i[Ai , Ω]
#
! "
2N
†
†
+
Re
Tr Ux,i Ωx+i Ux,i Ωx
λ
x
2
|Di Ω| 
Ω ∈ SU (N )
i
%
# a
$&
%
# a
$&
" # $3
b
b
!
! " # dp $3
θ −θ
dp
1
θ −θ
2
2
2 2
3
log at p + sin
=Λ +
log 1 + 2 2 sin
V (θ) =
2π
2
2π
at p
2
a!=b
a!=b
=Λ +Λ
3
!
a!=b
sin
2
"
a
θ −θ
2
2
= Λ + Λ |tr Ωclassical | +
3
S one−site
is non-renormalizable ...
need counter-terms...
b
#
+ ...
. . . , Ωab
classical

iθ a
=e
Bernard&Appelquist `80, Longhitano `80,
Banks&Ukawa `84, Gasser-Leutwyler ‘84,
Arkani-Hamed-Cohen-Georgi `01, Pisarski `06,
II.
Weak coupling analysis: L2,3,4 = oo, L1=1.
BB, JHEP, 0905.2406
What does this teach us?
•
Continuum limit in space with L1=1 (or for any D > 2L1 ).
•
need counter-terms
•
means treating theory as an Effective Field Theory (EFT), and at one-loop:
new Low Energy Constants (LEC).
!
!2
2
!
V (θ) −→ V (θ) + b1 |tr Ωclassical | + b2 tr Ωclassical !
2
Dim-reg hides this and implicitly sets b1=b2=0
Amusing: these are
(Mithat and Larry)’s terms
b1,2 > 0 fixes ZN
Z2 breaking.
II.
Weak coupling analysis: L2,3,4 = oo, L1=1.
BB, JHEP, 0905.2406
What does this teach us?
•
Continuum limit in space with L1=1 (or for any D > 2L1 ).
•
need counter-terms
•
means treating theory as an Effective Field Theory (EFT), and at one-loop:
new Low Energy Constants (LEC).
!
!2
2
!
V (θ) −→ V (θ) + b1 |tr Ωclassical | + b2 tr Ωclassical !
2
Dim-reg hides this and implicitly sets b1=b2=0
Amusing: these are
(Mithat and Larry)’s terms
b1,2 > 0 fixes ZN
EFT point of view is not necessary
In any case, large-N reduction defined at fix lattice cutoff:
But ZN-realization may depend on lattice action ...
Results I got cannot be anticipated in advance (from Kovtun et al.)
Z2 breaking.
In any case, we saw that :
Wilson fermions at weak coupling obey reduction
But really need a non-perturbative lattice study
•
Really interested in L1,2,3,4=1, but IR div’s.
•
What happens at
•
Non-perturbative effect (e.g. QEK and TEK model).
g2 N ! 1 − 3 ?
In any case, we saw that :
Wilson fermions at weak coupling obey reduction
But really need a non-perturbative lattice study
•
Really interested in L1,2,3,4=1, but IR div’s.
•
What happens at
•
Non-perturbative effect (e.g. QEK and TEK model).
g2 N ! 1 − 3 ?
•
Wilson gauge + Nf=1 Wilson fermions, Metropolis.
•
SU(N) with N=8,10,11,13,15.
•
A variety of of spacings and masses.
BB+S.Sharpe, 0906.3538
Goal : Map single-site theory in κ and g 2 N
Look for intact ZN.
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
What should we expect? L1,2,3,4=oo :
quarks are
light along
line
Continuum
physics
(1/g 2 N =)
strong-to-weak
lattice transition
strong-coupling/
lattice physics
YM
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
What should we expect? L1,2,3,4=1:
~0.04
Continuum
physics
(1/g 2 N =)
strong-to-weak
lattice transition
strong-coupling/
lattice physics
YM
quarks are
light along
line
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
What should we expect? L1,2,3,4=1:
~0.04
Continuum
physics
(1/g 2 N =)
strong-to-weak
lattice transition
1
strong-coupling/
lattice physics
YM
quarks are
light along
line
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
What should we expect? L1,2,3,4=1:
~0.04
Continuum
physics
(1/g 2 N =)
strong-to-weak
lattice transition
strong-coupling/
lattice physics
YM
quarks are
light along
line
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
What should we expect? L1,2,3,4=1:
quarks are
light along
line
~0.04
Continuum
physics
(1/g 2 N =)
strong-to-weak
lattice transition
2
strong-coupling/
lattice physics
YM
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
What should we expect? L1,2,3,4=1:
~0.04
Continuum
physics
(1/g 2 N =)
strong-to-weak
lattice transition
strong-coupling/
lattice physics
YM
quarks are
light along
line
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
What should we expect? L1,2,3,4=1:
~0.04
Continuum
physics
3
(1/g 2 N =)
strong-to-weak
lattice transition
strong-coupling/
lattice physics
YM
quarks are
light along
line
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
What should we expect? L1,2,3,4=1:
~0.04
Continuum
physics
(1/g 2 N =)
strong-to-weak
lattice transition
strong-coupling/
lattice physics
YM
quarks are
light along
line
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
Scan no. 1 : infinitely massive quarks
X-axis: Real(Polyakov)
Y-axis: Imag(Polyakov)
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
Scan no. 1 : infinitely massive quarks
X-axis: Real(Polyakov)
Y-axis: Imag(Polyakov)
b=0
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
Scan no. 1 : infinitely massive quarks
X-axis: Real(Polyakov)
Y-axis: Imag(Polyakov)
b=0
b ≈ 0.3
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
Scan no. 1 : infinitely massive quarks
X-axis: Real(Polyakov)
Y-axis: Imag(Polyakov)
b=0
b ≈ 0.3
b = 0.5
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
Scan no. 2 : decreasing the quark mass b=0.5 ,SU(10)
κ≈0
κ = 0.12
κ = 0.03
κ = 0.06
κ = 0.1475
κ = 0.16
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
Scan no. 2 : decreasing the quark mass b=0.5 ,SU(10)
κ≈0
κ = 0.12
κ = 0.03
κ = 0.06
κ = 0.1475
κ = 0.16
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
Scan no. 2 : decreasing the quark mass b=0.5 ,SU(10)
κ≈0
κ = 0.12
κ = 0.03
κ = 0.06
κ = 0.1475
κ = 0.16
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
Scan no. 2 : decreasing the quark mass b=0.5 ,SU(10)
κ≈0
κ = 0.12
κ = 0.03
κ = 0.06
κ = 0.1475
κ = 0.16
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
Scan no. 2 : decreasing the quark mass b=0.5 ,SU(10)
κ≈0
κ = 0.12
κ = 0.03
κ = 0.06
κ = 0.1475
κ = 0.16
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
Scan no. 2 : decreasing the quark mass b=0.5 ,SU(10)
κ≈0
κ = 0.12
κ = 0.03
κ = 0.06
κ = 0.1475
κ = 0.16
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
Scan no. 2 : decreasing the quark mass b=0.5 ,SU(10)
,SU(15)
κ≈0
κ = 0.03
κ=0
κ = 0.06
κ = 0.1275
κ = 0.1475
κ = 0.12
κ = 0.1475
κ = 0.06
κ = 0.09
κ = 0.155
κ = 0.16
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
Scan no. 2 : looking for the “critical” line
b=0.35:
1st transition
at kappa~0.15
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
Scan no. 2 : looking for the “critical” line
0.08
increasing kappa
decreasing kappa
0.06
0.04
Im(Polyakov)
0.02
0
-0.02
-0.04
-0.06
-0.08
0.08
0.1
0.12
0.14
kappa
0.16
0.06
0.18
0.2
increasing kappa
decreasing kappa
0.04
Re(Polyakov)
0.02
0
-0.02
-0.04
-0.06
-0.08
0.08
0.1
0.12
0.14
kappa
0.16
0.18
0.2
b=0.35:
1st transition
at kappa~0.15
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
Scan no. 2 : looking for the “critical” line
0.125
0.08
increasing kappa
decreasing kappa
0.06
0.04
Im(Polyakov)
0.02
0
-0.02
-0.04
-0.06
-0.08
0.08
0.1
0.12
0.14
kappa
0.16
0.06
0.18
0.2
increasing kappa
decreasing kappa
0.04
Re(Polyakov)
0.02
0
-0.02
-0.04
-0.06
-0.08
0.08
0.1
0.12
0.25
0.14
kappa
0.16
0.18
0.2
1stb=0.35:
transition
structure
at all b.
1st transition
Extending
from
at
kappa~0.15
kappa=0.25 to 0.125
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
All results consistent with phase diagram
and validity of reduction.
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
All results consistent with phase diagram
and validity of reduction.
But:
•
See long autocorrelation times there at very very weak coupling.
•
More order parameters for nontrivial breaking of ZN.
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
All results consistent with phase diagram
and validity of reduction.
But:
•
See long autocorrelation times there at very very weak coupling.
•
More order parameters for nontrivial breaking of ZN.
Perform long runs and measure order parameters of the form
tr [P1n1 P2n2 P3n3 P4n4 ] with ni ∈ [−5, 5]
14641 order parameters !
III. Results of non-perturbative MC lattice simulations
BB+S.Sharpe, 0906.3538
(1/g 2 N =)
All results consistent with phase diagram
and validity of reduction.
But:
•
See long autocorrelation times there at very very weak coupling.
•
More order parameters for nontrivial breaking of ZN.
Perform long runs and measure order parameters of the form
tr [P1n1 P2n2 P3n3 P4n4 ] with ni ∈ [−5, 5]
14641 order parameters !
Still no sign of breakdown of ZN, up to b=0.5-1.0
IV. Conclusions and future prospects.
Weak coupling with L2,3,4=oo, L1=1
•
Large-N volume reduction works for YM+Wilson adjoints fermions.
Non-perturbative lattice Monte-Carlo of Nf=1 case.
•
Large-N volume reduction works for YM+Wilson adjoints fermions.
This is exciting :
Eguchi-Kawai finally works
start extracting physics of QCD(Adj) and QCD(AS)
•
•
•
•
•
Mesons, baryons (see Carlos Q. Hoyos’s talk).
Realization of chiral symmetry (of both adjoint sea-quarks and valence fundamental).
Comparisons with RMT.
Static potential, string tensions.
Open to suggestions ...
A new UNEXPLORED physically relevant and rich theory
IV. Conclusions and future prospects.
Weak coupling with L2,3,4=oo, L1=1
•
Large-N volume reduction works for YM+Wilson adjoints fermions.
Non-perturbative lattice Monte-Carlo of Nf=1 case.
•
Large-N volume reduction works for YM+Wilson adjoints fermions.
This is exciting :
Eguchi-Kawai finally works
start extracting physics of QCD(Adj) and QCD(AS)
•
•
•
•
•
Mesons, baryons (see Carlos Q. Hoyos’s talk).
Realization of chiral symmetry (of both adjoint sea-quarks and valence fundamental).
Comparisons with RMT.
Static potential, string tensions.
Open to suggestions ...
A new UNEXPLORED physically relevant and rich theory
pretty cool...