## New perspectives on quark confinement or,

## QCD in a periodic box

Philip Argyres^{1} Mithat ¨Unsal^{2}

1Physics Dept., Univ. of Cincinnati, Cincinnati OH 45221-0011, USA

2Dept. of Physics and Astronomy, SFSU, San Francisco CA 94132, USA

Will discuss some new results from “Dynamics of gauge theories and new instanton, bion and renormalon effects”,arXiv:1204.xxxx ... but will mostly reviewM. ¨Unsal and collaborators’work.

## The quark confinement problem

Some QCD phenomenology...

QCD looks alot like electromagnetism at short distances:

QED QCD

charge “color”

photons gluons electrons quarks

But at long distances we don’t see quarks, only uncharged (color-neutral) particles.

## Quark potential model

This is easy to model: just add a linear potential on top of a 1/r Coulomb potential:

• Λ is the energy scale defined by linear potential.

• F =−Λ^{2} indepdent of r for r Λ^{−1}.

• Like a (relativistic) string with tension = Λ^{2}.

## Strings

• This string picture matches experiment.

• Quantize string rotational and vibrational modes

• Find, e.g. from WKB approximation,

M^{2}= Λ^{2}(J + n + 1)

• get Regge trajectories (plot from Shifman, Vainshtein 2008).

Figure 1:The plot shows M^{2}of various meson resonances which are believed to be built of

¯qq where q = u or d. The resonances at levels 2, 3 and some resonances at 4 level GeV^{2}are
taken from the Particle Data Group (PDG) compilation. Most of those at level 4 and all
resonances at level 5 GeV^{2}are taken from the compilation of resonances in p¯p annihilation
prepared by Glozman [2], see also [3]. In selecting the ¯qq resonances we followed Kaidalov’s
work [4] in discarding presumed four-quark states, gluonia or resonances built of ¯ss.

generators adds soft pions instead. Of course, we know for sure from the low-lying states that the chiral symmetry is realized in the Nambu–Goldstone mode in the hadron world.

The question is whether the linear realization of U(Nf)L×U(Nf)Rcould be re- stored for highly excited states yielding a part of degeneracy visible in Fig. 1. In this case one could speak of the asymptotic chiral symmetry restoration (χSR). On the other hand, if in higher excitations the Nambu–Goldstone mode persists, other dynamical reasons must be responsible for the spectral degeneracy. The issue of possible restoration of the full U(Nf)L×U(Nf)Rchiral symmetry attracted much at- tention lately mainly in connection with the inspiring works of Glozman and collab- orators [5–10]. The history of the topic of parity doubling on the Regge trajectories is presented in the review papers [11,12].

3

## The meaning of confinement: strings

• Confinement does notmean that we have an unbreakable string.

• If stretched far enough, it will store enough energy to pair-create quarks.

• If the quarks are too light, we will never see string-like confinement.

## The meaning of confinement: strings

• Confinement does notmean that we have an unbreakable string.

• If stretched far enough, it will store enough energy to pair-create quarks.

• If the quarks are too light, we will never see string-like confinement.

## The meaning of confinement: strings

• Confinement does notmean that we have an unbreakable string.

• If stretched far enough, it will store enough energy to pair-create quarks.

• If the quarks are too light, we will never see string-like confinement.

## The meaning of confinement: strings

• Confinement does notmean that we have an unbreakable string.

• If stretched far enough, it will store enough energy to pair-create quarks.

• If the quarks are too light, we will never see string-like confinement.

## The meaning of confinement: strings

• Confinement does notmean that we have an unbreakable string.

• If stretched far enough, it will store enough energy to pair-create quarks.

• If the quarks are too light, we will never see string-like confinement.

## The meaning of confinement: strings

• Confinement does notmean that we have an unbreakable string.

• If stretched far enough, it will store enough energy to pair-create quarks.

• If the quarks are too light, we will never see string-like confinement.

## The meaning of confinement: glueballs

• A clearer diagnostic is the existence of a mass gap.

• If the gluons were massless, they would mediate long-range
1/r^{2} forces, and even heavy charges could be separated.

• In the string picture this massiveness of the gluons is, vaguely, due to the string eating its tail

## The meaning of confinement: center symmetry

• A mass gap does not, by itself, imply string-like confinement of heavy charges, though. E.g., theHiggs mechanism gives a mass gap.

• A way of ensuring that charge confinement does occurs is if there is a global symmetry which under whichquarks(heavy charges) arechargedand the gluons and other light fields are neutral.

• The gluons then can not totally screen the quarks, and their flux must go off somewhere — the string. (‘t Hooft 1978-9)

• This global symmetry exists in certain gauge theories, and is called thecenter symmetry. E.g., for SU(N) it isZN.

• So unbroken center symmetry and a mass gap are sufficient to imply confinement (but center symmetry is not a necessary condition).

## The meaning of confinement: center symmetry

• A mass gap does not, by itself, imply string-like confinement of heavy charges, though. E.g., the Higgs mechanism gives a mass gap.

• A way of ensuring that charge confinement does occurs is if there is a global symmetry which under whichquarks(heavy charges) arechargedand the gluons and other light fields are neutral.

• The gluons then can not totally screen the quarks, and their flux must go off somewhere — the string. (‘t Hooft 1978-9)

• This global symmetry exists in certain gauge theories, and is called thecenter symmetry. E.g., for SU(N) it isZN.

• So unbroken center symmetry and a mass gap are sufficient to imply confinement (but center symmetry is not a necessary condition).

## The meaning of confinement: center symmetry

• A mass gap does not, by itself, imply string-like confinement of heavy charges, though. E.g., the Higgs mechanism gives a mass gap.

• A way of ensuring that charge confinement does occurs is if there is a global symmetry which under whichquarks(heavy charges) arechargedand the gluons and other light fields are neutral.

• The gluons then can not totally screen the quarks, and their flux must go off somewhere — the string. (‘t Hooft 1978-9)

• This global symmetry exists in certain gauge theories, and is called thecenter symmetry. E.g., for SU(N) it isZN.

• So unbroken center symmetry and a mass gap are sufficient to imply confinement (but center symmetry is not a necessary condition).

## The meaning of confinement: center symmetry

• A mass gap does not, by itself, imply string-like confinement of heavy charges, though. E.g., the Higgs mechanism gives a mass gap.

## The meaning of confinement: center symmetry

## The meaning of confinement: center symmetry

## The meaning of confinement: center symmetry

## Field theory models

The picture above was mostly phenomenology.

How is it derived from the known microscopic QCD field theory of gluons and quarks?

Gluon perturbation theory: find that the effective gauge coupling constant becomes large at some energy scale Λ

Nicely compatible with existence of a mass gap Λ (or string tension
Λ^{2}) but makes it astrong coupling problemwhere perturbation
theory no longer applies.

## Field theory models

The picture above was mostly phenomenology.

How is it derived from the known microscopic QCD field theory of gluons and quarks?

Gluon perturbation theory: find that the effective gauge coupling constant becomes large at some energy scale Λ

Nicely compatible with existence of a mass gap Λ (or string tension
Λ^{2})but makes it a strong coupling problemwhere perturbation
theory no longer applies.

## Field theory models

The picture above was mostly phenomenology.

How is it derived from the known microscopic QCD field theory of gluons and quarks?

Gluon perturbation theory: find that the effective gauge coupling constant becomes large at some energy scale Λ

Nicely compatible with existence of a mass gap Λ (or string tension
Λ^{2})but makes it a strong coupling problemwhere perturbation
theory no longer applies.

## Field theory models

The picture above was mostly phenomenology.

How is it derived from the known microscopic QCD field theory of gluons and quarks?

Nicely compatible with existence of a mass gap Λ (or string tension
Λ^{2})but makes it a strong coupling problemwhere perturbation
theory no longer applies.

## Field theory models

The picture above was mostly phenomenology.

How is it derived from the known microscopic QCD field theory of gluons and quarks?

^{2})but makes it a strong coupling problemwhere perturbation
theory no longer applies.

## Field theory models

The picture above was mostly phenomenology.

How is it derived from the known microscopic QCD field theory of gluons and quarks?

Nicely compatible with existence of a mass gap Λ (or string tension
Λ^{2}) but makes it astrong coupling problemwhere perturbation
theory no longer applies.

## Field theory models

The picture above was mostly phenomenology.

How is it derived from the known microscopic QCD field theory of gluons and quarks?

Flux lines: can try to use the (color-)electric flux lines as the basic variables. Would like to see if they are attracted to form string-like flux tubes:

These areWilson lines, and we only know how to compute with them on

the lattice. Successful, but numerical.

## Field theory models

The picture above was mostly phenomenology.

How is it derived from the known microscopic QCD field theory of gluons and quarks?

Flux lines: can try to use the (color-)electric flux lines as the basic variables. Would like to see if they are attracted to form string-like flux tubes:

These areWilson lines, and we only know how to compute with them on

the lattice. Successful, but numerical.

## Field theory models

The picture above was mostly phenomenology.

How is it derived from the known microscopic QCD field theory of gluons and quarks?

Flux lines: can try to use the (color-)electric flux lines as the basic variables. Would like to see if they are attracted to form string-like flux tubes:

These areWilson lines, and we only know how to compute with them on

the lattice. Successful, but numerical.

## Field theory models

The picture above was mostly phenomenology.

How is it derived from the known microscopic QCD field theory of gluons and quarks?

These areWilson lines, and we only know how to compute with them on

the lattice. Successful, but numerical.

## Field theory models

The picture above was mostly phenomenology.

How is it derived from the known microscopic QCD field theory of gluons and quarks?

These areWilson lines, and we only know how to compute with them on

the lattice. Successful, but numerical.

## Field theory models

The picture above was mostly phenomenology.

How is it derived from the known microscopic QCD field theory of gluons and quarks?

These areWilson lines, and we only know how to compute with them on

the lattice. Successful, but numerical.

## Field theory models

The picture above was mostly phenomenology.

How is it derived from the known microscopic QCD field theory of gluons and quarks?

These areWilson lines, and we only know how to compute with them on

the lattice. Successful, but numerical.

## Field theory models

The picture above was mostly phenomenology.

How is it derived from the known microscopic QCD field theory of gluons and quarks?

These areWilson lines, and we only know how to compute with them on

the lattice. Successful, but numerical.

## Dual superconductor

• The flux tube picture is reminiscent of superconductors where magnetic flux is confined to tubes and electric charges are screened.

• So (Mandelstam/Polyakov/‘t Hooft 1970s) proposed that QCD is a dual superconductorin which (color-)electric flux is confined to tubes and (color-)magnetic charges are screened.

• So we would need a magnetically charged scalar field to condense, analogous to superconductivity where Cooper pairs condense.

• But in 4d gauge theory magnetic monopoles

(1) need low energy U(1) gauge groups, which at weak coupling come from “adjoint” Higgs condensation hϕi 6= 0.

(2) are very heavy at weak coupling m_{M} ∼ hϕi/g.

• So idea looks hopelessly strongly-coupled.

## Dual superconductor

• The flux tube picture is reminiscent of superconductors where magnetic flux is confined to tubes and electric charges are screened.

• So (Mandelstam/Polyakov/‘t Hooft 1970s) proposed that QCD is a dual superconductorin which (color-)electric flux is confined to tubes and (color-)magnetic charges are screened.

• So we would need a magnetically charged scalar field to condense, analogous to superconductivity where Cooper pairs condense.

• But in 4d gauge theory magnetic monopoles

(1) need low energy U(1) gauge groups, which at weak coupling come from “adjoint” Higgs condensation hϕi 6= 0.

(2) are very heavy at weak coupling m_{M} ∼ hϕi/g.

• So idea looks hopelessly strongly-coupled.

## Dual superconductor

• The flux tube picture is reminiscent of superconductors where magnetic flux is confined to tubes and electric charges are screened.

• So (Mandelstam/Polyakov/‘t Hooft 1970s) proposed that QCD is a dual superconductorin which (color-)electric flux is confined to tubes and (color-)magnetic charges are screened.

• So we would need a magnetically charged scalar field to condense, analogous to superconductivity where Cooper pairs condense.

• But in 4d gauge theory magnetic monopoles

(1) need low energy U(1) gauge groups, which at weak coupling come from “adjoint” Higgs condensation hϕi 6= 0.

(2) are very heavy at weak coupling m_{M} ∼ hϕi/g.

• So idea looks hopelessly strongly-coupled.

## Dual superconductor

• But in 4d gauge theory magnetic monopoles

(2) are very heavy at weak coupling m_{M} ∼ hϕi/g.

• So idea looks hopelessly strongly-coupled.

## Dual superconductor

• But in 4d gauge theory magnetic monopoles

(2) are very heavy at weak coupling m_{M} ∼ hϕi/g.

• So idea looks hopelessly strongly-coupled.

## Instantons

• Another approach is to calculate semi-classicall (at weak
coupling) non-perturbative quantities∝ e^{−c}^{/g}^{2}.

• Though very small, they may give qualitatively new effects.

• These are tunnelling events, or instantons, analogous to tunnelling in QM.

• The amplitudes for these “events” in 4d space-time are calculated by a saddlepoint approximation.

## Instantons

• The saddlepoint configurations are Euclidean with fugacity
e^{−S} = e^{−8}^{π}^{2}^{/g}^{2} which controls the probability of these events
occuring per unit space-time volume.

• So the vacuum can be thought of as a Euclidean “gas” of
instantons with space-time density∼ e^{−8}^{π}^{2}^{/g}^{2}Λ^{4}.

• But this idea does notwork: the instanton gas free energy diverges uncontrollably and does not screen magnetic charge.

• A clue?: Euclidean saddlepoints control the rate of divergence of perturbation theory sums (Lipatov 1976), but actual QCD peturbation theory diverges much faster than the instanton saddlepoint would imply.

• These “IR renormalon divergences” would require saddlepoint configurations with actions∼ (1/N) × (instanton action), but no such saddlepoints are known!

• (‘t Hooft 1982) speculated that the elusive ”renormalon”

saddlepoint configuration is responsible for confinment.

## Changing the problem

• QCD is too hard so try to get “perspective” by changing the problem.

• There are many ways to do this:

change the matter content, ⇒ adjoint fermions change the gauge group, ⇒ any simple G, large N change the geometry, ⇒ periodic spatial circle add supersymmetry.

• We will use them all except supersymmetry...

## Changing the problem

• QCD is too hard so try to get “perspective” by changing the problem.

• There are many ways to do this:

change the matter content, ⇒ adjoint fermions change the gauge group, ⇒ any simple G, large N change the geometry, ⇒ periodic spatial circle add supersymmetry.

• We will use them all except supersymmetry...

## Changing the problem

• QCD is too hard so try to get “perspective” by changing the problem.

• There are many ways to do this:

change the matter content, ⇒ adjoint fermions change the gauge group, ⇒ any simple G, large N change the geometry, ⇒ periodic spatial circle add supersymmetry.

• We will use them all except supersymmetry...

## Changing the problem

• QCD is too hard so try to get “perspective” by changing the problem.

• There are many ways to do this:

• We will use them all except supersymmetry...

## Changing the problem

• QCD is too hard so try to get “perspective” by changing the problem.

• There are many ways to do this:

• We will use them all except supersymmetry...

## Changing the problem

• QCD is too hard so try to get “perspective” by changing the problem.

• There are many ways to do this:

• We will use them all except supersymmetry...

## Changing the problem

• QCD is too hard so try to get “perspective” by changing the problem.

• There are many ways to do this:

• We will use them all except supersymmetry...

## Changing the problem

• QCD is too hard so try to get “perspective” by changing the problem.

• There are many ways to do this:

• We will use them all except supersymmetry...

## Changing the problem

• QCD is too hard so try to get “perspective” by changing the problem.

• There are many ways to do this:

• We will use them all except supersymmetry...

## Changing the problem

• QCD is too hard so try to get “perspective” by changing the problem.

• There are many ways to do this:

• We will use them all except supersymmetry...

## Changing the problem

• QCD is too hard so try to get “perspective” by changing the problem.

• There are many ways to do this:

• We will use them all except supersymmetry...

## Changing the problem

• QCD is too hard so try to get “perspective” by changing the problem.

• There are many ways to do this:

• We will use them all except supersymmetry...

## Field theory on a small circle

• Periodic boundary conditions on a spatial circle of size L

• Momenta around the circle are quantized as p = n/L

• So modes with n6= 0 have energies E > 1/L.

• As L→ 0 freeze out all except 0-modes so the low energy theory is effectively 3d.

## Field theory on a small circle

• Periodic boundary conditions on a spatial circle of size L

• Momenta around the circle are quantized as p = n/L

• So modes with n6= 0 have energies E > 1/L.

• As L→ 0 freeze out all except 0-modes so the low energy theory is effectively 3d.

## Field theory on a small circle

• Periodic boundary conditions on a spatial circle of size L

• Momenta around the circle are quantized as p = n/L

• So modes with n6= 0 have energies E > 1/L.

• As L→ 0 freeze out all except 0-modes so the low energy theory is effectively 3d.

## Field theory on a small circle

• Periodic boundary conditions on a spatial circle of size L

• Momenta around the circle are quantized as p = n/L

• So modes with n6= 0 have energies E > 1/L.

• As L→ 0 freeze out all except 0-modes so the low energy theory is effectively 3d.

## Field theory on a small circle

• Periodic boundary conditions on a spatial circle of size L

• Momenta around the circle are quantized as p = n/L

• So modes with n6= 0 have energies E > 1/L.

• As L→ 0 freeze out all except 0-modes so the low energy theory is effectively 3d.

## QCD on a small circle

• By RG running, small circle size, L Λ^{−1}, means the gauge
coupling is still weak when the theory becomes effectively 3d.

• Confinement is a strong-coupling phenomenon in 4d, might
expect to only see it at energy scales lower than L^{−1} in the 3d
theory, where the 3d coupling gets strong.

• But something more interesting happens...

## QCD on a small circle

• By RG running, small circle size, L Λ^{−1}, means the gauge
coupling is still weak when the theory becomes effectively 3d.

• Confinement is a strong-coupling phenomenon in 4d, might
expect to only see it at energy scales lower than L^{−1} in the 3d
theory, where the 3d coupling gets strong.

• But something more interesting happens...

## QCD on a small circle

• By RG running, small circle size, L Λ^{−1}, means the gauge
coupling is still weak when the theory becomes effectively 3d.

• Confinement is a strong-coupling phenomenon in 4d, might
expect to only see it at energy scales lower than L^{−1} in the 3d
theory, where the 3d coupling gets strong.

• But something more interesting happens...

## QCD on a small circle

^{−1}, means the gauge
coupling is still weak when the theory becomes effectively 3d.

^{−1} in the 3d
theory, where the 3d coupling gets strong.

• But something more interesting happens...

## QCD on a small circle

^{−1}, means the gauge
coupling is still weak when the theory becomes effectively 3d.

^{−1} in the 3d
theory, where the 3d coupling gets strong.

• But something more interesting happens...

## Gauge holonomy

• Theory still keeps some memory of its 4d origin:

• The Wilson line wrapping the circle has N scalar 0-modes ~ϕ

• = the eigenvalues of the holonomy

Ω(x ) = exp{i R_{x}^{x +L}A}.

• Can think of as points on a circle,

• or as a point in a bounded N-dimensional region.

• Center symmetry acts geometrically on ~ϕ.

## Gauge holonomy

• Theory still keeps some memory of its 4d origin:

• The Wilson line wrapping the circle has N scalar 0-modes ~ϕ

• = the eigenvalues of the holonomy

Ω(x ) = exp{i R_{x}^{x +L}A}.

• Can think of as points on a circle,

• or as a point in a bounded N-dimensional region.

• Center symmetry acts geometrically on ~ϕ.

## Gauge holonomy

• Theory still keeps some memory of its 4d origin:

• The Wilson line wrapping the circle has N scalar 0-modes ~ϕ

• = the eigenvalues of the holonomy

Ω(x ) = exp{i R_{x}^{x +L}A}.

• Can think of as points on a circle,

• or as a point in a bounded N-dimensional region.

• Center symmetry acts geometrically on ~ϕ.

## Gauge holonomy

• Theory still keeps some memory of its 4d origin:

• The Wilson line wrapping the circle has N scalar 0-modes ~ϕ

• = the eigenvalues of the holonomy

Ω(x ) = exp{i R_{x}^{x +L}A}.

• Can think of as points on a circle,

• or as a point in a bounded N-dimensional region.

• Center symmetry acts geometrically on ~ϕ.

## Gauge holonomy

• Theory still keeps some memory of its 4d origin:

• The Wilson line wrapping the circle has N scalar 0-modes ~ϕ

• = the eigenvalues of the holonomy

Ω(x ) = exp{i R_{x}^{x +L}A}.

• Can think of as points on a circle,

• or as a point in a bounded N-dimensional region.

• Center symmetry acts geometrically on ~ϕ.

## Gauge holonomy

• Theory still keeps some memory of its 4d origin:

• The Wilson line wrapping the circle has N scalar 0-modes ~ϕ

• = the eigenvalues of the holonomy

Ω(x ) = exp{i R_{x}^{x +L}A}.

• Can think of as points on a circle,

• or as a point in a bounded N-dimensional region.

• Center symmetry acts geometrically on ~ϕ.

## Adjoint Higgs mechanism

• In the interior of the gauge cell, all the eigenvalues are different,

• so ~ϕ : G → U(1)^{N}.

• ⇒ N copies of E&M,

• stops the running of the coupling, and

• stable monopoles.

• On the boundaries of the gauge cell, some

eigenvalues coincide,

• so ~ϕ no longer fully abelianizes G , and

• 3d gauge coupling still runs to strong coupling.

## Adjoint Higgs mechanism

• In the interior of the gauge cell, all the eigenvalues are different,

• so ~ϕ : G → U(1)^{N}.

• ⇒ N copies of E&M,

• stops the running of the coupling, and

• stable monopoles.

• On the boundaries of the gauge cell, some

eigenvalues coincide,

• so ~ϕ no longer fully abelianizes G , and

• 3d gauge coupling still runs to strong coupling.

## Adjoint Higgs mechanism

• In the interior of the gauge cell, all the eigenvalues are different,

• so ~ϕ : G → U(1)^{N}.

• ⇒ N copies of E&M,

• stops the running of the coupling, and

• stable monopoles.

• On the boundaries of the gauge cell, some

eigenvalues coincide,

• so ~ϕ no longer fully abelianizes G , and

• 3d gauge coupling still runs to strong coupling.

## Adjoint Higgs mechanism

• In the interior of the gauge cell, all the eigenvalues are different,

• so ~ϕ : G → U(1)^{N}.

• ⇒ N copies of E&M,

• stops the running of the coupling, and

• stable monopoles.

• On the boundaries of the gauge cell, some

eigenvalues coincide,

• so ~ϕ no longer fully abelianizes G , and

• 3d gauge coupling still runs to strong coupling.

## Adjoint Higgs mechanism

• In the interior of the gauge cell, all the eigenvalues are different,

• so ~ϕ : G → U(1)^{N}.

• ⇒ N copies of E&M,

• stops the running of the coupling, and

• stable monopoles.

• On the boundaries of the gauge cell, some

eigenvalues coincide,

• so ~ϕ no longer fully abelianizes G , and

• 3d gauge coupling still runs to strong coupling.

## Adjoint Higgs mechanism

• In the interior of the gauge cell, all the eigenvalues are different,

• so ~ϕ : G → U(1)^{N}.

• ⇒ N copies of E&M,

• stops the running of the coupling, and

• stable monopoles.

• On the boundaries of the gauge cell, some

eigenvalues coincide,

• so ~ϕ no longer fully abelianizes G , and

• 3d gauge coupling still runs to strong coupling.

## Adjoint Higgs mechanism

• In the interior of the gauge cell, all the eigenvalues are different,

• so ~ϕ : G → U(1)^{N}.

• ⇒ N copies of E&M,

• stops the running of the coupling, and

• stable monopoles.

• On the boundaries of the gauge cell, some

eigenvalues coincide,

• so ~ϕ no longer fully abelianizes G , and

• 3d gauge coupling still runs to strong coupling.

## Adjoint Higgs mechanism

• In the interior of the gauge cell, all the eigenvalues are different,

• so ~ϕ : G → U(1)^{N}.

• ⇒ N copies of E&M,

• stops the running of the coupling, and

• stable monopoles.

• On the boundaries of the gauge cell, some

eigenvalues coincide,

• so ~ϕ no longer fully abelianizes G , and

• 3d gauge coupling still runs to strong coupling.

## QCD(adj) on a small circle

• The vacuum position of ~ϕ is determined dynamically by the theory:

• it develops a potential in perturbation theory.

• The 1-loop formula for the potential for QCD(adj) with n_{f}
Weyl fermions is

V (ϕ) = 8π^{2}

3L^{3}(1− nf) X

α∈Φ^{+}

B_{4}(α(ϕ)).

• For n_{f} = 0 (pure YM) the minimum is at ~ϕ = 0 and the
theory remains strongly coupled.

• For n_{f} = 1, V ≡ 0 (by supersymmetry).

• For nf > 1, V develops a minium somewhere away from 0.

## QCD(adj) on a small circle

Results (analytic and numerical):

*A**N*

*B**N*

*C**N* *D**N*

## QCD(adj) on a small circle

Results (analytic and numerical):

13 23

-13 13

*A*2

Φ_{1}
Φ_{2}

14 12

14 12

*C*_{2}

Φ_{1}
Φ2

16 13

16

*G*2

Φ1

Φ_{2}

12 1

12

*B*_{2}

Φ_{1}
Φ2

## QCD(adj) on a small circle

Results (analytic and numerical):

G → H

SU(N+1) ' AN → U(1)^{N} for N ≥ 1

SO(2N+1) ' BN → U(1)^{N−1}× SO(3) for N = 2, 3

→ SO(4) × U(1)^{N−3}× SO(3) for N ≥ 4

Sp(2N) ' CN → U(1)^{N} for N ≥ 3

SO(2N) ' DN → SO(4) × U(1)^{N−4}× SO(4) for N ≥ 4
E_{6} → SU(3)× SU(3) × SU(3)

E7 → SU(2)× SU(4) × SU(4)
E_{8} → SU(2)× SU(3) × SU(6)
F_{4} → SU(3)× SU(2) × U(1)

G2 → SU(2)× U(1)

All-order perturbative result

Above pattern of gauge symmetry breaking and the fact that center symmetry is not broken holds to all orders in p.t.

## How does QCD(adj) in a small box confine?

• Upshot: QCD(adj) with gauge groups SU(N) and Sp(N) and
n_{f} > 1 fermions have unbroken center symmetry and remain
weakly coupled.

• They have minima at h~ϕi ∼ 1/(NL)

• But do they confine?

• Only if they develop a mass gap for the remaining U(1)^{N}
photons at weak coupling.

• We have to look at the magnetic degrees of freedom (Polyakov) ...

## Monopoles on a small circle

• Rotate monopole

worldlines to be spacelike

• Gives a magnetically charged but point-like object in 3d space-time: a 3d monopole-instanton.

• Its action∼ mass×length:

S_{M} ∼ hϕi · L ∼ 1/N.

• Monopole-inst. action is independent of L!

• S_{M} is 1/N of 4d instanton
action — are they the
missing renormalon?

• ...not quite...

## Monopoles on a small circle

• Rotate monopole

worldlines to be spacelike

• Gives a magnetically charged but point-like object in 3d space-time: a 3d monopole-instanton.

• Its action∼ mass×length:

S_{M} ∼ hϕi · L ∼ 1/N.

• Monopole-inst. action is independent of L!

• S_{M} is 1/N of 4d instanton
action — are they the
missing renormalon?

• ...not quite...

## Monopoles on a small circle

• Rotate monopole

worldlines to be spacelike

• Gives a magnetically charged but point-like object in 3d space-time: a 3d monopole-instanton.

• Its action∼ mass×length:

S_{M} ∼ hϕi · L ∼ 1/N.

• Monopole-inst. action is independent of L!

• S_{M} is 1/N of 4d instanton
action — are they the
missing renormalon?

• ...not quite...

## Monopoles on a small circle

• Rotate monopole

worldlines to be spacelike

• Gives a magnetically charged but point-like object in 3d space-time: a 3d monopole-instanton.

• Its action∼ mass×length:

S_{M} ∼ hϕi · L ∼ 1/N.

• Monopole-inst. action is independent of L!

• S_{M} is 1/N of 4d instanton
action — are they the
missing renormalon?

• ...not quite...

## Monopoles on a small circle

• Rotate monopole

worldlines to be spacelike

• Gives a magnetically charged but point-like object in 3d space-time: a 3d monopole-instanton.

• Its action∼ mass×length:

S_{M} ∼ hϕi · L ∼ 1/N.

• Monopole-inst. action is independent of L!

• S_{M} is 1/N of 4d instanton
action — are they the
missing renormalon?

• ...not quite...

## Monopoles on a small circle

• Rotate monopole

worldlines to be spacelike

• Gives a magnetically charged but point-like object in 3d space-time: a 3d monopole-instanton.

• Its action∼ mass×length:

S_{M} ∼ hϕi · L ∼ 1/N.

• Monopole-inst. action is independent of L!

• S_{M} is 1/N of 4d instanton
action — are they the
missing renormalon?

• ...not quite...

## Monopoles on a small circle

• Monopoles &

anti-monopoles populate 3d vacuum with density

∼ L^{−3}e^{−8}^{π}^{2}^{/(Ng}^{2}^{)}
making it amagnetic plasma.

• This screens E&M waves, photons get a gap, realizes the dual superconductoridea (Polyakov 1977):

• Electric flux is confined to tubes of tension

∼ L^{−2}e^{−16}^{π}^{2}^{/(Ng}^{2}^{)}

## Monopoles on a small circle

• Monopoles &

anti-monopoles populate 3d vacuum with density

∼ L^{−3}e^{−8}^{π}^{2}^{/(Ng}^{2}^{)}
making it amagnetic plasma.

• This screens E&M waves, photons get a gap, realizes the dual superconductoridea (Polyakov 1977):

• Electric flux is confined to tubes of tension

∼ L^{−2}e^{−16}^{π}^{2}^{/(Ng}^{2}^{)}

## Monopoles on a small circle

• Monopoles &

anti-monopoles populate 3d vacuum with density

∼ L^{−3}e^{−8}^{π}^{2}^{/(Ng}^{2}^{)}
making it amagnetic plasma.

• This screens E&M waves, photons get a gap, realizes the dual superconductoridea (Polyakov 1977):

• Electric flux is confined to tubes of tension

∼ L^{−2}e^{−16}^{π}^{2}^{/(Ng}^{2}^{)}

## Monopoles on a small circle

• Problem: above scenario doesnotwork for QCD(adj)!

• Monopoles are not quite scalar operators: they carry fermion 0-modes.

• Monopole plasma then generates (tiny) multi-fermion interactions, not a photon mass — i.e. screens only fermions.

## Monopoles on a small circle

• Problem: above scenario doesnotwork for QCD(adj)!

• Monopoles are not quite scalar operators: they carry fermion 0-modes.

• Monopole plasma then generates (tiny) multi-fermion interactions, not a photon mass — i.e. screens only fermions.

## Monopoles on a small circle

• Problem: above scenario doesnotwork for QCD(adj)!

• Monopoles are not quite scalar operators: they carry fermion 0-modes.

• Monopole plasma then generates (tiny) multi-fermion interactions, not a photon mass — i.e. screens only fermions.

## Monopole–anti-monopole bound states

• But fermion 0-modes can cancel between monopoles and anti-monopoles.

• (Magnetic) Coulomb repulsion balances against fermion exchange attraction ...

• ... giving monopole–anti-monopole bound state: “bions”

(Unsal 2008)¨

• Note that, since there are many U(1)’s the monopoles carry a whole vector of magnetic charges.

• This means that bions can carry non-zero magnetic charge.

## Monopole–anti-monopole bound states

• But fermion 0-modes can cancel between monopoles and anti-monopoles.

• (Magnetic) Coulomb repulsion balances against fermion exchange attraction ...

• ... giving monopole–anti-monopole bound state: “bions”

(Unsal 2008)¨

• Note that, since there are many U(1)’s the monopoles carry a whole vector of magnetic charges.

• This means that bions can carry non-zero magnetic charge.

## Monopole–anti-monopole bound states

• But fermion 0-modes can cancel between monopoles and anti-monopoles.

• (Magnetic) Coulomb repulsion balances against fermion exchange attraction ...

• ... giving monopole–anti-monopole bound state: “bions”

(Unsal 2008)¨

• Note that, since there are many U(1)’s the monopoles carry a whole vector of magnetic charges.

• This means that bions can carry non-zero magnetic charge.

## Monopole–anti-monopole bound states

• But fermion 0-modes can cancel between monopoles and anti-monopoles.

• (Magnetic) Coulomb repulsion balances against fermion exchange attraction ...

• ... giving monopole–anti-monopole bound state: “bions”

(Unsal 2008)¨

• Note that, since there are many U(1)’s the monopoles carry a whole vector of magnetic charges.

• This means that bions can carry non-zero magnetic charge.

## Monopole–anti-monopole bound states

• But fermion 0-modes can cancel between monopoles and anti-monopoles.

• (Magnetic) Coulomb repulsion balances against fermion exchange attraction ...

• ... giving monopole–anti-monopole bound state: “bions”

(Unsal 2008)¨

• Note that, since there are many U(1)’s the monopoles carry a whole vector of magnetic charges.

• This means that bions can carry non-zero magnetic charge.

## Monopole–anti-monopole bound states

• Bions populate 3d vacuum with density

∼ e^{−2·8}^{π}^{2}^{/(Ng}^{2}^{)}

• making it a magnetic plasma.

• This screens E&M waves, photons get a gap, electric flux confined to tubes, realizing a dual superconductor.

Bion mechanism (Unsal)¨ Bion condensation drives confinement in QCD(adj) in a periodic box.

## Monopole–anti-monopole bound states

• Bions populate 3d vacuum with density

∼ e^{−2·8}^{π}^{2}^{/(Ng}^{2}^{)}

• making it a magnetic plasma.

• This screens E&M waves, photons get a gap, electric flux confined to tubes, realizing a dual superconductor.

Bion mechanism (Unsal)¨ Bion condensation drives confinement in QCD(adj) in a periodic box.

## Monopole–anti-monopole bound states

• Bions populate 3d vacuum with density

∼ e^{−2·8}^{π}^{2}^{/(Ng}^{2}^{)}

• making it a magnetic plasma.

• This screens E&M waves, photons get a gap, electric flux confined to tubes, realizing a dual superconductor.

Bion mechanism (Unsal)¨ Bion condensation drives confinement in QCD(adj) in a periodic box.

## Monopole–anti-monopole bound states

• Bions populate 3d vacuum with density

∼ e^{−2·8}^{π}^{2}^{/(Ng}^{2}^{)}

• making it a magnetic plasma.

Bion mechanism (Unsal)¨ Bion condensation drives confinement in QCD(adj) in a periodic box.

## Bion–anti-bion bound states

Can go one step further and calculate the contribution of bion–anti-bion topological molecules to QCD(adj) plasma

## Bion–anti-bion bound states

Can go one step further and calculate the contribution of bion–anti-bion topological molecules to QCD(adj) plasma

## Bion–anti-bion bound states

Need to use an extension of QM instanton–anti-instanton techniques to QFT which we call the

Bogomolny–Zinn-Justin prescription

... too long to fit in this box ...

## Bion–anti-bion bound states

These contribute to a divergence in perturbation theory at the same order as the 4d renormalon divergence.

Renormalon conjecture Bion–anti-bion tunnelling event is (continuously connected to) ’t Hooft’s “IR renormalon”

## Mass gap (string tensions) for QCD(adj)

• So QCD(adj) in a small box confines at weak coupling.

• Calculate the mass gap ...

• But this is in 3d ... what can we deduce about 4d physics?

• Extrapolate to large L ...

• ... gives a “phase diagram” of QCD(adj).

• But can we trust this extrapolation to large L since it goes through strong coupling?

## Mass gap (string tensions) for QCD(adj)

• So QCD(adj) in a small box confines at weak coupling.

• Calculate the mass gap ...

• But this is in 3d ... what can we deduce about 4d physics?

• Extrapolate to large L ...

• ... gives a “phase diagram” of QCD(adj).

• But can we trust this extrapolation to large L since it goes through strong coupling?

## Mass gap (string tensions) for QCD(adj)

• So QCD(adj) in a small box confines at weak coupling.

• Calculate the mass gap ...

• But this is in 3d ... what can we deduce about 4d physics?

• Extrapolate to large L ...

• ... gives a “phase diagram” of QCD(adj).

• But can we trust this extrapolation to large L since it goes through strong coupling?

## Mass gap (string tensions) for QCD(adj)

• So QCD(adj) in a small box confines at weak coupling.

• Calculate the mass gap ...

• But this is in 3d ... what can we deduce about 4d physics?

• Extrapolate to large L ...

• ... gives a “phase diagram” of QCD(adj).

• But can we trust this extrapolation to large L since it goes through strong coupling?

## Mass gap (string tensions) for QCD(adj)

• So QCD(adj) in a small box confines at weak coupling.

• Calculate the mass gap ...

• But this is in 3d ... what can we deduce about 4d physics?

• Extrapolate to large L ...

• ... gives a “phase diagram” of QCD(adj).

• But can we trust this extrapolation to large L since it goes through strong coupling?

## Mass gap (string tensions) for QCD(adj)

• So QCD(adj) in a small box confines at weak coupling.

• Calculate the mass gap ...

• But this is in 3d ... what can we deduce about 4d physics?

• Extrapolate to large L ...

• ... gives a “phase diagram” of QCD(adj).

• But can we trust this extrapolation to large L since it goes through strong coupling?

## Large N volume independence

• There is no a priori reason to trust the naive extrapolation from 3d to 4d.

• But another old idea helps for large gauge groups:

Eguchi-Kawai reduction

For SU(N) in limit N→ ∞: large class of observables are independent of Las long as center symmetry is unbroken.

Physical idea: as N gets large, lowest-momentum modes∝ 1/NL instead of 1/L because can “fold” NL-periodic modes into the gauge degrees of freedom. Thus effective size of box is NL, not L.

## Large N volume independence

• There is no a priori reason to trust the naive extrapolation from 3d to 4d.

• But another old idea helps for large gauge groups:

Eguchi-Kawai reduction

For SU(N) in limit N→ ∞: large class of observables are independent of Las long as center symmetry is unbroken.

Physical idea: as N gets large, lowest-momentum modes∝ 1/NL instead of 1/L because can “fold” NL-periodic modes into the gauge degrees of freedom. Thus effective size of box is NL, not L.

## Large N volume independence

• There is no a priori reason to trust the naive extrapolation from 3d to 4d.

• But another old idea helps for large gauge groups:

Eguchi-Kawai reduction

For SU(N) in limit N→ ∞: large class of observables are independent of Las long as center symmetry is unbroken.

Physical idea: as N gets large, lowest-momentum modes∝ 1/NL instead of 1/L because can “fold” NL-periodic modes into the gauge degrees of freedom. Thus effective size of box is NL, not L.

## Large N volume independence

• There is no a priori reason to trust the naive extrapolation from 3d to 4d.

• But another old idea helps for large gauge groups:

Eguchi-Kawai reduction

## Large N volume independence

• For SU(N) QCD(adj) since center symmetry is not broken so

Eguchi-Kawai reduction is realized (Kovtun, ¨Unsal, Yaffe 2007)

• Gives a semi-quantitative way to map out phase diagrams of many QCD-like theories (Poppitz, ¨Unsal 2009).

## Large N volume independence

• For SU(N) QCD(adj) since center symmetry is not broken so

Eguchi-Kawai reduction is realized (Kovtun, ¨Unsal, Yaffe 2007)

• Gives a semi-quantitative way to map out phase diagrams of many QCD-like theories (Poppitz, ¨Unsal 2009).

## Large N volume independence

• For SU(N) QCD(adj) since center symmetry is not broken so

Eguchi-Kawai reduction is realized (Kovtun, ¨Unsal, Yaffe 2007)

• Gives a semi-quantitative way to map out phase diagrams of many QCD-like theories (Poppitz, ¨Unsal 2009).

## Further directions

• What happens for other matter contents? (Poppitz- ¨Unsal)

• Even chiral matter? Yes ... (e.g. Shifman- ¨Unsal 2010)

• What is the fate of the theories which do not abelianize in 3d?

• Can the strong gauge dynamics in 3d be attacked by compactification on more circles, down to 2d? 1d? 0d?

• Is there a systematic way to extend the

Bogomolnyi–Zinn-Justin prescrption to all oders in the semi-classical expansion?

• Can large-N volume independence hold for Sp(N) and SO(N) groups?

• Is there a way of extrapolating to large volumes directly at finite N?

• Is there a more refined set of gauge theory order parameters than just Wilson and ‘t Hooft loops?