Lessons from QCD on a circle Aleksey Cherman INT, University of - - PowerPoint PPT Presentation

Lessons from QCD on a circle

Aleksey Cherman INT, University of Washington

SEWM 2018, Barcelona

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Aleksey Cherman INT, University of Washington

Lessons from QCD-like theories on a circle

SEWM 2018, Barcelona

QCD still has mysteries!

Don’t understand mechanisms of mass gap, confinement, chiral symmetry breaking in detail. Don’t fully understand phase diagram. Lots of formal and not so formal issues. Basic issue: strong coupling seems essential, but that’s exactly where our QFT tools fail. So what is to be done? Don’t understand why nuclear physics exists. Old idea: look at related QFTs, try to make progress.

Analytic approaches to QCD

Very few classes of 4d QFTs which are both solvable, and resemble QCD. Worth taking each one seriously! (1) AdS/CFT models — powerful, but large N, no asymptotic freedom. (2) SUSY models — nice, but have light scalars. This talk: new analytic approach using “adiabatic compactification” Works for finite N asymptotically free QFTs without scalars; gives insights into confinement, chiral symmetry breaking, etc. Hope for no phase transitions on way back to QCD. Absence of phase transitions between strong and weak coupling has very interesting implications.

Adiabatic continuity

Claim: there exist 4d SU(N) gauge theories without fundamental scalar fields with a controllable coupling parameter, with Simplest example: adjoint QCD Adjoint QCD = SU(N) gauge theory + NF Weyl adjoint fermions Who cares? (a) confinement already at weak coupling (b) no phase transitions on way to strong coupling First, statement is theoretically surprising. Second, QCD(Adj) is much closer to QCD than it looks. Third, related constructions for normal YM and QCD using Unsal-Yaffe double-trace deformations.

Unsal, Yaffe 2008 Unsal 2007; Kovtun, Unsal, Yaffe 2008; AC, Shifman, Unsal, 2018?

Adjoint QCD

Adjoint QCD — QCD(Adj) — is a close cousin to normal QCD. In fact, at large N, QCD(Adj) becomes equivalent to QCD(AS), in a common subsector. QCD(AS) = SU(N) + NF Dirac two-index anti-symmetric fermions.

For N = 3, QCD(AS) = QCD, due to So studying QCD(Adj) is no worse than studying the large N limit of normal QCD Expect confinement and spontaneous chiral symmetry breaking on R4 Has continuous SU(NF) chiral symmetry and a ZN center symmetry.

Armoni, Shifman, Veneziano, 2003

Idea: put QCD(Adj) on a circle

Break 4D Lorentz, but as little as possible!

R3 S1

This is not obvious.

Unsal, Yaffe, Shifman, … 2008-onward

Small S1 can cause phase transitions. If they appear, small circle physics completely different from large circle. If circle size L is small, might get weak coupling by asymptotic freedom Would happen if relevant coupling for deep IR is λ(1/L) Can phase transitions in L be prevented?

Self-Higgsing?

When YM compactified on S1, Polyakov loops become important Non-coincident eigenvalues for Ω ⇒ “broken” gauge group SU(N) → U(1)N-1 in long-distance 3D EFT

A3 can act as a (compact) adjoint Higgs field But we don’t get to choose eigenvalues: theory picks own vacuum Resulting mW ~ 1/L - would guarantee weak coupling in IR

Naive attempt

Simplest idea: anti-periodic (thermal) BCs

When LΛ >> 1, expect confinement <tr O> = 0. When LΛ << 1, <tr O> determined by GPY effective potential <A3> = 0, so 3D EFT = 3D SU(N) YM, strongly coupled. Small and large L separated by a phase transition. 1/L = temperature. This is just deconfinement!

Smoothness in L

Kovtun, Unsal, Yaffe, Shifman, … , 2007-onward

More devious: choose periodic BCs for fermions. View S1 as a spatial direction, or interpret circle path integral as rather than

What happens to L dependence?

Smoothness in L

Kovtun, Unsal, Yaffe, Shifman, … , 2007-onward

For large LΛ, BCs don’t matter, expect confinement. For LΛ << 1 , look at GPY potential <A3> ≠ 0 with all eigenvalues are distinct, ⟹ 3D EFT is weakly-coupled U(1)N-1 gauge theory. No deconfinement transition!

NB: when NF > 1 and finite N, works so long as m/Λ << 1

NF = 1 QCD(Adj) is supersymmetric; separate analysis.

Strongly coupled regime Semiclassically calculable regime

Flow for NLΛ ≪ 1 Flow for NLΛ ≫ 1 Λ (N L)-1 Q λ(1/NL)

1

λ

Coupling flow with adiabatic compactification

Lots of things have been worked out: mass gap, string tensions, effects of fundamental quarks, chiral symmetry breaking effects, theta dependence, large N surprises…

The plan

Pick two: Mass gap at weak coupling Large N surprises

Small L limit in perturbation theory

At long distances l >> N L ~ 1/mW due to the center-symmetric background holonomy. Light fields always include N - 1 “Cartan gluons” Small-L physics easiest to describe using 3D Abelian duality

(added fictitious p = 0 mode for notational simplicity; it decouples exactly.)

Adjoint fermions give light (mass ~ mq) Cartan quarks after Higgsing

Small L limit in perturbation theory

N - 1 Cartan gluons are classically gapless. σi shift symmetry ⟺ conservation of 3d magnetic charge. But there are no magnetic monopoles in perturbation theory. σi are massless to all orders in perturbation theory.

Finite-action field configurations

Since SU(N) → U(1)N-1, 4D BPST instanton breaks up into N ‘monopole-instantons’ ℳa with action SI/N = 8π2/λ λ ≪ 1 when NLΛ ≪ 1, dilute gas approximation justified at small L. Contrast with usual IR disasters with instantons in YM!

Lee, Yi; Kraan, van Baal; 1998

Induced interactions for σs:

Monopole-instantons carry both magnetic and topological charges Suppose adjoint fermions are massive. Then

Weak coupling confinement

Unsal, Yaffe, Shifman, Poppitz, Sulejmanpasic, …

Concrete realization of old Mandelstam, ’t Hooft, Polyakov dreams: mass gap driven by `condensation’ of magnetic monopoles. String tension also calculable, and is finite. Behaves just as expected from YM.

Poppitz, Erfan S. T., 1708.08821

Different from Seiberg-Witten confinement model.

Anber, Pellizzani, 1710.06509

Monopole-instantons induce potential for σ’s:

Tension depends only on N-ality of test quarks, as expected.

Massless adjoint fermions

finite topological charge ⇒ fermion zero modes. Induced interactions: If mq = 0, each monopole-instanton picks up 2 N NF zero modes

Monopole-instantons can’t generate a potential for σa if mq = 0 ∃ finite-action configurations with zero topological charge, non-zero magnetic charge, and action ~ 16π2/λ: the “magnetic bions” So massless fermions don’t destroy confinement. Locally-4D nature of theory vital for this. Fails in purely 3D theories.

Unsal 2007

What have we learned so far?

There is a theory closely resembling QCD which does NOT deconfine when put on a small circle. When circle is small enough, get weak coupling in IR. Mass gap becomes calculable. Can guarantee that symmetry realizations match between large L and small L regimes Can explore many things: spectrum and resonances, θ dependence, chiral symmetry breaking, etc. But focus on the first statement: it’s very surprising at large N.

Confinement for small L

The GPY potential is reliable provided LΛ << 1.

Generic observables weakly-coupled only if NLΛ << 1, L ~ 1/N But Polyakov loop can be calculated whenever LΛ << 1. So can be sure confinement holds even when 1/N << LΛ << 1 and N >> 1!

Kovtun, Unsal, Yaffe 2007 AC, Shifman, Unsal, 2018…

This is extremely surprising.

We must have phase transition at or below TH — deconfinement transition. Once L < LH = 1/TH, partition function becomes singular.

Large N Hagedorn instability

Hagedorn scaling Expected in any confining large N theory. Excitations = infinite number of free particles mass gap ~ Λ Hagedorn instability and small-L confinement are in conflict! Or are they? Signature of a string theory

Periodic BCs for fermions ⟹ working with twisted partition function QCD(Adj) has light fermion states at large N, in contrast to QCD(F)! There isn’t necessarily a conflict - but to avoid it we need a miracle.

Volume independence vs Hagedorn

‘All’ we need is enough cancellation between ρB and ρF

Basar, AC, Dorigoni, Unsal, 2013

How much is enough?

, not just

Expect Hagedorn scaling for both ρB and ρF. More precisely:

How much cancellation do we need?

ALL red terms must cancel EXACTLY to avoid an instability!

Basar, AC, Dorigoni, Unsal, 2013

This is a crazy thing to expect without a symmetry…

Is there some sort of emergent Bose-Fermi “symmetry” at large N? Thinking along these lines, at NF = 1, we rediscover supersymmetry. If NF > 1, “emergent symmetry” can’t be supersymmetry!

Emergent fermionic “symmetry”

NF (N2 - 1) microscopic fermions, only (N2 - 1) microscopic bosons.

Basar, AC, Dorigoni, Unsal, 2013

Coleman-Mandula, Haag-Lopuzhansky-Sohnius: SUSY is only game in town

Considers symmetries with local charges, constraints on non-trivial S-matrix S-matrix trivializes at large N. Still, whatever leads to “Bose- Fermi conspiracy” in QCD(Adj) must be quite exotic…

Nice to look at a regime where conspiracy plays out in full view Compactify space to S3 with radius R, and work on S3xS1

Closer look at the Bose-Fermi conspiracy

Small L with R3 x S1 not good enough: only get full IR control if L ~ 1/N When RΛ << 1, spectrum fully calculable. Physically, not possible to get to large distances! Get a free theory as RΛ → 0

Large N confined-phase partition functions

As λ(1/R) → 0, microscopic fields Aμ, ψi = matrix-valued harmonic oscillators Gauss law ⟹ physical states are color singlets Color singlets are built from color traces S3 is compact

+

• single-particle state

multi-particle state Large N:

Large N confined-phase partition functions

Grand-canonical partition functions are known in closed form:

Can check it by plotting log(dn) versus n Hagedorn phenomenon: level degeneracies dn are exponential in n

Level degeneracies in adjoint QCD

• ■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

Degeneracies

• Bosons

■ Fermions

10 20 30

n

5 10 15 20 25 30 35

Log dn

NF = 2

Eyeball ⇒ leading exponential growth of B and F states identical (Half-integral Bose-Fermi splitting due to S3 curvature couplings)

Expect the asymptotics of density of states to be described by an infinite series of exponentials, one for each ‘Regge trajectory’

Cancellation of Hagedorn instabilities

Can’t tell whether enough cancellations happen by eyeballing dn So look for poles of partition functions Z[q] in q ∈ [0,1] No singularities in [0,1] ⟹ complete cancellation of Hagedorn.

Singularities of NF = 1 thermal QCD(Adj)

Hagedorn singularities

Singularities of NF = 2 thermal QCD(Adj)

Hagedorn singularities

Singularities of NF = 1 twisted QCD(Adj)

No Hagedorn singularities

Singularities of NF = 2 twisted QCD(Adj)

No Hagedorn singularities

In a theory with volume independence for all L, all Hagedorn instabilities cancel for NF ≥ 1, as expected from general arguments.

Cancellation of Hagedorn instabilities

Enormous cancellations in twisted partition function of QCD(Adj) testify to very tight relations between B and F states. And it’s happening in QFTs which are manifestly not supersymmetric.

Consider (-1)F twisted partition function in a 𝓞 = 1 SUSY QFT on curved space M3xS1

Quantifying power of cancellations

Cancellation of “extensive” 1/L3 piece = SUSY vacuum energy cancellation

DiPietro, Komargodski 2014

Left-over piece scales like log Z for a 2D QFT, tied to anomalies. SUSY ⟹ extremely stringent cancellation, not even a single 4D particle’s worth of density of states uncancelled. Much more than is needed to avoid Hagedorn! NF = 1 adjoint QCD

Quantifying power of cancellations

Basar, AC, Dienes, McGady, Shifman, Unsal, Yamazaki 2014-2018

What happens in U(N) QCD(Adj) with NF >1 at large N? Naive expectation: just enough cancellation to avoid Hagedorn.

no SUSY ⟹ some particle species make non-cancelling contributions?

Quantifying power of cancellations

Basar, AC, Dienes, McGady, Shifman, Unsal, Yamazaki 2014-2018

What happens in U(N) QCD(Adj) with NF >1 at large N? Naive expectation: just enough cancellation to avoid Hagedorn.

no SUSY ⟹ some particle species make non-cancelling contributions? In fact NF > 1 cancellations are just powerful as SUSY!

Quantifying power of cancellations

Leading extensive `vacuum energy’ piece cancels without SUSY, but only at large N!

Who ordered this?

So what’s really going on? Kutasov-Seiberg, 1990: Upper bound on (-1)F-graded spacetime density of states for any tachyon-free consistent string theory: Can grow at most as fast as that of a 2D CFT! See also Dienes, 1990s, “misaligned SUSY” and spectral sum rules from modular invariance Large N gauge theory = some kind of string theory. From this point of view, vacuum energy cancellation ‘natural’ in adjoint QCD

Basar, AC, Dienes, McGady,

Important to study this further!

Conclusions

Some QCD-like theories, such as adjoint QCD, have smooth weak-coupling limits with confinement No need for fundamental scalars or giving up on asymptotic freedom. Just need a circle! Fun playground for exploring mass gap, chiral symmetry breaking, θ dependence, … Lack of phase transitions in LΛ has spectacular implications at large N Extremely constraining emergent Bose-Fermi conspiracy Strong hint that SUSY is not only game in town! The end