Higgs-confinement continuity
in light of particle-vortex statistics
Yui Hayashi (YITP, Kyoto U.)
熱場の量子論とその応用@KEK

August 28, 2023
arXiv:2303.02129 [hep-th]Dense QCD matter
𝑇Hadron gas

Nuclearsuperfluid?

Color superconducting" 𝑞𝑞 ≠ 0"

𝜇𝐵
same symmetry breaking pattern → No transition?

(“quark-hadron continuity”) [Schäfer-Wilczek ‘99]

Higgs-confinement continuity
An idea behind quark-hadron continuity:

Higgs-confinement continuity𝜅 ∼ (Higgs strength)[Osterwalder-Seiler ‘78][Fradkin-Shenker ‘79][Banks-Rabinovici ‘79]

For fundamental gauge-Higgs systems, theconfining regime (small 𝛽, 𝜅) and Higgs regime(large 𝛽, 𝜅) are connected by an analyticity region(without any transition).

Condensation of fundamental matter“ 𝜙 ≠ 0” does not distinguish phases

AnalyticityHiggs

confining

The CFL diquark condensation is in (anti-)fundamental representation𝛽 ∼ 1/𝑔2

When distinguishable? to introduce the recent discussion……
𝜅 ∼ (Higgs strength)e.g.) charge-𝒒 Abelian Higgs Model (𝒒 > 𝟏)
The Higgs/confining phases are distinguishedby perimeter-law/area-law of the Wilson loop
[1](equivalently ℤ𝑞 broken/unbroken)

𝑊 𝐶 𝑉(𝑆)[1]ℤ𝑞 broken

Confining[1]ℤ𝑞 unbroken

Another characterization: topological ordering
In the (low-energy effective theory of) Higgs phase, wehave two topological operators, showing
nontrivial mutual statistics:• Wilson loop: 𝑊(𝐶)• Vortex worldsheet: 𝑉(𝑆)
2𝜋𝑖
Link(𝐶,𝑆)=𝑒 𝑞
𝑊 𝐶Higgs

𝑉(𝑆)Coulomb𝛽 ∼ 1/𝑔2
Wilson loop

= 𝑒vortex

2𝜋𝑖𝑞

Recent discussion• Particle-vortex statistics in the CFL phase [Cherman-Sen-Yaffe ‘18]
Wilson loop

= 𝑒2𝜋𝑖3Non-Abelian CFL vortex

Does this nontrivial AB phase signal a quark-hadron transition?
• This AB phase does not mean topological order [Hirono-Tanizaki ‘18 ‘19]

how connected,concretely?

(∵ The CFL vortex is (partially) a global vortex, which does not become topological)
More constructively?
At least, the previous logic does not apply here → continuity?
• Still, it was conjectured that the AB phase can be an order parameter for a Higgsconfinement transition, by studying an Abelian toy model (detailed later) [ChermanJacobson-Sen-Yaffe ‘20]. → transition?

Main claim by a Fradkin-Shenker-like analysis
For superfluid fundamental gauge-Higgs systems, the AharonovBohm phase around the vortex is continuous (or constant, ifprotected by symmetry) in the strong-coupling and deep-Higgs
regions, connecting confining and Higgs regimes
𝜅Below, we illustrate this claim in thefollowing two lattice models analogous to:1) The Abelian toy model
2) Ginzburg-Landau model for CFL diquarkHiggs

conf𝛽 ∼ 1/𝑔2

An order parameter?: Aharonov-Bohm phase
In what follows, we evaluate the AB phase 𝑂Ω around a minimal superfluid vortex,
schematically defined by,𝑊 𝐶 𝑉 𝑆𝑂Ω : = lim
𝐶 →∞ 𝑊 𝐶𝑉(𝑆)asymptotically large loop linking with thevortex worldsheet S, Link 𝐶, 𝑆 = 1

Wilson loop= 𝑂Ω
a minimal superfluid vortex

Example 1: the Abelian toy model
used to argue a Higgs-confinement transition [Cherman-Jacobson-Sen-Yaffe ‘20].

Field contents:
3d compact U(1) gauge 𝑎 + charge-(±1) matters (𝜙+ , 𝜙− ) + neutral scalar 𝜙0Action:1
𝑆 = න 2 𝑑𝑎 2 + 𝐷𝜙+ 2 + 𝐷𝜙− 2 + 𝑑𝜙0 2 + 𝑉𝑐 𝜙+ + 𝑉𝑐 𝜙− + 𝑉0 𝜙0 + 𝜖𝜙+ 𝜙− 𝜙0 + 𝑐. 𝑐.
2𝑒𝑉0 𝜙0 : wine bottle potential → 𝜙0 condensation (superfluidity)
𝑉𝑐 𝜙± = 𝑚𝑐2 𝜙±24+ 𝜆𝑐 𝜙± : identical potential for charged matters (𝜙+ , 𝜙− )AB phase as an order parameter?:Higgs𝑂Ω = −1

from classical vortex
configurations

Confinement𝑂Ω = +1?

𝑚𝑐2 /𝑒 4(𝜙+ , 𝜙− ) decouple

The AB phase must be ±1 due to ℤ2 symmetry [𝜙± → 𝜙∓ , 𝑎 → −𝑎] → transition somewhere?

Example 1: the Abelian toy modelWilson loop

Result: the AB phase is -1 (constant!) in both regions (skip lattice details)• Deep Higgs region𝑑𝜑 −𝑑𝜑−In the deep Higgs limit, the gauge field 𝑎 is frozen to be +(with 𝜙± = 𝑣 𝑒 𝑖 𝜑± ).
2The minimal 𝜙0 vortex rotates (𝜙+ 𝜙− ) by 2𝜋 asymptotically.

𝑊 𝐶 𝑉(𝑆) ∼𝜙0 vortex
𝜅𝑑𝜙+ −𝑑𝜙−𝑖 ‫𝐶׬‬
2𝑒vortex ∼ −1Higgs

in accordance with [Cherman-Jacobson-Sen-Yaffe ‘20].

• Strong coupling region
Even if charged matters are heavy, an asymptotically large Wilson loop is dominatedby screened perimeter-law part, which can be affected by 𝜙0 vortex.

For example, in the “deep confining regime” (𝛽 → +0, small 𝜅),ℓ∈𝐶

𝑒 𝑖 𝜃1 + 𝑒 𝑖 𝜃2 = (𝑒 𝑖 𝜃1conf∼ ෑ [ 𝜙+ hopping ∗ + 𝜙− hopping ]𝑊 𝐶 𝑉(𝑆) = ෑ 𝑈ℓvortex𝑒 𝑖 (−𝜃1 +𝜃2) )|𝑒 𝑖 𝜃1 + 𝑒 𝑖 𝜃2 | → ∼ℓ∈𝐶𝑒vortex𝑖 ‫𝐶׬‬

𝑑𝜑+ −𝑑𝜑−2vortex ∼ −1 (matched!) (Actually, 𝑂Ω = −1 without 𝜅 expansion)𝛽Example 2: Ginzburg-Landau model for diquark(anti-)fundamental𝑡
CFL diquark Φ𝑎𝑖 ∼ 𝜖 𝑎𝑏𝑐 𝜖 𝑖𝑗𝑘 𝑞𝑏𝑗
𝐶𝛾 5 𝑞𝑐𝑘

Field contents:

We add superfluid “dibaryon”
for nuclear superfluid phase

SU(N) gauge 𝑎 + (𝑁 × 𝑁)-matrix-valued fundamental matter Φ + neutral scalar 𝜙0Action:
𝑆 = න 𝑓 2 + 𝐷Φ 2 + 𝐷𝜙0 2 + 𝑉 tr Φ† Φ + 𝑉0 𝜙0 + 𝜖 𝜙0∗ (det Φ) + 𝑐. 𝑐.
𝑉0 𝜙0 : wine bottle potential → 𝜙0 condensation (superfluidity)
By tuning 𝑉 tr 𝛷 † 𝛷 , this model has superfluid confining regime [nuclear superfluidity] and Higgs regime [CFL].

(apparent) mismatch of AB phase:
Wilson loop= 𝑒vortex

2𝜋𝑖𝑁, 𝑂Ω = ቐ+1 ?
𝑒2𝜋𝑖
𝑁Confining limit
Higgs limit

Example 2: Ginzburg-Landau model for diquark𝜅

We can perform the similar analysis on an analogous lattice model• Deep Higgs region
𝑊 𝐶 𝑉(𝑆)

Higgs

2𝜋𝑖
∼𝑒 𝑁

reproducing [Cherman-Sen-Yaffe ‘18]

• Strong coupling region
The AB phase is not a constant [at 𝑁 = 3] and trivial in the deep
confining limit (𝛽 → 0, small 𝜅)
𝑊 𝐶 𝑉(𝑆) ∼ 1
Still, the AB phase is continuous and smoothly interpolates

between 1 (𝜅 → +0) and 𝑒limit (𝛽 → 0):

2𝜋𝑖
𝑁(𝜅 → +∞) in the strong coupling

conf𝛽 ∼ 1/𝑔2e.g.) diquark condensation in dense QCD

Summary
In some superfluid gauge-Higgs systems,

Higgs

Wilson loop

= 𝑒Confinement

2𝜋𝑖
𝑁

Recently-debated issue:
Does the nontrivial AB phase
necessitate a transition?

vortexClaim: For fundamental superfluid gauge-Higgs systems,
the AB phase respects the Higgs-confinement continuity.
the quark-hadron continuity is still a possible scenario.