Bootstrapping the genesis of Nambu-Goldstone Bosons from Quark - - PowerPoint PPT Presentation

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Mar 10, 2024 47 likes •171 views

Bootstrapping the genesis of Nambu-Goldstone Bosons from Quark Gluon Plasma Yu Nakayama (Kavli IPMU, Caltech) in collaboration with Tomoki Ohtsuki (Kavli IPMU) arXiv:1404.0489 arXiv:1407.6195 with further updates From BCS theory to hadron

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Bootstrapping the genesis of Nambu-Goldstone Bosons from Quark Gluon Plasma

Yu Nakayama (Kavli IPMU, Caltech) in collaboration with Tomoki Ohtsuki (Kavli IPMU) arXiv:1404.0489 arXiv:1407.6195 with further updates

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From BCS theory to hadron mass

BSC mechanism = chiral condensation
Explains mass of fermions (with chiral symmetry)
Masslessness of pions (Nambu-Goldstone bosons)
It allows Landau-Ginzburg-Wilson paradigm of

phase transition… Nambu-san had been interested in condensed matter physics. Especially BCS theory fascinated him.

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To make it well-defined, consider massless flavors

(say up and down quarks)

At finite temperature, chiral symmetry is restored
How does Nambu-Goldstone bosons (dis)appear?
Is it first order or second order?
Huge controversies in lattice/theory community

What is the order of chiral phase transition in our real QCD?

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Various results in 2 flavor QCD

Claim Pisarski Wilczek (RG, 1-loop) 1st order (or O(4)) Karsch (lattice simulation) 2nd order Iwasaki et al (lattice simulation) 2nd order (O(4)?) D’Elia et al (lattice simulation) 1st order Ejiri et al (lattice simulation) 2nd order Aoki et al (lattice theoretical study) Cannot be O(4), 1st order. Delamotte et al (functional RG) 1st order Bonati et al (lattice simulation) 1st order Pelissetto et al (RG,6-loop) 2nd order (new fixed point) Grahl (functional RG) 2nd order Sato et al (RG with U(1) breaking) 2nd order Aharony et al (AdS/QCD, probe) 1st order Kiritstis et al (AdS/QCD, bulk) 3rd order

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Two main theoretical controversies

What is the effective symmetry that governs the chiral

phase transition?

(A) U(1)A is fully recovered (Pisarski-Wilczek, Cohen) (B) U(1)A is partially recovered (Aoki et al) (C) U(1)A is fully broken (Lee-Hatsuda)

Is there RG attractive fixed point with each

symmetries?(no fixed point  1st)

(A) Very controversial (no near d=4, but…)
(B) Not very much studied (no near d=4)
(C) Most probably O(4) Heisenberg

3d LG theory:

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Assume conformal invariance (rather than scale

invariance) is realized at RG fixed point

Ask for the consistency of 4pt functions
Give rigorous bounds on critical exponents for a given

symmetry

With more data, one may determine the existence of

fixed point

No explicit Lagrangian/Hamiltonian is necessary. Fully

non-perturbative and universal

Conformal bootstrap is a non-perturbative method to approach RG fixed point

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Schematic conformal bootstrap equations

Consider 4pt functions
Operator product expansions (OPE)
Crossing relations

Bootstrap machine No Maybe CFT Data

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BCS Superfluidity (Kos et al)

The existence of “kink” suggests RG fixed point

No Maybe Critical exponents for BCS superfulidity can be predicted:

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What to expect in conformal bootstrap?

Assume conformal invariant (conformal hypothesis)
Hypothesis: non-trivial behavior of the bound

indicates conformal fixed point

Try O(4) x O(2) (with deformations from possible

anomaly)

Pisarski-Wilzcek, Delamotte etc said no
Calabrese, Vicari etc said yes

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O(4) x O(2) fixed point (yes!)

If anomaly is suppressed 2nd order phase transition in QCD

No Maybe

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Summarizing O(4) x O(2) results and RG

We found a fixed point with O(4) x O(2) symmetry in bootstrap
Symmetry breaking pattern can be studied from bootstrap
This establishes the following RG picture

(A) U(1)A is fully recovered (Pisarski-Wilczek, Cohen) non-trivial fixed point exists: second order phase transition one-loop prediction by Pisarski-Wilczek is not trusted (B) U(1)A is partially recovered (Aoki et al)  Fixed point (A) is unstable under non-symmetric deformation with  first order phase transition (C) U(1)A is fully broken (Lee-Hatsuda)  Fixed point (A) is unstable under non-symmetric deformation with (this gives crossover exponent)  Second order phase transition with

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Future of conformal bootstrap

At strings 2013, Slava Rychkov said

“Conformal bootstrap is mathematically well-defined (convergent) so even if you cannot solve it, you can at least put it on a computer”

Perturbative QFT is not mathematically well-defined, so

if you cannot solve it, what can you do?

More applications to come: frustrated spin systems,