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 mass Nambu-san had been interested in condensed matter physics. Especially BCS theory fascinated him. • 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…
What is the order of chiral phase transition in our real QCD? • 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
Various results in 2 flavor QCD Claim Pisarski Wilczek (RG, 1-loop) 1st order (or O(4)) 2 nd order Karsch (lattice simulation) 2 nd order (O(4)?) Iwasaki et al (lattice simulation) 1 st order D’Elia et al (lattice simulation) 2 nd order Ejiri et al (lattice simulation) Cannot be O(4), 1 st order. Aoki et al (lattice theoretical study) 1 st order Delamotte et al (functional RG) 1 st order Bonati et al (lattice simulation) 2 nd order (new fixed point) Pelissetto et al (RG,6-loop) 2 nd order Grahl (functional RG) 2 nd order Sato et al (RG with U(1) breaking) 1 st order Aharony et al (AdS/QCD, probe) 3 rd order Kiritstis et al (AdS/QCD, bulk)
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) 3d LG theory: • Is there RG attractive fixed point with each symmetries?(no fixed point 1 st ) • (A) Very controversial (no near d=4, but…) • (B) Not very much studied (no near d=4) • (C) Most probably O(4) Heisenberg
Conformal bootstrap is a non-perturbative method to approach RG fixed point • 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
Schematic conformal bootstrap equations • Consider 4pt functions • Operator product expansions (OPE) • Crossing relations CFT Data No Bootstrap machine Maybe
BCS Superfluidity (Kos et al) No Maybe The existence of “ kink ” suggests RG fixed point Critical exponents for BCS superfulidity can be predicted:
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
O(4) x O(2) fixed point (yes!) No Maybe If anomaly is suppressed 2 nd order phase transition in QCD
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
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, quantum liquid, asymptotic safe/ non-renormalizable (Gross-Nuveu, NJL) etc …
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