What can conformal bootstrap tell about QCD chiral phase transition? Yu Nakayama （ Kavli IPMU & Caltech ） In collaboration with Tomoki Ohtsuki

My memory of Alyosha Alyosha had visited Tokyo university once a year when I was PhD student there “Higher Equations of Motion in N=1 Supersymmetric Liouville Field Theory” After the talk, we went to (famous?) eel restaurant “Izuei” near Hongo campus.  Picture of us around the time Pictures are removed due to copyright issues

Is this sea eel or freshwater eel? Pictures are removed due to copyright issues

Higher equation of motion in Liouville Alyosha demonstrated higher EOM in Liouville theory Related to norm of logarithmic primary operators Appear in residues of recursion relation of Virasoro conformal blocks

QCD chiral phase transition and conformal bootstrap

What is the order of finite temperature chiral phase transition in QCD? 1 st order?2 nd order?

Chiral phase transition in QCD Consider SU(N c ) gauge theory with N f massless quarks When N f < N f *  confinement, chiral symmetry breaking at zero temperature SU(N f ) L x SU(N f ) R x U(1)  SU(N f ) V x U(1) Increasing temperature  chiral symmetry will be restored For N f = 2, still on-going debates if it is first order or second order… Lattice simulation is very controversial

I’m talking about REAL QCD. No supersymmetry. No large N. No holography. Hopeless?

Conformal bootstrap Non-perturbative constraint on CFT Surprising success in d=2 (BPZ) –Completely solves minimal models, Liouville theory etc More astonishing success in d=3, 4… –Constraint on possible operator spectrum –Determines critical exponent in 3d Ising model –Can tell if a unitary CFT with particular properties really exists or not

3d Ising bootstrap Any unitary CFT cannot exist above the region Determination of conformal dimension is as good as or even better than any other methods (e.g. epsilon expansion) El-ShowkEl-Showk, Paulos, Poland, Rychkov, Simmons-Duffin, VichiPaulosPolandRychkovSimmons-DuffinVichi Figure is removed due to copyright issues. See fig 3 of arXiv:

So how come conformal bootstrap has anything to do with QCD phase transition?

Pisarski-Wilzcek argument Suppose finite temperature chiral phase transition in massless QCD were 2 nd order. Landau-Wilson theory:  3 dimensional fixed point with the symmetry breaking pattern of U(N f ) x U(N f )  U(N f ) with order parameter Landau-Ginzburg Effective Hamiltonian: 1-loop beta function in  only O(2N f 2 ) symmetric fixed point at Nothing to do with QCD. Therefore QCD phase transition cannot be 2 nd order!

Problems? Effects of anomaly for N f = 2 –Some debate if U(1) anomaly effect is relevant or irrelevant at chiral phase transition point Can we trust epsilon expansion or even effective Landau-Wilson Hamiltonian? –Calabrese, Vicari etc claim they found a U(2) x U(2) symmetric fixed point at 5 or 6-loop. O(1000) Feynman diagrams (not visible at 1-loop)! –If correct, could be 2 nd order –Again there are a lot of debates… Is the fixed point conformal?

Our strategy Assume conformal invariance at the hypothetical fixed point (Assume U(1) anomaly is suppressed) Fixed point must have U(2) x U(2) symmetry but not O(8) enhanced symmetry Only one relevant singlet deformation: temperature Does such a CFT exist?  apply conformal bootstrap

Conformal bootstrap Assume spectrum: say Find a linear operator s.t., for all O with the assumed spectrum (e.g. x, y) If there exists such an operator, the assumed spectrum is inconsistent as unitary CFT Repeat the analysis We use semi-definite programming Bootstrap equation:

Recursion relation for conformal blocks No closed formula in d=3. Use recursion relation similar to what Alyosha proposed (Kos, Poland, Simmons-Duffin)KosPolandSimmons-Duffin 3 series of null vectors. For instance Formula is still conjectural (d=3 and higher) Higher dimensional analogue of Liouville theory for the proof? We need to evaluate conformal blocks as precisely and fast as possible.

Results

What we could expect KosKos, Poland, and Simmons-DuffinPolandSimmons-Duffin Figure is removed due to copyright issues. See fig 3 of arXiv:

Results on our bound O(8) bound U(2) x U(2) bound U(2) x U(2) fixed point? O(8) fixed point

Enhanced spectrum We can read the operator spectrum once we assume CFT lives at the boundary of the bound SpinO(8)1 x 11 x 33 x 33 c x 3 c 1 c x 3 c

With extra assumptions… We can get rid of symmetry enhancement by demanding no O(8) Noether current We may assume anomalous dimensions in non-conserved current operator Can we say anything about the fixed point proposed by Calabrese, Vicari etc? Can we approach genuine U(2) x U(2) fixed point (if any) from conformal bootstrap?

More severe constraint U(2) x U(2) fixed point?

What we have learned Existence of CFT can be tested by conformal bootstrap in d>2 There is no U(2) x U(2) fixed point which is more strongly bound than O(8) fixed point May suggest 1st order chiral phase transition We barely excluded the fixed point proposed by Calabrese et al with no extra assumption Extra assumption on non-conserved current gives strong constraint on the critical exponent at their proposed fixed point (if any).