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 tocopyright issues

Is this sea eel or freshwater eel?

Pictures are removed due tocopyright issues

Higher equation of motion in Liouville• Alyosha demonstrated higher EOM in Liou

ville 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 tra

nsition in QCD?

1st order? 2nd order?

Chiral phase transition in QCD• Consider SU(Nc) gauge theory with Nf mas

sless quarks• When Nf < Nf* confinement, chiral sym

metry breaking at zero temperature

SU(Nf)L x SU(Nf)R x U(1) SU(Nf)V x U(1) • Increasing temperature chiral symmetry will be restored• For Nf = 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 t

heory 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 propert

ies 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-Showk, Paulos, Poland, Rychkov, Simmons-Duffin, Vichi

Figure is removed due to copyright issues. See fig 3 of arXiv:1203.6064

So how come conformal bootstrap has anything to do with

QCD phase transition?

Pisarski-Wilzcek argument• Suppose finite temperature chiral phase transitio

n in massless QCD were 2nd order.• Landau-Wilson theory: 3 dimensional fixed point with the symmetry b

reaking pattern of U(Nf) x U(Nf) U(Nf) with order parameter• Landau-Ginzburg Effective Hamiltonian:

• 1-loop beta function in

only O(2Nf2) symmetric fixed point at 　　　

　 Nothing to do with QCD.Therefore QCD phase transition cannot be 2nd order!

Problems?• Effects of anomaly for Nf = 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) s

ymmetric fixed point at 5 or 6-loop.

O(1000) Feynman diagrams (not visible at 1-loop)!– If correct, could be 2nd 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 inco

nsistent 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 propo

sed (Kos, Poland, Simmons-Duffin)

• 3 series of null vectors. For instance

• Formula is still conjectural (d=3 and higher)• Higher dimensional analogue of Liouville theory for t

he proof?

We need to evaluate conformal blocks as precisely and fast as possible.

Results

What we could expect

Kos, Poland, and Simmons-Duffin

Figure is removed due to copyright issues. See fig 3 of arXiv:1307.6856

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 spectrumWe can read the operator spectrum once we assume CFT lives at the boundary of the bound

Spin O(8) 1 x 1 1 x 3 3 x 3 3c x 3c 1c x 3c

0 1.8444 1.8445

0 1.1229 1.1226 1.1223 1.1224

0 3.3204 3.3194 3.3256 3.3197

1 2.0000 2.0000 2.0000 2.0000 2.0000

2 3.0000 3.0000

2 3.0194 3.0230 3.0771 3.0320

3 4.0288 4.0301 4.0316 4.0276 4.0260

4 5.0548 5.0577

4 5.0254 5.0254 5.0278 5.0277

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 bo

otstrap 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 give

s strong constraint on the critical exponent at their proposed fixed point (if any).