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).