Is QCD a string theory?

Schrödinger’s cat?

In preparation &

TDC Phys .Lett. B637 (2006) 81.

Is Large Nc QCD a string theory?

In preparation &

TDC Phys .Lett. B637 (2006) 81.Schrödinger’s cat?

Is Large Nc QCD a an approximate string theory for

sufficiently excited states?

In preparation &

TDC Phys .Lett. B637 (2006) 81.Schrödinger’s cat?

QCD and the

Hagedorn Spectrum

In preparation &

TDC Phys .Lett. B637 (2006) 81.Schrödinger’s cat?

Outline

• Introduction

• Theoretical reasons to expect a Hagedornspectrum in QCD

• Testing the Hagedorn spectrum through thermodynamics on the lattice.

Prehistory

• Modern string theory grew out of the failed attempt to treat strong interactions as a string theory.

• While the approach had various phenomenological successes, it was ultimately abandoned as the fundamental theory of strong interactions.– Phenomenological issues (a pesky massless spin-2 meson

etc.)

– Theoretical consistency (negative norm states, tachyons)

– Emergence of QCD as a viable field theory for strong interactions

• String theory reemerged, phoenix-likefrom the ashes of this failure, as a putative theory of everything

Phenomenological successes of

the hadronic string picture

• String theories naturally give rise to Regge

trajectories with Mn2=n mo

2; Lmax=n.

• String theories naturally give rise to a Hagedorn

spectrum for the density of hadrons:

ρ(M)~m-2B exp(M/TH) , where TH , the Hagedorntemperature is a mass parameter.

Thermodynamically it corresponds to an upper

bound on the temperature of hadronic matter.

Phenomenologically both appear to

work at some level

• Regge trajectories with Mn2=n mo

2; Lmax=n.

Get data from

the PARTICLE

DATA BOOK

• String theories naturally give rise to a Hagedornspectrum for the density of hadrons: ρ(M)~exp(M/TH) , where TH , the Hagedorntemperature is a mass parameter. Thermodynamically it corresponds to an upper bound on the temperature of hadronic matter.

Bugg 2004

PDG

Integrated # of mesons with mass less then m

From Broniowski, Florkowski and Glozman 2004

QCD Strings

• Theoretical side:

– Strings are thought to emerge in QCD as flux tubes.

• Some theory reasons why an area law for Wilson loops may be expected (eg. strong coupling limit, but is it relevant?)

• Lattice evidence for an area law in quenched QCD

• Lattice evidence in quenched QCD for static flux tubes

– All this evidence is essentially static

• In a real sense this is not a real probe of the hadronic string which is essentially dynamic.

• Key issue is to find ways to probe the dynamics of a stringy picture of QCD. Hadron spectroscopy does this.

Phenomenology and QCD Strings

• Hadronic strings are thought to emerge in QCD

as flux tubes only for highly excited states.

– Picture becomes valid when excitations are high

enough so that the length of the flux tube is much bigger than its width and string motion dominates

– Dynamics of flux tubes give excitation spectrum of hadrons.

• Mesons are open strings.

• Glueballs are closed strings.

qq

• Flux tubes in QCD are only a sensible

description for highly excited states.

– Restriction to highly excited states critical on the

stringy side as well

– Naïve string theory in four dimensions is diseased

• Tachyons

• Massless tensors

– However these diseased states are low-lying

– One might hope that a theory which approaches a

string theory in four dimensions might work for highly excited states

The Large Nc Limit and QCD Strings

• Phenomenology and theory of QCD strings are

both only clean at large Nc.

– Spectral results for hadrons (Regge, Hagedorn) depend

on hadron masses which are well defined if decays are suppressed.

• Meson & glueball widths are suppressed at large as Nc

-1 and Nc-2 respectively.

– Flux tubes break due to the creation of quark-antiquark

pairs. This destroys a simple string picture.

• This is suppressed by a factor of 1/Nc.

Does the hadron spectrum of large Nc QCD approach

that of a string theory for highly excited states?

• This is a well-posed question (or more precisely can be

made into one) and thus in principle is testable.– There is a potential issue in ordering of limits

• mhadron→ ¶ Nc→ ¶

• Idea is supposed to work at high enough mass so that tube “looks” string-like.

• The rate of QCD string breaking is

(Casher, Neuberger, Nussinov 1979)

Proportional to mass and inversely with Nc

– Assume here that Nc→ ¶ limit taken first so stringy description survives for high lying states.

c

bodyN

mm

σΛΓ − ~)(2

What features of a string spectrum do we expect if

large Nc QCD is “stringlike” for high excitations?

• An approximate Regge spectrum

– Mn2=c n with L≤n

• An approximate Hagedorn spectrum

– Exponentially growing spectrum

• Density of hadrons goes to infinity exponentially in the energy (up to power law corrections)

• Density of hadrons of any spin goes to infinity exponentially inthe energy (up to power law corrections)

– A weaker conditiontion: the density of hadrons of any

spin diverges asymptotically

• This divergence is some sense “exponentially large”

How can you tell?To paraphrase what Dorothy Parker said when

she heard that Calvin Coolidge was dead

A similar question was asked by the current speaker when he heard that a different former

Republican president had Alzheimer’s

How can you tell?

• The folklore is that QCD (and all confining

theories) will have a Regge Spectrum with an

exponentially growing density of hadrons

• So far as I know this folklore is based on the

notion that QCD does become an effective string

theory.

– I know of no demonstration from within QCD that is in

fact true.

– Thus a key issue is how can you tell if QCD really

does behave this way.

In principle the spectrum can be

determined via lattice simulataions• Lattice simulations with moderately large Nc are

certainly possible and have been done.

• However, extracting excited meson masses from

correlation functions becomes exponentially

difficult as one goes up in the spectrum.

– Direct tests of the high lying states in the spectrum of

large Nc QCD will be impractical for the foreseeable future.

• Can properties of the large Nc QCD spectrum of

excited hadrons be probed without direct

calcualtions of masses?

• I know of no way to probe the Regge spectrum

(beyond the obvious fact that it is not too bad

emprically)

• There are ways to see if QCD is consistent with

a Hagedorn spectrum

– Theoretical Arguments

• Need some relatively standard assumptions

• Can show that the weaker condition that the density of hadrons of any spin diverges asymptotically is correct

• Can show that the Hagedorn spectrum is consistent with this and arises naturally

– Thermodynamics which are testable on the lattice.

Theoretical Arguments

Why the weaker condition that the density of hadrons

of any spin diverges asymptotically still has strength.

• Note that this is qualitatively different from what one expects in a potential model for quark-antiquarkinteractions

• Potential models have level repulsion: degeneracies“never” happen except for states with different conserved quantum numbers.

• If one has a potential model which gives rise to a Reggespectrum, without degeneracies then density of hadrons of a given spin s goes as ρs(m)~m-1/2

• To get an exponentially rising number of states requireisdegeneracies. The number of states at each n must grow as en.

– In contrast, string theories have an infinite number of conservation laws and high levels of degeneracy

rV

Vpm

s

s

σ=

+= 222 ˆ

A simple relativistic potential model with a linear rising potential

This yields a Regge spectrum but has only one

state of given spin per n

If QCD has an infinite density of hadrons for each spin asymptotically it is very different from a potential model of this sort.

constant

Strategy for showing that that the density of hadrons of

any spin diverges asymptotically

• Theoretical inputs:

– Confinement in the sense of no colored states in the

physical space.

– Asymptotic freedom

• Manifest through “semi-local duality” in correlation functions

• Start with Witten’s strategy 1979 for showing

that there are an infinite number of

mesons/glueballs with any quantum number

– Trick is to generalize this to show that there is an infinite density of mesons/glueballs with any quantum

number

Witten’s argument:

Consider a correlation function of a local color singlet operator J

of dimension D and which at large Nc is “ color indivisible” in

having a single color trace. For simplicity consider a scalar

[ ]

)log(

)0()()(

2)2(2

22

2

222

QQconst

MQ

c

qQJxJTedxQ

D

n n

n

xiq

−

⋅

→

+−=

−≡≡Π

∑

∫

Large Nc

Large Q (Asymptotic

Freedom)

Two forms are incompatible if the sum over hadrons is finite since

the large Nc result with any finite number of poles drops off as

1/Q2 asymptotically .

Ergo there must be an • number of mesons with this quantum #.

hadrons

A modest generalization:

If one accepts “semi-local duality”, this also implies a nonzero

density of hadrons with this quantum number

Λ

∆+

+−=

+−≡Π

∫

∫

2

2

2

2

2

1)(

)()(

q

isq

sds

isq

sdsq

pert

s

s

ε

ρ

ε

ρ

In effect, as long as one is far from the position of the spectral

strength on scales of Λ2, so that you cannot really resolve the

individual spectral features in detail, the large Nc discrete spectrum

“looks” effectively like its perturbative value.

For a generic

complex q

Distance to singularity in

Π(q2)

Spectral function

When you are far enough

away---and have enough

beer---the correlations

functions seem perturbative

In effect when smeared over

scales of order Λ, the

spectral function

approaches the pertubative

one.

This is possible only if there number density of

hadrons is high enough so after smearing over

this scale it looks smooth.

This argument shows that at large Nc any color singlet

“indivisible” operator creates a non-zero density of

hadrons with the quantum #s of the operator.

The “weak condition” is that the density of hadrons of any

quantum number diverges as one goes asymptotically high

in the spectrum.

This can happen in one of two ways:

1. The density of hadrons created by a given operator

diverges.

2. There is a divergent number of operators with the same

quantum numbers but which couple to distinct hadrons.

Scenario 2 can be shown to happen; it is the analog of string

theories having an infinite number of conserved quantities

Multiple independent operators

• For simplicity focus on glueballs/closed strings

rather than meson/open strings

– Analogous arguments can be made for mesons and

yield same conclusions but they are more involved.

• Construct gauge invariants local “color-

indivisible” operators which contain n gluons:

( )ψωγδαβψωγδαβ GGGTrJn

L=,...,,

Color trace

This is “color indivisible”; given confinement it makes

only a single glueball at large Nc.

[ ]

'for as0

2 since'for)log(

)0()()(

2

2)44(

22'2',

nnQ

nDnnQQconst

qQJxJTedxQ

n

nnxiqnn

≠∞→→

==→

−≡≡Π

−

⋅

∫

A useful result:

Easy to see why it goes to zero for n≠n’: it necessarily

requires extra powers of as which go to zero at large Q

24Eg.

Where Lorentz have been suppressed

[ ]

baba

Q

qQJxJTedxQn

b

n

a

xiqn

ab

≠

∞→→

−≡≡Π ∫⋅

and operators ofset specfic ain ,for

as0

)0()()(

2

222

A second useful result:

There are many operators with fixed n. They differ due to

the Lorentz indices. For any fixed value of n, there are at

least 3n/n distinct operators whose mixed correlators vanish

as Q2→¶ .

Where a, b represent the Lorentz operators indices and the set

contains more than 3n/n distinct operators

To proceed, recall that at large space-like momentum

correlators look like those of free fields.

To choose operators a set of operators, go to any given Lorentz

frame and start by considering operators made from gluo-electric

fields only.

(One could equivalently picked color magnetic fields only)

( )zyxlji

GGGTrJ lji

n

lji

,, valueson the can take ,,,

0000,...,0,0

K

L=

Color trace

For large Q, each qluo-electric operator creates a single

polarized gluon with fixed color (specified by two indices) and

the direction of propagation ┴ to E.

The corrleator vanishes unless the second current annihilates a

gluon with the same momentum, color and polarization as made by

the current creating it. This requires the Lorentz index specifying

the direction of the E field to be the same in the annihilating current

as in the creating current for a gluon of the same color.

The color traces means that cyclic re-orderings of the colors do not

change the operators

( )0000,...,0,0 lji

n

lji GGGTrJ L=

Color trace

[ ] as0

)0()()(

2

',0',...,0',0',0,...,0,0

2

0',0',...,0',0';0,0,...,0,0

∞→→

≡Π ∫⋅

Q

JxJTedxQn

omlji

n

molji

xiqn

mljimlji

unless L

MMor

'

'

'

or

'

'

'

lm

ij

mi

mm

jj

ii

=

=

=

=

=

=

Thus

Thus, the maximum set of operators of this type whose cross-

correlators all vanish at large Q contains distinct operators with

n ordered labels which can take the values x, y, z and which are

distinct from all other operators with the same labels or cyclic

permutations

How big is such a set of operators for a given n?

This is in general a non-trivial combinatoric problem.

However it is easy to see* that when n is prime, the number of

elements of the, N, is given by N= (3n+3n-3) /n and that when n

is not prime the number of elements, N satisfies N > (3n+3n-3) /n .

Thus in general N > 3n/n which is of course exponentially large

* Thanks to Michael Cohen for pointing out how to see this

The Lorentz quantum numbers control the allowable quantum

numbers created by the operators. It is easy to see that there

must be an exponentially large number of states created by

operators in this class with any spin which is much less then n.

Associated with each operator in the class for any given n is a

nonzero density of hadrons with the appropriate quantum

number.

A crude argument:

Since at high Q the mixed correlators go to zero (both within

each set for fixed n and for operators of different n), at high

mass each distinct operator to good approximation couples only

to states associated with that operator and not others. As there

are an exponentially large number of operators with any spin

the density of hadrons with any spin diverges as one goes

sufficiently high in the spectrum for the perturbative region to

set in for an arbitrarily large class of operators.

Actually this is a bit of a swindle:

However a somewhat more sophisticated argument leads to the

same conclusion: the density of hadrons with any spin is

arbitrarily large as one goes sufficiently high in the spectrum.

The basic point is that in order to have k distinct operators have

no cross-correlators perturbatively requires at least k states per

“smearing bin” to which they can couple.

.This confirms the weak condition

How about showing that QCD has a Hagedorn

spectrum?

A natural scenario

• Suppose that for n>>1 , each class of operator enumerated above has a smeared spectral strength which does not approximate its perturbative value until s=const n2 or higher.

• Suppose further that below this value the spectral strength is greatly suppressed.

• A Hagedorn spectrum with exponentially growing spectrum automatically happens due to the exponentially large number of operators which set in with the mass.

Is such a scenario consistent with standard

pertubative results.

• Check this by setting the spectral strength to zero before some value and see how high up you need to go in mass before one reproduces approximately the perturbativeresult for a sensible observable.

• Cannot just look at correlator in momentum space as it is sensitive to the UV (it requires subtractons)

• A useful UV insensitive observable is the p=0 correlator as a function of time

tsesdst

−

∫≡Π )()( ρ

simple dimensional analysis

)1(2~)( −npert

n ssρ

Define

)34(

),34(

)(

)(),(

0

−Γ

−Γ=≡

∫

∫∞

−

∞−

n

tmn

esds

esdstmR

tspert

n

m

tspert

n

n

ρ

ρ

Rn tells us what fraction of the total perturbative result we have

at specfic value of t if we cut off the spectrum below m. It

quantifies how much cutting off can be accommodated without

doing serious violence to the perturbative result.

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

m t

R

n=2

n=3

n=4 n=5

n=6 n=7

n=8 n=9

n=10

Linear onset with n keeps the region in which the

value of t at which perturbation theory begins to

describe Π(t) . Consistent with a Hagedorn spectrum

Summary of Theoretical Argument

• By generalizing Witten’s argument to many

operators it is possible to show that the desnity

of hadrons of any spin diverges asympototically.

– This is consistent with the Hagedorn spectrum of a

string description and highly unnatural in a potential

model

• If the mass at which the perturbative spectral

function sets in, grows linearly with the number

of gluons in the operator then there spectrum is

of the Hagedorn form

Thermodynamic Probes

Review of Thermodynamics of

QCD at Large Nc

• Regime of interest: T~Nc0

• Baryons are heavy ~ O(Nc1) and thus irrelevant

thermodynamically for T~Nc0

• At small T, QCD should look like a gas of weakly

interacting mesons and glueballs (hadrons don’t

interact at large Nc): a confining hadronic phase exists

at large Nc

– At large T energy density scales as Nc2(gluon loops

dominate)

– Phase transition must occur (unlike for Nc =3 and nonzero

quark mass); as Nc→∞ any cross-over must become sharp.

Thermodynamic Functions for Hadronic

phase at Large Nc

Free gas thermodynamics for non-interacting particles

summed over species. Energy density and pressure

given by:

PPPP((((mmmmkkkk,,,, TTTT )))) ====((((mmmm2222kkkkTTTT

2222

2222ππππ2222

)))) ∑∑∑∑∞∞∞∞

nnnn====1111 KKKK2222

((((nnnnmmmmkkkk

TTTT

))))→→→→ mmmm3333////2222TTTT 5555////2222

22223333////2222ππππ3333////2222eeee−−−−mmmm////TTTT

εεεε((((mmmmkkkk,,,, TTTT )))) ====∫∫∫∫ dddd3333pppp((((2222ππππ))))3333

√√√√pppp2222++++mmmm2222

kkkk

eeee

√√√√pppp2222++++mmmm2222

kkkk////TTTT−−−−1111

→→→→ TTTT 3333////2222mmmm5555////2222kkkk

eeee−−−−mmmmkkkk////TTTT

((((2222ππππ))))3333////2222==== mmmmkkkk nnnnkkkk

where ρ(m) is the density of states of hadrons and

εεεε((((TTTT )))) ====∫∫∫∫∞∞∞∞0000

ddddmmmmρρρρ((((mmmm))))((((mmmm,,,, TTTT )))) ++++OOOO((((NNNN−−−−1111cccc ))))

PPPP ((((TTTT )))) ====∫∫∫∫∞∞∞∞0000

ddddmmmmρρρρ((((mmmm))))PPPP ((((mmmm,,,, TTTT )))) ++++OOOO((((NNNN−−−−1111cccc ))))

The arrow indicates the behavior for m� T .nk is the number density of the nth species.The O(N−1

c ) corrections arise from hadron-hadron interactions.

Hadronic Phase: ε ε ε ε , P ~ Nc0

Deconfined Phase: ε ε ε ε , P ~ Nc2

2

cN

ε

TcT

2

cN

P

TcT

Consequences of a Hagedorn Spectrum

– The Hagedorn Spectrum:

• Functional Form:

• Where A is a constant, TH is the Hagedorn temperature and D┴

is the number of effective transverse dimensions of the string.

ρρρρ((((mmmm))))→→→→ ρρρρaaaassssyyyy

((((mmmm)))) ==== AAAA ((((mmmm////TTTTHHHH))))−−−−2222BBBB eeeemmmm////TTTTHHHH

wwwwiiiitttthhhh BBBB ==== ((((DDDD⊥⊥⊥⊥ ++++ 3333))))////4444

A naïve QCD StringDDDD⊥⊥⊥⊥ ==== 2222

Large Nc in hadronic phase for T → TH

Combine large Nc thermodynamics (non-interacting hadrons) with Hagedorn spectrum (string theory). As T → TH , the system is dominated exponential growth in states.

εεεεaaaassssyyyy ==== ((((2222ππππ))))−−−−3333////2222AAAATTTT ((((9999−−−−DDDD⊥⊥⊥⊥))))////2222 TTTT7777////2222HHHH ××××

((((TTTTHHHH −−−− TTTT ))))((((DDDD⊥⊥⊥⊥−−−−6666))))////2222 ΓΓΓΓ((((6666−−−−DDDD⊥⊥⊥⊥

2222

))))iiiiffff DDDD⊥⊥⊥⊥ <<<< 6666

−−−− lllloooogggg ((((((((TTTTHHHH −−−− TTTT ))))////TTTTHHHH)))))))) iiiiffff DDDD⊥⊥⊥⊥ ==== 6666

PPPPaaaassssyyyy ==== ((((2222ππππ))))−−−−3333////2222AAAATTTT ((((9999−−−−DDDD⊥⊥⊥⊥))))////2222 TTTT5555////2222HHHH ××××

((((TTTTHHHH −−−− TTTT ))))((((DDDD⊥⊥⊥⊥−−−−4444))))////2222 ΓΓΓΓ((((4444−−−−DDDD⊥⊥⊥⊥

2222

))))iiiiffff DDDD⊥⊥⊥⊥ <<<< 4444

−−−− lllloooogggg ((((((((TTTTHHHH −−−− TTTT ))))////TTTTHHHH)))))))) iiiiffff DDDD⊥⊥⊥⊥ ==== 4444

As T → TH both energy density and pressure diverge if d┴ ≤4. This includes the navie string/simple flix tube of d┴=2).

This is means that TH is unreachable in the hadronicphase. This is a concrete realization in QCD of Hagedorn’s old idea of an unreachable hadronictemperature.

Note that the idea is only well defined at large Nc.

εεεε ∼∼∼∼ ((((TTTTHHHH −−−− TTTT ))))−−−−2222

PPPP∼∼∼∼ ((((TTTTHHHH −−−− TTTT ))))−−−−1111for d┴=2

• If one were to observe the energy density or pressure behaving this way (say in a lattice calculation extrapolated to large Nc) one would have a direct test of confinement ocuring via a stringy mechanism at large Nc.– Note this is different from the behavior of any typical stat

mech system which have

where α is the critical exponent. Here

• However, at large Nc, there will be a first order phase transition to a deconfined phase prior to TH.– A phase transition is expected on general grounds.

The system should become perturbative for temperatures which are large but order Nc

0. As one gets to this regime the free energy density is dominated by gluons (Nc

2) which indicates a break the hadronic phase (free energies ~Nc

0 )

– This is supported by numerical lattice studies (Teper & collaborators).

εεεε ≈≈≈≈ εεεε0000 ++++ CCCC((((TTTT −−−− TTTTcccc))))1111−−−−αααα wwwwiiiitttthhhh αααα ≤≤≤≤ 1111

α = 4− D⊥

2 → 3

Can one observe Hagedorn behavior in the thermodynamics to verify stringy behavior?

• Previous arguments would be bootless otherwise.

• Fortunately it is possible, at least in principle:

study the metastable hadronic phase above Tc.

• Lattice studies of metastable phase are practical:metastable phase of pure glue theory up to Nc=12has been studied on the lattice (B. Bringoltz & M. Teper, Phys.Rev. D73 (2006) 014517)

– Studies focused on string tension rather than thermodynamic variables (vanishing string tension assumed to occur at Hagedorn temp)

– Extrapolation to Th assuming a spinodal point (2nd order transition for metastable phase) with XYZ model critical exponents.

• If present argument is correct, the metastablephase does not have a spinodal point at TH

– Rather TH is energetically unreachable (at large

Nc)..

• However Bringoltz-Teper extrapolation based on

string tension having usaul stat mech critical

exponents may still be Kosher: the string tension is

NOT a local operator and does not correspond to

any thermodynamic quantity.

• Clearly lattice studies of the energy and pressure of

the metastable phase are needed to test this

picture.

• There is one important subtlety in doing this:the limits T→ TH and Nc→ ∞ do not commute.

• As T→ TH , in the hadronic phase the denistyof hadrons diverges and hadronic interactions become important even though they are suppressed in the 1/Nc expansion.

• The free gas expression then breaks down and and so does the Hagedorn behavior

• Presumably beyond the breakdown there is a spinodal point.

• However, as Nc gets large the size of the breakdown region goes to zero.

Log(εεεε)

T

Universal Hagedornbehavior

Nc dependent spinodal region

Tc

Essential to do numerical studies in the universal regime

Summary of Thermodynamic

probes

• At large Nc the Hagedorn temperature is above transition temperature for a first order phase transition.

• In principle thermodynamic studies on the lattice can probe the metastable hadronicphase above the transition temperature.

• Critical exponents near the spinodal point of the metastable phase test the effective transverse dimension of the QCD string