Effective string models for confining LGTs.

M. Caselle

STRONGnet Workshop; October 2011

Plan of the Talk

• A brief history of the effective string picture of confinement

• String corrections in Wilson loop expectation values.

• String corrections in Polyakov loop correlators.

• Flux tube thickness.

1

A brief history of effective string picture of confinement

1974-1979 Nielsen-Olesen, ’t Hooft, Wilson, Polyakov, Nambu....

Conjecture:

Two color sources in a confining gauge theory are bound together by athin flux tube, which can fluctuate like a massless string.

Seminal ideas:

• large N limit• use of duality to understand confinement• 3d gauge Ising model described by a “fermionic string” (Polyakov)• use LGT to test the conjecture

2

1980 Luscher, Symanzik, Weisz, Munster..

Universal signatures of the quantum fluctuations of the confining string:

• Universal quantum corrections (”Luscher term”) to the potential.

V (R) ∼ σR− (d− 2)π24R

• Logarithmic increase of the flux tube thickness.

These effects could in principle be observed in LGT

1982-1986 Alvarez, Pisarski, Olesen ....

Finite T version of universal string corrections

• Deconfinement driven by the quantum fluctuations of the string.• Effective string prediction for the ratio Tc/

√σ.

3

1985 Olesen, Alvarez ....

The flux tube as an effective string

• It can only describe the “large distance” (R > 1/√σ) behaviour of

the interquark potential• First indications that the corrections due to the anomaly appear at

high order in 1/R

1983-1987 Dietz and Filk + Nordita group + Torino group + deForcrand, et al.

Functional form of the effective string corrections

• Extension of the universal quantum corrections to other geometriesand to finite size shapes.• Higher order corrections.• Connection with CFT’s

4

1980-1990

First attempts to test effective string predictions with MC simulations

• Main problem: too large statistical and systematic errors.• Results: qualitative agreement with effective string predictions, but no

clearcut evidence.

1994-2001 Torino group + M. Hasenbusch and K.Pinn

Study of effective string corrections in the 3d gauge Ising model:

• test of the log increase of the flux tube width with the distance.• test of the effective string predictions for Wilson loops, Polyakov loops

and interfaces

5

1999-2001 Lucini and Teper, Necco and Sommer, Bali...

First unambiguous evidences (with standard MC algorithms) in favour ofa bosonic effective string theory in non-abelian LGT’s.

2002-2011 Luscher and Weisz, Teper and collaborators, Pepe,Lucini, Kuti, Drummond......

A new “multilevel” algorithm is proposed for non-abelian LGT’s.

• Exponential reduction of statistical errors: large interquark distancescan be reached• SU(N) LGT’s both in d = 3 and d = 4 are well described by a bosonic

effective string theory.• Unexpected: the transition from perturbative to string behaviour takes

place at surprisingly small distances.• Main result: the Nambu-Goto effective string works impressively well

both for the ground state and for the excited states

6

2004-2010 Torino group + M. Hasenbusch

• Using string inspired techniques the explicit expression of thecorrections due to the Nambu-Goto string are obtained to all ordersfor Polyakov correlators, interfaces and Wilson loops.• At the same time, high performance algorithms based on the duality

transformation for 3d gauge Ising model allow to succesfully test thesepredictions up to the next to leading order

2008-2011 Pepe, Wiese + Torino group

First studies of the behaviour of the flux tube thickness at the next toleading order both in SU(N) and in Ising. The results agree remarkablywell with the Nambu-Goto predictions

7

2004-2010 Luscher and Weisz, Drummond, Aharony andcollaborators

Universality theorems for the effective string

• Effective string corrections are universal up the next to leading orderin d > 3 and up to the third order in d = 3

8

Main contexts in which effective string contributionsmay appear

• Luscher type corrections to the interquark potential (lot of papers ...)

• Spectrum of flux tube excited states (Teper, Majumdar, Brandt ...)

• Finite T dependence of the string tension and prediction of Tc/√σ

• Flux tube thickness

• Low lying states of glueball spectrum (Lucini, Teper....)

• Hagedorn transition (Meyer + Torino group) (see Feo’s talk tomorrow)

9

String theory and Lattice Gauge Theories

Conjecture: Two color sources in a confining gauge theory are boundtogether by a thin flux tube, which can fluctuate like a massless string.

Main consequence:

• linearly rising potential

• quantum corrections to the interquark potential (Luscher term)

• log increase of the flux tube thickness

10

• Potential V (R) between two external, massive quark and anti-quarksources from Wilson loops

〈W (L,R)〉 ∼ e−LV (R) (large L)

V (R) = − limL→∞

1L

log 〈W (L,R)〉

In the limit of infinite mass quarks (pure gauge theory) we find thefamous “area law” for the Wilson loop

• Area law ↔ linear potential

〈W (L,R)〉 ∼ e−σRL; V (R) = σR+ . . .

σ is the string tension

11

Quantum corrections and effective models

The area law is exact in the strong coupling limit (β → 0). As thecontinuum limit (β → ∞) is approached quantum corrections becomeimportant.

• Leading correction for large R: The “Luscher term”

V (R) = σ R− π

24d− 2R

+O

(1R2

).

12

It can be obtained summing up the quantum fluctuations of the transversedegrees of freedom of the string treated as d−2 massless modes [Luscher,Symanzik and Weisz, (1980)]. This is the “gaussian approximation” ofthe effective string:

〈W (L,R)〉 ∼ e−σRL∫DXie−

σ2

Rdx0dx1{(∂0

~X)2+(∂1~X)2+int.s}

• Can also be derived assuming a “SOS picture” for the fluctuations of thesurface bordered by the Wilson loop → two-dimensional conformal fieldtheory of d− 2 free bosons

• Consequence: Using known results of CFT’s we can predict the behaviourof the Wilson loop for finite values of L.

13

< W (R,L) >= e−(σRL)Zq(R,L)Where Zq(R,L) is the partition function of d − 2 free massless scalarfields living on the rectangle defined by the Wilson loop: R× L

Zq(R,L) ∝[η(τ)√R

]−d−22

where η(τ) is the Dedekind η function and τ = iL/R.

• The L↔ R simmetry is ensured by this identity

η

(−1τ

)=√−iτ η(τ)

known as the “modular” transformation of the η.

14

• Defining: F (R,L) ≡ − log < W (R,L) > and expanding the Dedekindfunction

η(τ) = q124

∞∏n=1

(1− qn) ; q = e2πiτ ,

one finds

F (R,L) = σRL−(d− 2)[πL

24R+

14

logR]

+ ...

From which we find as anticipated:

V (R) = σ R− π

24d− 2R

+ ...

15

This result is universal: all confining gauge theories flow toward thisgaussian effective action and thus share the same behaviour for theinterquark potential (for large enough R) with no dependence on the gaugegroup and a trivial (linear) dependence on the space-time dimensions.

Finding a string description able to select among different gauge groups(hence finding the ”QCD string theory”) requires going beyond the gaussianapproximation

The simplest proposal is the Nambu-Goto action

16

The Nambu-Goto string

Action ∼ area of the surface spanned by the string in its motion:

S = −σ∫dξ0dξ1

√det gαβ (1)

where gαβ is the metric “induced” on the w.s. by the embedding:

gαβ =∂XM

∂ξα∂XN

∂ξβGMN (2)

ξα = world-sheet coords. (ξ0 = proper time, ξ1 spans the extension of thestring)

17

Connection with the gaussian model

• One can use the world-sheet re-parametrization invariance of the NGaction to choose the so called “physical gauge”:

– The w.s. coordinates ξ0, ξ1 are identified with two target spacecoordinates x0, x1

• One can study the 2d QFT for the d − 2 transverse bosonic fields withthe gauge-fixed NG action

Z =∫DXie−σ

Rdx0dx1

√1+(∂0

~X)2+(∂1~X)2+(∂0

~X∧∂1~X)2

= e−σRL∫DXie−

σ2

Rdx0dx1{(∂0

~X)2+(∂1~X)2+int.s}

18

The anomaly problem

• Problem: This gauge fixing is anomalous (unless we are in d = 26)

• “Effective” solution: It can be shown (Olesen, 1985) that the correctionsdue to the anomaly decay at least as 1/R3 thus one can trust theperturbative expansion at least up to the order 1/R2 i.e. the first orderbeyond the gaussian approximation.

• “Stringy” solution: The Nambu-Goto action can be rewritten (order byorder in 1/L) so as to be anomaly-free in any dimension (Polchinski andStrominger, 1991):

19

Polchinski and Strominger action:

Seff =1

4π

∫dτ+dτ−

[1a2

(∂+X · ∂−X)

+(D − 26

12

)(∂2

+X · ∂−X)(∂+X · ∂2−X)

(∂+X · ∂−X)2+O(L−3)

], (3)

where τ± are light-cone world-sheet coordinates, and a is a length scalerelated to the string tension.

As a consistency check, it can be shown (Luscher, Weisz, Drummond,Hari Dass, Mattlock, 2004) that the first perturbative correction beyondthe gaussian contribution in both frameworks is the same.

20

Finite Temperature LGTs

• Finite temperature can be realized by imposing periodic boundaryconditions in the (euclidean) “time” direction

• The (finite) temperature is given by the inverse of the lattice length inthe compactified time direction T = 1/L

• A new set of observables can be constructed: the Polyakov loop which isthe trace of the ordered product of timelike variable along a timelike axisof the lattice and behaves as an order parameter for the deconfinementphase transition

• The interquark potential is given by the correlator of two Polyakov loops.

V (R, T ) = − log(< P (x)P+(x+R) >) T = 1/L

21

22

Effective string corrections

〈P †(R)P (0)〉 = Z(L,R) ∼∫DXe−S[X] .

If we choose a Nambu-Goto action for the effective string (and set d = 2+1)then:

S[X] = σ

∫ L

0

dτ

∫ R

0

dς√

1 + (∂τX)2 + (∂ςX)2 .

23

Perturbative expansion in powers of 1/(σRL)

Z(L,R) = e−σRL · Z1 ·(

1 +F4

σR2+

F6

(σR2)2+ · · ·

)

The leading order of this expansion: Z1 corresponds to the partitionfunction of a free boson on a cylinder

There is no L ↔ R simmetry: the modular transformation of theDedekind function τ → −1

τ allows to relate the T and R dependences ofthe interquark potential, i.e. to predict the behaviour of σ as a function ofthe temperature T

In string theory this modular transformation is known as open ↔ closedstring duality

24

Z1 can be easily evaluated

< P (x)P+(x+R) >∝ e−(σRL)

[η(i

L

2R)]−(d−2)

where

η(τ) = q124

∞∏n=1

(1− qn) ; q = e2πiτ , τ = iL

2R

to be compared with analogous term in the Wilson case:

< W (R,L) >∝ e−(σRL)

[η(iLR)√R

]−d−22

In the large L/R limit (i.e. τ >> 1) the two expression give the sameLuscher term

25

Defining: F (R,L) ≡ − log < P (x)P+(x + R) > and expanding theDedekind function one finds

F (R,L) = σRL−(d− 2)[πL

24R+ ...

]From which we find as anticipated:

V (R) = σ R− π

24d− 2R

+O

(1R2

).

In the large R/L limit (i.e. τ << 1) instead we find

V (R) = σ R− (d− 2)π

6L2R+ ...

26

F4 was evaluated for the first time in (Dietz-Filk 1983):

F4 =π2L

1152σR3

[2E4

(iL

2R

)− E2

2

(iL

2R

)],

where Ek are Eisentein functions of order k and are defined as:

E2(τ) = 1− 24∞∑n=1

σ1(n)qn (4)

E4(τ) = 1 + 240∞∑n=1

σ3(n)qn (5)

q = e2πiτ , (6)

where σ1(n) and σ3(n) are, respectively, the sum of all divisors of n(including 1 and n), and the sum of their cubes.

27

We obtain in the large R limit

V (R) = σ R− (d− 2)π

6L2R− (d− 2)

π2

72σL4R+ ...

28

All orders calculations in the Nambu-Goto case

In the Polyakov loop case the Nambu Goto partition function can beevaluated exactly if one neglects the anomaly

F =12

∑k

cke−LEk(R) ,

where the coefficient ck are the partitions of integers (they come form theexpansion of the Dedekind function) and (Arvis 1982)

Ek(R) = σR

√1 +

2πσR2

(k − d− 2

24

)

29

Performing a L ↔ R transformation (i.e. going to the closed stringchannel)

F = 2πL(πσ

)d2−2 ∑

k

ckG (R;M(k))

where G (R;M) = propagator of a scalar field of mass M over the spatial distance ~R:

G(R;M) =1

2π

„M

2πR

«d−32

Kd−32

(MR) ,

the mass M(k) is that of a closed string state with k representing the total oscillator

number:

M2(k) = (σL)

2

»1 +

8π

σL2

„k −

d− 2

24

«–

30

Two important remarks

• From the lowest mass (k = 0, m = 1) in the closed string channel:

M2(1, 0) = (σL)2

[1− π(d− 2)

3σL2

]one can obtain an estimate for the deconfinement temperature (recallthat T = 1/L) which in this framework appears as a consequence of thetachionic state present in the NG string (Olesen, 1985):

Tc√σ

=

√3

π(d− 2)

This estimate turns out to be in very good agreement with MCsimulations for several LGT’s.

31

SU(N) : continuumN Tc/

√σ χ2/ndf

2 0.7091(36) 0.283 0.6462(30) 0.054 0.6344(42)[(81)] 3.7[1.0]6 0.6101(51) 0.028 0.5928(107) 0.003

NG 0.691

Table 1: Data for SU(N) in d=4, (from B.Lucini, M.Teper and U. Wenger,JHEP0502:033,2005.

32

• In the large R and small L limit the NG partition function reduces toa single Bessel function. This means that the the NG effective stringpredicts the following behaviour for the Polyakov loop correlator:

〈P (0)P †(R)〉 ∼ Kd−32

(MR) ,

with M ∼ σL. This is exactly the limit in which dimensional reductionoccurs (“Svetitsky-Yaffe” conjecture). From the QFT approach to spinmodels we know that at high temperature and large distance whatevermodel we study the spin spin correlator will be dominated by the stateof lowest mass whose propagator in d′ dimensions is given by

G(R) ∼ Kd′−22

(mR)

Since d′ = d − 1 this result exactly coincides with what we obtain withthe NG string.

33

Comparison with MC simulations

Duality and a new algorithm: the snake algorithm allow high precisionsimulations for very large values of R and L in the gauge Ising model.

All the above predictions can be tested with precision δGG which in some

cases reaches 10−4.

In the comparison there is no free parameter. The figures are not theresult of a fitting procedure.

The agreement at large distance with NG is impressive. At shorterdistances deviations appear.

34

-0.004

-0.002

0

0.002

0.004

10 20 30 40 50 60 70 80 90

Gam

ma-

Gam

ma_

NG

R

Ising, L=80

Polyakov loop correlators in the (2+1) dimensional gauge Ising model at T = Tc/10

(corresponding for this β at L = 80). 10 < R < 80 is the interquark distance. In the

figure is plotted the deviation of Γ (the ratio G(R+ 1)/G(R) of two correlators shifted

by one lattice spacing) with respect to the Nambu-Goto string expectation ΓNG (which

with this definition of observables corresponds to the straight line at zero).

35

0

0.005

0.01

0.015

0.02

8 10 12 14 16 18 20 22 24

Gam

ma-

Gam

ma_

LO

L

R=32

Polyakov loop correlators in the (2+1) dimensional gauge Ising model at a fixed interquark

distance R = 32 varying inverse-temperature size (8 < L < 24). In the figure is plotted

the deviation of Γ with respect to the asymptotic free string expectation ΓLO (which with

this definition of observables corresponds to the straight line at zero). The curve is the

Nambu-Goto prediction for this observable.

36

Universality of the effective string corrections

The deep reason behind this impressive agreement are the universalitytheorems for the quartic (Luscher and Weisz 2004) and sextic (Aharony andKarzbrun 2009) corrections to the interquark potential.

In 3d the Nambu-Goto action is correct up to O(1/R6). In particulr,contributions due to the anomaly appear only at order O(1/R8) !

37

The flux tube thickness.

The flux density in presence of a pair of Polyakov loops is:

< Fµ,ν(x0, x1, h,R) >=< P (0, 0)P+(0, R)Uµ,ν(x0, x1, h) >

< P (0, 0)P+(0, R) >− < Uµ,ν >

where x0 denotes the timelike direction, x1 is the direction of the axisjoining the two Polyakov loops and h denotes the transverse direction.

38

PÖ

P

UΜΝHx,hLh

x

39

To evaluate the flux tube thickness we fix x2 = R/2 to minimizeboundary effects. (thanks to the periodic b.c. in the “temperature”direction we can instead choose any value of x0)

In the x1 direction the flux density shows a gaussian like shape, thewidth of this gaussian is the “flux tube thickness”: w(R,L). w(R,L)only depends on the interquark distance R and on the lattice size in thecompactified timelike direction L, i.e. on the inverse temperature of themodel

By tuning L we can thus study the flux tube thickness in the vicinity ofthe deconfinement transition

40

41

w2HxLx

FΜΝHx,hL

42

Shape of the flux density generated by a 30 × 30 Wilson loop in the Ising gauge model

(at β = 0.7460). The dashed line is the gaussian fit.

43

Effective string predictions for the flux tube thickness.

In the Nambu-Goto framework one should sum over all the surfacesbordered by the Polyakov loops and the plaquette with a weight proportionalto the surface area.

Fµν(x, h) =∑surf

e−σArea[surf]

w2(x) =∫

dh h2 Fµν(x, h)∫dh Fµν(x, h)

=∫

dh h2∑

surf e−σArea[surf]∫

dh∑

surf e−σArea[surf]

44

45

If we assume the size of the plaquette to be negligible with respect tothe other scales, perform a “physical” gauge fixing and denote as h0 thetransverse coordinate of the plaquette then:

w2(~x0) =

∫dh0 h0

2∫h(~x0)=h0

[Dh(~x)] e−σS[h]∫dh0

∫h(~x0)=h0

[Dh(~x)] e−σS[h]

which can be rewritten as

w2(~x0) =∫

[Dh(~x)] h(~x0)2e−σS[h]∫[Dh(~x)] e−σS[h]

≡ 〈h2(~x)〉

with S[h] = σ∫

d2x√

1 + (Oh)2

46

This expectation value is singular and must be regularized using forinstance a point splitting prescription.

σw2(~x) = 〈h(~x+ ~ε)h(~x)〉 ≡ G(~x+ ~ε; ~x)

The U.V. cutoff ε has a natural physical meaning: ε ∼ plaquette size.

Dealing with the whole NG action turns out to be too difficult. Weresort again to the free boson approximation (”SOS picture”)

S[h] ' σ∫

d2x [1 + 1/2(Oh)2]

47

Then G(~x+~ε; ~x) is the Green function of the Laplacian on the cylinder.

which can be written (choosing the reference frame so as to have thetwo loops located in ±R/2) as:

G2(z; z0) = − 12π

log∣∣∣∣θ1 [π(z − z0)/2R]θ2 [π(z + z0)/2R]

∣∣∣∣with q = e−πL/2R

Setting z0 = z + ε and performing an expansion in ε one finds:

σw2(z) = − 12π

logπ|ε|R

+1

2πlog |2θ2 (π<z/R) /θ′1|

48

A similar calculation in the Wilson loop case gives

σw2(z) = − 12π

logπ|ε|R

+1

2πlog∣∣∣∣2θ2(π<z/R)θ4(iπ=z/R)

θ′1 θ3(πz/R)

∣∣∣∣with q = e−πL/R.

In both cases the dominant term diverges as 12π logR

Both results assume L >> R

49

Performing a dual tranformation we can obtain the behaviour for R >>L i.e. in the high T regime. In this case the Green function can be writtenas:

G(z; z0) = − 12π

log∣∣∣∣θ1 [π(z − z0)/L]θ4 [π(z − z0)/L]

∣∣∣∣− Imz Imz0

LRq = e−2πR/L

and we have:

σw2(z) = − 12π

logπ|ε|L

+1

2πlog |θ4(2πi=z/L)/θ′1| −

(=z)2

LR

Expanding in powers of R/L we find

σw2(z) = − 12π

log2π|ε|L

+R

4L

This time the R dependence is linear !!

50

Summary of the result for the Polyakov loop correlator

• Low temperature

w2 ∼ 12πσ

log(R

Rc) + . . . (L >> R >> 0)

• High temperature (but below the deconfinement transition)

w2 ∼ 12πσ

(πR

2L+ log(

L

2π|ε|) + . . .) (R >> L)

Log increase of the flux tube width at zero temperature but Linearincrease near the deconfinement transition!

51

Comparison with MC simulations.

The 3d gauge Ising model is perfectly suited for studying the flux tubewidth.

Thanks to duality one can create a “vacuum” containing the Wilsonloop or the Polyakov loop correlators by simply frustrating the links in thedual lattice orthogonal to a surface bordered by the loops.

Any choice of the surface is equivalent.

52

Measuring the energy operator (which is the dual of the plaquetteoperator) in this vacuum one can thus evaluate the ratio:

< P (0, 0)P+(0, R)Uµ,ν(x0, x1, h) >< P (0, 0)P+(0, R) >

for any value of R and L with the same statistical uncertainty of theexpectation value of the plaquette in the usual vacuum < Uµ,ν >.

53

We used Wilson loops to test the low T predictions and Polyakov loopcorrelators for the high T ones.

• Low T we tested several values of β in the range 0.6543 ≤ β ≤ 0.7516and several sizes of the Wilson loops (ranging from 122 to 642) so asto test a wide range of R

√σ values. The log fit of σw2 as a function

of R√σ shows a very good χ2 with an angular coef. 0.150(5) to be

compared with 1/2π ∼ 0.15915...

• High T we studied the model at β = 0.75180 which corresponds toaTc = 1/8 and a2σ = 0.0105(2) and tested lattice sizes 9 ≤ L ≤ 16 i.e.temperatures ranging from T = Tc

2 to T = 89Tc

54

Squared width of the flux tube in units of sigma for the ZZ2 gauge theory. The open

symbols are Wilson loop data while the black circles refer to the (dual) Ising interface.

55

56

8 10 12 14 16 180.000

0.002

0.004

0.006

0.008

0.010

L

Σ

57

Near the deconfinement transition the linear fit has very good χ2, butthe ang.coef. shows deviations with respect to the expected value 1/(4σL).This is likely to be the signature of higher order (universal) contributions inthe effective string action.

This conjecture can be tested using the 2 loop calculation recentlyobtained by Gliozzi, Pepe and Wiese.

w2nlo =

(1 +

4πf(τ)σr2

)w2lo(r/2)− f(τ) + g(τ)

σ2r2,

f(τ) =E2(τ)− 4E2(2τ)

48,

g(τ) = iπτ

(E2(τ)

12− qddq

)(f(τ) +

E2(τ)16

) +E2(τ)

96,

The dominant term in the R >> L limit (i.e. τ → 0) turns out to be

58

again linear in R:

σw2nlo ∼

πR

24σ2L3+ · · ·

Combining together the leading and next to leading corrections we end upwith the following expression:

σw2 =R

4σL

(1 +

π

6σL2+ · · ·

)

Extending these calculations to higher orders is too difficult. Howeverdimensional reduction arguments suggest the following conjecture for thesquare width, resummed to all orders for the Nambu-Goto string:

59

σw2 =R

4m=

R

4σL√

1− π3σL2

Assuming this expression and expanding up the third order (i.e. keepingonly the universal terms) we find:

σw2 =R

4σL

(1 +

π

6σL2+

π2

24(σL2)2+ · · ·

)

The montecarlo data nicely agree with this conjecture

60

0

2

4

6

8

10

0.4 0.5 0.6 0.7 0.8 0.9 1

k

T/T_c

61

Conclusions

• The effective string approach describes remarkably well the interquarkpotential, both at zero and at finite temperature, both in the 3d gaugeIsing model and in non-abelian SU(N) LGTs. Recent MC results are soprecise that also next to leading corrections (which have been proved tobe universal) can be succesfully tested

• Assuming the Nambu-Goto action for the effective string a value forTc/√σ can be obtained which is in good but not exact agreement with

MC simulations

• The universality theorems explain why the Nambu-Goto action works sowell, despite its anomaly. Corrections due to the anomaly arise only atvery high orders in 1/σR2 (or equivalently in 1− T/Tc)

62

• The effective string predicts a logarithmic increase of the flux tube widthat low temperature and a linear increase at high T (but still in theconfining regime)

• This scenario is confirmed by MC simulations in the 3d gauge Isingmodel both at low and at high T, but an increasing discrepancy in thecoefficient of the linear term appears as Tc is approached if one truncatesthe Nambu-Goto action at the first order. Including the next to leadingorder contribution improves the agreement.

• Dimensional reduction also predicts a linear increase of the flux tubethickness at high temperature, with a T dependence of the coefficientwhich suggests how to resum to all orders the Nambu-Goto action.

• This conjecture agrees with the two loop calculation of Gliozzi Pepe andWiese.

63

• It is interesting to notice that at a difference with respect to the oneloop approximation this expression for the amplitude of the linear growthdivereges as the deconfinement temperature is approached.

64

Open issues

• Boundary terms

• Intrinsic thickness of the effective string

• Effective string versus dual superconductor models of confinement:

Nambu-Goto string 6= Abrikosov vortices

• Vortex configurations and the behaviour of space-like Wilson loops athigh temperature

65

References

M. Billo’ and M.C.

Polyakov loop correlators from D0-brane interactions in bosonic string theory

JHEP 0507 (2005) 038

M. Billo’, M.C. and L. Ferro

“The partition function of interfaces from the Nambu-Goto effective string theory.”

JHEP 0602: (2006) 070

M.C., M. Hasenbusch and M.Panero

“Comparing the Nambu–Goto string with LGT results.”

JHEP 0503 (2005) 026

“High precision monte carlo simulations of interfaces in the three-dimensional Ising

model: a comparison with the Nambu-Goto effective string model.”

JHEP 0603 (2006) 084

A. Allais and M.C

“Linear increase of the flux tube width at high temperature.”

JHEP 0901:073,2009

66

M. Billo’, M.C. and R. Pellegrini

New numerical results and novel effective string predictions for Wilson loops

ArXiv:1107.4356

67

The snake algorithm.

Goal: compute the ratio G(R)/G(R+ 1).

Proposal: Use duality and factorize the ratio of partition functions insuch a way that for each factor the partition functions differ just by thevalue of J〈ij〉 at a single link

ZL×RZL×(R+1)

=ZL×R,0ZL×R,1

...ZL×R,MZL×R,M+1

...ZL×R,L−1

ZL×R,L,

where L×R,M denotes a surface that consists of a L×R rectangle witha M × 1 column attached.

68

Figure 1: Sketch of the surface denoted by L×R,M . In the example, L = 6, R = 8

and M = 2. The circles indicate the links that intersect the surface.

69

Each of the factors can be written as expectation value in one of thetwo ensembles:

ZL×R,M+1

ZL×R,M=

∑si=±1 exp(−βHL×R,M(s)) exp(−2βsksl)

ZL×R,M,

where < k, l > is the link that is added going from L×R,M to L×R,M+1.

Further improvement: hierarchical updates.

Result: the error show no dependence on R !!

70

Dimensional reduction and the Svetitsky-Yaffe conjecture

• ”weak form of the S-Y conjecture:”

The high temperature behaviour of a (d+1) LGT with gauge group Gcan be effectively described by a spin model in d dimensions with (global)symmetry group C (the Center of G).

• “strong form of the S-Y conjecture:”

If both the spin model and the LGT have continuous phase transitionsthen they share the same universality class.

The mapping between the LGT and the effective spin model is based onthe following identifications

71

For all the values L ≥ 10 the linear fit has very good χ2, but the ang.coef. shows deviations with respect to the expected value 1/(4σL). This is likely to be the signature of higher order (universal) contributions in the effective string action LGT spin model

Low T confining phase High T symmetric phase

Polyakov loop (“C-odd”) spin operator

Plaquette operator (“C-even”) energy operator

Thermal perturbation energy perturbation

string tension (σ/T ) mass of the theory

Polyakov loop correlator spin-spin correlator

72

These mappings should be intended in a “renormalization group sense”i.e. the Polyakov loop operator is mapped into a linear combination of allthe (C-odd) operators in the spin model. For instance, in the 2d Ising casethe whole conformal family of the spin operator. This combination will bedominated by the relavant(s) operator(s). In the Ising case only one: thespin operator.

In the case of the plaquette operator the mapping will be in general alinear combination of the energy and the identity families.

73

Agreement between 2d spin model estimates andeffective string predictions.

With the Nambu-Goto effective string we obtain for the Polyakov loopcorrelator:

⟨P (0, 0)P (0, R)+

⟩=∞∑n=0

|vn|2(Enπ

)K0(EnR).

This expression is expected to be reliable in the large distance limit. Inthis limit only the lowest state (n = 0) survives and we end up with a singleK0 function:

74

limR→∞

⟨P (0, 0)P (0, R)+

⟩∼ K0(E0R).

The spin-spin correlator of any 2d spin model in the symmetric phase isgiven by

limR→∞

⟨σ(0, 0)σ(0, R)+

⟩∼ K0(mR).

The two expression coincide, they are universal (no dependence on the

symmetry group) and allow to identify m with E0 = σL{

1− π3σL2

}1/2.

which, at the first order in 1/L becomes

m↔ σL = σ/T

75

Effective spin model description of the the flux tubethickness.

Following the S-Y mapping the flux density in the Ising LGT in (2+1)dimensions becomes the ratio of connected correlators:

〈σ(x1)ε(x2)σ(x3)〉〈σ(x1)σ(x3)〉

in the high temperature phase of the 2d Ising model in zero magneticfield.

The large distance behaviour of this correlator can be evaluated usingthe Form Factor approach.

76

Width of the flux tube

The width of the flux tube evaluated at the midpoint between the twospins is given by

w2(r) =r2

2K0(2mr)

∫ ∞−∞

dxx2

1 + x2e−2mr

√1+x2

.

where R ≡ 2r is the distance between the two spins.

77

In the large R limit we thus obtain

w2 ' R

4m+ . . .

to be compared with the effective string result:

w2 ∼(

R

4σL+ log(

L

2π) + . . .

)(R >> L)

linear increase of the width in both cases but with a different T

dependence of the coefficent m = E0 = σT

√1− πT 2

3σ = σL√

1− π3σL2

This discrepancy is due to the free bosonic approx in the stringcalculation.

78

References

M. Billo’ and M.C.

Polyakov loop correlators from D0-brane interactions in bosonic string theory

JHEP 0507 (2005) 038

M. Billo’, M.C. and L. Ferro

“The partition function of interfaces from the Nambu-Goto effective string theory.”

JHEP 0602: (2006) 070

M.C., M. Hasenbusch and M.Panero

“Comparing the Nambu–Goto string with LGT results.”

JHEP 0503 (2005) 026

“High precision monte carlo simulations of interfaces in the three-dimensional Ising

model: a comparison with the Nambu-Goto effective string model.”

JHEP 0603 (2006) 084

M.C and M. Zago

“A new simulation strategy for the study of the interquark potential in abelian LGTs.”

In preparation

79