On the effective string theory of confining flux tubes Michael Teper - PowerPoint PPT Presentation

On the effective string theory of confining flux tubes Michael Teper (Oxford) - GGI 2012 • Flux tubes and string theory : effective string theories - recent analytic progress fundamental flux tubes in D=2+1 fundamental flux tubes in D=3+1 higher representation flux tubes • Concluding remarks 1

gauge theory and string theory ↔ A long history ... • Veneziano amplitude • ’t Hooft large- N – genus diagram expansion • Polyakov action • Maldacena ... AdS/CFT/QCD ... at large N , flux tubes and perhaps the whole gauge theory can be described by a weakly-coupled string theory 2

calculate the spectrum of closed flux tubes − → close around a spatial torus of length l : • flux localised in ‘tubes’; long flux tubes, l √ σ ≫ 1 look like ‘thin strings’ • at l = l c = 1 /T c there is a ‘deconfining’ phase transition: 1st order for N ≥ 3 in D = 4 and for N ≥ 4 in D = 3 • so may have a simple string description of the closed string spectrum for all l ≥ l c • most plausible at N → ∞ where scattering, mixing and decay, e.g string → string + glueball, go away • in both D=2+1 and D=3+1 Note: the static potential V ( r ) describes the transition in r between UV (Coulomb potential) and IF (flux tubes) physics; potentially of great interest as N → ∞ . 3

Some References recent analytic work: Luscher and Weisz, hep-th/0406205; Drummond, hep-th/0411017. Aharony with Karzbrun, Field, Klinghoffer, Dodelson, arXiv:0903.1927; 1008.2636; 1008.2648; 1111.5757; 1111.5758 recent numerical work: closed flux tubes: Athenodorou, Bringoltz, MT, arXiv:1103.5854, 1007.4720, ... ,0802.1490, 0709.0693 open flux tubes and Wilson loops: Caselle, Gliozzi, et al ..., arXiv:1202.1984, 1107.4356, ... also Brandt, arXiv:1010.3625; Lucini,..., 1101.5344; ...... 4

historical aside: for the ground state energy of a long flux tube, not only l →∞ E 0 ( l ) = σl but also the leading correction is ‘universal’ E 0 ( l ) = σl − π ( D − 2) 1 l + O (1 /l 3 ) 6 the famous Luscher correction (1980/1) 5

calculate the energy spectrum of a confining flux tube winding around a spatial torus of length l , using correlators of Polyakov loops (Wilson lines): τ →∞ � l † n,p ⊥ c n ( p ⊥ , l ) e − E n ( p ⊥ ,l ) τ p ( τ ) l p (0) � = � ∝ exp {− E 0 ( l ) τ } in pictures ✻ ✻ x l † l p − → ↑ p l → t ❄ ❄ ✛ ✲ τ a flux tube sweeps out a cylindrical l × τ surface S · · · integrate over these world dSe − S eff [ S ] � ∝ sheets with an effective string action cyl = l × τ 6

⇒ � c n ( p ⊥ , l ) e − E n ( p ⊥ ,l ) τ = � l † � dSe − S eff [ S ] p ( τ ) l p (0) � = n,p ⊥ cyl = l × τ where S eff [ S ] is the effective string action for the surface S ⇒ the string partition function will predict the spectrum E n ( l ) – just a Laplace transform – but will be constrained by the Lorentz invariance encoded in E n ( p ⊥ , l ) Luscher and Weisz; Meyer 7

this can be extended from a cylinder to a torus (Aharony) � e − E n ( p,l ) τ = e − E n ( p,τ ) l = Z w =1 � � dSe − S eff [ S ] torus ( l, τ ) = n,p n,p T 2 = l × τ where p now includes both transverse and longitudinal momenta ↔ ‘closed-closed string duality’ 8

Example: Gaussian approximation: � τ � l 0 dx 1 S G,eff = σlτ + 0 dt 2 ∂ α h∂ α h ⇒ dSe − S G,eff [ S ] = e − σlτ | η ( q ) | − ( D − 2) n e − E n ( τ ) l = : q = e − πl/τ � Z cyl ( l, τ ) = � cyl = l × τ 1 24 � ∞ n =1 (1 − q n ) in terms of the Dedekind eta function: η ( q ) = q ⇒ open string energies and degeneracies E n ( τ ) = στ + π 1 � � n − 24 ( D − 2) τ – the famous universal Luscher correction(1981) Also : modular invariance of η ( q ) → closed string energies, ˆ E n ( l ) = σl + 4 π 1 24 ( D − 2) } + O (1 /l 3 ) l { n − 9

So what do we know today? any S eff ⇒ ground state energy � 1 − { π ( D − 2) } 2 σl 3 − { π ( D − 2) } 3 � σl − π ( D − 2) 1 1 l →∞ E 0 ( l ) = σ 2 l 5 + O l 7 6 l 72 432 with universal terms: � 1 � ◦ O Luscher correction, ∼ 1980 l � 1 � ◦ O Luscher, Weisz; Drummond, ∼ 2004 l 3 � 1 � ◦ O Aharony et al, ∼ 2009-10 l 5 and similar results for E n ( l ), but only to O (1 /l 3 ) in D = 3 + 1 ∼ simple free string theory : Nambu-Goto in flat space-time up to O (1 /l 7 ) 10

Nambu-Goto free string theory D Se − κA [ S ] � spectrum (Arvis 1983, Luscher-Weisz 2004) : � 2 . � 2 πq E 2 ( l ) = ( σ l ) 2 + 8 πσ � � N L + N R − D − 2 + 2 24 l p = 2 πq/l = total momentum along string; N L , N R = sum left and right ‘phonon’ momentum: N L = � N R = � n L ( k ) k, n R ( k ) k, N L − N R = q k> 0 k> 0 so the ground state energy is: � 1 / 2 � 1 − π ( D − 2) 1 E 0 ( l ) = σl 3 σl 2 11

a n L ( k ) a n R ( k ) P = ( − 1) number phonons state = � | 0 � , k − k k> 0 lightest p = 0 states: | 0 � a 1 a − 1 | 0 � a 2 a − 2 | 0 � , a 2 a − 1 a − 1 | 0 � , a 1 a 1 a − 2 | 0 � , a 1 a 1 a − 1 a − 1 | 0 � · · · lightest p � = 0 states: a 1 | 0 � P = − , p = 2 π/l a 2 | 0 � P = − , p = 4 π/l a 1 a 1 | 0 � P = + , p = 4 π/l ⇒ 12

⇒ lightest states with p = 0 solid lines: Nambu-Goto 14 12 E √ σ 10 8 6 4 2 0 1 2 3 4 5 6 7 8 l √ σ gs : P=+ . ex1 : P=+. ex2: 2 × P=+ and 2 × P=- 13

So what does one find numerically? results here are from: • D = 2 + 1 Athenodorou, Bringoltz, MT, arXiv:1103.5854, 0709.0693 • D = 3 + 1 Athenodorou, Bringoltz, MT, arXiv:1007.4720 • higher rep Athenodorou, MT, in progress and we start with: D = 2 + 1 , SU (6) , a √ σ ≃ 0 . 086 i.e N ∼ ∞ , a ∼ 0 14

lightest 8 states with p = 0 P = +( • ) , P = − ( ◦ ) 14 12 E √ σ 10 8 6 4 2 0 1 2 3 4 5 6 7 8 l √ σ solid lines: Nambu-Goto ground state → σ : only parameter 15

lightest levels with p = 2 πq/l, 4 πq/l P = − 12 E 10 √ σ 8 6 4 2 0 1 2 3 4 5 6 l √ σ Nambu-Goto : solid lines 16

Now, when Nambu-Goto is expanded the first few terms are universal e.g. ground state � 1 � 1 − π ( D − 2) 2 E 0 ( l ) = σl 3 σl 2 � 1 − { π ( D − 2) } 2 σl 3 − { π ( D − 2) } 3 σl − π ( D − 2) 1 1 � l>l 0 = σ 2 l 5 + O l 7 6 l 72 432 √ √ σ = 3 /π ( D − 2); and also for excited states for l √ σ > l n √ σ ∼ � where l 0 8 πn ⇒ is the striking numerical agreement with Nambu-Goto no more than an agreement with the sum of the known universal terms? 17

NO! universal terms: solid lines Nambu-Goto : dashed lines 14 12 E √ σ 10 8 6 4 2 0 1 2 3 4 5 6 7 8 l √ σ 18

= ⇒ • NG very good down to l √ σ ∼ 2, i.e energy fat short flux ‘tube’ ∼ ideal thin string • NG very good far below value of l √ σ where the power series expansion diverges, i.e. where all orders are important ⇒ universal terms not enough to explain this agreeement ... • no sign of any non-stringy modes, e.g. µ ∼ M G / 2 ∼ 2 √ σ E ( l ) ≃ E 0 ( l ) + µ where e.g. = ⇒ ... in more detail ... 19

but first an ‘algorithmic’ aside – calculating energies • deform Polyakov loops to allow non-trivial quantum numbers • block or smear links to improve projection on physical excitations • variational calculation of best operator for each energy eigenstate • huge basis of loops for good overlap on a large number of states C ( t ) ≃ c n e − E n ( l ) t already for small t • i.e. for example: 20

Operators in D=2+1: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 21

0 . 9 0 . 8 aE eff ( n t ) 0 . 7 0 . 6 0 . 5 0 . 4 0 . 3 0 . 2 0 . 1 0 0 2 4 6 8 10 12 14 16 18 20 n t abs gs l = 16 , 24 , 32 , 64 a ( ◦ ); es p=0 P=+ ( • ); gs p = 2 π/l, P = − ( ⋆ ); gs, es p = 0 , P = − ( ⋄ ) 22

lightest P = − states with p = 2 πq/l : q = 0 , 1 , 2 , 3 , 4 , 5 a q | 0 � 20 18 E √ σ 16 14 12 10 8 6 4 2 0 1 2 3 4 5 6 l √ σ ( ap ) 2 → 2 − 2 cos( ap ) : dashed lines Nambu-Goto : solid lines 23

ground state deviation from various ‘models’ D = 2 + 1 0 . 02 E 0 − E model σl 0 − 0 . 02 − 0 . 04 − 0 . 06 1 2 3 4 5 6 l √ σ model = Nambu-Goto, • , universal to 1 /l 5 , ◦ , to 1 /l 3 , ⋆ , to 1 /l , +, just σl , × lines = plus O (1 /l 7 ) correction 24

= ⇒ ◦ for l √ σ � 2 agreement with NG to � 1 / 1000 moreover ◦ for l √ σ ∼ 2 contribution of NG to deviation from σl is � 99% despite flux tube being short and fat ◦ and leading correction to NG consistent with ∝ 1 /l 7 as expected from current universality results 25

8 χ 2 7 n d f 6 5 4 3 2 1 0 1 3 5 7 9 11 13 γ χ 2 per degree of freedom for the best fit E 0 ( l ) = E NG c ( l ) + 0 l γ 26

first excited q = 0 , P = + state D = 2 + 1 0 . 2 E − E NG √ σ 0 − 0 . 2 − 0 . 4 − 0 . 6 − 0 . 8 − 1 1 2 3 4 5 6 l √ σ fits: � − 2 . 75 c c 1 + 25 . 0 � - dotted curve; - solid curve ( l √ σ ) 7 ( l √ σ ) 7 l 2 σ 27

q = 1 , P = − ground state SU (6) , D = 2 + 1 0 . 3 0 . 2 E − E NG √ σ 0 . 1 0 − 0 . 1 − 0 . 2 − 0 . 3 − 0 . 4 − 0 . 5 1 2 3 4 5 6 l √ σ fits: � − 2 . 75 c c 1 + 25 . 0 � solid curve; : dashed curve ( l √ σ ) 7 ( l √ σ ) 7 l 2 σ 28