Effective string description of the interquark potential in the 3D U(1) Lattice Gauge Theory. 1 Davide Vadacchino 1 Michele Caselle 1 Marco Panero 2 Roberto Pellegrini 1 1Universit` a degli Studi di Torino/INFN Sezion di Torino 2Instituto de F´ ısica Te´ orica UAM/CSIC, Universidad Aut´ onoma de Madrid Lattice 2014 Columbia University, NYC 1 Caselle, Panero, Pellegrini, and Vadacchino (2014)
Outline ◮ The 3D U(1) lattice gauge model: 1. Confinement. 2. Dual formulation and gauge/string duality. ◮ Numerical results on the interquark potential: deviations with respect to Nambu-Goto. ◮ Conclusions and future directions.
The U (1) lattice gauge model in 3 D Definition � � � � µ,ν U ⋆ ν,µ U ⋆ S = β 1 − Re U x ,µ U x + a ˆ x + a ˆ x ,ν x ∈ Λ 1 ≤ µ<ν ≤ 3 where Λ is a 3 D euclidean spacetime lattice and U x ,µ = exp [ ia ϑ µ ( x + a ˆ µ/ 2)] ∈ U (1) Since the model is abelian µ,ν U ⋆ ν,µ U ⋆ Re U x ,µ U x + a ˆ x ,ν = cos (∆ µ ϑ x ,ν − ∆ ν ϑ x ,µ ) = cos ϑ x ,µν x + a ˆ Adopting discrete differential forms notation � π d ( ϑ ) e − β � c 2 (1 − cos d ϑ ) � Z = − π c 1 with c 1 and c 2 links and plaquettes on Λ.
The U (1) lattice gauge model in 3 D The weak coupling approximation Taking the periodicity of Z into account in the β ≫ 1 approximation e − 2 π 2 β ( q , ∆ − 1 q ) � Z = Z sw Z top = Z sw { q } where Z top describes a coulomb like gas of topological excitations, Z sw describes spin-waves. ◮ The model is always confining in 3 D 2 ◮ In the semiclassical approximation c σ 8 π 2 β e − π 2 β v (0) , e − π 2 β v (0) , � m 0 = c 0 σ ≥ v (0) = 0 . 2527 � 2 π 2 β the bounds are saturated and c σ = 8 , c 0 = 1. ◮ The ratio √ σ = 2 π c 0 m 0 (2 πβ ) 3 / 4 e − π 2 v (0) β/ 2 , √ c σ can be tuned at will by an appropriate choice of β , in contrast to the general Yang-Mills case. 2 (G¨ opfert and Mack, 1981, Polyakov, 1977)
The U (1) lattice gauge model in 3 D The dual formulation of the model The dual model is a globally Z symmetric spin model 3 {∞} � � Z = I | d ⋆ l | ( β ) , ⋆ c 1 { ⋆ l = −∞} where ◮ I α Bessel functions of order α ◮ ⋆ c 1 are links of the dual lattice ⋆ Λ. ◮ ⋆ l is an integer valued scalar field, and d ⋆ l differences at neighboring dual sites. The advantage is twofold: ◮ Physical insight into the confinement mechanism : dual superconductor scenario. ◮ Ease in numerical computation. 3 (Savit, 1980)
The U (1) lattice gauge model in 3 D The confinement mechanism and gauge/string duality The dual superconductor scenario of confinement 4 : ◮ Condensation of magnetic monopoles drives confinement of electric charges. ◮ The dynamics of flux tubes should be described by string like degrees of freedom: no proven gauge/string duality in the general case. In the U(1) LGT, however, an heuristic proof exists 5 e 2 d 2 ξ √ g d 2 ξ √ gK 2 � � S Pol = c 1 e 2 m 0 + c 2 m 0 NG Rigidity where c 1 and c 2 are two undetermined constants. ◮ At tree level, the rigidity term doens’t contribute to the interquark potential. ◮ If c 1 = σ and c 2 = α then � σ/α = m ∼ m 0 . and the rigidity correction is dominant in the β → ∞ limit. 4 (Polyakov, 1977) 5 (Antonov, 1998, Polyakov, 1997)
The U (1) lattice gauge model in 3 D Inclusion of Polyakov lines in the partition function The interquark potential V ( R ) can be extracted from � G ( R ) = � P ⋆ ( R ) P (0) � = e − N t V ( R ) ∝ [ DX ] e − S eff [ X ] where S eff is the effective string action and P ( x ) Polykakov lines. In the dual formulation, Polyakov lines P ( x ) are easily included in Z {∞} � � Z R = e − β N l I | d ⋆ l + ⋆ n | ( β ) L + L − L { ⋆ l = −∞} ⋆ c 1 where ⋆ n is integer valued and nonvanishing only on links dual to a surface bounded by the lines. x x + R Thus in the dual formulation G ( R ) = Z R Z which, however, is hard to measure because of an exponentially decaying signal-to-noise ratio.
Simulation of the The U (1) LGT in 3 D Snake algorithm and hierarchical update The problem can be circumvented using the snake algorithm 6 : Z L d − 1 · · · Z 1 G ( R + 1) = Z R +1 = Z R +1 R R Z L d − 1 Z L d − 2 G ( R ) Z R Z R R R where R L Z L d − i +1 � � I | d ∗ l +1 | ( β ) R = Z L d − i I | d ∗ l | ( β ) R R , L d − i R + 1 are L d independent local observables. And efficiency of the computation can be improved by updating the lattice hierarchically around the local observable 6 (de Forcrand, D’Elia, and Pepe (2001))
Simulation of the U (1) LGT in 3 D The general setting and the measured quantity We obtained high precision estimates of Q ( R ) = − 1 log G ( R + 1) = V ( R + 1) − V ( R ) G ( R ) N t ◮ The dual model was simulated at several values of β on lattices L 2 xN t chosen to avoid finite size effects: � 64 a , for β < 2 . 4 N t , L = 128 a , for β ≥ 2 . 4 ◮ Q ( R ) was probed in the range 1 / √ σ < R < L / 2 ◮ A single site metropolis update algorithm was used with Jacknife error estimation.
Simulation of the U (1) LGT in 3 D Preliminary measurements The data was fitted asymptotically with � π V NG ( R ) = σ R 1 − 12 σ R 2 using σ as free parameter in the range [ R min a , La / 2]. 1 / √ σ √ σ σ a 2 β L , N t R min 1 . 7 0 . 122764(2) 64 3 a 11 a 1 . 9 0 . 066824(6) 64 4 a 17 a 2 . 0 0 . 049364(2) 64 5 a 20 a 2 . 2 0 . 027322(2) 64 6 a 26 a 2 . 4 0 . 015456(7) 128 8 a 34 a ◮ At low β , NG describes the data for a wide range of Ra ◮ As β grows, the deviations from NG grow: at β = 2 . 2 only 6 degrees of freedom can be fitted! a / √ σ, R min a � � Deviations should be detectable in the range
Simulation of the U (1) LGT in 3 D Deviations with respect to NG Deviations ( Q ( R ) − Q NG ( R )) a with respect to NG at β = 2 . 2 on a (64 a ) 3 lattice. 0.0005 0.0004 ( Q ( R ) − Q NG ( R )) a 0.0003 0.0002 0.0001 0.0000 1 2 3 4 5 6 p σ R √ σ = 4 . 3. The best fit value of σ a 2 = 0 . 027322(2) was obtained with R min
Deviations with respect to NG How to explain them? In general S eff = S NG + S b + S 2 , K Up to the resolution of our data S NG ≃ S cl . + σ � � ∂ α X · ∂ α X − 1 � 4 ( ∂ α X · ∂ α X ) 2 d 2 ξ , 2 � S b ≃ b 2 d ξ 0 [ ∂ 1 ∂ 0 X · ∂ 1 ∂ 0 X ] , � (∆ X ) 2 S 2 , K ≃ α For each we can compute the L.O. contribution to V ( R ) perturbatively 7 � σ ∞ π 3 1 V r ( R ) = − m K 1 (2 nmR ) � V b ( R ) = − b 2 R 4 , , m = 60 2 π n 2 α n =1 7 (Aharony and Field, 2011, Bill´ o et al., 2012, Klassen and Melzer, 1991, Nesterenko and Pirozhenko, 1997)
Deviations with respect to NG Rigidity and boundary at LO The boundary correction V b alone can’t describe the deviations: √ σ . ◮ χ 2 R ∼ 1 only for very large values of R min ◮ The best fit values of b 2 have the wrong scaling behaviour: b 2 σ 3 / 2 = 0 . 033(3) , β = 1 . 7 b 2 σ 3 / 2 = 0 . 62(6) , β = 2 . 4 ◮ A complementary test with the potential A V ( R ) = R B with A , B free parameters shows that b � = 4. Fitting with the rigidity correction V r works much better: ◮ Good fits are obtained already at small distances: √ σ = 2 . 15 ma = 0 . 112(2) , χ 2 r = 1 . 03 , R min √ σ = 4 . 3 for NG. to be compared with R min ◮ The best fit value of m scales with m 0 .
Deviations with respect to NG Rigidity at NLO The NLO correction due to the rigidity contribution can be computed in the large D limit 8 � π D � 2 3 V 2 ( R ) = − 20 m σ R 4 24 and in the general case 9 � π � 2 3 V ′ 2 ( R ) = − ( D − 2)( D − 10) 20 m σ R 4 24 ◮ This contribution is detected within the precision of our data and contributes to the best fit value of ma . ◮ It is entangled to the boundary correction, which then cannot be neglected! 8 (Braaten et al., 1987) 9 (German and Kleinert, 1989)
Deviations with respect to NG 3 parameters fit of the data V ( R ) = V NG ( R ) + V r ( R ) + V ′ 2 ( R ) + V b ( R ) using σ , m and b 2 as free parameters results in the best fit values σ a 2 = 0 . 027318(2) , ma = 0 . 11(1) , b 2 σ 3 / 2 = 0 . 005(1) , √ σ = 1 . 65. with χ 2 r = 1 . 2 and R min 0.0007 0.0006 0.0005 ( Q ( R ) − Q NG ( R )) a 0.0004 0.0003 0.0002 0.0001 0.0000 1 2 3 4 5 p σ R In the plot: The deviations ( Q ( R ) − Q NG ( R )) a and the curve Q r ( R ) + Q ′ 2 ( R ) + Q b ( R ) calculated with the best fit values for σ , m and b 2 .
Determination of ma The same analysis for the other couplings leads to: β ma m 0 a m / m 0 1 . 7 0 . 28(9) 0 . 88(1) 0 . 32(10) 1 . 9 0 . 25(4) 0 . 56(1) 0 . 45(7) 2 . 0 0 . 17(2) 0 . 44(1) 0 . 39(4) 2 . 2 0 . 11(1) 0 . 27(1) 0 . 41(4) 2 . 4 0 . 06(2) 0 . 20(1) 0 . 30(10) ◮ Takes into account the interplay between σ , m , and b 2 in the error. ◮ m scales with m 0 as predicted by Polyakov. Our estimate of the rigidity parameter is m / m 0 = 0 . 35(10) .
Conclusions ◮ The strong deviations with respect to NG observed in the U (1) LGT in 3 D can be explained by the addition of a rigidity term to the effective string action, as predicted by Polyakov. This contribution becomes dominant in the limit β → ∞ . ◮ Future directions: 1. Try to disentangle the NLO rigidity contribution from the boundary correction. 2. Study the behaviour intrinsic width of the string and compare with predictions of the string with rigidity. 3. Finite temperature behaviour of the interquark potential.
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