String Dynamics in Yang-Mills Theory

String Dynamics in Yang-Mills Theory Uwe-Jens Wiese Albert Einstein Center for Fundamental Physics Institute for Theoretical Physics, Bern University Workshop on Strongly Interacting Field Theories Jena, October 1, 2010

Outline Yang-Mills Theory Systematic Low-Energy Effective String Theory Lüscher-Weisz Multi-Level Simulation Technique String Width at Zero and at Finite Temperature String Breaking and String Decay Exceptional Confinement in G(2) Yang-Mills Theory Conclusions

Yang-Mills theory G µ (x) = igg a µ(x)t a, a {1, 2,..., n G }, G µν (x) = µ G ν (x) ν G µ (x) + [G µ (x), G ν (x)], S[G µ ] = d 4 1 x 4g 2 Tr[G µνg µν ]

Yang-Mills theory G µ (x) = igg a µ(x)t a, a {1, 2,..., n G }, G µν (x) = µ G ν (x) ν G µ (x) + [G µ (x), G ν (x)], S[G µ ] = d 4 1 x 4g 2 Tr[G µνg µν ] Main sequence gauge groups group SU(N) Sp(N) SO(2N + 1) SO(4N) SO(4N + 2) n G N 2 1 2N 2 + N 2N 2 + N 8N 2 2N 8N 2 + 6N + 1 rank N 1 N N 2N 2N + 1 center Z(N) Z(2) Z(2) Z(2) Z(2) Z(4)

String tension from Polyakov loop correlators t = β = 1/T t = 0 0 ( r β ) Φ( x) = Tr P exp dt G 0 ( x, t), Φ(0) Φ(r) exp ( βv (r)), 0 V (r) σr

String tension from Polyakov loop correlators t = β = 1/T t = 0 0 ( r β ) Φ( x) = Tr P exp dt G 0 ( x, t), Φ(0) Φ(r) exp ( βv (r)), 0 V (r) σr σ (0.4GeV) 2 10 5 N

String tension from Polyakov loop correlators t = β = 1/T t = 0 0 ( r β ) Φ( x) = Tr P exp dt G 0 ( x, t), Φ(0) Φ(r) exp ( βv (r)), 0 V (r) σr σ (0.4GeV) 2 10 5 N As strong as a cm-thick steel cable, but 13 orders of magnitude thinner.

How many people can be lifted by a Yang-Mills elevator?

How many people can be lifted by a Yang-Mills elevator? String tension σ Gravity F = Mg

How many people can be lifted by a Yang-Mills elevator? String tension σ Gravity F = Mg Mg = σ 10 5 N, g 10m/sec 2 M 10 4 kg 100 People

Low-energy effective string theory 0 r

Low-energy effective string theory 0 r The height variable h(x, t) points to the fluctuating string world-sheet in the (d 2) transverse dimensions ( h(0, t) = h(r, t) = 0).

Low-energy effective string theory 0 r The height variable h(x, t) points to the fluctuating string world-sheet in the (d 2) transverse dimensions ( h(0, t) = h(r, t) = 0). S[ h] = β dt r 0 0 dx σ 2 µ h µ h, µ {0, 1},

Low-energy effective string theory 0 r The height variable h(x, t) points to the fluctuating string world-sheet in the (d 2) transverse dimensions ( h(0, t) = h(r, t) = 0). S[ h] = β 0 dt r 0 dx σ 2 µ h µ h, µ {0, 1}, π(d 2) V (r) = σr + O(1/r 3 ), 24r wlo(r/2) 2 = d 2 2πσ log(r/r 0) M. Lüscher, K. Symanzik, P. Weisz, Nucl. Phys. B173 (1980) 365 M. Lüscher, Nucl. Phys. B180 (1981) 317 M. Lüscher, G. Münster, P. Weisz, Nucl. Phys. B180 (1981) 1

Effective string theory for d = 3 at the 2-loop level β r S[h] = dt dx σ [ µ h µ h 1 ] 2 8σ ( µh µ h) 2 0 0 Even to the next order, the action agrees with the one of the Nambu-Goto string. O. Aharony and E. Karzbrun, JHEP 0906 (2009) 012 w 2 (r/2) = h(r/2, t) h(r/2 + ɛ, t + ɛ ) ( = 1 + 4πf (τ) σr 2 f (τ) = E 2(τ) 4E 2 (2τ), ( 48 E2 (τ) g(τ) = iπτ q d 12 dq E 2 (τ) = 1 24 n=1 ) wlo(r/2) 2 f (τ) + g(τ) σ 2 r 2, ) ( f (τ) + E ) 2(τ) + E 2(τ) 16 96, n q n, q = exp(2πiτ), τ = iβ/2r 1 qn F. Gliozzi, M. Pepe, UJW, arxiv:1006.2252 [hep-lat]

Two-link operators x 0 x 1 r T (t) αβγδ = U 0 (0, t) αβ U 0(r, t) γδ

Two-link operators x 0 x 1 r T (t) αβγδ = U 0 (0, t) αβ U 0(r, t) γδ Multiplication law {T (t)t (t + a)} αβγδ = T (t) αλγɛ T (t + a) λβɛδ

Two-link operators x 0 x 1 r T (t) αβγδ = U 0 (0, t) αβ U 0(r, t) γδ Multiplication law {T (t)t (t + a)} αβγδ = T (t) αλγɛ T (t + a) λβɛδ Polyakov loop correlator Φ(0) Φ(r) = {T (0)T (a)... T (β a)} ααγγ M. Lüscher and P. Weisz, JHEP 0109 (2001) 010

Sub-lattice expectation value [T (t)... T (t )] = 1 DU sub T (t)... T (t ) exp( S[U] sub ) Z sub

Sub-lattice expectation value [T (t)... T (t )] = 1 DU sub T (t)... T (t ) exp( S[U] sub ) Z sub Φ(0) Φ(r) = {[T (0)T (a)]... [T (β 2a)T (β a)]} ααγγ

Simulations in (2 + 1)-d SU(2) Yang-Mills theory S[U] = 1 g 2 Tr[U µ (x)u ν (x + ˆµ)U µ (x + ˆν) U ν (x) ], x,µ,ν Φ(0)Φ(r) = 1 DU Φ(0)Φ(r) exp( S[U]) exp( βv (r)) Z

Simulations in (2 + 1)-d SU(2) Yang-Mills theory S[U] = 1 g 2 Tr[U µ (x)u ν (x + ˆµ)U µ (x + ˆν) U ν (x) ], x,µ,ν Φ(0)Φ(r) = 1 DU Φ(0)Φ(r) exp( S[U]) exp( βv (r)) Z 1.4 V(R) = a + b R + c/r spin 1/2 1.2 1 0.8 0.6 V (r) = σr π 24r +... 0.4 0.2 0 2 4 6 8 10 12 14 16 There is very good agreement with the effective string theory. The value of the string tension obtained on a 54 2 48 lattice at 4/g 2 = 9 is σ = 0.025897(15)/a 2.

Computation of the string width P(x) = Tr[U 1 (x)u 0 (x + ˆ1)U 1 (x + ˆ0) U 0 (x) ], C(x 2 ) = Φ(0)Φ(r)P(x) P(x) Φ(0)Φ(r) 0.8858 fit data 0.88575 0.8857 0.88565 0.8856 0.88555 0.8855 0.88545 0 2 4 6 8 10 12 14 16 18

Computation of the string width P(x) = Tr[U 1 (x)u 0 (x + ˆ1)U 1 (x + ˆ0) U 0 (x) ], C(x 2 ) = Φ(0)Φ(r)P(x) P(x) Φ(0)Φ(r) 0.8858 fit data 14 0.88575 0.8857 12 10 0.88565 0.8856 0.88555 0.8855 0.88545 0 2 4 6 8 10 12 14 16 18 8 6 4 2 fit w 2 (r/2) 5 10 15 20 25 There is very good agreement with the effective string theory at the two-loop level. The values of the low-energy parameters are σ = 0.025897(15)/a 2, r 0 = 2.26(2)a = 0.364(4)/ σ 0.2 fm. F. Gliozzi, M. Pepe, UJW, Phys. Rev. Lett. 104 (2010) 232001

String width at finite temperature w 2 (r/2) = 1 ( ) β 2πσ log + r 4r 0 2β +... At finite temperature, the effective string theory predicts a linear increase of the width as one separates the static sources. 20 18 16 14 12 10 8 6 4 5 10 15 20 25 Again, there is excellent agreement with the effective string theory at the two-loop level, now without any adjustable parameters. F. Gliozzi, M. Pepe, UJW, to be published

String breaking

String breaking and string decay (strand rupture)

String breaking and string decay (strand rupture) Static potential between triplet (Q = 1) charges in (2+1)-d SU(2) Yang-Mills theory

String breaking and string decay (strand rupture) Static potential between triplet (Q = 1) charges in (2+1)-d SU(2) Yang-Mills theory V(R) spin 1 2 1.5 1 0.5 0 2 4 6 8 10 12 14 16 M. Pepe and UJW, Phys. Rev. Lett. 102 (2009) 191601

String decay 4 3.5 3 2.5 0.4 0.35 0.3 0.25 0.2 fit Q = 1/2 Q = 3/2 2 1.5 1 fit Q = 1/2 Q = 3/2 2 4 6 8 10 12 14 16 0.15 0.1 0.05 0 2 4 6 8 10 12 14 Potential and force between quadruplet (Q = 3/2) charges

String decay 4 3.5 3 2.5 0.4 0.35 0.3 0.25 0.2 fit Q = 1/2 Q = 3/2 2 1.5 1 fit Q = 1/2 Q = 3/2 2 4 6 8 10 12 14 16 0.15 0.1 0.05 0 2 4 6 8 10 12 14 Potential and force between quadruplet (Q = 3/2) charges 5 4.5 4 0.7 0.6 0.5 0.4 fit Q = 1 Q = 2 3.5 0.3 3 2.5 2 fit Q = 1 Q = 2 2 4 6 8 10 12 14 16 0.2 0.1 0-0.1 2 4 6 8 10 12 14 Potential and force between quintet (Q = 2) charges

Constituent gluon model: E Q,n (r) = σ Q r c Q r + 2M Q,n ( ) E1,0 (r) A H 1 (r) =, A E 0,1 (r) ( ) E3/2,0 (r) B H 3/2 (r) =, B E 1/2,1 (r) E 2,0 (r) C 0 H 2 (r) = C E 1,1 (r) A, 0 A E 0,2 (r)

Constituent gluon model: E Q,n (r) = σ Q r c Q r + 2M Q,n ( ) E1,0 (r) A H 1 (r) =, A E 0,1 (r) ( ) E3/2,0 (r) B H 3/2 (r) =, B E 1/2,1 (r) E 2,0 (r) C 0 H 2 (r) = C E 1,1 (r) A, 0 A E 0,2 (r) Q σ Q a 2 σ Q /σ 4Q(Q + 1)/3 1/2 0.06397(3) 1 1 1 0.144(1) 2.25(2) 8/3 3/2 0.241(5) 3.77(8) 5 2 0.385(5) 6.02(8) 8 We observe deviations from Casimir scaling.

Masses and mass differences: Q,n = M Q 1,n+1 M Q,n Q M Q,0 a M Q 1,1 a M Q 2,2 a Q,0 a Q 1,1 a 1/2 0.109(1) 1 0.37(3) 1.038(1) 0.67(3) 3/2 0.72(5) 1.32(5) 0.60(5) 2 1.04(3) 1.71(3) 2.42(1) 0.67(3) 0.71(3) Constituent gluon mass: M G = 0.65(4)/a = 2.6(2) σ 1 GeV 0 + glueball mass: M 0 + = 1.198(25)/a H. Meyer and M. Teper, Nucl. Phys. B668 (2003) 111

Exceptional Lie Group G(2) SO(7) Ω ab Ω ac = δ bc, T abc = T def Ω da Ω eb Ω fc, T 127 = T 154 = T 163 = T 235 = T 264 = T 374 = T 576 = 1

Exceptional Lie Group G(2) SO(7) Ω ab Ω ac = δ bc, T abc = T def Ω da Ω eb Ω fc, T 127 = T 154 = T 163 = T 235 = T 264 = T 374 = T 576 = 1 1/ 3 6 1/2 3 0 2 7 1 1/2 3 4 5 1/ 3 3 1/2 0 1/2 {7} = {3} {3} {1},

Exceptional Lie Group G(2) SO(7) Ω ab Ω ac = δ bc, T abc = T def Ω da Ω eb Ω fc, T 127 = T 154 = T 163 = T 235 = T 264 = T 374 = T 576 = 1 1/ 3 1/2 3 0 1/2 3 1/ 3 2 4 6 3 7 1 5 U+ V+ 3 / 2 X+ 1 / 3 Y+ Z 0 T T+ 1/ 2 3 Y Z + X 3 / 2 V U 1/2 0 1/2 1 1/2 0 1/2 1 {7} = {3} {3} {1}, {14} = {8} {3} {3} The fact that G(2) has a trivial center causes exceptional confinement. K. Holland, P. Minkowski, M. Pepe, UJW, Nucl. Phys. B668 (2003) 207

Casimir Scaling in G(2) Yang-Mills Theory {14} {14} {14} = {1} {7} 5 {14} 3 {27} 2 {64} 4 {77} 3 {77 } {182} 3 {189} {273} 2 {448}

Casimir Scaling in G(2) Yang-Mills Theory {14} {14} {14} = {1} {7} 5 {14} 3 {27} 2 {64} 4 {77} 3 {77 } {182} 3 {189} {273} 2 {448} Three gluons can screen a single quark.

Casimir Scaling in G(2) Yang-Mills Theory {14} {14} {14} = {1} {7} 5 {14} 3 {27} 2 {64} 4 {77} 3 {77 } {182} 3 {189} {273} 2 {448} Three gluons can screen a single quark. V st.a t 4.5 4 3.5 3 2.5 2 1.5 1 0.5 {7} {14} {27} {64} {77} {77 } 0 0 1 2 3 4 5 6 7 r/a s L. Liptak and S. Olejnik, Phys. Rev. D78 (2008) 074501 B. H. Wellegehausen, A. Wipf, and C. Wozar, arxiv:1006.2305

String Breaking in 3-d G(2) Yang-Mills Theory {14} {14} {14} = {1} {7} 5 {14} 3 {27} 2 {64} 4 {77} 3 {77 } {182} 3 {189} {273} 2 {448}

String Breaking in 3-d G(2) Yang-Mills Theory {14} {14} {14} = {1} {7} 5 {14} 3 {27} 2 {64} 4 {77} 3 {77 } {182} 3 {189} {273} 2 {448} One or two gluons cannot screen a fundamental quark to a smaller charge, but three can. B. H. Wellegehausen, A. Wipf, and C. Wozar, arxiv:1006.2305

Conclusions In Yang-Mills theory there is a systematic low-energy effective string theory (analogous to chiral perturbation theory in QCD), which describes the dynamics of the Goldstone modes of the spontaneously broken translation symmetry of the string world-sheet. The effective theory has been tested at the two-loop level with very high precision using the Lüscher and Weisz multi-level simulation technique, which led to the determination of low-energy parameter r 0. Strings connecting static charges in higher representations may decay before they break completely. A constitutent gluon model accounts for the brown muck surrounding a screened color charge. The exceptional group G(2) is the smallest one with a trivial center. One observes Casimir scaling at the few percent accuracy level. A fundamental {7} quark can bescreened by three adjoint {14} gluons. The fundamental G(2) string breaks by the simultaneous creation of six gluons without string decay.

Yang-Mills elevator as a metaphor for our workshop

Yang-Mills elevator as a metaphor for our workshop String tension σ Gravity F = Mg

Yang-Mills elevator as a metaphor for our workshop String tension σ Gravity F = Mg Thanks for organizing a very nice workshop that contributes to quickly elevate us to a higher level of understanding of strongly interacting field theories!