String Dynamics in Yang-Mills Theory

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1 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

2 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

3 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 Collaboration: Ferdinando Gliozzi (INFN Torino) Michele Pepe (INFN Milano)

4 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

5 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

6 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 ]

7 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)

8 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) Exceptional gauge groups group G(2) F (4) E(6) E(7) E(8) n G rank center {1} {1} Z(3) Z(2) {1}

9 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

10 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) N

11 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) N As strong as a cm-thick steel cable, but 13 orders of magnitude thinner.

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

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

14 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

15 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

16 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

17 Low-energy effective string theory 0 r

18 Low-energy effective string theory 0 r

19 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).

20 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},

21 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 V (r) = σr r 0 dx σ 2 µ h µ h, µ {0, 1}, π(d 2) 24r + O(1/r 3 ),

22 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)

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).

23 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

24 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) 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

25 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) 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: [hep-lat]

26 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

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

28 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

29 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

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

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

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

33 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

34 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

35 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

36 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

37 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/ V (r) = σr π 24r

38 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/ V (r) = σr π 24r There is very good agreement with the effective string theory. The value of the string tension obtained on a lattice at 4/g 2 = 9 is σ = (15)/a 2.

39 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) fit data

40 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) fit data fit w 2 (r/2)

41 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) fit data fit w 2 (r/2) There is very good agreement with the effective string theory at the two-loop level. The values of the low-energy parameters are σ = (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)

42 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.

43 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

44 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 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

45 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

46 String breaking

47 String breaking and string decay (strand rupture)

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

49 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 M. Pepe and UJW, Phys. Rev. Lett. 102 (2009)

50 String decay fit Q = 1/2 Q = 3/ fit Q = 1/2 Q = 3/ Potential and force between quadruplet (Q = 3/2) charges

51 String decay fit Q = 1/2 Q = 3/ fit Q = 1/2 Q = 3/ Potential and force between quadruplet (Q = 3/2) charges fit Q = 1 Q = fit Q = 1 Q = Potential and force between quintet (Q = 2) charges

52 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)

53 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/ (3) (1) 2.25(2) 8/3 3/ (5) 3.77(8) (5) 6.02(8) 8 We observe deviations from Casimir scaling.

54 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/ (1) (3) 1.038(1) 0.67(3) 3/2 0.72(5) 1.32(5) 0.60(5) (3) 1.71(3) 2.42(1) 0.67(3) 0.71(3)

55 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/ (1) (3) 1.038(1) 0.67(3) 3/2 0.72(5) 1.32(5) 0.60(5) (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

56 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/ (1) (3) 1.038(1) 0.67(3) 3/2 0.72(5) 1.32(5) 0.60(5) (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

57 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/ (1) (3) 1.038(1) 0.67(3) 3/2 0.72(5) 1.32(5) 0.60(5) (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 A simple constituent gluon model describes the data rather well.

58 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

59 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

60 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/ / / 3 3 1/2 0 1/2 {7} = {3} {3} {1},

61 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 1/ 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}

62 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 1/ 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

63 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}

64 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.

65 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 {7} {14} {27} {64} {77} {77 } r/a s L. Liptak and S. Olejnik, Phys. Rev. D78 (2008) B. H. Wellegehausen, A. Wipf, and C. Wozar, arxiv:

66 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}

67 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.

68 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:

69 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: The fundamental string breaks by the simultaneous creation of six gluons, without intermediate string decay.

70 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

71 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.

72 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.

73 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.

74 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.

75 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.

76 Yang-Mills elevator as a metaphor for our workshop

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

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

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

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

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

82 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!

G2 gauge theories. Axel Maas. 14 th of November 2013 Strongly-Interacting Field Theories III Jena, Germany

G2 gauge theories. Axel Maas. 14 th of November 2013 Strongly-Interacting Field Theories III Jena, Germany G2 gauge theories Axel Maas 14 th of November 2013 Strongly-Interacting Field Theories III Jena, Germany Overview Why G2? Overview Why G2? G2 Yang-Mills theory Running coupling [Olejnik, Maas JHEP'08,