Possible string effects in 4D from a 5D anisotropic gauge theory in a mean-field background Nikos Irges, Wuppertal U. Based on N.I. & F. Knechtli, arxiv:0905.2757 to appear in NPB + work in progress Corfu, Sept. 2009
Background. J. M. Drouffe and J. B. Zuber, Phys. Rept. 02 (983) 2. J. Zinn-Justin, Quantum Field Theories and Critical Phenomena 3. I. Montvay and G. Muenster, Quantum Fields on a Lattice 4. M. Creutz, Phys. Rev. Lett. 43 (979), 553 5. B. Svetitsky and L.G. Yaffe, Nucl. Phys. B20 (982) 423 6. Y.K. Fu and H.B. Nielsen, Nucl. Phys. B236 (984), 67 7. S. Ejiri, J. Kubo and M. Murata, Phys. Rev. D62 (2000), 05025 8. M. Luscher and P. Weisz, JHEP 0407:04, 2204
In 5 Dimensions... M 5 = E 4 S A M {A µ,a 5 } 5D gauge field SU(2) +φ(y +2πR) Z: 4D gauge field h: 4D Hi#s S S +ψ(y +2πR) φ(y) = n φ n e iny/r ψ(y) = n ψ x e iny/r 4D Georgi-Glashow (G-G) model + KK
The Anisotropic Lattice 4D slice U(n M,µ) - L β 5 β 4 N 5 U(n M, 5) β 4 = β N β 5 = βγ 5 γ L = 2πR γ l γ = anisotropy parameter
The mean-field expansion L, β, γ parameters: (we keep ) N 5 = L Z = DU DV DHe (/N )Re[trH(U V ) ]e S G[V ] Z = DV DHe Seff[V,H], Seff = S G [V ]+u(h) + (/N )RetrHV e u(h) = DUe (/N )RetrUH
To 0 th order The background is determined by V = u H H H = S G[V ] V V The free energy F (0) = N ln(z[v,h]) = S eff[v,h] N
To st order H = H + h V = V + v Seff = Seff[ V, H]+ 2 ( δ 2 Seff δh 2 h 2 +2 δ2 Seff V,H δhδv hv + δ2 Seff V,H δv 2 ) v 2 V,H δ 2 Seff δh 2 δ 2 Seff δv 2 δ 2 Seff δv δh h 2 = h i K (hh) ij h j = h T K (hh) h V,H v 2 = v i K (vv) ij v j = v T K (vv) v V,H v 2 = v i K (vh) ij h j = v T K (vh) h V,H
The Gaussian fluctuations z = Dv Dhe S(2) [v,h] S (2) [v, h] = 2 ( ) h T K (hh) h +2v T K (vh) h + v T K (vv) v z = (2π) h /2 (2π) v /2 det[( + K (hh) K (vv) )] Z () = Z[V,H] z = e S eff [V,H] z The free energy to first order F () = F (0) N ln(z) =F (0) + [ ( 2N ln det + K (hh) K (vv)) ] 2 FP
Observables O[V ]=O[V ]+ δo δv O = Z = O[V ]+ 2 Dv δ 2 O δv 2 <v i v j >= z v + δ 2 O V 2 δv 2 v 2 +... V Dv Dhv 2 e S(2) [v,h] z V Dh ( O[V ]+ 2 Dv δ 2 O δv 2 ) v 2 V e (Seff[V,H]+S (2) [v,h]) Dh v i v j e S(2) [v,h] =(K ) ij K = K (vh) K (hh) K (vh) + K (vv) < O >= O[V ]+ 2 tr { δ 2 O δv 2 } K V st order master formula
Correlators C(t) =< O(t 0 + t)o(t 0 ) > < O(t 0 + t) >< O(t 0 ) >= C (0) (t)+c () (t)+ C (0) (t) = 0 < O(t 0 + t)o(t 0 ) >= O (0) (t 0 + t)o (0) (t 0 )+ 2 tr { δ 2 (O(t 0 + t)o(t 0 )) δ 2 v K } + C () (t) = 2 tr { δ (,) (O(t 0 + t)o(t 0 )) δ 2 v K } = 2 tr { δ(,) (O(t 0 + t)o(t 0 )) δ 2 v K } C () (t) = λ c λ e E λt E 0 = m H, E = m H,
To 2nd order S eff = S eff [V,H]+ 2 + 6 ( δ 3 S eff δh 3 h3 + δ3 S eff δv 3 ( δ 2 S eff δh 2 h2 +2 δ2 S eff δhδv hv + δ2 S eff δv 2 v3 ) + 24 v2 ) ( δ 4 S eff δh 4 h4 + δ4 S eff δv 4 v4 ) + O[V ]=O[V ]+ δo δv v + 2 δ 2 O δv 2 v2 + 6 ( δ 2 ) O δ 3 O δv 3 v3 ++ 24 δ 4 O δv 4 v4 + 2nd order master formula < O >= O[V ]+ 2 + 24 ( δ 4 ) O i,j,l,m δv 4 ijlm δv 2 ij ( K ) ij ((K ) ij (K ) lm +(K ) il (K ) jm +(K ) im (K ) jl ) C (2) (t) = 24 ( δ 4 O c ) (t) ) ((K ) ij (K ) lm +(K ) il (K ) jm +(K ) im (K ) jl i,j,l,m δv 4 ijlm
Lattice Observables (Polyakov loops) The scalar The vector t0+t t0+t t! A n5 t0 k t0! A m = lim t ln C () (t) C () (t ) n5=0 n5=n5 m = lim t ln C (2) (t) C (2) (t )
The Wilson loop x or x 5 t t t : e Vt < O W >
Mean-field parametrization The SU(2) model U(n, M) =u 0 (n, M) + i k u k (n, M)σ k u α R,u α u α = V (n, M) =v 0 (n, M) + i k H(n, M) =h 0 (n, M) i k v k (n, M)σ k, h k (n, M)σ k, v α C h α C The propagator (in momentum space) K(p,M, α ; p,m, α )=δ p p δ α α C M M (p, α ) C M M (p, α ) = [Aδ M M + B M M ( δ M M )] [ A = ( δ α 0)+ ] δ α 0 2βv 2 0 cos (p b 2 b N)+ ξ sin2 (p M /2) N M ( ) ( ) ( ) ( p B M M = 4βv 0 [δ 2 α 0 cos M p cos M p +( δ α 0) sin M p sin M 2 2 2 2 ( I2 (h 0 ) b = (I 2 )) 2 (h 0 ) h 0 h 0 I (h 0 ) I (h 0 ) I 3(h 0 ) b 2 = v 0 h 0 )]
The static potential V (r) = 2 log(v 0 ) +3 p M 0,p 0 =0 4 2v 2 0 L 3 N 5 p M 0,p 0 =0 (2 cos(p Nr) 2) N 0 4 (2 cos(p N r) + 2) C00 (p, 0) N 0 C 00 (p, ) The scalar mass C () H (t) = N (P (0) 0 ) 2 p 0 cos (p 0t) p 5 ( ) (N5) (p 5) 2 K (p 0, 0,p 5), 5, 0; (p 0, 0,p 5), 5, 0 (m 5) (n 5 )= m 5 r=0 δ n5 r, ˆr = r +/2 v 0 (ˆr)
The vector mass C (2) V (0) (t) =2304(P N 2 0 ) 4 (v 0 (0)) 4 p k sin 2 (p k) ( ) 2 K (t, p, ) K ((p 0, p ), 5, α) = p 5,p 5 (N 5) (p 5) (N 5) ( p 5)K (p, 5, α; p, 5, α) K (t, p, α) = p 0 e ip 0 t K ((p 0, p ), 5, α) The free energy F () = F (0) + 2N p ln [ ( det + K (hh) α =0 ) ( (vv) K α =0 det + K (hh) α 0 K (vv) α 0 ) 3 ] 2 FP
A few remarks on the calculation. We fix the Lorentz gauge; We pick up a Fadeev-Popov determinant but no ghost loops at this order; Gauge dependent free energy but no worries 2. Torons appear always as 0/0 --> 0/0=0 regularization 3. Formulas are generalized to the anisotropic lattice --> 2 independent Wilson loops since the background is different along 4D and the 5th dimension 4. Analytical formulas are computed numerically 5. Our working assumption: whenever non-trivial, physical Keep this in mind
The phase diagram the mean-field background compact 4D x µ v 0 4 D E C v 05 x 5 3 2 confined (v 0 = v 05 = 0) O N non-compact 5D F I N (v 0 0, v 05 0) d-compact 4D E D layered 4D (v 0 0, v 05 = 0).68 2 3
compact 4D The d-compact phase is stable: 4 D E C!=2.3, "=0.25 3 2 confined O N non-compact 5D F I N d-compact 4D E D 0 layered 4D.68 2 3 0 F () 0 20 30 40 0 0.2 0.4 0.6 0.8 0 0.5 v 5 v The compact phase (for small ) is unstable to this order β
compact 4D The W is massless or perhaps exponentially light: 4 3 2 confined D E C O N non-compact 5D F I N 0.4!=2.36, "=0.25 layered 4D d-compact 4D E D 0.35.68 2 3 0.3 0.25 a 4 m W 0.2 0.5 0. 0.05 mf data linear fit a 4 m W (0)=0.000(5) 0 0 0.005 0.0 0.05 0.02 0.025 0.03 0.035 /L
compact 4D The d-compact phase is separated from the layered phase by a 2nd order phase transition: "=/4 4 3 2 confined D E C O N non-compact 5D F I N E D d-compact 4D.6 layered 4D.68 2 3.4 am H.2 0.8 0.6 0.4 ν =/2 agrees with Svetitski & Yaffe: dim reduction to 4d Ising 0.2 0 mf data fit #=0.5 2.5 2.2 2.25 2.3 2.35 2.4 2.45 2.5!
The Static Potential
The static potential on the isotropic lattice: 0.66!=2.36, "=, L=N 5 =00 0.65 0.64 0.63 av(r) 0.62 0.6 0.6 mf data 4d Coulomb 5d Coulomb 0.59 0.58 0.57 0 0 20 30 40 50 r/a
The static potential (Wilson loop) in the d-compact phase along the extra dimension:. 0.9 Potential along the extra dimension,!=2.36, "=0.25 L=32, N 5 =32 L=64, N 5 =64 L=64, N =32 5 It doesn t scale with L: a 4 V(r) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 0 5 0 5 r/a 5 0 2!=0.55, "=0.625 The 5D force vanishes in the infinite volumecontinuum limit: F 5 (r)/f 4 (r) 0 0 0 L=00 L=26 0 0 2 L=50 L=200 L=250 L=300 0 0.05 0. 0.5 0.2 0.25 0.3 r/l
We are looking at an array of non-interacting 4D hyperplanes 4D Wilson loop - 5D Force --> 0
The short distance static potential in the d-compact phase along the 4d hyperplanes: β 4 large and β 5 small 0.65!=2.36, "=0.25, L=N 5 =200 0.645 a 4 V(r) 0.64 0.635 0.63 mf data 4d Coulomb 5d Coulomb 0.625 0.62 0 20 30 40 50 r/a 4 In the stability region r=0...50 the potential is clearly 4d Coulomb, large Yukawa excluded dimensional reduction in the d-compact phase
We are looking at an array of non-interacting 4D hyperplanes where gauge interactions are localized. 4D Georgi-Glashow on each of the branes 4D Wilson loop - G G G G G G G G G G 5D Force --> 0 The localization is perfect in the infinite L limit
compact 4D Remember: 4D Georgi-Glashow model + KK 4 3 2 confined D E C O N non-compact 5D F I N d-compact 4D E D layered 4D Higgs.68 2 3 Gauge coupling Confined phase We are somewhere around here Higgs coupling
We are looking at an array of non-interacting 4D hyperplanes where gauge interactions are localized. 4D G-G on each of the branes. The 4D Wilson loop at large r must describe a string 4D Wilson loop - 5D Force --> 0
What are we trying to do (preliminary analysis)? In the Luescher & Weisz paper: Our formalism: # ' & '-./02345*6."7!"##6.#7!"8%# 97$!! 97$%8 97$#! 97%!! 97%#! 97&!!.!+(,* % %! $ c(r)!!"$ d'('%!!$.!!"!#!"$!"$#!"%!"%#!"& ()*!!"&!!"% '023456478,9!:!"##9":!")%# d'(')!!"%%!!"%!!"%'!!"%)!!"# $ r [fm] V (r) =µ + σr + c r + d r 2 + l log(r) -.*/!!"%(!!"& ;:$!!!!"&% ;:$%)!!"&' ;:$#! ;:%!!!!"&) ;:%#! ;:&!!!!"&( ;<5=->5*'0 ;<5=->5*#0!!"'!!"!#!"$!"$#!"%!"%#!"& *+,
Systematics are not under control yet... e.g. if the /r^2 or the log term is not included in the ansatz:!!!"!'!!"#!!"#' /0(2*3.#.456 /0(2*3.$.456 #)$.( %.78,(-)7( 925:+;5(.&7 925:+;5(.'7. ' & %./(0*2-#-345./(0*2-$-345 - +,(-!!"$!+(,* $ $!!"$' #!!"%!!"%'!!!"&.!!"#!"$!"%!"&!"' ()*!# -!!"#!"$!"%!"&!"' ()* also, if we move along the line of the phase transition,...the value of the Luescher term shifts a bit...
Conclusions. The non-perturbative regime of 4d gauge theories can be probed analytically by the mean-field expansion in 5d 2. The phase diagram has a 2nd order phase transition where the system reduces dimensionally to an array of non-interacting 3-branes and where the continuum limit can be taken 3. In a dimensionally reduced phase there are effects of confinement and of the associated string 4. Controlling the systematics is in progress I would like to thank the Alexander von Humboldt Foundation for finacial supportx
compact 4D The continuum limit can be taken:!=0.25, "=.44333 0.038 4 3 2 confined D E C O N non-compact 5D F I N 0.037 d-compact 4D E D 0.036 layered 4D.68 2 3 0.035 0.034 q 2 0.033 0.032 0.03 0.03 0.029 0.028 0 0.005 0.0 0.05 0.02 /L 2 The physical charge at fixed ρ = m W m H
The long distance static potential in the d-compact phase along the 4d hyperplanes: 0.76 0.75!=2.36, "=0.25, L=N 5 =200 mf data linear + log linear + log + /r 2 0.74 L(x) 0.73 0.72 0.7 0.7 0.69 50 60 70 80 90 x=r/a 4 Luescher term L(x) =a 4 [V (r)+π/(8r)] and fit to σx + µ + c 2 ln(x)
Comparison Mean-Field vs Monte Carlo: (sample) 0.48!=2.3, L=N 5 =6 0.46 a 4 V(r) 0.44 0.42 0.4 MC data, "=4 mf data, "=2 mf data, "=4 0.38 Comparison Mean-Field vs Monte Carlo: 0.36 0 2 (sample) 4 6 8 0 r/a 4 MC data generated by M. Luz
The raw data... 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0 0 20 30 40 50 60 70 80 90 00
Yes, it scales with the lattice size... 0.65 Potential in the 4d hyperplanes,!=2.36, "=0.25 0.6 0.55 a 4 V(r) 0.5 0.45 0.4 0.35 L=32, N 5 =32 L=64, N 5 =64 L=64, N 5 =32 0.3 0 5 0 5 r/a 4
No, it can not be 5D Coulomb or Yukawa: 0 Potential in the 4d hyperplanes c 5d (r) 0.2 0.4 0.6 0.8.2.4.6.8 L=00 L=25 L=50 L=200 2 0 20 40 60 80 00 r/a 4