PoS(Lattice 2010)246 http://pos.sissa.it/ mass? W ∗ Speaker. As long as a Higgsbreaking boson remains is of interest. not The observed, question addressed themechanisms, here design is which whether of there deconfine alternatives are SU(2) for possibly dynamical electroweak attriplet. symmetry zero Results for temperature a and model generate withare a joint presented local in massive U(2) a vector transformations limit, boson of which SU(2) does not and involve U(1) any vector unobserved fields fields. ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c
The XXVIII International Symposium on LatticeJune Field 14-19, Theory 2010 Villasimius, Sardinia, Italy Department of Physics, Florida State University, Tallahassee, FL 32306, USA Bernd A. Berg Dynamical PoS(Lattice 2010)246 (2.1) (1.1) (1.2) Bernd A. Berg ]. 3 ) 1) phase. In the x . ( 1 ∗ , ν V = ν ) a e , B , ˆ β ν b µ µν with Polyakov loop scatter + B F x 1 b ( b µν ∗ µ F ] preceded our non-perturbative ig . For aligned U(1) matrices the 1 U ) Tr ) + 2 a ( 1 2 µ ˆ µ one may expect a SU(2) deconfining B − ν SU + λ ∂ x em ∈ µν ( − F ν 9) and Coulomb ( ν V ) generates dynamically a non-perturbative V . em B ) µν 0 x µ F 1.2 ( ∂ = 1 4 µ 2 e U − = β = b µν F center symmetry. ew ReTr 2 , L 2 2 matrices and λ µ boson through spontaneous symmetry breaking is explicit in , a ], which is demonstrated in Fig. × = ν 4 ew ∂ W µν add − xL S 4 ν a d , µ are SU(2) gauge fields. Z ∂ µ µν add = = S B S µν ∑ em µν F = ). taken as diagonal 2 ) add 1.1 1 S ( ] is not what is wanted (e.g., masses of spin 0 and 2 states are lower than for spin 1), 2 U are U(1) and µ 0 ∈ a U This mass generation for the The self interaction due to the commutator ( Motivated by the behavior of the U(1) Polyakov loop in the Coulomb phase, we add to the In Euclidean field theory notation the action of the electroweak gauge part of the standard Typical textbook introductions of the standard model emphasize at this point that the theory SU(2) matrices become aligned toophase and transition for due to large breaking enough of the Z with plots for coupling constants in the confined ( and, fermions would be confined,causes which a is deconfining not phase the transition, case.glueballs so break that Coupling up fermions a are into Higgs liberated, field elementaryhas a in indeed massive been photon the vector observed stays usual bosons. in massless way pioneering and Such lattice gauge a theory confinement-Higgs (LGT) transition investigations [ perturbation theory. Are therenew possibly physics? alternative dynamical The model mechanisms, considered whichU(1) here do LGT is not with motivated involve by the the Wilson deconfining action phase [ transition of pure U(1) and SU(2) Wilson actions model reads 2. SU(2) alignment Dynamical W mass? 1. Introduction where contains four massless gauge bosonsthe and theory so introduce that the only Higgs onethe gauge mechanism introduction boson, as of the the a photon, Higgs stays vehicle particle massless. tounderstanding in Such electroweak modify of presentations interactions reflect [ non-Abelian that gaugespectrum theories. of ( In fact, massless gluons aremass not gap, in and the the SU(2) physical spectrumto consists set of massive a glueballs. mass Thedoes lightest scale. not glueball constitute can an be Choosing used ansatz forspectrum for it, [ an electroweak e.g., theory 80 by two GeV main and reasons: coupling The fermions SU(2) glueball is admissible. This Coulomb phase the effective potential of the Polyakov loop is similar to that of a Higgs field. PoS(Lattice 2010)246 (2.3) (2.2) , values. is a pure ) 9 (center) . ) ˆ κ ν 0 ˆ Φ ν . The gauge − = ) Bernd A. Berg − n e 2 n ( ( } β ( ν + ν V SU )] ) U a ˆ ) ν ˆ ˆ ν ν 0.2 − and µ + − lattice at n B ) ) x ( n ν 4 1 ( a ∗ ( µ ( (after gauge fixing) give ∂ ˆ ∗ ν µ Φ b V which would be obtained by U ) ) g V + x µ µν add ˆ ) ν ( x − ˆ S B ν ν ( − ν µ 0.1 ∗ V ) from the following gauge invari- µ . The vacuum value of − ) A ˆ B µ ) ∞ U x ˆ µ µ 1 ( ) ∼ ∂ ( + x + a → + ( n U g ) ( Φ µ n κ a ∗ ( ν ∈ U ∗ ν = )][ V { α 0 a i U ˆ e ν , Re(P add ) + µν ) + n ) + contributions of F 2 ReTr ( n a ( 3 4 ∗ ( ν 2 λ ˆ µ a , ∗ ν V SU ) + U + ˆ ν ) } x ∈ ˆ add ] ( ν µν -0.1 0 the + 2 g F Φ ) + n ) 1 ( → n the gluon field. The interaction leads to similarities with the a add ∗ µν µ ( ˆ − a µ µ ∗ F V µ 1) ) B Φ V + ) we perform the the London limit for the potential and fix the ˆ ) , where µ + = x Tr ˆ ( µ Φ a Φ + 2.1 2 ν -0.2 λ g [( + n V α ( ) − i . n ν a Tr 2 e ( 0 − ˆ V g ν e µ κ = 0.2 0.1 / -0.2 -0.1 U 1 µ
+
we need the contribution to the action, which comes from the summed up
→
=
6= a x ∑ Im(P ) = add ) ν ν ( ∑ Φ e L n b + ( β 2 2 λ β µν add . To obtain ( 2 matrix scalar field that is charged with respect to µ Φ is the photon and S [ g V . Similar results are expected from simulations at sufficiently large finite 1 × α + i 1 µ − ) = a e n = ( 1 (ring), = . Scatter plot for U(1) Polyakov loops with the Wilson action on a 12 is a 2 µ = Φ , 1 µ t Φ Φ A = V e In the classical continuum limit Our Monte Carlo procedure proposes the usual U(1) and SU(2) changes. For the update of This interaction can be obtained in the London limit ( β gauge transformations are where staples (using now lattice units a SU(2) matrix where SU(2) Higgs model. However, there isapplying no the explicit London mass limit term to conventional Higgs coupling. gauge to ant expression: Figure 1: Dynamical W mass? and PoS(Lattice 2010)246 3 . 2 (3.2) = b β 5 as shown × Bernd A. Berg 1 ] from fits to correlation lattices, which are shown 7 N 0.9 t (ordered o and disordered d starts). N U(1) o U(1) d λ 0.8 0 (3.1) , SU(2) o SU(2) d Mixed o Mixed d t . N → lattice 3 4 ) ) N x 2 0.7 ( / ] of Wilson loops up to size 5 µ 1 6 λ k V , and signals are only followed up to distance ( i 2 This is well suited for updates with a biased τ + . 0.6 ) 4 n Tr 4 sin a strong first-order transition is found as shown in ( i µ − − λ estimates on lattice as function of U 1 Actions on 12 Actions on 1 in the Coulomb phase of pure U(1) LGT and 2 k 4 0.5 . N 1 E / ) = π = = x 2 e ( β µ 0.4 , = i 1 V ], which gives acceptance rates larger than 95% for U(1) as well 2 photon k 5 , m 1 k 0.3 E 0
0.1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
0.01 0.3 0.2 0.1 0.4
0.001 0.15 0.05 0.45 0.35 0.25
C(t) κ 1/2 Data and fits for photon energy SU(2) and U(1) string tensions on a 12 Figure 4: with that of the labeling. Figure 3: Dynamical W mass? PoS(Lattice 2010)246 . 5 6 64 lattice. , we have 3 κ Bernd A. Berg is given in Fig. 9 on a 12 . λ 0 25 = 1 λ 20 =0.9 0.8 λ 15 mass estimate at N=12, 0.6 λ W t [a] m 10 mass spectrum as function of 6 0.4 W glueball + 5 0 0.2 SU(2) theory, which exhibits a zero-temperature deconfining W vector boson ⊗ 0 0 1 Sketch of the glueball and vector boson mass spectrum. 0.1
0.01 0.2 0.1 3.2 1.6 0.8 0.4
0.001 0.05
am C(t) 9 (ordered start). The signal is beautifully strong and can be followed beyond . 0 Figure 6: = . A sketch of the glueball and a λ 20 = Momentum zero correlation function and fit for t To the extent that similar results hold also for finite (sufficiently large) values of Figure 5: They are zero in the disordered phase and strong in the ordered phase. For our largest lattice Fig. shows them at distance 4. Summary and conclusions constructed a gauge-invariant U(1) Dynamical W mass? PoS(Lattice 2010)246 , 165 , 370 Bernd A. Berg 140 , 95 (1987). 199 , 327 (1983). 19 , , ) ) a a ˆ µ ˆ µ + + x x ( ( α becomes unphysical, because it α i i e e ) Φ ) a a ˆ ˆ µ µ + + , 94 (1984); P. Majumdar, Y. Koma, and M. , 294 (1981); H. Kühnelt, C.B. Lang, and G. x x ( ( 52 1 1 104 − − g g ) ) x x ( 7 ( µ µ V U , 114506 (2005). ) ) x x 71 ( ( g g ) ) , 441 (1985); W. Langguth and I. Montvay, Phys. Lett. B x x ( ( SU(2) symmetry α α SU(2) scalar matrix field i i ⊗ 150 − − , 1264 (1967); A. Salam, Proceedings of the 8th Nobel Symposium, ⊗ e e ]. 19 9 U(1) → → , , 2445 (1974). , 2308 (1980). = [FS10], 16 (1984); M. Tomiya and T. Hattori, Phys. Lett. B 8 ) ) boson mass dynamically. However, by dimensional reasons this theory , 273 (2004). 10 21 x x ( ( This work has in part been supported by DOE grant DE-FG02-97ER- W µ µ 230 677 V U 135 (1985). Koma, Nucl. Phys. B simulation program, which has meanwhile beensome corrected. of This the accounts figures for here the and difference corresponding between ones of this reference. (1984); I. Montvay, Phys. Lett. B N. Svartholm (editor), Almqvist and Wiksell, Stockholm 1968. indicating a high mass: K. Ishikawa, G. Schierholz and M. Teper, Z. Phys. C Vones, Nucl. Phys. B Improvements focused on spin 0 and 2: C. Michael and M. Teper, Phys. Lett. B Acknowledgments: In this way all remnants of the scalar field disappear without destroying invariance of the In the London limit the U(1) [4] K. Wilson, Phys. Rev. D [5] A. Bazavov and B.A. Berg, Phys. Rev. D [9] B.A. Berg, arXiv:0910.4742. [8] B.A. Berg, arXiv:0909.3340. Monte Carlo results reported in this reference suffer from a bug in the [6] M. Creutz, Phys. Rev. D [7] B.A. Berg and C. Panagiotakopoulos, Phys. Rev. Lett. [1] S. Weinberg, Phys. Rev. Lett. [2] In early lattice calculations no useful correlations beyond one lattice spacing were found for spin 1, [3] C.B. Lang, C. Rebbi, and M. Virasoro, Phys. Lett. B 41022 and by the German Humboldtfor Foundation. The their author kind thanks Wolfhard hospitality Jankediscussions. and at his group Leipzig University and Holger Perlt andReferences Arwed Schiller for useful action, so that a localof U(2) the invariance scalar of bosons (fermionic) matter theto fields clarify additional whether can action this be term allows kept. isthe for SU(2) After now a Higgs elimination quantum (as model, continuum desired) where limit. dimensionless theinvolved, but The and interaction acquires situation term it a is is dimension remains kind dimensionless in as of the long opposite London as limit. to the scalar boson is Dynamical W mass? transition and generates a which was first proposed in [ with scalar bosons has presumably no physical quantum continuum limit. takes its vacuum valuepresent and model they does can not be absorbed fluctuate,vector by fields extending while into the its a gauge local transformations gauge U(2) of transformations the U(1) survive. and SU(2) In the