K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B493:413, 1997  Home/Search   Document Not in Database   Summary   Related Articles   Check

....J(3 2 ; 1; 1; 1) up to the part. Using the representation (9) it can be easily calculated, and we arrive at the following result: I(3 Gamma 2 ; 1; 1; 1) 4 Gamma2 Gamma 2 (1 ) ae 1 2 Gamma 4 ln (m 1 m 2 m 3 ) oe O( 34) This corresponds to the result presented in [23], eq. 110) Note that simple results for three dimensional triangles were presented in [24] For n = 4 Gamma 2 , eq. 30) gives: I(4 Gamma 2 ; 1; 1; 1) 3 Gamma3 i 1 2 (m 2 1 m 2 2 m 2 3 ) 1 Gamma Gamma( Gamma1 2 ) Gamma(2 Gamma ) J(2 2 ; 1; 1; 1) 35) Using Feynman ....

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B425 (1994) 67.

....bosonic modes contribute to the effective action. In contrast to QCD, dimensional reduction should work for the electroweak theory around and above the transition temperature because g 2 is small. For the electroweak phase transition this approach has been pioneered by Farakos et al. see e.g. [17, 18]) This program aims 2 at exploring the accuracy of dimensional reduction at various Higgs masses by comparing various parameters of the transition with those of 4 dimensional lattice and perturbative approaches. Perturbation theory is necessary to relate the 4 dimensional continuum theory to ....

.... g 4 3 12 h 3 g 2 3 Gamma 6 h 2 3 9 3 g 2 3 Gamma 12 2 3 Theta ln 3T 3 Gamma 0:348725 (4) with L b = ln 2 4 T 2 Gamma 2(ln(4) Gamma fl E ) L f = L b ln 16; s 2 = 4ff s g 2 0 : 5) Note that we have changed the notation as compared to [18] to stay in direct contact with the notations used in [7] The strong coupling is chosen ff s = 0:125 and the weak coupling is g 2 0 = 4 p 2GF M 2 W with the Fermi constant GF = 1:16639 Theta 10 Gamma5 GeV Gamma2 . For definiteness, we use as the 3 dimensional renormalization scale 3 ....

[Article contains additional citation context not shown here]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B458 (1996) 90

....bosonic modes contribute to the effective action. In contrast to QCD, dimensional reduction should work for the electroweak theory around and above the transition temperature because g 2 is small. For the electroweak phase transition this approach has been pioneered by Farakos et al. see e.g. [17, 18]) This program aims 2 at exploring the accuracy of dimensional reduction at various Higgs masses by comparing various parameters of the transition with those of 4 dimensional lattice and perturbative approaches. Perturbation theory is necessary to relate the 4 dimensional continuum theory to ....

....sizes are needed in order to obtain a reasonable extrapolation to the infinite volume limit. Besides of jumps in the quadratic and quartic Higgs condensates also a discontinuity in the expectation values of E link appears being a good indicator for the phase transition as well. It is known (see [17]) that the sum of expectation values of the various Higgs operators is constant according to a sort of Schwinger Dyson equation Gamma3fi H hE link i (1 Gamma 2fi R )hae 2 i 2fi R hae 4 i = C (21) independent of the couplings fi H , fi G and 3 =g 2 3 . We have checked this sum rule ....

[Article contains additional citation context not shown here]

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B425 (1994) 67; B442 (1994) 317

....within a pure SU(2) gauge Higgs version of the theory. This model has become a test field to control the validity of perturbative predictions over a broad range of Higgs masses. At present, one is interested to see whether the first order transition ends somewhere around a Higgs mass MH 100 GeV [15, 16]. Lattice simulations are not only able to describe both phases starting from first principles but, moreover, make it possible to put both phases into coexistence near the phase equilibrium. Thus one is able to measure directly quantities like latent heat, surface tension, condensates etc. ....

....deviation from perturbative predictions at M H = 70 GeV, i.e. additional weakening of the phase transition with increasing Higgs mass. At still larger Higgs masses (MH MW = 80 GeV) the phase transition has been suggested recently to be of second order [15] or to be a smooth crossover [16] in accordance with predictions from the study of gap equations [14] The first case would be difficult to discriminate from a phase transition being very weakly first order. In any case, however, one will observe the turn over to a very weak or continuous transition at higher Higgs mass. 6 ....

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, CERNTH -96-126 (1996), hep-ph/9605288

....doublet. This model represents whole classes of 4d theories but might not be sufficient to describe all 4d variants generically. So far, only this simple effective theory has been studied by lattice techniques, for several Higgs self couplings 3 in units of the 3d gauge couplings squared g 2 3 [5, 7, 6, 8]. Here we extend our analysis [6] to a higher coupling value 3 g 2 3 = 1 8 i M H 80 GeV j 2 (1) namely 3 =g 2 3 0:095703, referred to as M H = 70 GeV, and compare with previous results at this smaller coupling 3 =g 2 3 0:023926 (M H = 35 GeV) As expected the first order ....

....3 g 2 3 = 1 8 i M H 80 GeV j 2 (1) namely 3 =g 2 3 0:095703, referred to as M H = 70 GeV, and compare with previous results at this smaller coupling 3 =g 2 3 0:023926 (M H = 35 GeV) As expected the first order nature has become weaker but is still evident. Note that Ref. [7] uses another parametrization for the ratio (1) such that our M H = 70 GeV (35 GeV) would correspond to m H = 72:17 GeV (38:92 GeV) in their notation. In the following we use the abbreviation h = MH =MW to denote the ratio of the supposed Higgs mass to the real W mass. 2 Relation of the ....

[Article contains additional citation context not shown here]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B466 (1996) 189

....of the non perturbative features can be simplified by combining perturbation theory and non perturbative methods. Indeed, the momentum scale p T can be integrated out perturbatively, resulting in an effective theory for length scales larger than 1=T . This is called dimensional reduction [12,28 34]. The effective theory is essentially a threedimensional (3D) super renormalizable SU(2) gauge theory with fundamental and adjoint Higgs fields. The effective theory can be studied with analytic [14 20] and lattice [25 27] methods with less effort than the original four dimensional theory. ....

....The theories to be discussed are the SU(2) and U(1) gauge theories with Higgs fields in fundamental and adjoint representations. The SU(2) theory with both fundamental and adjoint Higgs fields, or only with a fundamental Higgs field, is relevant for the cosmological electroweak phase transition [12,31,32]. The SU(2) theory with just an adjoint Higgs field is relevant for studies of dimensional reduction of the pure SU(2) gauge theory [38] The U(1) gauge theory with a fundamental Higgs field could be relevant for numerical studies of superconductivity [39] The plan of the paper is the following. ....

[Article contains additional citation context not shown here]

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 425 (1994) 67 [hep-ph/9404201].

....of EWPT, experienced during the last year. Large scale studies of the full finite temperature system receive renewed interest [4,5] The range of the Higgs mass explored to date is below 50 GeV, where the first order signal is strong. In simulations of the reduced (static) sector of the model [6,7] careful implementation of the curves of equal physics is the most sensitive part of the interpretation of lattice data. 2. Continuum limit of the lattice model Our goal with the present investigation was to study effects exerted by scalar fluctuations on the transition. Magnetic and electric ....

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B425 (1994) 67

....of EWPT, experienced during the last year. Large scale studies of the full finite temperature system receive renewed interest [4,5] The range of the Higgs mass explored to date is below 50 GeV, where the first order signal is strong. In simulations of the reduced (static) sector of the model [6,7] careful implementation of the curves of equal physics is the most sensitive part of the interpretation of lattice data. 2. Continuum limit of the lattice model Our goal with the present investigation was to study effects exerted by scalar fluctuations on the transition. Magnetic and electric ....

....Linde s renormalisation conditions at T = 0 [10] When compared with other (simpler) renormalisation schemes significant differences were found for mH (T = 0) 30GeV . In order to ensure comparability with results from simulations of the more complete version of the 3 dimensional reduced model [6] we have chosen to give finally for a value corresponding to mH 35GeV . has been tuned at fixed Theta to the transition point indicated by the fast variation of the order parameters O 1 = 1 L 3 P x x x ; O 2 = 1 3L 3 P x;e x x e ; O 3 = 1 L 3 q f( P x x ....

[Article contains additional citation context not shown here]

K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B407 (1993) 356

....transition, for example, lattice simulation [14, 15, 16, 17, 18, 19] C.J.T. method [20, 21] expansion [22] effective three dimensional theory e mail address: inagaki hetsun1.phys.kobe u.ac.jp y e mail address: ogure icrr.u tokyo.ac.jp z e mail address: joe hep th.phys.s.u tokyo.ac. jp [23, 24, 25, 26, 27], gap equation method [28] non purturbative renormalization group method[29, 30, 31, 32, 33] and so on. All the same we still need another method to study the phase transition since they are applicable to limited situations. In Ref. 34] a new non perturbative approach was suggested to avoid the ....

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B442 (1995) 317.

....here that at temperatures T TR the electroweak symmetry is restored , and that the U(1) hypercharge magnetic field is massless at that time. No term like m 2 Y b 2 is generated in any order of perturbation theory in abelian gauge theory at high temperature [10] the lattice study in [11] confirmed this expectation for SU(2) ThetaU(1) EW theory beyond perturbation theory) If the hypercharge magnetic fields survive until the time of the EW phase transition (T 100 GeV) they will give rise to ordinary magnetic fields because of electroweak mixing. In the rest of this paper we ....

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B493:413, 1997

.... accepted that baryogenesis is not possible in the minimal Standard Model (MSM) The reason is that the sphaleron constraint, which is needed to ensure the survival of the baryon number against sphaleron induced erasure, cannot be satisfied in the MSM, no matter what the mass of the Higgs boson is [1], because the radiative correction induced to the effective action by the top quark makes the transition too weakly first order. While this result was first borne out from a detailed lattice calculation, it can actually be robustly seen already at the level of the simple 1 loop effective ....

....From (3) one can easily find that at the critical temperature, when the two minima of the potential become degenerate OE T = 2ffi eff : 8) In MSM OE=T has a maximum value of 0.55 when mH = 0 and decreases monotonically as mH increases. However, avoiding the sphaleron wash out requires that [1] OE T min 1:0 Gamma 1:5 ; 9) and therefore it follows from (8) that baryogenesis is not possible in MSM. Note that this conclusion is robust, because one expects that for accepted values of OE=T 1 in Eq. 9) the effective potential approach is quite reliable. Let us now address ....

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposnikov, Nucl. Phys. B466 (1996) 189.

....should turn out to be too weakly first order for spectacular effects, and modified theories have to be considered. The electroweak effective potential at high temperatures has been treated carefully in two loop order of perturbation theory [16] 19] Comparison with lattice nonperturbative results [4, 8, 9, 10] gives good agreement for the Higgs and W boson masses in the broken phase up to zero temperature Higgs masses of about 70 GeV. The hot electroweak phase appears as a case of a confining gauge system. The Higgs and W correlation masses have recently been explained by a bound state model within ....

....well with the values of these quantities measured on the lattice. The situation for the interface tension is quite different. It is more difficult to measure on the lattice. While one lattice group finds good agreement [9] another group reports large deviations at intermediate Higgs masses [4]. Furthermore the perturbative values of the interface tension calculated from oe = Z A S d q 2 V ( r) 1) in the literature do not include important perturbative effects, as we will argue in this paper. The interface tension is the free energy per area of interface between the hot and ....

[Article contains additional citation context not shown here]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, CERN-TH/95-263, hep-lat/9510020.

....Perturbation Theory The electroweak standard theory at the transition temperature allows the high temperature expansion. To good accuracy it can be described by a local effective 3 dimensional SU(2) Higgs model whose parameters can be calculated from the fundamental 4 dimensional theory [3]. All infrared problems appearing in the unbroken phase are covered by this effective theory. The nature of the phase transition depends substantially on the static Matsubara frequencies. Calculating the one loop effective action of a scalar background field in the derivative expansion using a ....

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, CERN-TH/95-226, hep-ph/9508379.

....is known at present about the size and functional form of quantum corrections to dynamical, real time quantities. The best we can offer is an educated guess based on a much better understood relation between the static properties of the quantum theory and of the effective dimensinally reduced one [12, 13]. The full thermal path integral, where the bosonic fields obey periodic boundary conditions in the imaginary time direction, reduces in the high temperature limit to a partition function of a three dimensional statistical mechanical problem. This partition function inherits the UV thermal cutoff ....

K. Farakos, K. Kajantie, K. Rummukainen and M.E. Shaposhnikov, Nucl.Phys. B425 (1994) 67; Nucl.Phys.B442 (1995) 317; Phys.Lett.B336 (1994) 494.

....which are related to the heavy quark potential at finite temperature. Observables, which project onto states with the correct quantum numbers have been discussed in Ref. 7] and have recently been used to analyse the high temperature phase of QCD within the framework of dimensional reduction [8, 9]. In how far the definition of a screening mass through gauge invariant operators agrees with the pole mass has to be further analysed. The experience made with similar concepts in gauge Higgs models shows that there is the possibility that gauge invariant operators only project onto a ....

....as well as the color singlet potential we find m e (T ) q 1:70(2) g(T ) T . This result shows that the screening mechanism is highly non perturbative even for temperatures as large as 14000 T c . This observation is in accordance with studies of screening in dimensionally reduced 3d QCD [8, 9]. Our simulation of the gluon correlation functions at finite momenta still suffer from insufficient statistics. We find a modification of the energy momentum dispersion relation of a free particle, but we are not yet able to quantify its temperature dependence. The improvement of the action does ....

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, hepph /9704416, April 1997.

....is very small. Sec. 11 is the conclusions. 2 The EW phase transition in the Standard Model The thermodynamics of the electroweak phase transition in the full Standard Model has been extensively studied in the literature ( 2] and references therein) Perturbative studies exist up to 2 loop level [23, 24, 14, 25]. Non perturbative lattice studies rely on perturbative 2 loop dimensional reduction [12 15] and have been performed for a wide range of Higgs masses [18 22] The Higgs mass bound in terms of the parameters of the 3d SU(2) Higgs model was derived in [19] The Higgs mass bound arises as follows. ....

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 425 (1994) 67.

....the phase transition is not known at present. The temperature dependence of the effective potential has been studied in perturbation theory [6, 7] as well as in lattice simulations [8, 9] If the mass of the Higgs is not too high (less than MW ) the phase transition is supposed to be first order [10]. In this case the transition from the symmetric vacuum with massless particles to the broken symmetry phase would proceed via bubble nucleation. This phenomenon as well as its cosmological aspects, has been studied by various groups [11, 12, 13] Part of the basic information needed in developing ....

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Phys. Lett. B336, 494 (1994).

....[1] Many subjects, as e.g. the question of baryogenesis [2, 3] are still controversial [4, 5] Even the nature of the phase transition is not known at present. The temperature dependence of the effective potential has been studied in perturbation theory [6, 7] as well as in lattice simulations [8, 9]. If the mass of the Higgs is not too high (less than MW ) the phase transition is supposed to be first order [10] In this case the transition from the symmetric vacuum with massless particles to the broken symmetry phase would proceed via bubble nucleation. This phenomenon as well as its ....

K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B407, 356 (1993).

....been discussed recently in various aspects [1] Many subjects, as e.g. the question of baryogenesis [2, 3] are still controversial [4, 5] Even the nature of the phase transition is not known at present. The temperature dependence of the effective potential has been studied in perturbation theory [6, 7] as well as in lattice simulations [8, 9] If the mass of the Higgs is not too high (less than MW ) the phase transition is supposed to be first order [10] In this case the transition from the symmetric vacuum with massless particles to the broken symmetry phase would proceed via bubble ....

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B425, 67 (1994).

....absent, b) one may also integrate out the temporal component A 0 of the gauge field, so its mass scale gT also disappears. Thus there are two mass scales less and this reduces substantially the computer time needed to get reliable results. The model that has already been studied along these lines [5, 8] has been based on SU(2) with one complex Higgs doublet. For work on the same model in asymmetric four dimensional lattices, see [15] The issues studied have been the order and the characteristics (critical temperature, latent heat, surface tension, correlation lengths) of the phase transition, ....

....theory are renormalization group invariant because the theory is supernormalisable. The mass parameter m 2 3 contains a linear and a logarithmic divergence. It is convenient to use the new set of parameters (g 2 3 ; x; y) rather than the set (g 2 3 ; 3 ; m 2 3 ) x; y are defined by [8]: x = 3 g 2 3 (10) y = m 2 3 (g 2 3 ) g 4 3 (11) It is evident that x is just proportional to the ratio of the squares of the scalar and vector masses; on the other hand, y is related to the temperature. The parameters x; y can be expressed in terms of the four dimensional parameters as ....

[Article contains additional citation context not shown here]

K.Kajantie, M.Laine, K.Rummukainen and M.Shaposhnikov, Nucl.Phys.B466 (1996) 189

....because of severe infrared divergencies. One promising approach has been to reduce the four dimensional model at finite temperature to an effective model in three dimensions. This can be done if the couplings are small and the temperature is much larger than any other mass scale in the theory [1, 2, 3]. The parameters of the reduced theory are related to the ones of the original model through perturbation theory. The reduced theory has some advantages over the original one from the computational point of view [5, 6, 7] It is super renormalizable and yields transparent relations between the ....

....modes A (n 6= 0; k) n 6= 0; k) are thus suppressed by the factor (2 nT ) 2 relative to the static A (n = 0; k) and (n = 0; k) modes. The method of dimensional reduction consists in integrating out the non static modes in the action and deriving an effective action [2, 3]. An important remark is that the mass of the adjoint Higgs field is of order gT; which is large compared to g 2 T; the typical scale of the theory. Thus one can go on one step further and integrate it out using perturbation theory [5, 6, 7] The effective action may then be written in the ....

K.Kajantie, K.Rummukainen and M.Shaposhnikov, Nucl.Phys. B407 (1993) 356.

....related solely to the broken phase, away from the critical temperature, could be calculated. The problem is that there may be dependent vacuum parts in these quantities as in the value of pressure. However, when one is comparing perturbation theory with lattice calculations, can be fixed [16], and the vacuum parts are not a problem. To illustrate the convergence of these calculations, we note that for mH = 80 GeV, 100 GeV and T = 60 GeV, the h 0 , h 1 and h 2 contributions to h Phi y Phii are [25:47, 3:76 and 0:39] Theta 10 3 GeV 2 (for these parameters, T c ....

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Phys. Lett. B 336 (1994) 494.

....Higgs masses, mH mW 80 GeV. This case seems to be the most interesting one for cosmological applications, because the phase transition at least in this region is of first order. First Monte Carlo results on the electroweak phase transition using a 3d effective theory have already been given in [1, 3], see also [14, 15] In comparison with [1, 3] we considerably extend numerical calculations. This makes it possible to determine for the first time or more accurately than previously a number of finite T quantities such as latent heat, correlation lengths, interface tension, magnitude of the ....

....to be the most interesting one for cosmological applications, because the phase transition at least in this region is of first order. First Monte Carlo results on the electroweak phase transition using a 3d effective theory have already been given in [1, 3] see also [14, 15] In comparison with [1, 3] we considerably extend numerical calculations. This makes it possible to determine for the first time or more accurately than previously a number of finite T quantities such as latent heat, correlation lengths, interface tension, magnitude of the higher order perturbative terms, etc. An essential ....

[Article contains additional citation context not shown here]

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Phys. Lett. B 336 (1994) 494 [hep-ph/9405234].

....phase transition, see [1 4] In the present paper, we study with lattice simulations the dominant SU(2) Higgs part of the theory, defined by the Lagrangian L = 1 4 F a ij F a ij (D i OE) y (D i OE) m 2 3 OE y OE 3 (OE y OE) 2 : 1. 1) The procedure of dimensional reduction [2], 6 13] allows one to compute perturbatively the relationship between the temperature T and the physical parameters of the underlying 4d electroweak theory or its extensions, and the parameters of the 3d theory. Concrete formulae for the SU(2) Higgs model and MSM can be found in [5] see also ....

....of our results to cosmology. The conclusions and proposals for future work are in Section 13. Readers not interested in details of lattice simulations could check tables 5 7, and go directly to Section 10, which contains a summary of the non perturbative results. In this paper we use results from [2] and [4] All references to specific formulae from these papers are indicated by I and II followed by the number of the corresponding expression. 2 3d theory in continuum and on lattice To make the paper self contained, we summarize here the essential properties of the 3d theory in continuum and ....

[Article contains additional citation context not shown here]

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 425 (1994) 67 [hep-ph/9404201].

.... 3 kari trek.physics.indiana.edu 4 mshaposh nxth04.cern.ch 1 Introduction The 3d SU(2) ThetaU(1) Higgs model is a universal theory for the description of the electroweak phase transition in the standard electroweak theory and many extensions thereof, including the MSSM [1 5] (for a motivation of the study of the electroweak phase transition, see [1 4] In the present paper, we study with lattice simulations the dominant SU(2) Higgs part of the theory, defined by the Lagrangian L = 1 4 F a ij F a ij (D i OE) y (D i OE) m 2 3 OE y OE 3 (OE y OE) 2 : ....

.... 1 Introduction The 3d SU(2) ThetaU(1) Higgs model is a universal theory for the description of the electroweak phase transition in the standard electroweak theory and many extensions thereof, including the MSSM [1 5] for a motivation of the study of the electroweak phase transition, see [1 4]) In the present paper, we study with lattice simulations the dominant SU(2) Higgs part of the theory, defined by the Lagrangian L = 1 4 F a ij F a ij (D i OE) y (D i OE) m 2 3 OE y OE 3 (OE y OE) 2 : 1.1) The procedure of dimensional reduction [2] 6 13] allows one to ....

[Article contains additional citation context not shown here]

K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 407 (1993) 356.

....a function of mH and T in Fig. 7, and the parameters x and y are given in Fig. 8. The phase diagram of the theory with the parameters x, y, together with the values of the latent heat, surface tension and correlation lengths in units of g 2 3 , have been studied with lattice MC simulations in [50]. Finally, one has to account for the effect of the U(1) subgroup on the EW phase transition. Since there are no lattice simulations available for the 3d SU(2) ThetaU(1) Higgs model, the best one can do is to estimate the effect of the U(1) subgroup perturbatively. In Fig. 9, we display the ....

....display the percentual perturbative effect of the U(1) subgroup on the critical temperature T c , the vacuum expectation value of the Higgs field v, the latent heat L, and the surface tension oe as a function of the physical Higgs mass. Using the non perturbative values for the case g 0 = 0 from [50], one can then derive results for the full Standard Model. 5.4 The effect of higher order operators In Landau gauge in the Standard Model, the dominant 6 dimensional OE 6 operators related to the integration over the superheavy scale are O (6) g 2 = 3i(3) 16 384 4 g 6 OE 6 T 2 ; ....

[Article contains additional citation context not shown here]

K. Farakos, K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, in preparation. (a) (b)

....this theory was carried out in [44, 45, 46] The procedure of dimensional reduction will give an infrared safe connection between the parameters of the underlying 4d theory and those of the 3d theory. The latter can then be studied by non perturbative means, such as lattice Monte Carlo simulations [47, 48, 10, 49, 50]. ....

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Phys. Lett. B 336 (1994) 494.

....this theory was carried out in [44, 45, 46] The procedure of dimensional reduction will give an infrared safe connection between the parameters of the underlying 4d theory and those of the 3d theory. The latter can then be studied by non perturbative means, such as lattice Monte Carlo simulations [47, 48, 10, 49, 50]. ....

K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 407 (1993) 356.

....the idea of dimensional reduction has been around for quite a long time [3, 4] However, some concrete analytical results for the construction of the 3d effective theory have appeared only recently. They are relevant for the description of the high temperature electroweak phase transition [2] [7] [13] and high temperature QCD [14] 19] The aim of the present paper is the formulation of the general rules of dimensional reduction in a constructive way. Namely, we present a set of 1 loop and 2 loop Feynman diagrams with the results of their computation which can be used for dimensional ....

....1 loop and 2 loop Feynman diagrams with the results of their computation which can be used for dimensional reduction in any gauge field theory. As an example we construct the 3d effective theory corresponding to the Standard Model of electroweak interactions. New elements here in comparison with [7] are the inclusion of fermions, and the direct relation of the parameters of the effective theory to the physical parameters of the EW theory (the physical Z and W boson, Higgs particle and top quark masses, the muon lifetime and the temperature) We also discuss the strategy for the derivation of ....

[Article contains additional citation context not shown here]

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 425 (1994) 67.

....the thermodynamics of Yang Mills fields [1] A direct way to compute static equilibrium quantities at high temperature would be to do lattice Monte Carlo simulations in the 4d high temperature theory. However, in many interesting cases the use of the full 4d theory is difficult, if not impossible [2]. These obstacles invoke a demand for a formalism which can solve in a constructive way the problems mentioned. Since a finite temperature equilibrium field theory is equivalent to a zero temperature Euclidean field theory with compact 4th dimension, the idea of the 4d 3d dimensional reduction ....

....In fact, the idea of dimensional reduction has been around for quite a long time [3, 4] However, some concrete analytical results for the construction of the 3d effective theory have appeared only recently. They are relevant for the description of the high temperature electroweak phase transition [2], 7] 13] and high temperature QCD [14] 19] The aim of the present paper is the formulation of the general rules of dimensional reduction in a constructive way. Namely, we present a set of 1 loop and 2 loop Feynman diagrams with the results of their computation which can be used for ....

[Article contains additional citation context not shown here]

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 442 (1995) 317.

....numerically below, and it will turn out that the phase diagram is the standard tricritical one, as shown in the left part of Fig. 2. 4 Relation to 4d and to Z(N) symmetry The relation between x; y and 4d physics (T; MS ; N; N f ) is worked out explicitly to leading next to leading order in [4, 21]. For N = 3; N f = 0 the answer is x = 3 8 2 g 2 (4 e Gammafl E Gamma3=11 T ) 3 11 1 ln(5:371T= MS ) 25) y j y dr (x) 3 8 2 x 9 16 2 O(x) 26) As discussed, the 4d finite T theory without matter in the fundamental representation x Volumes 0.10 12 3 0.15 16 3 ....

K. Kajantie, M. Laine, J. Peisa, A. Rajantie, K. Rummukainen and M. Shaposhnikov, Phys. Rev. Lett. 79 (1997) 3130 [hep-ph/9708207].

....of finite temperature QCD, and thus we assume that h 3 = h 5 = 0. In the absence of h 3 ; h 5 , the theory in Eq. 1) is symmetric under A 0 GammaA 0 . This symmetry was called R parity in [6] In terms of 4d physics, this symmetry is related to the usual discrete transformations CT, P [14, 15]. However, it should be clearly stated that the breaking of the A 0 GammaA 0 symmetry to be discussed below, does certainly not imply spontaneous breaking of any of the discrete symmetries of finite temperature QCD, since the broken phases are not physical from the point of view of QCD [4] Due ....

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Phys. Lett. B 423 (1998) 137 [hep-ph/9710538].

....at m 2 3 0 and a broken phase at m 2 3 0 at any 3 . The problem now is to determine the phase diagram in the full quantum theory. This question has previously been answered with numerical lattice Monte Carlo simulations for a number of theories: SU(2) fundamental representation Higgs [7] [10] SU(2) ThetaU(1) fundamental Higgs [11] and SU(2) adjoint Higgs [12, 4] The case of U(1) fundamental scalar Higgs (the Ginzburg Landau theory) has also been extensively studied [13] For small Higgs self coupling, 3 g 2 3 , the tree level second order transition is, in all ....

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Phys. Rev. Lett. 77 (1996) 2887 [hep-ph/9605288].

.... case is interesting since the 3d SU(3) adjoint Higgs theory is an effective theory of finite temperature QCD in the weak coupling domain [1] The requirement of small couplings means that this effective theory is accurate only in the limit T AE MS , not in the phase transition region T T c [2, 3, 4]. For pure gauge SU(3) theory, this is related to the fact that the phase transition at T = T c has to do with the breaking of the Z(3) symmetry. This symmetry is lost in the effective theory, but some traces of it remain, as will be discussed below. The second interesting aspect of the 3d SU(3) ....

....is to determine the phase diagram in the full quantum theory. This question has previously been answered with numerical lattice Monte Carlo simulations for a number of theories: SU(2) fundamental representation Higgs [7] 10] SU(2) ThetaU(1) fundamental Higgs [11] and SU(2) adjoint Higgs [12, 4]. The case of U(1) fundamental scalar Higgs (the Ginzburg Landau theory) has also been extensively studied [13] For small Higgs self coupling, 3 g 2 3 , the tree level second order transition is, in all cases studied, radiatively changed into a first order transition. Its strength decreases ....

[Article contains additional citation context not shown here]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 503 (1997) 357 [hep-ph/9704416].

....and m h much too small, as can be seen from the m h = 60 GeV curve. Hence the e#ect proposed in [38] does not take place close to our reference point. 6 It should also be noted than in the Standard Model there is a critical Higgs mass above which the phase transition ceases to be of first order [33]. In the MSSM, on the contrary, there exists an upper bound on m h , and in some cases (e.g. in the vicinity of our reference point) all possible Higgs masses result in a first order transition. 100.0 150.0 200.0 250.0 300.0 m A 0.00 0.02 0.04 0.06 0.08 tanb=2, A t =100 tanb=2, A t =50 ....

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Phys. Rev. Lett., in press [hep-ph/9605288].

.... reduction step is free of IR problems, and the resulting super renormalizable 3d theory can then be studied with high precision Monte Carlo simulations [18 22] The non perturbative investigations of the electroweak phase transition in the Standard Model have revealed the following pattern [19]. As long as the transition is strong enough for baryogenesis, the IR problems are not very dramatic and e#ective potential studies do produce a reasonable estimate of the properties of the phase transition. 2 Upon completion of this work, three more papers on the same subject appeared [37 39] ....

....of the formulas to the electroweak phase transition are not discussed. When the transition gets weaker, non perturbative e#ects become large. However, even if non perturbative e#ects are small for stronger transitions, it is nevertheless interesting to note that prior to the lattice study in [19] many perturbative studies stated that baryogenesis was possible up to mH # 45 GeV. In [19] it was discovered that practically no Higgs mass is possible. While this e#ect is mostly related to vacuum renormalization instead of non perturbative IR e#ects, it nevertheless proves that it is ....

[Article contains additional citation context not shown here]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 466 (1996) 189 [hep-lat/9510020].

.... of the electroweak phase transition in the full Standard Model has been extensively studied in the literature ( 2] and references therein) Perturbative studies exist up to 2 loop level [23, 24, 14, 25] Non perturbative lattice studies rely on perturbative 2 loop dimensional reduction [12 15], and have been performed for a wide range of Higgs masses [18 22] The Higgs mass bound in terms of the parameters of the 3d SU(2) Higgs model was derived in [19] The Higgs mass bound arises as follows. Assume that there in some underlying physical 4d theory in which the electroweak phase ....

....in the real time dynamics of the phase transition (whether the Universe reheats back to T c after the nucleation period, etc. For the Standard Model, the parameters m 2 3 , # 3 and g 2 3 have been calculated in terms of temperature and the physical zero temperature parameters of the theory in [15]. Then one may solve for the critical temperature from the condition m 2 3 (m h , T c ) 0, 2.3) and use this T c in the estimate of x = # 3 (m h , T c ) g 2 3 (m h , T c ) 2.4) Eq. 2.3) does not give T c exactly (it corresponds to resummed 1 loop accuracy) but this does not matter ....

[Article contains additional citation context not shown here]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 458 (1996) 90.

....simulations. The method can be applied to any system exhibiting critical behaviour, provided that it is possible to perform Monte Carlo simulations on the critical point itself. This includes, e.g. the endpoints of the 1st order lines in the 3d SU(2) adjoint Higgs theory (where the line ends [39, 40]) the 3d SU(3) adjoint Higgs theory (where the line turns into a second order line after a tricritical point) or in the U(1) Higgs theory. On the other hand, the two flavour 4d finite temperature chiral transition in QCD occurs at the limit m q 0, which is not directly accessible with standard ....

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 503 (1997) 357 [hep-ph/9704416].

....these show the existence of a line of first order phase transitions T c = T c (mH ) mH mH;c , which ends at T c (mH;c ) and turns into a crossover at mH mH;c . When approaching the endpoint along the first order line, the mass of one of the scalar excitations seems to go to zero suggesting [10] that ultimately all other masses could be integrated out, leaving near (mH;c ; T c (mH;c ) a final effective theory S crit [OE 0 ] containing only one scalar degree of freedom OE 0 . To be more precise, the electroweak theory with a Higgs doublet contains a massless vector excitation, ....

....repeat the study at higher values of fi G (we have used fi G = 5, 8 and 12) Since the mass m 0 is expected to vanish at the endpoint x c , a natural question arising is whether the effective theory in Eq. 1) could be further reduced leaving only the lightest excitations in the final action [22, 23, 10]. Knowing the effective theory would also imply knowledge of the universality class of the second order transition at x c . Unfortunately, no systematic perturbative derivation of such an effective action has been found. One problem is that in perturbation theory the vector excitations are ....

[Article contains additional citation context not shown here]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 466 (1996) 189 [hep-lat/9510020].

.... T . This works for equilibrium phenomena in the high T small coupling limit. Furthermore, all masses gT can also be integrated out. Hereby one obtains a 3d effective theory S[B i ; A a i ; OE k ] i; a = 1; 2; 3; k = 1; 4, with SU(2) ThetaU(1) symmetry and a fundamental doublet OE [7]. The superrenormalizable 3d theory provides a very good approximation to high T 4d physics. The accuracy of the effective description has been discussed in detail in [7] further corrections to the effective action can also be computed. The previous steps can be performed perturbatively, but ....

....theory S[B i ; A a i ; OE k ] i; a = 1; 2; 3; k = 1; 4, with SU(2) ThetaU(1) symmetry and a fundamental doublet OE [7] The superrenormalizable 3d theory provides a very good approximation to high T 4d physics. The accuracy of the effective description has been discussed in detail in [7]; further corrections to the effective action can also be computed. The previous steps can be performed perturbatively, but further progress is only possible with numerical lattice Monte Carlo techniques (for reviews, see [8, 9] In terms of SM physics, these show the existence of a line of first ....

[Article contains additional citation context not shown here]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 458 (1996) 90 [hep-ph/9508379]; Phys. Lett. B, in press [hep-ph/9710538].

....1 kari nordita.dk 2 tsypin td.lpi.ac. ru 3 keijo.kajantie cern.ch 4 mikko.laine cern.ch 5 mshaposh nxth04.cern.ch 1 Introduction The finite temperature phase transition in the Standard Model (SM) is known to be a first order transition for small and a crossover for large Higgs masses [1]. In between there is a critical region at about mH = 75 GeV [2, 3] see also [4] The purpose of this paper is to study this critical region in a detailed manner. We shall show that the universality class of the endpoint is that of the three dimensional (3d) Ising model. We also obtain the value ....

....been determined. The number 0.002 on the vertical axis is symbolic, as the figure is scale invariant in this direction. As to the functional form of the effective theory, the fact that there is only one light physical scalar degree of freedom, the Higgs particle, naturally leads to the suggestion [1, 10] that the corresponding MS continuum effective field theory is S = Z d 3 x 1 2 i OE i OE h I OE 1 2 m 2 I OE 2 1 4 I OE 4 : 10) We discuss the renormalisation and discretization of this theory in the Appendix. The theory in Eq. 10) is characterized by the scale I ....

[Article contains additional citation context not shown here]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Phys. Rev. Lett. 77 (1996) 2887 [hep-ph/9605288].

....be integrated out. It is also important to note that all other CP violating operators, such as those related to F F (see, e.g. 24] result in higher order operators whose contributions to Higgs sector CP violation are suppressed relative to the effects arising within the theory in Eq. 2. 1) [36]. The expressions for the couplings in Eq. 2.1) arising from dimensional reduction are summarized in Appendix A.7 (see also [9, 10, 11, 21, 22, 23] Here it is sufficient to know the leading parametric behaviour of the couplings: omitting multiplicative numerical factors and replacing logarithms ....

....theory is still useful and accurate in the phase transition region (for weak coupling) and fi 6= 0 does not change the properties of the system qualitatively. As a side remark, let us note that, for 6 O(g 2 ) the accuracy estimate O(g 2 ) above differs from the one, O(g 3 ) in [33, 36]. The reason is that with the present type of interactions, there can be reducible (left) diagrams with a heavy internal line, in additional to irreducible ones (right) In the Standard Model case considered in [33, 36] in contrast, there are only irreducible diagrams, whose contribution is ....

[Article contains additional citation context not shown here]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Phys. Lett. B 423 (1998) 137 [hep-ph/9710538].

....in Sec. 2 are presented in the Appendix. 2 The effective 3d theory for H 1 ; H 2 ; U To discuss the physics of finite temperature CP violation, we shall first construct the corresponding effective 3d theory, using the method of dimensional reduction applied to systems with phase transitions [33]. The motivation for this approach is that it implements automatically the resummations needed at finite temperatures and is thus the simplest way of seeing which infrared (IR) problems can be cured and which remain non perturbative. The 3d theory may also allow for lattice simulations, as ....

....theory is still useful and accurate in the phase transition region (for weak coupling) and fi 6= 0 does not change the properties of the system qualitatively. As a side remark, let us note that, for 6 O(g 2 ) the accuracy estimate O(g 2 ) above differs from the one, O(g 3 ) in [33, 36]. The reason is that with the present type of interactions, there can be reducible (left) diagrams with a heavy internal line, in additional to irreducible ones (right) In the Standard Model case considered in [33, 36] in contrast, there are only irreducible diagrams, whose contribution is ....

[Article contains additional citation context not shown here]

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B 458 (1996) 90 [hep-ph/9508379].

....the typical correlation lengths. Without any external magnetic field, the electroweak phase transition in the SU(2) ThetaU(1) Minimal Standard Model is of the first order for small Higgs masses [2] The transition weakens with increasing mH so that the first order line has a second order endpoint [3] of Ising type at mH;c 72 GeV [4] Beyond that there is only a crossover. The two phases of the system, the symmetric and the broken (or Higgs) phase, are thus analytically connected. If there is an external magnetic field, one expects the transition to be significantly stronger [5] This is ....

....larger fields, an instability may take place. However, finite temperature perturbation theory cannot to be trusted in the regime of large experimentally allowed Higgs masses, mH 80 GeV. Indeed, as mentioned, the first order electroweak phase transition turns into a crossover for mH 80 GeV [3, 8, 9, 4] in the absence of a magnetic field, in contrast to the perturbative prediction. Thus the effects of external magnetic fields should also be studied non perturbatively, and this is our objective here. We do observe significant non perturbative effects. It is instructive to compare the present ....

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Phys. Rev. Lett. 77 (1996) 2887 [hep-ph/9605288].

....and the temperature is much larger than any other mass scale in the theory [1, 2, 3] The parameters of the reduced theory are related to the ones of the original model through perturbation theory. The reduced theory has some advantages over the original one from the computational point of view [5, 6, 7]. It is super renormalizable and yields transparent relations between the (dimensionful) continuous parameters and the lattice ones. Moreover, the number of mass scales is drastically reduced: a) the scale T; present in four dimensions is evidently absent, b) one may also integrate out the ....

....modes in the action and deriving an effective action [2, 3] An important remark is that the mass of the adjoint Higgs field is of order gT; which is large compared to g 2 T; the typical scale of the theory. Thus one can go on one step further and integrate it out using perturbation theory [5, 6, 7]. The effective action may then be written in the form: S 3D eff [A i ( x) 3 ( x) Z d 3 x[ 1 4 F ij F ij jD i 3 j 2 m 2 3 3 3 3 ( 3 3 ) 2 ] 5) The index 3 in (5) denotes the 3D character of the theory. The relations between the 4D and 3D parameters are (up ....

[Article contains additional citation context not shown here]

K.Farakos, K.Kajantie, K.Rummukainen and M.Shaposhnikov, Phys.Lett B336 (1994) 494.

....and the temperature is much larger than any other mass scale in the theory [1, 2, 3] The parameters of the reduced theory are related to the ones of the original model through perturbation theory. The reduced theory has some advantages over the original one from the computational point of view [5, 6, 7]. It is super renormalizable and yields transparent relations between the (dimensionful) continuous parameters and the lattice ones. Moreover, the number of mass scales is drastically reduced: a) the scale T; present in four dimensions is evidently absent, b) one may also integrate out the ....

....modes in the action and deriving an effective action [2, 3] An important remark is that the mass of the adjoint Higgs field is of order gT; which is large compared to g 2 T; the typical scale of the theory. Thus one can go on one step further and integrate it out using perturbation theory [5, 6, 7]. The effective action may then be written in the form: S 3D eff [A i ( x) 3 ( x) Z d 3 x[ 1 4 F ij F ij jD i 3 j 2 m 2 3 3 3 3 ( 3 3 ) 2 ] 5) The index 3 in (5) denotes the 3D character of the theory. The relations between the 4D and 3D parameters are (up ....

[Article contains additional citation context not shown here]

K.Farakos, K.Kajantie, K.Rummukainen and M.Shaposhnikov, Nucl.Phys. B442 (1995) 317.

....and the temperature is much larger than any other mass scale in the theory [1, 2, 3] The parameters of the reduced theory are related to the ones of the original model through perturbation theory. The reduced theory has some advantages over the original one from the computational point of view [5, 6, 7]. It is super renormalizable and yields transparent relations between the (dimensionful) continuous parameters and the lattice ones. Moreover, the number of mass scales is drastically reduced: a) the scale T; present in four dimensions is evidently absent, b) one may also integrate out the ....

....absent, b) one may also integrate out the temporal component A 0 of the gauge field, so its mass scale gT also disappears. Thus there are two mass scales less and this reduces substantially the computer time needed to get reliable results. The model that has already been studied along these lines [5, 8] has been based on SU(2) with one complex Higgs doublet. For work on the same model in asymmetric four dimensional lattices, see [15] The issues studied have been the order and the characteristics (critical temperature, latent heat, surface tension, correlation lengths) of the phase transition, ....

[Article contains additional citation context not shown here]

K.Farakos, K.Kajantie, K.Rummukainen and M.Shaposhnikov, Nucl.Phys. B425 (1994) 67.

No context found.

K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B493:413, 1997

No context found.

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B407 (1993) 356

No context found.

K. Farakos, K. Kajantie, K. Rummukainen and M. Shaposhnikov, Nucl.Phys. B 425 (1994) 67.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC