The Electroweak Phase Transition

The Electroweak Phase Transition – Standard and “Beyond” 1

Michael G. Schmidt

arXiv:hep-ph/9708322v1 12 Aug 1997

Institut f¨ r Theoretische Physik u Universit¨t Heidelberg a Philosophenweg 16, D-69120 Heidelberg, Germany

Abstract. We ?rst review why a strongly ?rst-order phase transition needed for baryogenesis is excluded in the electroweak standard model. We also comment on some intriguing e?ects in the strongly interacting hot phase. In the MSSM with a light stop a strongly ?rst-order phase transition can be achieved. It possibly proceeds in two stages.

1. Introduction Heating up the electroweak matter of the standard model (SM) a phase transition (PT) is expected [1] at a temperature Tc ? 1015 K (? 100 GeV) since the positive plasma mass2 ? (gw T )2 of the Higgs ?eld switches the sign of the Higgs ?eld mass2 . In this simple picture the Higgs mechanism is suspended at high temperatures, the Higgs-?eld VeV becomes zero; naively in the “hot phase” the transversal W -bosons would be massless. Such a PT should have occurred in the early universe at about 10?12 sec after the big bang. Most of the interest in it in the last years came from the observation that the electroweak interactions violate the baryon number B: At T = 0 the instanton tunneling between topologically di?erent vacua 2 2 related to B-violation [2] is an immeasurably small e?ect ? e?8π /gw (unless perhaps strongly enhanced in multi-W production [3]). For T below Tc a
1 Talk presented at the workshop “Beyond the Desert”, Accelerator and Nonaccelerator Approaches, Castle Ringberg, Tegernsee, June 8-14, 1997

1

B-violating thermodynamical transition between states with neighbouring topological quantum numbers Ncs via an unstable 3-dimensional sphaleron con?guration [4, 5] is Boltzmann-suppressed and has a rate/volume ΓHiggs B /
Phase

? (αw T )4 e?S3 /T .

(1)

S3 is the sphaleron action, which can be rescaled as v(T )/gw · 2πA where v(T ) is the T -dependent Higgs VeV and A a number (? 3) only slowly varying with the electroweak parameters. For the large T of the hot phase it is argued that there is no Boltzmann suppression and that (2) ΓHot Phase ? Kαn T 4 . w B / (The prefactor K and the power n are under discussion [6]). Baryon minus lepton number B ? L is conserved in electroweak interactions. If it is zero in the very early universe, it stays zero cooling down to the PT. During the equilibrium period before the PT B as well as L is washed out in this case. The observed baryon asymmetry ?B/nγ ? 10?10 should then be produced during the PT. Indeed the three necessary criteria of Sakharov [7] for producing a B-asymmetry may be ful?lled in the SM [8]: (i)baryon-number violation, (ii) C, CP violation, and (iii) nonequilibrium. Concentrating on the last point in this talk we see that a ?rst-order PT is needed. The most attractive way to produce a B-asymmetry is the “charge transport mechanism” [9]. The walls of the expanding bubbles of condensing electroweak matter contain a C, CP violating phase factor and like a diaphragm produce axial charge which is transported into the hot phase in front of the bubble. There it is converted into a baryonic charge by the “hot sphaleron”-transition (2) before the Higgs phase bubble takes over. Thus ?rst the B-asymmetry should be produced in the PT, then it should quickly “freeze out” in the Higgs phase. Therefore the baryon number violating rate in the Higgs phase should be small compared to the inverse 2 Hubble time H ? Tc /mP lanck ? e?40 Tc (for Tc ? 100 GeV) from where it follows [8] that v(Tc )/Tc ≥ 1.3 in the Boltzmann factor in (1). Thus one needs a strongly ?rst-order PT.

2. The electroweak phase transition in the SM To learn about the electroweak PT one conveniently inspects an equilibrium quantity, the e?ective Higgs potential. The weak coupling gw is small but at high temperatures T the relevant expansion parameter g3 is the 3?2 2 2 2 dimensional gauge coupling g3 ? gw T divided by some infrared (IR) scale 2 mIR , g3 = g3 /mIR , and is not small for small mIR . The most elegant way ?2 2

of separating the infrared problem is the reduction from a 4- to a timeindependent 3-dimension e?ective theory containing the light degrees of freedom. Restricting the form of the e?ective action (derivative expansion) the “matching” [10, 11] of 4- and 3-dimensional amplitudes allows to ?x its parameters. This procedure is purely perturbative and has been carried through to 2-loop order. This loop order is necessary if one wants a few 3 percent accuracy (0(gw )) [12]. In a ?rst step all n = 0 Matsubara modes with mn = 2πnT thus including all fermion modes (n half integer) are integrated out in the above sense.In a second step also the longitudinal gauge bosons which have obtained a Debye mass ? gw T in the ?rst step are integrated out. One ends up with the e?ective Langrangian 1 a2 w w (3) F + (Di H)+ (Di H) + m2 H + H + λH3 (H + H)2 . 3 4 ik The U(1) part and Weinberg mixing can be neglected in this discussion without loosing an essential point. The SU(2)-Yang-Mills Lagrangian and w 2 2 the covariant derivative Di contain the gauge coupling g3 = gw T (1 + ...). The Higgs parameters m2 (T ) and λH3 (T ) depend on the temperature. 3 They can be made dimensionless in the ratios Lef f = 3 y= m2 (T ) 3 , 2 (g3 )2 x= λ2 3 (T ) H . 2 g3 (4)

2 y is related to T ? Tc and x(? λT /gw in terms of 4-dimensional quantities) determines the nature of the phase transition. Lef f characterizes a whole 3 class of theories. The speci?c properties of the 4-dimensional theory only enter via the computation of y, x. Lef f as it stands as a tree level theory would give a 2nd-order PT. 3 But of course it has to be studied in higher perturbative order and more than that it has to be treated as a potentially strongly interacting QFT like QCD because of its IR behaviour. Thus the most secure way is to discretize it on a lattice and to discuss the results of lattice calculations [12, 13]. Still perturbation theory can provide some interesting insights. The simple one-loop W -boson exchange graph in a constant Higgs ?eld background φ (3-dimensional, with φ4 = T 1/2 φ) shown below contributes the well-known term 3 1 (g 2 T φ+ φ)3/2 = ?E(T φ+ φ)3/2 (5) V3φ = ? 24π w

3

to V3ef f leading already to a ?rst-order PT. At the Higgs phase minimum v(T ) perturbation theory in 2-loop order (?gs. 1,2) indeed compares very 2 well with lattice results if g3 = g3 /v(T ) ? cx is su?ciently small [12, 13]. ?2 Even the critical temperature Tc which one obtains by comparing the Higgs φ = v(T ) and the φ = 0 IR sensitive minimum, can be determined quite reliably this way for x<0.08. Most sensitive is the surface tension of the ? critical bubble where we observe [12], [18] (?g. 3) strong deviations of the perturbative calculation from lattice results for x>0.05. ? An e?ective potential V ef f (φ) and also the Z-factor of the kinetic term is needed if one wants to calculate critical bubbles [19] and sphaleron ?eld con?gurations [5]. Thus some perturbative expression enlarged by some nonperturbative piece is desirable. Most excitingly, lattice calculations [18] show a “crossover” behaviour (qualitatively predicted in ref. [21], [22]) in this phase diagram for x> 0.11 ? (corresponding to mH >mW ): The ?rst order PT fades away at such values ? of x. Of course a strongly ?rst-order PT with v(Tc )/Tc > 1 avoiding the sphaleron erasure of B-asymmetry requires x< 0.03 ? 0.04 much below the ? crossover. Indeed, a careful perturbative analysis of x in terms of the physical Higgs mass mH and the top mass gives [10] x? m4 1 m2 top H +c 4 8 m2 mW W (6)

and shows that the above limit excludes SM baryogenesis for any mH ! Still the analysis of the SM at high T based on the Lagrangian (3) and its lattice regularization leads to results very interesting by themselves: In the hot phase there is a con?ning linear potential – about the same as in pure YM theory [13]. On the lattice one can measure a rich spectrum (?g. 4) of W -balls and of “Higgs hadrons” which are QCD-type bound states of Higgses rather than quarks. One should keep in mind that the masses of these bound states are 3-dimensional correlation masses. These masses can also be calculated [23] in a relativistic bound-state model with a con?ning potential and compare very well with lattice results. One can call spin 0 and spin 1 states Higgses and massive W -bosons respectively, but one should emphasize that the hot phase is not another Higgs phase although there are no massless vector bosons as in the naive picture mentioned in the introduction. It would be interesting to have a complete model of these bound states in the whole phase diagram (y versus x) and to compare it with lattice results. It is also desirable to have a concrete model [25] of how the perturbative e?ective potential is modi?ed by a nontrivial vacuum structure including gauge-?eld condensates at small values of φ. This is particularly important if the perturbative potential and its Higgs minimum are small as they are in the case of x values in the crossover region and beyond. 4

Vef f 2 gw v 4

0.003 0.0025 0.002 0.0015 0.001 0.0005 0

ξ=0 ξ=1 ξ=2

0

0.2

0.4

0.6

0.8

?/v

1

Figure 1. The 1- (lower) and 2-loop (upper curves) e?ective potential at Tc for x = 0.12 in units of v(T ) in di?erent ξ-covariant background gauges (from ref. [14]). The phase transition becomes stronger and the gauge dependence diminishes in the 2-loop results.
2.5
2 g3

gw v

2

ξ=0 ξ=1 ξ=2

1.5

1

0.5

0

0

0.02

0.04

0.06

0.08

0.1

x

0.12

2 Figure 2. g3 (Tc )/(gw v(Tc )) ? gw Tc /v(Tc ) as a function of x (1-loop: upper, 2-loop: lower curves) (ref. [14]); for x > 0.04 one has v(Tc )/Tc < 1.

5

10
2 gw σ 2 (g3 )3

1

10

0

10

?1

10

?2

0.00

0.02

0.04

0.06

0.08

x

0.10

Figure 3. from ref. [15]) The perturbatively calculated interface tension σ (including Z-factor e?ect and gauge variations) vs. x compared to lattice data from ref. [12] (squares), ref. [16] (triangles) and ref. [17] (circles).

Figure 4. (ref. [20]) Lattice results for the correlation masses for 0++ (J P C ), 1?? and 2++ operators (at x = 0.0239). Dark points indicate purely ”gluonic” operators. Whereas in the Higgs phase only H, W and multiple (2W ,...) are seen, in the hot phase one observes con?nement and a completely di?erent massive spectrum. 6

We conclude that the high T electroweak standard theory does not provide a ?rst-order phase transition which is strong enough for baryogenesis and that it even vanishes for mH >mW . It is also very questionable if stan? dard CP violation is large enough. Still a lot of know-how also concerning B-asymmetry production has accumulated. Thus if one does not want to go back to (B?L) violating asymmetry production in GU theories, it might be attractive to consider variants of the SM.

3. Variants of the SM, the MSSM with a light stop It is widely accepted that the SM is an e?ective theory to be embedded in a deeper theory including the Planck scale. It is not clear, however, if at all and how it should be varied at the electroweak scale considering the great experimental success of the SM. Taking a pragmatic view baryogenesis at the electroweak scale requires a strongly ?rst-order PT with v(Tc )/Tc > 1. In 1-loop order one has v(T )/T ? E/λT , where E is de?ned in eq. (5). This can be increased by (i) increasing the E-prefactor of the “φ3 -term” having more light bosons in the loop. (below (5)) (ii) decreasing the coupling λT = λ3 /T . There are further points to be mentioned: (iii) The 2-loop contributions to V3 are very important and, in the SM, lead to a considerable strengthening of the PT (?g. 1). (iv) A delay in the PT towards lower Tc strengthens the PT. (v) The rescaled sphaleron factor A mentioned after Eq. (1) may be increased in some variants. (vi) Models with a tree level φ3 -type term (like in NMSSMs) may be interesting. Here we concentrate on the ?rst point and argue [24]-[30] that in the Minimal Supersymmetric SM (MSSM) the φ3 -type term in the e?ective ? potential is increased if tR , the superpartner of the right handed top, is rather light. This enhancement is due to a diagram as shown below eq. (5) but now with a stop in the loop. Its coupling to the φ2 background is given by the Yukawa coupling h2 . To strengthen this term ht should be t large and the thermal mass m2 3 ? m2 + cT 2 u u (7)

? ? of tR should be small in the φ=0-phase. The physical tR mass is m2R = ? t 2 2 2 2 mu + mtop , where mu is the SUSY-breaking scalar mass and mtop ? h2 φ2 . t In renormalisation group equations starting with some m2 at the Planck u scale one indeed observes a decrease in m2 much stronger than for other u particles. Large ht means rather small tan β = v1 /v2 . It is also convenient 7

0.08
mtop=175 mQ=mD=300

m=
h

0.06
m =7
h

70, m

U

=15

0

5, m

U

0.04
mh =7

x

=50
=50

0, m

U

strong enough for baryogenesis

0.02

tanβ=2, mU=150 tanβ=2, mU=100 tanβ=2, mU=50

0.00 100.0

150.0

200.0 mA

250.0

300.0

2 Figure 5. (from ref. [28]). The parameter x = λ3 /g3 dependent on the axial mass mA , the Higgs mass mk (resp. tan β) and the SUSY breaking mu in the MSSM.

to make one Higgs doublet heavy postulating a large axial Higgs mass ma . The Higgs mass mH is then ?xed by tan β and ma . If now the SUSY-partner particles and one Higgs doublet combination are heavy enough, they can be “integrated out”, and one ends up at the same type of 3-dimensional theory (3) but with di?erent relations between 4- and 3-dimensional parameters [10, 28, 29, 30]. It turns out that one can get v(Tc )/Tc > 1 for mH <70/75 GeV (?g. 5). One can even go further ? and discuss a very small or even negative m2 . However, to do this properly u ? [31], the stop ?eld U = tR cannot be integrated out. It has to be included in the 3-dimensional Lagrangian adding to (3) a term 1 A2 s s G + (Di U )+ (Di U ) + m2 3 U + U + λU3 (U + U )2 + γ3 H + HU + U u 4 ik (8) s where G is the SU(3) Yang-Mills ?eld strength and Di the color covariant derivative, m2 3 and λu3 are the mass2 and the coupling like in (3), and the u last term is a mixing with, at tree level, γ3 ? T h2 sin2 β. We have calculated t the corresponding perturbative potential. 2-loop e?ects are important since Lstop = 3

8

? only in this order gluon and Higgs exchange in a tR -loop come into play.

g

~R t
?u This leads to a much stronger PT. Going with ?m2 3 = m2 to 70 GeV, u the upper limit for mH to obtain v(Tc )/Tc > 1 becomes mH ≤ 100 GeV (?g. 6), still avoiding an unstable Higgs vacuum at T = 0. Here we have ? put At = 0 (see [24] for its de?nition) for simplicity and in order to obtain maximal e?ect.
105.0
t ~97 G 2 (m H anβ=1 eV) (m H~ 92 Ge V)

5 tanβ=

95.0

Tc

strong enough for baryogenesis

~82 G 3 (m H tanβ=

eV)

85.0
Tc χ Tc χ?>φ Tc
φ

75.0 155.0

160.0 m~
tR

165.0

Figure 6. (from ref. [31]) The critical temperatures Tc of the three transitions 0 → φ, 0 → χ, χ → φ for tan β = 3, 12 (thin lines) and tan β=5 (thick lines). The two-stage transition would occur to the left of the crossing point of the three critical curves. Also the boundary x<0.04 for strongly ? ?rst-order PT 0 → φ is indicated. 9

0.015

2?loop, ?=T, ζ=0 2?loop, RG?improved 2?loop, ζ=1 1?loop, ?=T, ζ=0 2?l Tc =92.43 GeV

0.010

Tc =94.89 GeV

ζ=1

V(χ)/T

4

0.005

Tc =93.30 GeV

RG

Tc =97.93 GeV

1?l

0.000 0.0

0.5

1.0 χ/T

1.5

2.0

0.010
2?loop, ?=T, ξ=0 2?loop, RG?improved 2?loop, ξ=1 1?loop, ?=T, ξ=0 Tc =92.64 GeV Tc =92.43 GeV
2?l ξ=1

V(φ)/T

4

0.005

Tc =94.13 GeV

RG

Tc =96.50 GeV

1?l

0.000 0.0

0.5

1.0 φ/T

1.5

2.0

Figure 7. (from ref. [31]) The 1- and 2-loop e?ective potentials in <U3 >=χ and <H>=φ directions (tan β = 5). Here mtR = 158.3 GeV is chosen ? 2?loop such that the cricital temperatures Tc of the two-phase transitions are equal. 10

Most interestingly we also can obtain [31] a two-stage (?gs. 6, 7) PT in a certain range of mtR ? 155 ? 160 GeV. There is a ?rst-order PT to the ? “colored” minimum (strictly speaking, in a gauge theory < U + U >= 0 and not < U >= 0!) and then at lower T a transition to the Higgs vacuum. This delays the second B-asymmetry generating PT towards lower T and thus increases [31] v(T )/T . But now one has to make sure that the transition rate does not become too small. Fortunately, in the intermediate phase the sphaleron is Boltzmann-unsuppressed contrary to the 2-Higgs-two-stage PT considered previously [32] and thus allows strong B-violation. All this is based on perturbation theory and should be checked by lattice calculations. Experience tells us that perturbative calculations are o.k. for a strongly ?rst-order PT. Observing gauge and ?-dependence in our calculation [32] (?g. 7) as well as noting the steep tree level potential between the minima, doubts are, however, allowed. In conclusion of the last part it can be said that the MSSM variant of the SM allows a strongly ?rst-order PT for Higgs masses as large as 95 GeV. Even a two-stage PT is possible. Given such a strong PT it is of interest to develop further the machinery of producing B-asymmetry in front of expanding bubbles which also requires the discussion of CP violation in the model. I would like to thank D. B¨deker, H. G. Dosch, P. John, J. Kripfganz, M. o Laine and A. Laser for enjoyable collaborations on various topics reviewed in the report and also B. Bergerho?, O. Philipsen and C. Wetterich for useful discussions.

References
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[9] A. G. Cohen, D. B. Kaplan, A. E. Nelson, Ann. Rev. Nucl. Part. Sci 43 (1993) 27, and references quoted there [10] K. Kajantie, M. Laine, K. Rumukainen, M. Shaposhnikov, Nucl. Phys. B458 (1996) 90 [11] E. Braaten, A. Nieto, Phys. Rev. D51 (1995) 6990 [12] K. Kajantie, M. Laine, K. Rumukainen, M. Shaposhnikov, Nucl. Phys. B466 (1996) 189 [13] E. M. Ilgenfritz, J. Kripfganz, H. Perlt, A. Schiller, Phys. Lett. B356 (1995) 561; M. G¨rtler, E. M. Ilgenfritz, J. Kripfganz, H. Perlt, A. Schiller, Nucl. u Phys. B483 (1997) 383; hep-lat/9605042 [14] J. Kripfganz, A. Laser, M. G. Schmidt, Phys. Lett. B351 (1995) 266 [15] J. Kripfganz, A. Laser, M. G. Schmidt, Z. Physik C73 (1997) 353 [16] Z. Fodor, J. Hein,K. Jansen, A. Jaster, I. Montray, F. Czikor, Phys. Lett. B334 (1994) 405 [17] F. Csikor, Z. Fodor, J. Hein, J. Heitger, Phys. Lett. B357 (1995) 156 [18] K. Kajantie, M. Laine, K. Rumukainen, M. Shaposhnikov, Nucl. Phys. B466 (1996) 189; Phys. Rev. Lett. 77 (1996) 2887; F. Karsch, T. Neuhaus, A. Patkos, J. Rank, Nucl. Phys. B53 (Proc. Suppl.) (1997) 623; M. G¨rtler, R. M. Ilgenfritz, A. Schiller, UL-NTZ 10/97, hep-lat/9704013 u [19] A. Laser, J. Kripfganz, M. G. Schmidt, Nucl. Phys. B433 (1995) 467 [20] O. Philipsen, M. Teper, H. Wittig, Nucl. Phys. B469 (1996) 445; hep-ph/9708309 [21] M. Reuter, C. Wetterich, Nucl. Phys. B408 91993) 91: B. Bergerho?, C. Wetterich, Proceedings “Int. School of Astrophysics” (Erice 1996), HD-THEP-96-51, and references quoted there [22] W. Buchm¨ller, O. Philipsen, Nucl. Phys. B443 (1995) 47 u [23] H. G. Dosch, J. Kripfganz, A. Laser, M. G. Schmidt, Nucl. Phys. B, to appear (hep-ph/9612450) [24] M. Carena, M. Quiro?, C. E. M. Wagner, Phys. Lett. B380 (1996) 81; s M. Carena, C. E. M. Wagner, hep-ph/9704347 [25] A. Laser, M. Reuter, M. G. Schmidt, work in progress [26] D. Delepine, J. M. G?rard, R. Gonzalez Felipe, J. Weyers, Phys. Lett. B386 e (1996) 183 [27] J. R. Espinosa, Nucl. Phys. B475 (1996) 273 [28] M. Laine, Nucl. Phys. B481 (1996) 43 [29] J. M. Cline, K. Kainulainen, CERN-TH/96-76 (hep-ph/9605235) [30] M. Losada, RU-96-25 (hep-ph/9605266), hep-ph/9612337; G. R. Farrar, M. Losada, RU-96-26 (hep-ph/9605266)

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[31] D. B¨deker, P. John, M. Laine, M. G. Schmidt, Nucl. Phys. 497 (1997) 387 o [32] D. Land, E. D. Carlson, Phys. Lett. B292 (1992) 107; A. Hammerschmitt, J. Kripfganz, M. G. Schmidt, Z. Physik C64 (1994) 105

13

0.015

2?loop, ?=T, ζ=0 2?loop, RG?improved 2?loop, ζ=1 1?loop, ?=T, ζ=0 2?l Tc =92.43 GeV

0.010

Tc =94.89 GeV

ζ=1

V(χ)/T

4

0.005

Tc =93.30 GeV

RG

Tc =97.93 GeV

1?l

0.000 0.0

0.5

1.0 χ/T

1.5

2.0

0.010
2?loop, ?=T, ξ=0 2?loop, RG?improved 2?loop, ξ=1 1?loop, ?=T, ξ=0 Tc =92.64 GeV Tc =92.43 GeV
2?l ξ=1

V(φ)/T

4

0.005

Tc =94.13 GeV

RG

Tc =96.50 GeV

1?l

0.000 0.0

0.5

1.0 φ/T

1.5

2.0

0.08
mtop=175 mQ=mD=300

m=
h

0.06
m =7
h

70, m

U

=15

0

5, m

U

0.04
mh =7

x

=50
=50

0, m

U

strong enough for baryogenesis

0.02

tanβ=2, mU=150 tanβ=2, mU=100 tanβ=2, mU=50

0.00 100.0

150.0

200.0 mA

250.0

300.0

105.0
12 tanβ= 97 G (m H~ eV) 5 92 G (m H~ eV)

tanβ=

95.0

Tc

strong enough for baryogenesis

82 G (m H~ 3 tanβ=

eV)

85.0
Tc χ Tc χ?>φ Tc
φ

75.0 155.0

160.0 m~
tR

165.0


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