Electroweak Phase Transition and Sphaleron
Eibun Senaha (Natl Taiwan U) April 7, 2017
ACFI Workshop: Making the Electroweak Phase Transition (Theoretically) Strong
@Umass-Amherst Outline
• part1: EWPT and sphaleron in a complex-extended SM (cxSM) CW Chiang, M. Ramsey-Musolf, E.S., in progress
• part2: Band structure effect on sphaleron rate at
high-T. K. Funakubo, K. Fuyuto, E.S., arXiv:1612.05431 Introduction
- Standard perturbative treatment of EWPT is gauge-dependent.
T=TC TC: T at which Veff has degenerate minima
vC: minimum of Veff at TC
- Gauge-independent methods: vC
(1) vC and TC are determined by
(2) Patel-Ramsey-Musolf (PRM) scheme [JHEP07(2011)029]
vC and TC are determined separately.
We analyze EWPT and sphaleron in the cxSM using 2 methods. SM with a complex scalar (cxSM)
H: SU(2)-doublet Higgs, S: SU(2)-singlet Higgs
2 We assume all parameters are real. m , , 2, b2, d2, a1, b1
v0, vS0, mH1 , mH2 , ↵, mA, a1 SM with a complex scalar (cxSM)
H: SU(2)-doublet Higgs, S: SU(2)-singlet Higgs
2 We assume all parameters are real. m , , 2, b2, d2, a1, b1
v0, vS0, mH1 , mH2 , ↵, mA, a1 246GeV SM with a complex scalar (cxSM)
H: SU(2)-doublet Higgs, S: SU(2)-singlet Higgs
2 We assume all parameters are real. m , , 2, b2, d2, a1, b1
v0, vS0, mH1 , mH2 , ↵, mA, a1 246GeV 125GeV SM with a complex scalar (cxSM)
H: SU(2)-doublet Higgs, S: SU(2)-singlet Higgs
2 We assume all parameters are real. m , , 2, b2, d2, a1, b1
v0, vS0, mH1 , mH2 , ↵, mA, a1 246GeV 125GeV Patterns of PT
Because of 2 fields, there are many patterns of phase transitions. 3
(a) (b) minima at A and B. In this case, the EWPT at T = TC will be first order. Note, however, that an oversized pos- itive 2 may render B at T = 0 metastable, since the energy di↵erence between A and B phases, which is sup- posed to be positive, can become negative. Therefore, there should be an upper bound on the magnitude of 2, as will be discussed below. For negative 2, in contrast, TC is always raised which may preventv ¯C /TC becoming asu cient size. In fact, SFOEWPT cannot be found for (c) (d) 2 < 0 in our numerical analysis. The situation can be more complex in the presence of non-vanishing a1 and the remaining zero-temperature and thermal loop e↵ects encoded in the one-loop e↵ective potential:
Ve↵ ('; T )=V0(')+V1('; T ) (15)
where ' =(', 'S);
4 2 T m¯ j FIG. 1: Patterns of symmetry breaking at finite temperature. V ('; T )= n V (¯m2)+ I , 1 j CW j 2⇡2 B,F T 2 j ! X (16) where ' and 'S denote the neutral doublet and singlet background fields, respectively, and and nj count the degrees of freedom for particle species j, andm ¯ j are '-dependent masses. The Coleman-Weinberg 2 2 2 2 3g + g y potential VCW and IB,F (a ) are respectively given by [16, ⌃ = + 2 + 2 1 + t T 2 ⌃¯ T 2 , (11) H 8 24 16 4 ⌘ H 17] 2 4 2 T 2 2 m m ⌃ =( + d ) ⌃¯ T . (12) VCW(m )= ln c , (17) S 2 2 12 ⌘ S 64⇡2 µ2 ✓ ◆ For m < 0 and b < 0, T = 0 minima will exist for 2 1 2 px2+a2 2 2 IB,F (a )= dx x ln 1 e , (18) non-zero v0 and vS0. In the limit of vanishing a1, at su - 0 ⌥ Z ⇣ ⌘ ciently high-T the only minimum of the theory occurs at where c =3/2 for scalars and fermions and 5/6 for gauge the origin, denoted by “O”. As T decreases, one generi- bosons, and µ is the renormalization scale. cally expects that a secondary minimum at ' v˜ =0 s S For a1 = 0, the high-T minimum will no longer lie will first appear, since ⌃¯ < ⌃¯ . At a temperature⌘ 6 T , 6 S H 1 at the origin but will be shifted by a1 along the 'S the minimum atv ˜S will become the global minimum in- direction. Alternately, EWSB may occur directly from dicated by “A” in Fig. 1. As T further decreases, an ad- the origin to point B, as Type (d) in Fig. 1. However, ditional minimum at (' v = 0, ' v = 0) develops, ⌘ 6 S ⌘ S 6 this phase transition is not first order sincev ˜S is zero becoming the global minimum at temperature T2
Because of 2 fields, there are many patterns of phase transitions. 3
(a) (b) minima at A and B. In this case, the EWPT at T = TC will be first order. Note, however, that an oversized pos- itive 2 may render B at T = 0 metastable, since the energy di↵erence between A and B phases, which is sup- posed to be positive, can become negative. Therefore, there should be an upper bound on the magnitude of 2, as will be discussed below. For negative 2, in contrast, TC is always raised which may preventv ¯C /TC becoming asu cient size. In fact, SFOEWPT cannot be found for (c) (d) 2 < 0 in our numerical analysis. The situation can be more complex in the presence of non-vanishing a1 and the remaining zero-temperature and thermal loop e↵ects encoded in the one-loop e↵ective potential:
Ve↵ ('; T )=V0(')+V1('; T ) (15)
where ' =(', 'S);
4 2 T m¯ j FIG. 1: Patterns of symmetry breaking at finite temperature. V ('; T )= n V (¯m2)+ I , We will focus on type (a) PT. 1 j CW j 2⇡2 B,F T 2 j ! X (16) where ' and 'S denote the neutral doublet and singlet background fields, respectively, and and nj count the degrees of freedom for particle species j, andm ¯ j are '-dependent masses. The Coleman-Weinberg 2 2 2 2 3g + g y potential VCW and IB,F (a ) are respectively given by [16, ⌃ = + 2 + 2 1 + t T 2 ⌃¯ T 2 , (11) H 8 24 16 4 ⌘ H 17] 2 4 2 T 2 2 m m ⌃ =( + d ) ⌃¯ T . (12) VCW(m )= ln c , (17) S 2 2 12 ⌘ S 64⇡2 µ2 ✓ ◆ For m < 0 and b < 0, T = 0 minima will exist for 2 1 2 px2+a2 2 2 IB,F (a )= dx x ln 1 e , (18) non-zero v0 and vS0. In the limit of vanishing a1, at su - 0 ⌥ Z ⇣ ⌘ ciently high-T the only minimum of the theory occurs at where c =3/2 for scalars and fermions and 5/6 for gauge the origin, denoted by “O”. As T decreases, one generi- bosons, and µ is the renormalization scale. cally expects that a secondary minimum at ' v˜ =0 s S For a1 = 0, the high-T minimum will no longer lie will first appear, since ⌃¯ < ⌃¯ . At a temperature⌘ 6 T , 6 S H 1 at the origin but will be shifted by a1 along the 'S the minimum atv ˜S will become the global minimum in- direction. Alternately, EWSB may occur directly from dicated by “A” in Fig. 1. As T further decreases, an ad- the origin to point B, as Type (d) in Fig. 1. However, ditional minimum at (' v = 0, ' v = 0) develops, ⌘ 6 S ⌘ S 6 this phase transition is not first order sincev ˜S is zero becoming the global minimum at temperature T2
where (a)
Approximate formulas:
- large positive δ2 (negative α) gives larger vC/TC. - However, too large positive δ2 (negative α) leads to unstable vacuum. Leading order analysis
where (a)
Approximate formulas:
>0
- large positive δ2 (negative α) gives larger vC/TC. - However, too large positive δ2 (negative α) leads to unstable vacuum. Leading order analysis
An example: 250 mH = 230 GeV, vS0 = 40 GeV, 2 LO 3 TC a1 = (110 GeV) v¯LO 200 C
- vC and TC are determined 150 numerically. ] GeV [ - smaller α (large δ2)gives 100 larger vC/TC. 50 - EW vacuum becomes metastable for a small alpha. 0 -> upper bound on vC/TC -22 -21 -20 -19 -18 -17 -16 -15 ↵ [ ] - Stronger upper bound on vC/TC comes from bubble nucleation (see later.) Leading order analysis
An example: 250 mH = 230 GeV, vS0 = 40 GeV, 2 LO 3 TC a1 = (110 GeV) v¯LO 200 C
- vC and TC are determined 150 numerically. ] GeV [ - smaller α (large δ2)gives 100 larger vC/TC. 50 - EW vacuum becomes EW vacuum is metastable metastable for a small alpha. 0 -> upper bound on vC/TC -22 -21 -20 -19 -18 -17 -16 -15 ↵ [ ] - Stronger upper bound on vC/TC comes from bubble nucleation (see later.) 6
high-T 200 full one-loop effective potential Veff rather than V0 . As discussed in Ref. [20], this procedure, as well as the 180 conventional method for computing T , introduces an C 160 unphysical gauge dependence. In what follows, we will perform a gauge-invariant computation. We also address 140 the impact of the µ-dependence by implementing a RG- 120 improved analysis. These and other theoretical issues 100 associated with the BNPC and ΓN are discussed below. [GeV] 80
60
V. GAUGE-INVARIANT METHOD BEYOND 40 TC (RGI) THE LEADING ORDER v¯C (RGI) 20 TC v¯C 0 Here, we delineate the gauge-invariant treatment for 50 100 150 200 250 300 EWPT and sphaleron rate. Determination of TC andv ¯C µ [GeV] using the high-T effective potential is obviously gauge independent. Beyond this order, however, the potential FIG. 4: Renormalization scale dependence on TC andv ¯C barrier inherently depends on the gauge fixing parameter, in S1. The dotted curves are calculated based on the orig- which may lead to the gauge-dependent TC and vC as in inal PRM scheme to (!) while the solid ones are the RG- the ordinary method. Nevertheless, the gauge-invariant improved version. O TC can still be obtained by use of a method advocated - PRM scheme - in Ref. [20] (PRM scheme). In thisNLO method, analysisTC is deter- mined by the following degeneracy condition O(hbar)VI. NUMERICAL ANALYSIS TC vC
V (v(1); T ) V (v(2); T )=0, (24) A. S1 Case eff 0 C − eff 0 C
vC = minimum of high-TFig. 4potential shows atT TCC and the correspondingv ¯C with and where Veff is the effective potential whose ! expansion follows the Nielsen-Fukuda-Kugo identity [21, 22], and without the RG improvement. The latter corresponds to e.g. (1) (2) 250 the results in the PRM scheme to (!). The input pa- 0 v v0 denotes the minimum of V0(ϕEEW)whilev0 that of O ES rameters are the same as in the left plot of Fig 2. The V0(ϕ) but with-5x10 the7 electroweak broken VEV’s taken to 200 solid curves represent the former and the dashed ones the be zero. In this paper, we will confine ourself to the one- -1x108
] vC latter. One can see that T has a significant renormal- 4 C loop order so that Veff is approximated by Eq. (15). 150 -1.5x108 ization scale dependence before the RG improvement. In Since the renormalization scale µ exists in VCW in 8 this case,v ¯C can vary from zero to nonzero, making it -2x10 VEVs [GeV] Eq. (16), TC can suffer from a significant amount of the100 µ Vacuum Energies [GeV difficult to reach a conclusion whether the EWPT is of dependence. In-2.5x10 this8 paper, we will use a scheme in which 50 first-order or not. After the RG improvement, however, V0(ϕ) in the degeneracy-3x108 condition (24) is replaced by the the renormalization scale dependence is substantially al-
8 RG-improved-3.5x10 one V0(ϕ) in order to alleviate the µ 0de- leviated, and the EWPT is seen to be strongly first-order. 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 T [GeV] TC T [GeV] TC pendence. More explicitly, we replace all the parameters In Fig. 5, TC andv ¯C are plotted as functions of α, in V0 with the running! ones where the solid curves correspond to the PRM scheme with RG improvement, while the dashed ones the LO
V0(ϕ, ϕS) gauge-invariant method. One can see that TC gets lower 2 2 2 2 2 as α decreases in both cases. This implies that since m (µ ) 2 λ(µ ) 4 δ2(µ ) 2 2 b2(µ ) 2 2 2 δ2 = 2(m m )sαcα/(v0vS0), a positive δ2 is more fa- != ϕ + ϕ + ϕ ϕS + ϕS H1 H2 4 16 8 4 vorable for SFOEWPT− as also seen from the analytic for- d (µ2) b (µ2) 2 4 √ 2 1 2 mula Eq. (14). For α ! 22◦, however, phase A becomes + ϕS + 2a1(µ )ϕS + ϕS. (25) − 16 4 the global minimum, yielding an upper bound δ2 ! 2.6. It should be noted that the differences between the LO Here, we use the one-loop β functions [13] to evaluate and NLO results become more and more prominent as α the running parameters, and the RG effects on ϕ and ϕS approaches to zero. Moreover, the tree-potential barrier are ignored as they are negligible. Note that the other would disappear at around α 18.5◦, which signifies a parameters appearing in Eq. (24) remain unchanged in second-order EWPT in the LO≃− calculation. In such small order not to spoil the gauge independence to this order. α regions, the loop-induced barrier would be important Similarly, V0(ϕ) in Eq. (10) remains as is since the renor- for the realization of SFOEWPT. Our NLO calculation malization scale is not involved by construction. In what shows that the EWPT is still first-order for α 18.5◦, " − follows, we explicitly demonstrate how our prescription which persists to α =0◦. However, this could be an arti- works. fact since higher-order contributions such as (!2)terms O μ dependence
PRM scheme is gauge independent but scale dependent.
origin: 200 180
160
140
120
100 Different scales give different [GeV] orders of phase transition: 80 60
40 nd μ 2 order for ≲ 160 GeV 20
st 0 1 order for μ ≳ 160 GeV 50 100 150 200 250 300 [GeV] Improved-RPM scheme
idea: μ dependence is reduced by renormalization group eq.
our scheme: Improved-RPM scheme
200
180
160
140
120
100 [GeV] 80
60
40 TC (RGI) v¯C (RGI) 20 TC v¯C 0 50 100 150 200 250 300 µ [GeV]
μ dependence is significantly reduced by the RG improvement. In this example, phase transition is 1st order. LO vs. NLO
250 TC v¯C LO TC v¯LO 200 C
150 ] GeV [ 100
50
0 -22 -21 -20 -19 -18 -17 -16 -15
α [◦] LO vs. NLO
250 LO TC TC 150 ] GeV [ 100 50 0 -22 -21 -20 -19 -18 -17 -16 -15 α [◦] LO vs. NLO 250 LO TC TC 150 ] GeV [ 100 50 0 -22 -21 -20 -19 -18 -17 -16 -15 α [◦] LO vs. NLO 250 LO TC TC 150 ] GeV [ 100 LO TC 50 TC 0 -22 -21 -20 -19 -18 -17 -16 -15 246 GeV α [◦] LO vs. NLO 250 LO TC TC 150 ] GeV [ 100 LO TC 50 TC 0 -22 -21 -20 -19 -18 -17 -16 -15 246 GeV α [◦] LO vs. NLO 250 LO TC TC 150 ] GeV [ 100 LO TC 50 TC 0 -22 -21 -20 -19 -18 -17 -16 -15 246 GeV α [◦] LO vs. NLO α=-20.5o 250 LO TC TC 150 ] GeV [ 100 LO TC 50 TC 0 -22 -21 -20 -19 -18 -17 -16 -15 246 GeV α [◦] In the following, α=-20.5o is taken. LO vs. NLO benchmark point: high T minima of V ('i; T ) 250 m = 230 GeV, v = 40 GeV, v¯(T ) H2 S0 v T 3 ¯S ( ) ↵ = 20.5 , a1 = (110 GeV) v˜S (T ) 200 NLO Leading Order: LO 150 [GeV] 100 Next-to-Leading Order: 50 0 0 50 100 150 200 T [GeV] LO vs. NLO benchmark point: high T minima of V ('i; T ) 250 m = 230 GeV, v = 40 GeV, v¯(T ) H2 S0 v T 3 ¯S ( ) ↵ = 20.5 , a1 = (110 GeV) v˜S (T ) 200 NLO Leading Order: LO 150 [GeV] 100 Next-to-Leading Order: 50 0 0 50 100 150 200 T [GeV] How about nucleation temperature? Onset of PT - TC is not onset of the PT. - Nucleation starts somewhat Veff below TC. “Not all bubbles can grow” T=Tc 0 expand? or shrink? T=TN 0 50 100 150 200 250 300 volume energy vs. surface energy v [GeV] ∝(radius)3 ∝(radius)2 There is a critical value of radius -> critical bubble Nucleation temperature - Nucleation rate per unit time per unit volume 2 where Ecb(T ) is the energy of the critical bubble at temperature T . Note that this is [A.D. Linde, NPB216 (’82) 421] a rate per unit volume. We define the nucleation temperature TN as the temperature at which the rate of nucleation of a critical bubble within a horizon volume is equal to the Hubble parameter- Definition at thatof nucleation temperature. temperature Since the (T horizonN) scale is roughly given by 1 3 H(T )− , the nucleation temperature is defined by 3 ΓN (TN )H(TN )− = H(TN ). (4.2) Since the Hubble parameter at temperature T is 2 1/2 2 8πG 8π π 4 1/2 T H(T )= ρ(T )= 2 g (T )T 1.66 g (T ) , (4.3) ! 3 "3mP 30 ∗ # ≃ ∗ mP where g (T ) is the effective massless degrees of freedom at T defined by ∗ 7 g (T )= θ(T mB(T ))gB + θ(T mF (T ))gF , (4.4) ∗ − 8 − $B $F with gB and gF being intrinsic degrees of freedom of boson B and fermion F , respectively, the definition of TN (4.2) is reduced to E (T ) 3/2 T 4 cb N Ecb(TN )/TN 2 N e− =7.59 g (TN ) 4 , (4.5) " 2πTN # ∗ mP or Ecb(TN ) 3 Ecb(TN ) TN log = 152.59 2 log g (TN ) 4 log . (4.6) TN − 2 TN − ∗ − 100GeV A Derivatives of the effective potential Here we summarize the first and second derivatives of the effective potential, which are used in the tadpole conditions, the equations of motion and matrix elements in the relax- ation algorithm. A.1 tadpole conditions The tadpole conditions are imposed in the vacuum at zero temperature. iθ0 1 v0d e 0 Φd 0 = , Φu 0 = (A.1) ⟨ ⟩ √2 % 0 & ⟨ ⟩ √2 % v0u & 2In [3], the author only evaluated the contribution from translational zero modes which is proportional 3/2 4 to Ecb , but not that from rotational zero modes and those from nonzero modes. He multiplied T with the prefactor on the dimensional ground. We expect a factor of T 3 comes from the 3-dimensional translational zero modes, while the remaining factor of T may have its origin in the negative mode corresponding to the instability of the critical bubble. Any way, this uncertainty in the prefactor will have small effect on the estimation of the nucleation temperature, which is mainly determined by the exponent. 3One may think that only one bubble nucleated within a horizon volume cannot convert the whole region into the broken phase. In this sense, this definition of nucleation temperature will give an upper bound of temperature at which the phase transition begins to proceed. 6 7 250 105 TC v¯C LO TC v¯LO 200 C 104 150 T )/ T 3 ( 10 [GeV] 100 sph E 50 102 0 101 -22 -21 -20 -19 -18 -17 -16 -15 0 20 40 60 80 100 α [◦] T [GeV] S3(T)/T - LO case - FIG. 5: TC andv ¯C as a function of α in S1. The solid curves FIG. 7: Esph(T )/T as a function of T , where Esph(T )iscal- are obtained by the PRM scheme with RG improvement while culated using the high-T effective potential in Eq. (10). From the dashed ones by the high-T effective potential in Eq. (10). LO LO benchmark point: right to left, the dots mark for Esph(TC )/TC =61.31, Esph(TN )/TN =74.23 and Esph(TC )/TC =78.00, where LO 500 TC =90.4GeV,TN =84.9GeVandTC =83.1GeV. mH2 = 230 GeV, vS0 = 40 GeV, 3 ↵ = 20.5 , a1 = (110 GeV) 400 S1. The dotted line satisfies the condition in Eq. (19), from which we obtain S3(TN )/TN = 152.01 and TN = 84.9 GeV, which is closer to TC = 83.1 GeV obtained 300 TN = 84.9 GeV T in the PRM scheme. The degrees of supercooling is )/ LO LO LO T ( thus (TC TN )/TC 6.1%, where TC = 90.4 GeV 3 − ≃ S3(TN ) S and is one order of magnitude larger than that in the = 152.01 GeV 200 minimal supersymmetric SM (MSSM) case (see, e.g., TN 152.01 GeV Refs. [23, 24]). 100 It is found that the supercooling becomes larger as LO α decreases, and eventually the condition of Eq. (19) TC TN 6.1% 84.9 GeV cannot be fulfilled for α ! 21.4◦, rendering a stronger LO 0 − TC ' lower bound on α than the vacuum condition mentioned 80 82 84 86 88 90 LO above. For the critical α, we obtain TC = 78.1 GeV and cf., MSSM: O(0.1)% T [GeV] LO LO TN = 47.3 GeV, leading to (TC TN )/TC 39.4%. In such a large supercooling case,− it is worth≃ studying FIG. 6: S3(T )/T vs. T for α = 20.5◦ in S1. S3(T )is TC = 83.1 GeV; TN (LO) evaluated= 84.9 byGeV, use ofT theC (LO) high- T=e ff90.4ective− potentialGeV in Eq. (10). how the EWPT develops. If it mostly proceeds via the The dotted horizontal line satisfies the condition in Eq. (19). bubble nucleations rather than expansions, the EWBG LO may not work since the baryon asymmetry is normally In this case, we have TC =90.4GeVandTN =84.9GeV, with the latter closer to TC calculated in the PRM scheme. facilitated with the help of the bubble expansions. Fig. 7 displays E (T )/T against T for α = 20.5◦ sph − in S1, where Esph(T ) is estimate based on the high-T dropped here may not be neglected in the small α re- effective potential in Eq. (10). From right to left, the LO LO gion. As emphasized in Ref. [20], such a fake SFOEWPT three dots mark the results for Esph(TC )/TC = 61.31, to (!) calculation can also happen in the SM. Even Esph(TN )/TN = 74.23 and Esph(TC )/TC = 78.00 using O LO though a quantitative statement awaits a more precise the values of TC , TN and TC given above. analysis in that region, the negatively large α (positively In Fig. 8, the dimensionless sphaleron energy (T )is large δ) does show the generic feature of SFOEWPT in plotted as a function of T . Apparently, (T ) decreasesE as this model. T increases, showing that the temperatureE dependence of As discussed in Sec. IV, the actual beginning of the Esph(T ) is not fully embodied in Ω(T ), and a na¨ıve scal- EWPT is somewhat below the temperature at which the ing formula Esph(T )=Esph(0)v(T )/v0 is no longer valid, effective potential has two degenerate minima. Here, especially when T approaches TC (for earlier studies, see we calculate S3(T )tofindTN using the high-T effec- Refs. [24–26]). As in Fig. 7, the three dots correspond tive potential in Eq. (10). In Fig. 6, the solid curve to (T LO)=1.82, (T )=1.86 and (T )=1.87 from E C E N E C shows S (T )/T as a function of T for α = 20.5◦ in right to left. 3 − No nucleation case 106 - α=-22.0o 105 - VC/TC = (209.1GeV)/(65.52GeV) = 3.2 104 - To o st ro n g 1 st-order EWPT 3 may not be consistent!! 10 o 102 - No nucleation forα<-21.4 0 10 20 30 40 50 60 Sphaleron Sphaleron解を求める σφαλεροs (sphaleros) “ready to fall” saddle point = least-energy path上のmaximum-energy configuration [F.R.Klinkhamer and N.S.Manton, PRD30, 2212 (1984)] Energy least-energy path/gauge trf. = noncontractible loop sphaleron vacuum highest symmetry↕ config. configuration NCS=1 space vacuum NCS=0 ⋆ 4次元SU(2) gauge-Higgs doublet系 ⋆ 2 2 1 a aµν µ v = F F +(D Φ)† D Φ λ Φ†Φ L −4 µν µ − − 2 ! " a D Φ =(∂ igA ) Φ, F = ∂ A ∂ A ig[A ,A ], A = Aa τ µ µ − µ µν µ ν − ν µ − µ ν µ µ 2 —SphaleronTransition— 39/47 Sphaleron in SU(2) gauge-Higgs system Sphaleron解を求める How dosaddle we find point a= saddleleast-energy point path configuration?上のmaximum-energy configuration sphaleron -> use of a noncontractible loop. Energy least-energy path/gauge trf. = noncontractible loop vacuum highest symmetry↕ config. configuration NCS=1 space vacuum NCS=0 ⋆ 4次元SU(2) gauge-Higgs doublet系 ⋆ 2 2 1 a aµν µ v = F F +(D Φ)† D Φ λ Φ†Φ L −4 µν µ − − 2 ! " a D Φ =(∂ igA ) Φ, F = ∂ A ∂ A ig[A ,A ], A = Aa τ µ µ − µ µν µ ν − ν µ − µ ν µ µ 2 —SphaleronTransition— 39/47 Manton’s ansatz [N.S. Manton, PRD28 (’83) 2019] Energy functional Manton’s ansatz [N.S. Manton, PRD28 (’83) 2019] Energy functional Manton’s ansatz [N.S. Manton, PRD28 (’83) 2019] Energy functional input: Sphaleron energy Equations of motion for the sphaleron with the boundary conditions: Esph(T)/T in cxSM 7 250 105 TC v¯C LO TC v¯LO 200 C 104 150 T )/ T 3 ( 10 [GeV] 100 sph E 50 102 0 101 -22 -21 -20 -19 -18 -17 -16 -15 0 20 40 60 80 100 α [◦] T [GeV] LO Esph(TC ) Esph(TN ) Esph(T ) FIG. 5: TC andv ¯C as a function of α in S1. The solid curves FIG. 7: Esph(T )/T as a function of T , where Esph(T )iscal-C = 78.00, = 74.23, LO = 61.31, are obtained by the PRM scheme with RG improvement while TculatedC using the high-TTeNffective potential in Eq.T (10).C From LO LO the dashed ones by the high-T effective potential in Eq. (10). right to left, the dots mark for Esph(TC )/TC =61.31, Esph(TN )/TN =74.23 and Esph(TC )/TC =78.00, where LO 500 TC =90.4GeV,TN =84.9GeVandTC =83.1GeV. 400 S1. The dotted line satisfies the condition in Eq. (19), from which we obtain S3(TN )/TN = 152.01 and TN = 84.9 GeV, which is closer to TC = 83.1 GeV obtained 300 T in the PRM scheme. The degrees of supercooling is )/ LO LO LO T ( thus (TC TN )/TC 6.1%, where TC = 90.4 GeV 3 − ≃ S 200 and is one order of magnitude larger than that in the minimal supersymmetric SM (MSSM) case (see, e.g., Refs. [23, 24]). 100 It is found that the supercooling becomes larger as α decreases, and eventually the condition of Eq. (19) cannot be fulfilled for α ! 21.4◦, rendering a stronger 0 lower bound on α than the− vacuum condition mentioned 80 82 84 86 88 90 LO above. For the critical α, we obtain TC = 78.1 GeV and T [GeV] LO LO TN = 47.3 GeV, leading to (TC TN )/TC 39.4%. In such a large supercooling case,− it is worth≃ studying FIG. 6: S3(T )/T vs. T for α = 20.5◦ in S1. S3(T )is evaluated by use of the high-T effective− potential in Eq. (10). how the EWPT develops. If it mostly proceeds via the The dotted horizontal line satisfies the condition in Eq. (19). bubble nucleations rather than expansions, the EWBG LO may not work since the baryon asymmetry is normally In this case, we have TC =90.4GeVandTN =84.9GeV, with the latter closer to TC calculated in the PRM scheme. facilitated with the help of the bubble expansions. Fig. 7 displays E (T )/T against T for α = 20.5◦ sph − in S1, where Esph(T ) is estimate based on the high-T dropped here may not be neglected in the small α re- effective potential in Eq. (10). From right to left, the LO LO gion. As emphasized in Ref. [20], such a fake SFOEWPT three dots mark the results for Esph(TC )/TC = 61.31, to (!) calculation can also happen in the SM. Even Esph(TN )/TN = 74.23 and Esph(TC )/TC = 78.00 using O LO though a quantitative statement awaits a more precise the values of TC , TN and TC given above. analysis in that region, the negatively large α (positively In Fig. 8, the dimensionless sphaleron energy (T )is large δ) does show the generic feature of SFOEWPT in plotted as a function of T . Apparently, (T ) decreasesE as this model. T increases, showing that the temperatureE dependence of As discussed in Sec. IV, the actual beginning of the Esph(T ) is not fully embodied in Ω(T ), and a na¨ıve scal- EWPT is somewhat below the temperature at which the ing formula Esph(T )=Esph(0)v(T )/v0 is no longer valid, effective potential has two degenerate minima. Here, especially when T approaches TC (for earlier studies, see we calculate S3(T )tofindTN using the high-T effec- Refs. [24–26]). As in Fig. 7, the three dots correspond tive potential in Eq. (10). In Fig. 6, the solid curve to (T LO)=1.82, (T )=1.86 and (T )=1.87 from E C E N E C shows S (T )/T as a function of T for α = 20.5◦ in right to left. 3 − T-dependence of Esph(T) 4⇡v¯(T ) Esph(T )= (T ) g2 E If T-dependence comes from v(T) only, one has v¯(T ) Esph(T )=Esph(0) v0 Is this scaling law valid? T-dependence of Esph(T) 8 2 LO T [GeV] TC =84.3 TC =99.8 TN =96.6 4⇡v¯(T ) v¯(T )[GeV] 193.0 167.0 173.5 Esph(T )= (T ) Esph(T )/T 84.36 61.67 66.02 g2 E 1.95 (T ) 1.92 1.92 1.92 E TABLE I: VEVs and sphaleron energies at the different tem- 1.9 LO If T-dependence comes from peratures, TC , TC and TN for δ2 =0.55 in S2, where the ) last two are calculated by use of the high-T effective poten- T ( v(T) only, one has E tial (10). 1.85 v¯(T ) δ 0.5, the Higgs potential is unbounded from below, Esph(T )=Esph(0) 1.8 2 ! i.e., √−λd < δ (λ =0.52, d =0.5). For 0.5 v0 2 − 2 2 − ! δ2 ! 0.34, there is no phase A since TC is too large to 1.75 satisfy Eq. (26). For δ2 " 0.77, phase A becomes the Is this scaling law valid? 0 20 40 60 80 100 global minimum at T = 0. In this case, the EWPT never T [GeV] happens. As in the S1 case, we also evaluate S3(T ), Esph(T ) and FIG. 8: (T )=g2Esph(T )/(4πv¯) as a function of T .The ,fixingδ2 =0.55. The results are summarized in Table E LO E three dots correspond to (TC )=1.82, (TN )=1.86 and I. One can see that as in the S1 case, the degree of the su- E E (TC )=1.87 from right to left. percooling is one order of magnitude larger than the typ- E LO LO ical MSSM value [23, 24], i.e.,(TC TN )/TC 3.2%. 250 − ≃ TC However, one distinctive feature of S2 is that is inde- v¯C E LO pendent of T , implying that Esph(T )=Esph(0)¯v(T )/v0 TC v¯LO as in the SM. This is due to the fact that there is no 200 C singlet Higgs contribution to Esph sincev ¯S = 0. Before closing this section, we make a comment on the 150 DM mass mA dependence in SFOEWPT. In both S1 and S2 cases, the vacuum energy of phase B increases as mA increases and surpasses the vacuum energy of phase A [GeV] 100 at some critical value, and hence TC cannot be defined in the NLO calculation. As will be discussed in the next section, a relatively large mA would be required to obtain 50 the correct DM abundance, which could be in conflict with the realization of SFOEWPT. 0 -0.4 -0.2 0 0.2 0.4 0.6 VII. DARK MATTER δ2 In this model, the pseudoscalar particle A can be a can- FIG. 9: TC and VEV’s as functions of δ2.Forδ2 0.5, ! − didate for DM. We will study its properties in S1 and S2. we have √λd2 < δ2,whichmakestheHiggspotentialun- − We use micrOMEGAs [27, 28] to calculate the relic den- bounded from below. For 0.5 ! δ2 ! 0.34, the phase A is − sity of A, ΩA, and its spin-independent scattering cross absent since TC is too high to satisfy Eq. (26). For δ2 " 0.77, N phase A turns into the global minimum at T =0. section with nucleon N, σSI. The observed DM relic density is [29] Ω h2 =0.1186 0.0020 . (27) B. S2 Case DM ± We will use the central value as a constraint on Ω h2. In this case, there is no constraint among b , b , δ and A 1 2 2 If the relic abundance of A is less than the observed d from the tadpole condition of Eq. (6), as opposed to 2 one, σN should be scaled as the S1 case. Note that the nontrivial vacuum phase A in SI ϕS exists if N N ΩA σ˜SI = σSI . (28) 2 2 ΩDM v˜S(T )= b1 + b2 +2ΣS(T ) (26) # $ −d2 ! " Recently, the LUX Collaboration has updated their is positive, which implies that b1 + b2 must be negative experimental results [30]. After combining the previous N as long as ΣS(T ) is positive. LUX data, the minimum exclusion limit becomes σSI 46 2 ≃ Fig. 9 shows the dependence of EWPT on δ . For 1.1 10− cm at around 50 GeV of the DM mass. 2 × T-dependence of Esph(T) 8 2 LO T [GeV] TC =84.3 TC =99.8 TN =96.6 4⇡v¯(T ) v¯(T )[GeV] 193.0 167.0 173.5 Esph(T )= (T ) Esph(T )/T 84.36 61.67 66.02 g2 E 1.95 (T ) 1.92 1.92 1.92 E TABLE I: VEVs and sphaleron energies at the different tem- 1.9 LO If T-dependence comes from peratures, TC , TC and TN for δ2 =0.55 in S2, where the ) last two are calculated by use of the high-T effective poten- T ( v(T) only, one has E tial (10). 1.85 v¯(T ) δ 0.5, the Higgs potential is unbounded from below, Esph(T )=Esph(0) 1.8 2 ! i.e., √−λd < δ (λ =0.52, d =0.5). For 0.5 v0 2 − 2 2 − ! δ2 ! 0.34, there is no phase A since TC is too large to 1.75 satisfy Eq. (26). For δ2 " 0.77, phase A becomes the Is this scaling law valid? 0 20 40 60 80 100 global minimum at T = 0. In this case, the EWPT never T [GeV] happens. As in the S1 case, we also evaluate S3(T ), Esph(T ) and No, it breaks down especiallyFIG. 8: (Twhen)=g2E Tsph (approachesT )/(4πv¯) as a function TC. of T .The ,fixingδ2 =0.55. The results are summarized in Table E LO E three dots correspond to (TC )=1.82, (TN )=1.86 and I. One can see that as in the S1 case, the degree of the su- E E (TC )=1.87 from right to left. percooling is one order of magnitude larger than the typ- E LO LO ical MSSM value [23, 24], i.e.,(TC TN )/TC 3.2%. 250 − ≃ TC However, one distinctive feature of S2 is that is inde- v¯C E LO pendent of T , implying that Esph(T )=Esph(0)¯v(T )/v0 TC v¯LO as in the SM. This is due to the fact that there is no 200 C singlet Higgs contribution to Esph sincev ¯S = 0. Before closing this section, we make a comment on the 150 DM mass mA dependence in SFOEWPT. In both S1 and S2 cases, the vacuum energy of phase B increases as mA increases and surpasses the vacuum energy of phase A [GeV] 100 at some critical value, and hence TC cannot be defined in the NLO calculation. As will be discussed in the next section, a relatively large mA would be required to obtain 50 the correct DM abundance, which could be in conflict with the realization of SFOEWPT. 0 -0.4 -0.2 0 0.2 0.4 0.6 VII. DARK MATTER δ2 In this model, the pseudoscalar particle A can be a can- FIG. 9: TC and VEV’s as functions of δ2.Forδ2 0.5, ! − didate for DM. We will study its properties in S1 and S2. we have √λd2 < δ2,whichmakestheHiggspotentialun- − We use micrOMEGAs [27, 28] to calculate the relic den- bounded from below. For 0.5 ! δ2 ! 0.34, the phase A is − sity of A, ΩA, and its spin-independent scattering cross absent since TC is too high to satisfy Eq. (26). For δ2 " 0.77, N phase A turns into the global minimum at T =0. section with nucleon N, σSI. The observed DM relic density is [29] Ω h2 =0.1186 0.0020 . (27) B. S2 Case DM ± We will use the central value as a constraint on Ω h2. In this case, there is no constraint among b , b , δ and A 1 2 2 If the relic abundance of A is less than the observed d from the tadpole condition of Eq. (6), as opposed to 2 one, σN should be scaled as the S1 case. Note that the nontrivial vacuum phase A in SI ϕS exists if N N ΩA σ˜SI = σSI . (28) 2 2 ΩDM v˜S(T )= b1 + b2 +2ΣS(T ) (26) # $ −d2 ! " Recently, the LUX Collaboration has updated their is positive, which implies that b1 + b2 must be negative experimental results [30]. After combining the previous N as long as ΣS(T ) is positive. LUX data, the minimum exclusion limit becomes σSI 46 2 ≃ Fig. 9 shows the dependence of EWPT on δ . For 1.1 10− cm at around 50 GeV of the DM mass. 2 × T-dependence of Esph(T) 8 2 LO T [GeV] TC =84.3 TC =99.8 TN =96.6 4⇡v¯(T ) v¯(T )[GeV] 193.0 167.0 173.5 Esph(T )= (T ) Esph(T )/T 84.36 61.67 66.02 g2 E 1.95 (T ) 1.92 1.92 1.92 E TABLE I: VEVs and sphaleron energies at the different tem- 1.9 LO If T-dependence comes from peratures, TC , TC and TN for δ2 =0.55 in S2, where the ) last two are calculated by use of the high-T effective poten- T ( v(T) only, one has E tial (10). 1.85 v¯(T ) δ 0.5, the Higgs potential is unbounded from below, Esph(T )=Esph(0) 1.8 2 ! i.e., √−λd < δ (λ =0.52, d =0.5). For 0.5 v0 2 − 2 2 − ! δ2 ! 0.34, there is no phase A since TC is too large to 1.75 satisfy Eq. (26). For δ2 " 0.77, phase A becomes the Is this scaling law valid? 0 20 40 60 80 100 global minimum at T = 0. In this case, the EWPT never T [GeV] happens. As in the S1 case, we also evaluate S3(T ), Esph(T ) and No, it breaks down especiallyFIG. 8: (Twhen)=g2E Tsph (approachesT )/(4πv¯) as a function TC. of T .The ,fixingδ2 =0.55. The results are summarized in Table E LO E three dots correspond to (TC )=1.82, (TN )=1.86 and I. One can see that as in the S1 case, the degree of the su- E E ∵ presence of vS(T). (TC )=1.87 from right to left. percooling is one order of magnitude larger than the typ- E LO LO ical MSSM value [23, 24], i.e.,(TC TN )/TC 3.2%. 250 − ≃ TC However, one distinctive feature of S2 is that is inde- v¯C E LO pendent of T , implying that Esph(T )=Esph(0)¯v(T )/v0 TC v¯LO as in the SM. This is due to the fact that there is no 200 C singlet Higgs contribution to Esph sincev ¯S = 0. Before closing this section, we make a comment on the 150 DM mass mA dependence in SFOEWPT. In both S1 and S2 cases, the vacuum energy of phase B increases as mA increases and surpasses the vacuum energy of phase A [GeV] 100 at some critical value, and hence TC cannot be defined in the NLO calculation. As will be discussed in the next section, a relatively large mA would be required to obtain 50 the correct DM abundance, which could be in conflict with the realization of SFOEWPT. 0 -0.4 -0.2 0 0.2 0.4 0.6 VII. DARK MATTER δ2 In this model, the pseudoscalar particle A can be a can- FIG. 9: TC and VEV’s as functions of δ2.Forδ2 0.5, ! − didate for DM. We will study its properties in S1 and S2. we have √λd2 < δ2,whichmakestheHiggspotentialun- − We use micrOMEGAs [27, 28] to calculate the relic den- bounded from below. For 0.5 ! δ2 ! 0.34, the phase A is − sity of A, ΩA, and its spin-independent scattering cross absent since TC is too high to satisfy Eq. (26). For δ2 " 0.77, N phase A turns into the global minimum at T =0. section with nucleon N, σSI. The observed DM relic density is [29] Ω h2 =0.1186 0.0020 . (27) B. S2 Case DM ± We will use the central value as a constraint on Ω h2. In this case, there is no constraint among b , b , δ and A 1 2 2 If the relic abundance of A is less than the observed d from the tadpole condition of Eq. (6), as opposed to 2 one, σN should be scaled as the S1 case. Note that the nontrivial vacuum phase A in SI ϕS exists if N N ΩA σ˜SI = σSI . (28) 2 2 ΩDM v˜S(T )= b1 + b2 +2ΣS(T ) (26) # $ −d2 ! " Recently, the LUX Collaboration has updated their is positive, which implies that b1 + b2 must be negative experimental results [30]. After combining the previous N as long as ΣS(T ) is positive. LUX data, the minimum exclusion limit becomes σSI 46 2 ≃ Fig. 9 shows the dependence of EWPT on δ . For 1.1 10− cm at around 50 GeV of the DM mass. 2 × Summary of the 1st part - We have evaluated vC and TC using GI methods in the cxSM. -μ dependence can be alleviated by the RG improvement. - vC/TC is greater than the LO result. LO Esph(TC ) Esph(TC ) > LO TC TC - Around phase transition point, TC is subject to the large theoretical errors. -> higher-order corrections are needed. Band structure effect on B preservation criteria based on the collaborators with Koichi Funakubo (Saga U), Kaori Fuyuto (UMass-Amherst) Ref. arXiv:1612.05431 B preservation criteria modified by band effect? B preservation criteria modified by band effect? If yes, modified! B preservation criteria modified by band effect? If yes, modified! EWBG-viable region must be re-analyzed!! B+L violation - (B+L) is violated by a chiral anomaly in EW theories. Vacuum transition (instanton) [’t Hooft, PRL37,8 (1976), PRD14,3432 (1976)] Transition rate at finite-E instanton-based [Ringwald, NPB330,(1990)1, Espinosa, NPB343 (1990)310] E⤴ ⟹ σ(E)⤴ - But, instanton-based calculation is not valid at E>Esph Bounce is more appropriate (transition between the finite-E states) [Aoyama, Goldberg, Ryzak, PRL60, 1902 (’88)] -> Reduced model. [Funakubo, Otsuki, Takenaga, Toyoda, PTP87,663(’92), PTP89,881(’93)] [H. Tye, S. Wong, PRD92,045005 (’15)] Tye-Wong’s work [H. Tye, S. Wong, PRD92,045005 (2015)] F(E) Tye-Wong (instanton calculus) E0≃15 TeV F(E) = 0 for E>Esph (Tye-Wong) ∵ a band structure Q: Does the band affect sphaleron process at finite-T? Reduced model [Aoyama, Goldberg, Ryzak, PRL60, 1902 (1988)] [Funakubo, Otsuki, Takenaga, Toyoda, PTP87, 663 (1992), PTP89, 881 (1993)Sphaleron] 解を求める [H. Tye, S. Wong, PRD92,045005 (2015)] saddle point = least-energy path maximum-energy configuration Let us promote μ to a dynamical variable: 上の sphaleron μ ⇒ μ(t) Energy least-energy path/gauge trf. = noncontractible loop vacuum highest symmetry↕ config. configuration NCS=1 μ(-∞)=0, μ(+∞)=π: vacuum, space μ(tsph)=π/2: sphaleron vacuum NCS=0 - We construct a reduced model by adopting Non-contractible loop a Manton’s ansatz. (least energy path) ⋆ 4次元SU(2) gauge-Higgs doublet系 ⋆ 2 2 1 a aµν µ v = F F +(D Φ)† D Φ λ Φ†Φ L −4 µν µ − − 2 ! " a D Φ =(∂ igA ) Φ, F = ∂ A ∂ A ig[A ,A ], A = Aa τ µ µ − µ µν µ ν − ν µ − µ ν µ µ 2 —SphaleronTransition— 39/47 Comparison with Tye-Wong’s work Some differences between our work and Tye-Wong’s (TW’s). Method for band A0 Sphaleron mass structure WKB w/ 3 this work A0≠0 μ-dependent connection formulas Schroedinger eq. A0=0 μ-independent Tye-Wong numerically We use the Manton’s ansatz with Unlike the previous studies, our method is fully gauge invariant. N.B. If A0=0 is naively adopted with the Manton’s ansatz, an unwanted divergence would appear in DΦ at the region r=∞. -> some prescription is needed!! [Aoyama, Goldberg, Ryzak, PRL60, 1902 (1988)] Comparison with Tye-Wong’s work Some differences between our work and Tye-Wong’s (TW’s). Method for band A0 Sphaleron mass structure WKB w/ 3 this work A0≠0 μ-dependent connection formulas Schroedinger eq. A0=0 μ-independent Tye-Wong numerically We use the Manton’s ansatz with Unlike the previous studies, our method is fully gauge invariant. N.B. If A0=0 is naively adopted with the Manton’s ansatz, an unwanted divergence would appear in DΦ at the region r=∞. -> some prescription is needed!! [Aoyama, Goldberg, Ryzak, PRL60, 1902 (1988)] Comparison with Tye-Wong’s work Some differences between our work and Tye-Wong’s (TW’s). Method for band A0 Sphaleron mass structure WKB w/ 3 this work A0≠0 μ-dependent connection formulas Schroedinger eq. A0=0 μ-independent Tye-Wong numerically We use the Manton’s ansatz with Unlike the previous studies, our method is fully gauge invariant. N.B. If A0=0 is naively adopted with the Manton’s ansatz, an unwanted divergence would appear in DΦ at the region r=∞. -> some prescription is needed!! [Aoyama, Goldberg, Ryzak, PRL60, 1902 (1988)] Comparison with Tye-Wong’s work Some differences between our work and Tye-Wong’s (TW’s). Method for band A0 Sphaleron mass structure WKB w/ 3 this work A0≠0 μ-dependent connection formulas Schroedinger eq. A0=0 μ-independent Tye-Wong numerically We use the Manton’s ansatz with Unlike the previous studies, our method is fully gauge invariant. N.B. If A0=0 is naively adopted with the Manton’s ansatz, an unwanted divergence would appear in DΦ at the region r=∞. -> some prescription is needed!! [Aoyama, Goldberg, Ryzak, PRL60, 1902 (1988)] Classical action f, h are determined by the EOM for the sphaleron. Classical action where c.f., TW’s: Msph = 17.1 TeV. With same normalization, Msph(ours) -> 23.0 TeV. Number of band edges are affected by the size of Msph (see later). Band structure Esph=9.08 TeV Esph=9.11 TeV this work Units: TeV Tye-Wong Band Centre E Band Width Band Centre E Band Width 14.054 0.0744 ? ? 13.980 0.0741 ? ? ⫶ ⫶ ⫶ ⫶ 9.072 0.0104 9.113 0.0156 9.044 4.85x10-3 9.081 7.19x10-3 9.012 1.61x10-3 9.047 2.62x10-3 ⫶ ⫶ ⫶ ⫶ 0.1015 1.88x10-199 0.1027 ~10-177 0.03383 1.31x10-202 0.03421 ~10-180 Band gaps still exist E>Esph due to nonzero reflection rate. Band structure Esph=9.08 TeV Esph=9.11 TeV this work Units: TeV Tye-Wong Band Centre E Band Width Band Centre E Band Width 14.054 0.0744 ? ? 13.980 0.0741 ? ? Esph ⫶ ⫶ ⫶ ⫶ 9.072 0.0104 9.113 0.0156 9.044 4.85x10-3 9.081 7.19x10-3 9.012 1.61x10-3 9.047 2.62x10-3 ⫶ ⫶ ⫶ ⫶ 0.1015 1.88x10-199 0.1027 ~10-177 0.03383 1.31x10-202 0.03421 ~10-180 Band gaps still exist E>Esph due to nonzero reflection rate. Band structure Esph=9.08 TeV Esph=9.11 TeV this work Units: TeV Tye-Wong Band Centre E Band Width Band Centre E Band Width 14.054 0.0744 ? ? 13.980 0.0741 ? ? Esph ⫶ ⫶ ⫶ ⫶ 9.072 0.0104 9.113 0.0156 9.044 4.85x10-3 9.081 7.19x10-3 9.012 1.61x10-3 9.047 2.62x10-3 ⫶ ⫶ ⫶ ⫶ 0.1015 1.88x10-199 0.1027 ~10-177 0.03383 1.31x10-202 0.03421 ~10-180 # of band Band gaps still exist E>Esph due to nonzero reflection rate. Band structure Esph=9.08 TeV Esph=9.11 TeV this work Units: TeV Tye-Wong Band Centre E Band Width Band Centre E Band Width 14.054 0.0744 ? ? 13.980 0.0741 ? ? Esph ⫶ ⫶ ⫶ ⫶ 9.072 0.0104 9.113 0.0156 9.044 4.85x10-3 9.081 7.19x10-3 9.012 1.61x10-3 9.047 2.62x10-3 ⫶ ⫶ ⫶ ⫶ 0.1015 1.88x10-199 0.1027 ~10-177 0.03383 1.31x10-202 0.03421 ~10-180 # of band Band gaps still exist E>Esph due to nonzero reflection rate. Band structure Esph=9.08 TeV Esph=9.11 TeV this work Units: TeV Tye-Wong Band Centre E Band Width Band Centre E Band Width 14.054 0.0744 ? ? 13.980 0.0741 ? ? Esph ⫶ ⫶ ⫶ ⫶ 9.072 0.0104 9.113 0.0156 9.044 4.85x10-3 9.081 7.19x10-3 9.012 1.61x10-3 9.047 2.62x10-3 ⫶ ⫶ ⫶ ⫶ 0.1015 1.88x10-199 0.1027 ~10-177 0.03383 1.31x10-202 0.03421 ~10-180 # of band Band gaps still exist E>Esph due to nonzero reflection rate. Band structure Esph=9.08 TeV Esph=9.11 TeV this work Units: TeV Tye-Wong Band Centre E Band Width Band Centre E Band Width 14.054 0.0744 ? ? 13.980 0.0741 ? ? Esph ⫶ ⫶ ⫶ ⫶ 9.072 0.0104 9.113 0.0156 9.044 4.85x10-3 9.081 7.19x10-3 9.012 1.61x10-3 9.047 2.62x10-3 ⫶ ⫶ ⫶ ⫶ 0.1015 1.88x10-199 0.1027 ~10-177 0.03383 1.31x10-202 0.03421 ~10-180 # of band Band gaps still exist E>Esph due to nonzero reflection rate. Transition factor tunneling factor instanton calculus band picture 0 sum of band widths up to E Δ(E) ≃ -50 energy (E) -100 (E) ∆ 10 log Band picture: -150 - State of density is restricted. -200 - Exponential suppression at E≪Esph is due to the tiny -250 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 band width. E [GeV] Vacuum decay rate at finite-T Ordinary case: [Affleck, PRL46,388 (1981)] ≃14 GeV ≃0.42 ≃0.51 Band case: Impact of band For simplicity, we use the band structure obtained before. blue: ordinary case red: band case For T=100 GeV, Γ/ΓA = 1.06. How about B-number preservation criteria? Impact of band For simplicity, we use the band structure obtained before. typical EWBG region blue: ordinary case red: band case For T=100 GeV, Γ/ΓA = 1.06. How about B-number preservation criteria? Baryon number preservation criteria (T ) Including the band effect, Baryon number preservation criteria (T ) Including the band effect, band effect Baryon number preservation criteria (T ) Including the band effect, band effect Baryon number preservation criteria (T ) Including the band effect, band effect Band effect has little effect on the B preservation criteria. Summary of the 2nd part • We have discussed the band effect on the sphaleron processes at T≠0. • At T≃100 GeV, sphaleron process is virtually unaffected. -> no impact on EWBG. Backup Eigenvalue problem Hamiltonian: Band energy is determined by solving [N.L.Balazs, Ann.Phys.53,421 (1969)] with 3 connection formulas depending on energy. over-barrier parabolic potential linear potential Δ(B+L)≠0 process [Funakubo, Otsuki, Takenaga, Toyoda, PTP87, 663 (1992), PTP89, 881 (1993)] transition amplitude: path integral using coherent state |φ,π> ∵ appropriate for describing classical configuration - tunneling suppression appears in the intermediate process. - overlap issue: suppressions from This point is not properly discussed in the work of Tye and Wong. Δ(B+L)≠0 process [Funakubo, Otsuki, Takenaga, Toyoda, PTP87, 663 (1992), PTP89, 881 (1993)] transition amplitude: path integral using coherent state |φ,π> ∵ appropriate for describing classical configuration - tunneling suppression appears in the intermediate process. - overlap issue: suppressions from This point is not properly discussed in the work of Tye and Wong. overlap factor inner product between n particle state and coherent state: 1.0 2 2 0.8 - cross section ∝ |α1| …|αn| 0.6 2 0.4 - |α| has a peak at k=mW. 0.2 100 200 300 400 Sphaleron at colliders W Case1: 2 -> sphaleron Sphaleron For |p1|=|p2|≃Esph/2 W Creation of sphaleron from the 2 energetic particles is difficult. Sphaleron at colliders W Case1: 2 -> sphaleron Sphaleron For |p1|=|p2|≃Esph/2 W Creation of sphaleron from the 2 energetic particles is difficult. Case2: 2 -> n W -> sphaleron 80 W bosons W Sphaleron n≃80 since Esph/√2mW ⫶ phase space factor: W difficult to produce about 80 W bosons.