Remaining issues
- Is O(hbar) calculation in PRM sufficient?
In the SM case, O(hbar
2) + ring contributions cannot be neglected!
JHEP07(2011)029
20 10 0 10 20
70 75 80 100 140
Gauge Parameter, CriticalTemperature,TC(GeV)
0.035 (mH 65 GeV)
Landau:
78.0 GeV latt: 126.8 GeV
104.2 GeV
70.6 GeV 120
: :
Figure 5. A comparison of the critical temperatures as computed using the following methods: the standard method (solid line); the gauge-independent methods described in the text, derived from the full theory (dashed lines); and by performing lattice simulations (arrow). Note that the lattice result is much higher than the perturbative estimations and is displayed on a separate scale.
Also included are lattice results, that yield a critical temperature of 126.8 GeV, in-dependent of ξ by construction. Our estimate of the higher-order contributions included in (5.15) leads to a substantially larger value of TC, suggesting that the difference between the non-perturbative and O(!) perturbative results arises in part from the omission of higher-order contributions. In addition, we note that the precise definition of TC as ob-tained from the lattice studies differs from the one we have employed here as well as in other perturbative analyses (For a discussion of the lattice determinations, see, e.g., refs. [4, 6, 7]).
We speculate that part of the difference between the lattice and perturbative results may also be due to this difference in definition.
While the gauge-independent perturbative estimation of TC falls below the lattice value, it is interesting that the dependence on the relevant couplings follows the trend observed in non-perturbative studies. To illustrate, we plot the one-loop TC as a function of the Higgs quartic self-coupling λ in figure 6. We observe that increasing λ increases TC in agreement with our qualitative expectations in (3.11). As we discuss shortly, this trend implies that the efficiency of sphaleron-induced baryon number washout increases with λ and, thus, with the value of the Higgs boson mass. This trend is also observed in non-perturbative studies as well as in earlier gauge-dependent perturbative analyses.
We now turn our attention to the sphaleron scale, ¯v(T ), which we plot in figure 7.
We observe that in the vicinity of the TC obtained at O(!) in the full theory, ¯v(T ) drops rapidly to zero. This behavior makes the perturbative estimate of the sphaleron rate at the critical temperature highly sensitive to small changes in TC. Therefore, statements about the efficacy of baryon number preservation are susceptible to large uncertainties. To illustrate, we first consider the value of this scale at the one-loop TC in the full theory,
– 33 –
O(~)
SM case
[H.Patel, M.Ramsey-Musolf, JHEP,07(2011)029]
Remaining issues
- Is O(hbar) calculation in PRM sufficient?
In the SM case, O(hbar
2) + ring contributions cannot be neglected!
JHEP07(2011)029
20 10 0 10 20
70 75 80 100 140
Gauge Parameter, CriticalTemperature,TC(GeV)
0.035 (mH 65 GeV)
Landau:
78.0 GeV latt: 126.8 GeV
104.2 GeV
70.6 GeV 120
: :
Figure 5. A comparison of the critical temperatures as computed using the following methods: the standard method (solid line); the gauge-independent methods described in the text, derived from the full theory (dashed lines); and by performing lattice simulations (arrow). Note that the lattice result is much higher than the perturbative estimations and is displayed on a separate scale.
Also included are lattice results, that yield a critical temperature of 126.8 GeV, in-dependent of ξ by construction. Our estimate of the higher-order contributions included in (5.15) leads to a substantially larger value of TC, suggesting that the difference between the non-perturbative and O(!) perturbative results arises in part from the omission of higher-order contributions. In addition, we note that the precise definition of TC as ob-tained from the lattice studies differs from the one we have employed here as well as in other perturbative analyses (For a discussion of the lattice determinations, see, e.g., refs. [4, 6, 7]).
We speculate that part of the difference between the lattice and perturbative results may also be due to this difference in definition.
While the gauge-independent perturbative estimation of TC falls below the lattice value, it is interesting that the dependence on the relevant couplings follows the trend observed in non-perturbative studies. To illustrate, we plot the one-loop TC as a function of the Higgs quartic self-coupling λ in figure 6. We observe that increasing λ increases TC in agreement with our qualitative expectations in (3.11). As we discuss shortly, this trend implies that the efficiency of sphaleron-induced baryon number washout increases with λ and, thus, with the value of the Higgs boson mass. This trend is also observed in non-perturbative studies as well as in earlier gauge-dependent perturbative analyses.
We now turn our attention to the sphaleron scale, ¯v(T ), which we plot in figure 7.
We observe that in the vicinity of the TC obtained at O(!) in the full theory, ¯v(T ) drops rapidly to zero. This behavior makes the perturbative estimate of the sphaleron rate at the critical temperature highly sensitive to small changes in TC. Therefore, statements about the efficacy of baryon number preservation are susceptible to large uncertainties. To illustrate, we first consider the value of this scale at the one-loop TC in the full theory,
– 33 –
O(~)
O(~2) + ring corr.
SM case
[H.Patel, M.Ramsey-Musolf, JHEP,07(2011)029]
Remaining issues
- Is O(hbar) calculation in PRM sufficient?
In the SM case, O(hbar
2) + ring contributions cannot be neglected!
JHEP07(2011)029
20 10 0 10 20
70 75 80 100 140
Gauge Parameter, CriticalTemperature,TC(GeV)
0.035 (mH 65 GeV)
Landau:
78.0 GeV latt: 126.8 GeV
104.2 GeV
70.6 GeV 120
: :
Figure 5. A comparison of the critical temperatures as computed using the following methods: the standard method (solid line); the gauge-independent methods described in the text, derived from the full theory (dashed lines); and by performing lattice simulations (arrow). Note that the lattice result is much higher than the perturbative estimations and is displayed on a separate scale.
Also included are lattice results, that yield a critical temperature of 126.8 GeV, in-dependent of ξ by construction. Our estimate of the higher-order contributions included in (5.15) leads to a substantially larger value of TC, suggesting that the difference between the non-perturbative and O(!) perturbative results arises in part from the omission of higher-order contributions. In addition, we note that the precise definition of TC as ob-tained from the lattice studies differs from the one we have employed here as well as in other perturbative analyses (For a discussion of the lattice determinations, see, e.g., refs. [4, 6, 7]).
We speculate that part of the difference between the lattice and perturbative results may also be due to this difference in definition.
While the gauge-independent perturbative estimation of TC falls below the lattice value, it is interesting that the dependence on the relevant couplings follows the trend observed in non-perturbative studies. To illustrate, we plot the one-loop TC as a function of the Higgs quartic self-coupling λ in figure 6. We observe that increasing λ increases TC in agreement with our qualitative expectations in (3.11). As we discuss shortly, this trend implies that the efficiency of sphaleron-induced baryon number washout increases with λ and, thus, with the value of the Higgs boson mass. This trend is also observed in non-perturbative studies as well as in earlier gauge-dependent perturbative analyses.
We now turn our attention to the sphaleron scale, ¯v(T ), which we plot in figure 7.
We observe that in the vicinity of the TC obtained at O(!) in the full theory, ¯v(T ) drops rapidly to zero. This behavior makes the perturbative estimate of the sphaleron rate at the critical temperature highly sensitive to small changes in TC. Therefore, statements about the efficacy of baryon number preservation are susceptible to large uncertainties. To illustrate, we first consider the value of this scale at the one-loop TC in the full theory,
– 33 –
(*) high-T expanded 2-loop effective potential is used.
O(~)
O(~2) + ring corr.
SM case
[H.Patel, M.Ramsey-Musolf, JHEP,07(2011)029]
Remaining issues
- Is O(hbar) calculation in PRM sufficient?
In the SM case, O(hbar
2) + ring contributions cannot be neglected!
JHEP07(2011)029
20 10 0 10 20
70 75 80 100 140
Gauge Parameter, CriticalTemperature,TC(GeV)
0.035 (mH 65 GeV)
Landau:
78.0 GeV latt: 126.8 GeV
104.2 GeV
70.6 GeV 120
: :
Figure 5. A comparison of the critical temperatures as computed using the following methods: the standard method (solid line); the gauge-independent methods described in the text, derived from the full theory (dashed lines); and by performing lattice simulations (arrow). Note that the lattice result is much higher than the perturbative estimations and is displayed on a separate scale.
Also included are lattice results, that yield a critical temperature of 126.8 GeV, in-dependent of ξ by construction. Our estimate of the higher-order contributions included in (5.15) leads to a substantially larger value of TC, suggesting that the difference between the non-perturbative and O(!) perturbative results arises in part from the omission of higher-order contributions. In addition, we note that the precise definition of TC as ob-tained from the lattice studies differs from the one we have employed here as well as in other perturbative analyses (For a discussion of the lattice determinations, see, e.g., refs. [4, 6, 7]).
We speculate that part of the difference between the lattice and perturbative results may also be due to this difference in definition.
While the gauge-independent perturbative estimation of TC falls below the lattice value, it is interesting that the dependence on the relevant couplings follows the trend observed in non-perturbative studies. To illustrate, we plot the one-loop TC as a function of the Higgs quartic self-coupling λ in figure 6. We observe that increasing λ increases TC in agreement with our qualitative expectations in (3.11). As we discuss shortly, this trend implies that the efficiency of sphaleron-induced baryon number washout increases with λ and, thus, with the value of the Higgs boson mass. This trend is also observed in non-perturbative studies as well as in earlier gauge-dependent perturbative analyses.
We now turn our attention to the sphaleron scale, ¯v(T ), which we plot in figure 7.
We observe that in the vicinity of the TC obtained at O(!) in the full theory, ¯v(T ) drops rapidly to zero. This behavior makes the perturbative estimate of the sphaleron rate at the critical temperature highly sensitive to small changes in TC. Therefore, statements about the efficacy of baryon number preservation are susceptible to large uncertainties. To illustrate, we first consider the value of this scale at the one-loop TC in the full theory,
– 33 –
2-loop effective potential w/o high-T expansion is necessary!
(*) high-T expanded 2-loop effective potential is used.
O(~)
O(~2) + ring corr.
SM case
[H.Patel, M.Ramsey-Musolf, JHEP,07(2011)029]
Remaining issues
- Is O(hbar) calculation in PRM sufficient?
In the SM case, O(hbar
2) + ring contributions cannot be neglected!
JHEP07(2011)029
20 10 0 10 20
70 75 80 100 140
Gauge Parameter, CriticalTemperature,TC(GeV)
0.035 (mH 65 GeV)
Landau:
78.0 GeV latt: 126.8 GeV
104.2 GeV
70.6 GeV 120
: :
Figure 5. A comparison of the critical temperatures as computed using the following methods: the standard method (solid line); the gauge-independent methods described in the text, derived from the full theory (dashed lines); and by performing lattice simulations (arrow). Note that the lattice result is much higher than the perturbative estimations and is displayed on a separate scale.
Also included are lattice results, that yield a critical temperature of 126.8 GeV, in-dependent of ξ by construction. Our estimate of the higher-order contributions included in (5.15) leads to a substantially larger value of TC, suggesting that the difference between the non-perturbative and O(!) perturbative results arises in part from the omission of higher-order contributions. In addition, we note that the precise definition of TC as ob-tained from the lattice studies differs from the one we have employed here as well as in other perturbative analyses (For a discussion of the lattice determinations, see, e.g., refs. [4, 6, 7]).
We speculate that part of the difference between the lattice and perturbative results may also be due to this difference in definition.
While the gauge-independent perturbative estimation of TC falls below the lattice value, it is interesting that the dependence on the relevant couplings follows the trend observed in non-perturbative studies. To illustrate, we plot the one-loop TC as a function of the Higgs quartic self-coupling λ in figure 6. We observe that increasing λ increases TC in agreement with our qualitative expectations in (3.11). As we discuss shortly, this trend implies that the efficiency of sphaleron-induced baryon number washout increases with λ and, thus, with the value of the Higgs boson mass. This trend is also observed in non-perturbative studies as well as in earlier gauge-dependent perturbative analyses.
We now turn our attention to the sphaleron scale, ¯v(T ), which we plot in figure 7.
We observe that in the vicinity of the TC obtained at O(!) in the full theory, ¯v(T ) drops rapidly to zero. This behavior makes the perturbative estimate of the sphaleron rate at the critical temperature highly sensitive to small changes in TC. Therefore, statements about the efficacy of baryon number preservation are susceptible to large uncertainties. To illustrate, we first consider the value of this scale at the one-loop TC in the full theory,
– 33 –
2-loop effective potential w/o high-T expansion is necessary!
- How do we calculate T
Nin the PRM scheme?
T
Nis more important than T
C. (*) high-T expanded 2-loop effective potential is used.
O(~)
O(~2) + ring corr.
SM case
[H.Patel, M.Ramsey-Musolf, JHEP,07(2011)029]
- We have evaluated the gauge fixing parameter (ξ) dependence on the 1
st-order phase transitions.
- ξ dependence can propagate to nucleation temperature, gravitational waves and can be sizable!
- High-order contributions (hbar^2+ring diagrams) are necessary to get more reliable results.
-