Gauge Dependence of the High-Temperature 2-Loop Effective Potential for the Higgs Field

M. Laine

REVIEW

Gauge Dependence of the High-Temperature 2-Loop Effective Potential for the Higgs Field

M. Laine

High Energy Physics - Phenomenology · Submitted 9 Nov 1994 · 17 references

Pith's verdict: A careful, honest two-loop check of gauge independence whose durable product is the appendix formula, with one compressed algebraic step in the middle.

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arXiv IDhep-ph/9411252 (version 2)

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CitationM. Laine. "Gauge Dependence of the High-Temperature 2-Loop Effective Potential for the Higgs Field." arXiv:hep-ph/9411252 (1994). https://pith.science/paper/RLSYFGTD

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This one does what the abstract says, and it is worth your time if you work on the finite-temperature electroweak sector. Laine computes the high-temperature two-loop effective potential for the Higgs in a general covariant gauge, from the 3d reduced theory, and shows explicitly that the pressure at the broken minimum is gauge-independent at order h-bar squared. The conceptual statement is old — Nielsen, Fukuda-Kugo, both cited — so the contribution is the explicit check plus the complete two-loop expression in the appendix. That appendix is the thing people will actually use; it is the kind of formula you cite rather than rederive, and the note that graphs (g3), (e2), (g4), (b), (e1) drop out for pure SU(2)-Higgs without the A0 field is a nice reproducibility touch. The defense of the covariant gauge over R_xi — that R_xi picks a different gauge for each value of phi, which may not be legitimate — is a good point and I think correct. Section 4's cross-check against the Buchmuller-Fodor-Hebecker composite source is a genuinely useful consistency test, not padding. And Section 5 is honest in a way that is rare: Tc is not calculable to order h-bar squared, the latent heat picks up an imaginary part, and the author says so plainly instead of dressing it up. That narrows the program rather than widening it, and the paper deserves credit for saying it. The soft spot is Section 3. The cancellation between the finite gauge-dependent piece of V2 and -(V'1)^2/(2V''0) is dispatched with 'it can easily be seen from eq. 23.' It is a short algebra check and I believe it, but it is the load-bearing step and it should be displayed. The m2 -> 0+ handling of the Goldstone IR is a prescription, not a resummation; the scaling argument for phi2 is stated but thin. Neither of these breaks the claim — the Nielsen identity plus the explicit divergence cancellation in Eq. 7 make it credible — but a referee should ask for the two lines. Section 6's use of gauge dependence as a proxy for truncation error is heuristic; treat it as an estimate, not a bound. Overall: solid, narrow, honest. Send it out.

Referee report

Significance

This is a careful and useful technical paper. The explicit two-loop effective potential in a general covariant gauge, given in full in the Appendix, is a concrete deliverable that other groups can reuse, and the check that the pressure at the broken minimum is ξ-independent to O(hbar^2) tests the Nielsen-identity expectation at a level not previously displayed. The negative results of §5 — that T_c is not calculable to O(hbar^2) and that L_2 has an imaginary part — are honest and genuinely useful to the field, as is the pragmatic use of gauge dependence as an error diagnostic in §6. There are no free parameters and no fitted quantities; the loop functions I, D_VVS and D_SSV are quoted from the published ref. [3]. The internal consistency between the [25.47, 3.76, 0.39]×10^3 GeV^2 series of §5 and the 29.64×10^3 GeV^2 quoted in §6 is a good sign.

Major objections (3)

[§3, Eqs. (10), (12), (14)] The load-bearing step for the paper's central claim is asserted rather than shown. The sentence 'In addition to cancelling all the divergences, the term in eq. 14 also cancels the finite gauge-dependent piece of V2 in eq. 10' is what makes the word 'explicitly' in the abstract and in the §3 conclusion literally true, but no algebra is displayed. Since both V_1 (Eq. 23) and V_2 (Appendix) are given in full, the author should display the ξ-dependent part of V_2 evaluated at ϕ_0 — or at least its leading terms in small ξ, e.g. the ξ^{3/4} and ξ pieces — and show that it equals (V'_1)^2/(2V''_0). Without this the reader must take the cancellation on faith, and the paper's stated contribution is a verification.
[§3, Eqs. (11)-(13) and the m_2 → 0+ prescription] The Goldstone propagator of Eq. (11) is infrared singular as m_2 → 0, and the paper controls it by a limiting prescription and then drops the ϕ_1 V'_0 and ϕ_2 V'_0 terms of Eq. (10). The claim that ϕ_2 diverges only as -ξ/m_2, so that ϕ_2 V'_0 ∝ ξ m_2^2 vanishes at the minimum, is inferred from the cancellation rather than established independently. The author should derive the m_2-scaling of ϕ_1 and ϕ_2 directly from Eqs. (9) and (23), and state explicitly why the m_2 → 0+ limit commutes with the loop expansion, or add a caveat if it does not. A line showing the cancellation among the ξ^{5/2}m_2^{-1}, ξ^{7/4}m_2^{-1/2} and ξ^2 log(m_2) pieces quoted above Eq. (12) would also strengthen the argument considerably.
[§4, symmetric phase] The Legendre machinery of Eqs. (15)-(16) breaks down in the symmetric phase because w_0(J) = 0, and the paper disposes of the case in two sentences: 'Nevertheless, it is still true that w(0) as calculated from the loop expansion equals the value of the conventional effective potential at ϕ = 0.' Since §4's claim is that the two definitions of the effective potential give the same pressure, the symmetric-phase case needs at least a sketch of how the loop expansion of w(J) is organized when the tree-level term vanishes. This is secondary to the broken-minimum result but is part of the 'same physics' conclusion of §4.

Minor objections (8)

[§5, Eqs. (19)-(21)] The vanishing of L_1 is counterintuitive — the one-loop Landau-gauge potential has a barrier between the minima — and the paper itself flags this. A short derivation, or the reason Δp_1(T_0) = T_0(Δp'_1 - Δp_1 Δp''_0/Δp'_0) at T_0, would remove the reader's doubt. The comparison with ref. [2] ('behaves qualitatively like the g^4, λ^2-curve in Fig. 6 of ref. [2], but the numerical value is much smaller') should be made quantitative or dropped.
[§2, Eq. (4) and the covariant vs R_ξ discussion] The assertion that with the R_ξ gauge 'one is choosing a different gauge for each different value of ϕ in the effective potential, which might not be allowed' is a plausibility statement and deserves either a reference or a short argument. As written it reads as a preference rather than a demonstrated obstruction.
[§2, Eq. (7)] The vacuum terms 3g_3^2 m_3^2 + 6g_3^2 m_D^2 and the resulting μ-dependence of the value of V at the minimum are acknowledged, but a one-line statement of which counterterm in the four-dimensional theory removes them would help the reader follow the argument of §3.
[§5, Eq. (21)] The imaginary part of L_2 is reported without comment. A sentence on its physical interpretation — instability of the symmetric minimum, or breakdown of the expansion at this order — would be useful.
[§6] The use of gauge dependence as a measure of the missing higher-order terms is a heuristic, not a controlled error estimate. The text does say it 'gives support to the argument', but a more explicit statement that this is a plausibility diagnostic rather than a bound would be appropriate.
[Figures 1 and 2] The graph labels (a)-(x5) are referred to by name in the Appendix, so they must be legible in the published version; in the version I read the captions are partly garbled. The caption of Fig. 2 should also state explicitly that the straight line carries only Higgs particles, as the surrounding text does.
[Appendix] The shorthand m_± for m_{R±} is introduced in a single line and then used throughout; a reminder at the head of the graph list would help. The useful remark that graphs (g3), (e2), (g4), (b) and (e1) are absent for pure SU(2)-Higgs theory could be moved to the start of the Appendix.
[Notation and cross-references] Equation references alternate between 'eq. 1' and '(1)'; the section cross-references in the running text are garbled in places ('Sec. 2Method of calculation'). The Pauli matrices τ^a should be stated to be 2×2 in the line where they are introduced. In Eq. (5), a check that the A_i propagator reduces to the standard covariant-gauge form at ξ = 0 would reassure the reader about the ξ-dependent denominator.

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The paper

The paper claims

The pressure at the broken Higgs minimum is gauge-independent through two loops.

The reading

This paper asks whether a gauge-dependent object can still yield a physical number. The author computes the high-temperature two-loop effective potential for the Higgs field in a general covariant gauge, using a reduced three-dimensional theory for the long-wavelength modes, and shows that the value at the broken minimum, minus the pressure, is independent of the gauge parameter $xi$ through order $hbar^2$. The mechanism is a cancellation: gauge-dependent divergences and finite pieces from individual two-loop diagrams are offset by the shift of the minimum, specifically the term $-(V_1')^2/(2V_0'')$. The paper also checks that a manifestly gauge-independent effective potential built from a source for $Phi^daggerPhi$ gives the same pressure, and uses the residual gauge dependence of a renormalization-group-improved calculation as a rough error bar in the broken phase. It concludes that the equation of state is under control at this order, while the critical temperature and latent heat of the electroweak transition are not reliably calculable in powers of $hbar$ because of infrared divergences.

What carries the argument

The load-bearing device is the order-by-order shift of the minimum in the loop expansion. Writing $V=V_0+hbar V_1+hbar^2 V_2$ and $phi_{min}=phi_0+hbarphi_1+hbar^2phi_2$, the value at the minimum is $V_0+hbar V_1+hbar^2[V_2-tfrac12 (V_1')^2/V_0'']$ evaluated at $phi_0$, with terms proportional to $V_0'$ dropped. The term $-tfrac12 (V_1')^2/V_0''$ is what cancels the finite gauge-dependent part of $V_2$ and the Goldstone infrared divergence. The calculation uses the decomposition of the gauge-dependent Goldstone propagator into two Landau-gauge-like poles with $R_pm^2=1/2pmsqrt{1/4-xi(m_T/m_2)^2}$, reducing the two-loop integrals to standard Landau-gauge values. The appendix gives the complete two-loop potential, which is the concrete object other workers can check.

Extended reading notes

Core claim

The central claim is that the pressure of electroweak matter at high temperature can be extracted gauge-independently from the conventional gauge-dependent effective potential $V(phi)$ at order $hbar^2$. Working in the effective 3d SU(2)-Higgs theory with an $A_0$ field and a general covariant gauge, the author computes all non-vanishing two-loop vacuum diagrams and shows that the would-be gauge-dependent pieces cancel in $V(phi_{min})$. The cancellation involves both the explicit $xi$-dependent terms in the two-loop potential and the displacement of the minimum from its classical location; the latter contributes $-(V_1')^2/(2V_0'')$, which cancels the finite gauge-dependent remainder of $V_2$ and the infrared-divergent piece as $m_2^2to0$. The same value follows from the gauge-independent generating functional with source $2JPhi^daggerPhi$, so the two definitions agree for the equation of state at this order. The author is explicit that this does not make the phase transition's critical temperature or latent heat calculable to $hbar^2$.

Load-bearing premise

The argument rests on regulating the Goldstone infrared divergence by taking $m_2^2to0^+$ and dropping the correction terms proportional to $V_0'$; if that limiting procedure does not faithfully represent an expansion about the true minimum, the claimed gauge cancellation at order $hbar^2$ is not established.

Editorial extensions

If this is right

If the claim holds, the standard gauge-dependent effective potential can be used for the high-temperature equation of state at two loops without introducing a gauge artifact at that order.
The gauge-independent composite-source effective potential and the conventional effective potential agree for the pressure at order $hbar^2$, providing a consistency check between two formalisms.
The difference between the symmetric and broken phase pressures, and hence the critical temperature and latent heat, is gauge- and scale-independent at order $hbar^2$, although the paper argues these quantities are not reliably computable because of infrared problems.
Deep in the broken phase, away from the critical temperature, renormalization-group-improved perturbation theory appears convergent, and the size of gauge dependence can serve as an estimate of missing higher-order terms.
The explicit two-loop expression in the appendix allows independent checks and extension to other gauges, models, or observables.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The cancellation mechanism suggests that gauge dependence of the effective potential is not only an obstacle but also a diagnostic: its size can estimate the error from truncated higher-order terms, as the paper does in the broken phase.
The failure to compute the critical temperature and latent heat at order $hbar^2$ is a symptom of the known infrared problem of high-temperature field theory; dimensional reduction to three dimensions does not by itself cure the Goldstone mode near the transition.
The distinction between gauge dependence at the minimum and away from it may generalize to other composite-source effective potentials; the equality of pressures could be tested at three loops or nonperturbatively.
If the limiting procedure for the Goldstone infrared divergence is subtle, a lattice calculation of the pressure in the broken phase could test the two-loop gauge-independence claim independently.

Original abstract

The high-temperature limit of the 2-loop effective potential for the Higgs field is calculated from an effective 3d theory, in a general covariant gauge. It is shown explicitly that a gauge-independent result can be extracted for the equation of state from the gauge-dependent effective potential. The convergence of perturbation theory is estimated in the broken phase, utilizing the gauge dependence of the effective potential.

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The high-temperature limit of the 2-loop effective potential for the Higgs field is calculated from an effective 3d theory, in a general covariant gauge. It is shown explicitly that a gauge-independent result can be extracted for the equation of state from the gauge-dependent effective potential. The convergence of perturbation theory is estimated in the broken phase, utilizing the gauge dependence of the effective potential.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The high-temperature limit of the 2-loop effective potential for the Higgs field is calculated from an effective 3d theory, in a general covariant gauge.	method 0.67	0.92	0.06	asserted 1.00
s1	It is shown explicitly that a gauge-independent result can be extracted for the equation of state from the gauge-dependent effective potential.	result 0.99	0.92	0.72	asserted 1.00
s2	The convergence of perturbation theory is estimated in the broken phase, utilizing the gauge dependence of the effective potential.	contribution 0.52	0.88	0.12	asserted 0.91

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

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Figures

**Figure 1.** The non-vanishing 2-loop graphs. Dashed line is the scalar propagator, wiggly line is the Aa i -propagator, solid line is the Aa 0-propagator, and double line is the ghost propagator. The notations are the same as in ref. [3]. 1

**Figure 2.** The general structure of the connected reducible diagrams that are not included in the calculation of the e ective potential, but must be included, if one is calculating the partition function in the loop expansion without using the e ective potential. Only Higgs particles can propagate on the straight line

Figures reproduced from arXiv:hep-ph/9411252 by the authors.

References

Works cited (17)

[3]

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

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P. Arnold and O. Espinosa, Phys. Rev. D 47 (1993) 3546

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Z. Fodor and A. Hebecker, DESY-94-025, hep-ph 9403219

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[4]

R. Jackiw, Phys. Rev. D 9 (1974) 1686

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[5]

N.K. Nielsen, Nucl. Phys. B 101 (1975) 173

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[6]

R. Fukuda and T. Kugo, Phys. Rev. D 13 (1976) 3469

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[7]

G.A. Vilkovisky, Nucl. Phys. B 234 (1984) 125

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W. Buchm¨ uller, Z. Fodor and A. Hebecker, Phys. Lett. B 331 (1 994) 131

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[9]

M. Laine, Phys. Lett. B 335 (1994) 173

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[10]

T. Appelquist and R. Pisarski, Phys. Rev. D 23 (1981) 2305

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S. Nadkarni, Phys. Rev. D 27 (1983) 917

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A. Jakov´ ac and A. Patk´ os, Phys. Lett. B 334 (1994) 391

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[13]

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

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A. Jakov´ ac, K. Kajantie and A. Patk´ os, Phys. Rev. D 49 (19 94) 6810

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P. Arnold, Phys. Rev. D 46 (1992) 2628

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[16]

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

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[17]

C. Ford, D.R.T. Jones, P.W. Stephenson and M.B. Einhorn, Nucl. Phys. B 395 (1993) 17. 9 Appendix We present here the 2-loop eﬀective potential of the theory deﬁn ed by eqs. 3 and 4. The tree-level part is V0(ϕ) = 12 m23ϕ2 + 14 λ3ϕ4 . (22) With the functions R± deﬁned in eq. 6, the 1-loop contribution to the eﬀective potential is V1(ϕ) = − 112π [6m3T + 3m3...

View on Pith Pith review arXiv 1993

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How the literature uses it

36 arXiv papers cite this in 98 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 69 sentences without a confident label

Contests it 2 papers

The next two orders in the expansion are [ We remind the reader that prime is used to denote the temperature derivative, $V_0' ≡d V_0/dT$. ] [equation] Similarly, expanding the latent heat [equation] where [equation] We emphasize that in consistent EFT expansions we do not encounter the problem reported in [ref] , whereby the condition $Δp_0(T_0) = 0$ leads to a vanishing mass parameter $μ^2_3(T_0) = 0$ which then results in spurious IR divergences at two-loop order for $T_2$.

Perturbative effective field theory expansions for cosmological phase transitions · Strict perturbative expansions · contest 0.73
Thus, it appears to us that the breakdown of the $ħ$ expansion as obtained in Ref. [ref] is a consequence of expanding $T_C$ itself.

Baryon Washout, Electroweak Phase Transition, and Perturbation Theory · High-$T$ effective theory · contest 0.79

Builds on it 1 paper

Expanding the potential and its minima as [equation] and generalizing the analysis of [ref] to the case of two background fields gives the $ħ$ expansion" [equation] All derivatives are to be evaluated at the tree-level minimum $(v_0, x_0)$.

Thermodynamics of a two-step electroweak phase transition · Gauge-invariant effective potential to two loops · 0.95

Uses its method or tool 5 papers

When determining the critical temperature ($$) and other thermodynamic quantities of interest, we resort to a technique described in [ref] rather than expanding $$ explicitly in $ħ$ as in [ref] .

Combining thermal resummation and gauge invariance for electroweak phase transition · Introduction · 0.92
Evolution of the different minima is tracked using the gauge-invariant approach described in [ref] ; details of the calculation can be found in the Appendix.

Thermodynamics of a two-step electroweak phase transition · Effective theory at high temperature · 1.00
The extremal conditions [equation] can be solved either with a numerical minimization algorithm, or perturbatively by expanding around leading order minima [ref] (sometimes called the “$ħ$ expansion”).

Investigating two-loop effects for first-order electroweak phase transitions · Thermal effective potential · 0.98

Compares with it 1 paper

Inclusion of the $𝒪(ħ)$ terms yields a contribution $T_1$ proportional to the product[In the notation of [ref] these are denoted $h_3$ and $m_D$, respectively.] of $H^ab_ij$ and $Π^ab(T)$.

Baryon Washout, Electroweak Phase Transition, and Perturbation Theory · High-$T$ effective theory · 0.86

Cites it as support 6 papers

Notably, this expansion is gauge invariant order by order [ref] , since it satisfies the Nielsen-Fukuda-Kugo identities within the EFT [ref] , and the construction of the EFT through dimensional reduction is gauge invariant [ref] .

Perturbative effective field theory expansions for cosmological phase transitions · Strict perturbative expansions · 0.98
This would lead to an IR divergence for the singlet condensate already at two-loop level [ref] (also cf. [ref] ), and further signals that interpreting this point as the physical critical temperature is incorrect.

Combining thermal resummation and gauge invariance for electroweak phase transition · Critical temperature · 0.97
The focus on small gauge parameters is also supported by various observations in specific models, in which gauge-invariant analyses appear to agree well with the Landau gauge [ref] .

Theoretical uncertainties for cosmological first-order phase transitions · Gauge dependence · 1.00

Cites it as background 8 papers

This has been discussed in Refs. [ref] .

Theoretical uncertainties for cosmological first-order phase transitions · Dimensionally-reduced recipe for thermodynamics · 0.96
Concerns were raised that the direct minimisation method led to gauge-dependent results [ref] , while strict $ħ$-expansions appeared to lead to IR divergences [ref] .

Perturbative effective field theory expansions for cosmological phase transitions · Discussion · 0.84
Following earlier studies [ref] , Ref. [ref] addressed the possibility of producing a consistent, order-by-order gauge-invariant result in the perturbative expansion of the $V_eff$ and $S_eff$.

Phase Transitions and Gauge Artifacts in an Abelian Higgs Plus Singlet Model · Introduction · 0.81

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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