Complex Effective Potentials and Critical Bubbles*
한글 요약:
1. 1반전 에이전트의 복소해가 나타나는 조건 하에서, 힙스 필드의 1반전 에이전트를 계산할 때 두 가지 방법(method A, B)이 등장한다. method A는 에이전트의 실제 물리량을 얻기 위해 실수부만 고려하는 방법이며, method B는 에이전트의 복소해가 나타나는 조건 하에서, 적절한 정의域으로 수정하여 계산하는 방법이다.
2. 이 두 방법은 일련의 조건 하에서 다른 결과를 나타낸다. method A는 실제 물리량을 얻기 위한 가장 간단한 방법이지만, method B는 복소해가 나타나는 조건 하에서는 더 정확한 결과를 제공한다.
3. 에이전트의 실제 물리량을 구하기 위한 가장 적합한 방법은 method B인 것으로 판명되었다. 이 방법은 에이전트의 복소해가 나타나는 조건 하에서 적절한 정의域으로 수정하여 계산하는 것을 포함한다.
4. 이러한 연구는 힙스 메커니즘에서 복소해가 나타나는 현상을 연구하는데 도움이 될 것이다.
Complex Effective Potentials and Critical Bubbles*
arXiv:hep-ph/9206243v1 23 Jun 1992CALT-68-1797DOE RESEARCH ANDDEVELOPMENT REPORTComplex Effective Potentials and Critical Bubbles*David E. Brahm1California Institute of Technology, Pasadena, CA 91125AbstractThe Higgs contribution to the effective potential appears to be complex. Howdo we interpret this, and how should we modify the calculation to calculate physicalquantities such as the critical bubble free energy?June, 1992*To appear in Proceedings, Yale Workshop on Baryon Number Violation at the Electroweak Scale, March1992 (World Scientific).
Work supported in part by the U.S. Dept. of Energy under Contract No.
DEAC-03-81ER40050.1Email: brahm@theory3.caltech.edu
1. A Toy Model with a Complex Effective PotentialSuppose we wish to calculate the 1-loop finite-temperature effective potential V for atheory with a single scalar (I’ll call it the Higgs), whose tree potential V0 (Fig.
1) is of theformV0(φ) = µ22σ2 φ2(φ −σ)2 −ǫφ2σ2(1)The effective Higgs mass (at zero external momen-tum) ism2(φ, T) = V ′′0 (φ) + µ22σ2 T 2(2)The last term of eq. (2) is the Higgs’s “self-plasma-mass” (SPM); let us choose our parameters to makethe SPM small, so m2 < 0 over roughly (1−1/√3)/2 < φ/σ < (1+1/√3)/2.The 1-loop contribution of the Higgs to V can be calculated from the vacuum-to-vacuumgraph of Fig.
2a:V = V0 + V1 + T 42π2 I(m/T) ≈V0 + T 224 (V ′′0 ) −T12π (V ′′0 )3/2 + · · ·(3)or from the tadpole graph of Fig. 2b:V ′ = V ′0 + V ′1 + (V ′′′0 )T 224 F(m/T) ≈V0 + T 224 (V ′′′0 ) −T8π (V ′′′0 )(V ′′0 )1/2 + · · ·(4)where V1 is the T-independent 1-loop resultV1 =14π2Zdk k2√k2 + m2 = m464π2lnm2Λ2−32(5)and (writing x = k/T, y = m/T)I(y) ≡Z ∞0dx x2 ln1 −e−√x2+y2,F(y) ≡6I′(y)/(π2y)(6)I have expanded in small m/T.
Eqs. 3 and 4 give identicalresults.For m2 < 0 the potential appears to be complex.
Howare we to interpret the imaginary part of the potential? Howshould we modify V to get a real quantity to plot and use incalculations?
The naive answer, which I’ll call Method A,is simply to take the real part of V . Several much fanciermethods can be found in the literature[1].1
2. Relation to the Standard ModelThe toy model can, with minor modifications, represent the Standard Model after inte-grating out gauge bosons and fermions.
Now {µ, σ, ǫ} depend on T (and are simply relatedto the usual {λT, E, D}[2]). The contribution of the gauge bosons and fermions to the Higgsplasma mass is still given correctly by eq.
(2), as can be verified by direct calculation of Feyn-man diagrams. Goldstone bosons double the SPM of eq.
(2), and themselves have a squaredmass m2χ = V ′0/φ + SPM which becomes negative over 12 < φ/σ < 1 (for small SPM). Thetadpole calculation eq.
(4) must be used, using the 3-Higgs coupling of the original theory(6λφ for a λφ4/4 theory) in place of (V ′′′0 ), to avoid overcounting diagrams. None of thesemodifications seem relevant to the questions about imaginary parts.3.
Homogeneous and Inhomogeneous FieldsAt T = 0, Weinberg and Wu showed that the imaginary part represents the rate of decayof an unstable homogeneous field configuration to an inhomogeneous state[3]. Whether thisis true at finite T remains to be shown.For calculating percolation rates, however, we are more often interested in the freeenergy Ec of the critical bubble, an extremal configuration stable against any fluctuation inφ(x) except overall growth or shrinkage (the “breathing mode”):Ec =Zd3x"V (φ(x)) + 12(∇φ)2 + ATm3dm2dφ ∇φ2+ BTm9dm2dφ ∇φ4+ · · ·#(7)Here A, B · · · come from derivative corrections to the action[4,5].4.
Im{V }Im{V }Im{V } Does Not Represent Bubble Growth/ShrinkageOne might suppose that the contribution of Im{V } to Ec represents the instabilityof the breathing mode. We can disprove this hypothesis by examining a thin-wall bubble[ǫ ≪µ2σ2/4 in eq.
(1)], for which[6]R = 2S1ǫ ,S1 =Zdφ√2V = 2µσ29√3 ,δ = 1/µ(8)where R is the bubble radius and δ is its thickness. The contribution of Im{V } to Ec is∼R2, since V is only complex in the bubble wall, and the wall profile is independent of ǫ forǫ→0.
The breathing mode imaginary contribution to Ec, on the other hand, is independentof R, as can be seen by calculating[5,7,8]Ec = E0 +Xn,l(2l + 1)hωn,l2+ T ln1 −e−ωn,l/T i(9)2
where E0 is the tree-level energy, and ω2n,l is the eigenvalue of [−∇2 + V ′′0 (φ(x))] whoseeigenfunction has n radial nodes and angular dependence Y ml (θ, φ). Eq.
9 is just the standardthermodynamic result for the free energy of a system of harmonic oscillators. The radialpart χ(r)/r of the eigenfunction satisfies−d2dr2 + l(l + 1)r2+ V ′′0 (φ(r)) −ω2n,lχn,l(r) = 0(10)and we see[6] that states bound to the wall approximately satisfyω2n,l = ω2n,0 + l(l + 1)R2(11)The breathing mode eigenvalue ω20,0 can thus be found from the translational mode eigenvalueω20,1 = 0:ω20,0 = −2R2 ,[Ec]0,0 ≈i√2R + iπT2+ T ln √2RT!
(12)One can argue[8] that eq. (9) breaks down for unstable fluctuations (inverted harmonic os-cillators[9]), but even so it does not appear that Im{Ec} grows as R2.
Thus the contributionof Im{V } to Ec must be canceled by the derivative corrections of eq. (7).Such cancellation is plausible, since odd powers of m in the derivative expansion givecomplex terms, and a similar cancellation is known to occur to restore gauge invariance[10,11].However, the divergences as m→0 get increasingly worse, so this expansion seems inappro-priate for finding Im{Ec}.5.
Removal of Long-Wavelength ModesThe integral in eq. (6) comes from a sum over Fourier modes of Higgs field fluctuations,and the integrand is only complex for long wavelength modes (x < |y|, or k < |m|).
Atthe hump (φ = σ/2) for instance, m2 = −µ2/2, so only modes of wavelength λ > 2√2π/µcontribute to Im{V }. This is several times the bubble wall thickness[12] δ = 1/µ.This suggests Method B[11] for altering the calculation of V , namely changing the lowerlimit of integration in eq.
(6) to Im{y}. Several schemes discussed in ref.
[1] are in a similarspirit. Note that Methods A and B are equivalent for the T-independent part eq.
(5), butnot for eq. (6), since the integrand of the latter in the region 0 < x < Im{y} is complex, notpure imaginary.3
6. What’s the Best Method?To decide on a “best” method of calculating physical quantities from a complex V , wemust decide what “best” means.
It could mean that when we put our modified V into eq. (7)and set A = B = 0, we reproduce the correct Ec.
Alternately, it could mean the methodby which eq. (4) gives the correct new degenerate minimum of V , as determined by eq.
(3)(Method A satisfies this criterion).We usually bury our heads in the sand at this point, claiming the Higgs sector contri-bution to V is small in the Standard Model anyway. As experimental limits on the Higgsmass creep upward, however, it becomes increasingly important to address these questions.References:1.
A. Okopinska, Phys. Rev.
D36 (1987) 2415, & refs. therein;I. Roditi, Phys.
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G.W. Anderson & L.J.
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Hsu, CALT-68-1705/HUTP-91-A063 (Dec. 1991);C.G. Boyd et al., CALT-68-1795/HUTP-92-A027/EFI-92-22 (June 1992).12.
G. Anderson, private communication.4
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