Tricritical Point in Quantum Phase Transitions of the Coleman-Weinberg Model at Higgs Mass

Miguel C N Fiolhais∗ LIP, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal

Hagen Kleinert† Freie UniversitätBerlin, Institut fürTheoretische Physik Arnimallee 14, D-14195 Berlin, Germany (Dated: June 28, 2013) Abstract The tricritical point, which separates first and second order phase transitions in three-dimensional superconductors, is studied in the four-dimensional Coleman-Weinberg model, and the similarities as well as the differences with respect to the three-dimensional result are exhibited. The position of the tricritical point in the Coleman-Weinberg model is derived and found to be in agreement with the Thomas-Fermi approximation in the three-dimensional Ginzburg-Landau theory. From this we deduce a special role of the tricritical point for the Standard Model Higgs sector in the scope of the latest experimental results, which suggests the unexpected relevance of tricritical behavior in the electroweak interactions.

The following paper is published in Phys. Lett. A: http://www.sciencedirect.com/science/article/pii/S0375960113005768

INTRODUCTION this way, the GL theory has become what may be called the Standard Model of superconductive phenomenology. Ever since the formulation in 1964 of the electroweak A similar Ginzburg-Landau-like scalar field theory spontaneous symmetry breaking mechanism [1–5] to ex- with quartic interaction is successful in unifying the weak plain how elementary particles acquire mass, supercon- and the electromagnetic interactions, so that it has be- ductivity and high-energy physics became intimately con- come the Standard Model of particle physics, also called nected. For instance, the Ginzburg-Landau (GL) the- the Higgs Model. ory [1], proposed in 1950 to provide a local macroscopic An important new aspect that arises at the transi- description of superconductivity, makes use of a quar- tion from the three-dimensional GL theory to the (3+1)- tic potential of the same type that reappeared in the dimensional scalar electrodynamical Higgs model is that Higgs model1. In fact, the GL theory is the three- the field possess canonical commutation rules. These call dimensional predecessor of what is now called (3+1)- for the existence of a particle associated with each field. dimensional scalar quantum electrodynamics, that was After all, this is the logic which led to the discovery of studied in detail by Coleman and Weinberg [6, 7]. Be- pions as the quantum of the forces of nuclear physics. In fore the work of Ginzburg and Landau, the London the- particle physics, it induced an intensive search for a Higgs ory explained the existence of a finite penetration depth particle for many years. The recent discovery of a new of magnetic fields into a superconductor, the Meissner- signal in the 124-126 GeV mass region by the ATLAS Ochsenfeld effect [8]. GL extended this theory by a local and CMS collaborations at the Large Hadron Collider complex scalar order field φ(x), whose gradient terms in [10, 11] is a hopeful candidate for such a particle. the energy density produces a finite length scale of fluc- In this Letter we want to put this mass value into con- tuations of the order field, the so-called coherence length text with a known fact in superconductivity, that the su- ξ. In their theory, the Meissner-Ochsenfeld effect was perconductive phase transition may occur in two different arXiv:1306.6568v1 [hep-ph] 27 Jun 2013 explained by a local mass term of the vector potential, orders: a second order if the GL parameter κ lies deep in whose size is proportional to |φ|2. the type-II regime, and a first order in the type-I regime. The GL theory possesses two length scales, the London For a long time, this issue was a matter of theoretical con- penetration depth λ , and the coherence length ξ. The troversy after it had been argued by Halperin, Lubensky, L and Ma (HLM) [12] that superconductors should really competition√ between the two is ruled by the GL parameter κ ≡ λ / 2ξ. This serves to distinguish two types of arise in a first-order transition. The issue was finally set- L √ tled by the calculation of a tricritical point near the divid- superconductors,√ type-I with κ > 1/ 2 and type-II with κ < 1/ 2. The second type, possess bundles of vortices ing line between type-II to type-I superconductivity. The approximate value of κ where this happens was predicted which confine magnetic flux in tubes of radius λL [9]. In √ to be κtr ≈ 0.81/ 2 [13–15], a value later confirmed√ by Monte Carlo simulations to lie at κ = 0.76/ 2 ± 0.04 [16]. The important point in the theory was that the 1 With the short name “Higgs” we abbreviate collectively all au- mass term of the electromagnetic potential was reliable thors that contributed from [1–5]. only as long as it was big, which is the case in the type-I

1 regime. If it is small, the mass is destroyed by fluctuating τ drops below zero, the ground state of the potential, 1 2 g 4 vortex lines [17, 18]. The precise position of the tricriti- V (ψ) = 2 τ|ψ| + 4 |ψ| , is obtained for an infinite number cal point is unknown and should be determined by Monte of degenerate states satisfying: Carlo simulations as described in [19]. τ The calculation of HLM had an interesting parallel hψi2 = ρ2 = − , (2) 0 g in (3+1)-dimensional scalar QED, where Coleman and Weinberg2 calculated that a massless field would acquire and corresponds to a second-order phase transition. Af- a mass from the fluctuations of the electromagnetic field. ter the spontaneous symmetry breaking, i.e. fixing the In the language of superconductive, this implies that gauge to θ(x) = 0, the Hamiltonian becomes, scalar QED has a first-order phase transition3. After 2 2 the calculation of the tricritical point in superconductive 1 2 e ρ H(ψ, ∇ψ, A, ∇A) = (∇ρ) + V (ρ) + A2 it was proposed that a similar tricritical point should 2 2 come up in (3+1)-dimensional QED [20]. The Coleman- 1 2 + (∇ × A) . (3) Weinberg result was derived without considering the fluc- 2 tuating vortex sheets which are the (3+1)-dimensional The mass term of the vector field, m = eν, which ap- analogs of the vortex lines in superconductors. These A peared with the spontaneous symmetry breaking, and should modify the CW-result in the small e2 regime. One the scalar field mass term, can be associated with two should therefore expect a tricritical value of κ also in characteristic lengths of a superconductor, the London (3+1)-dimensional scalar QED, and the present Letter penetration length, λ = 1/m = 1/eρ , and the coher- gives further support for this expectation with experi- √ L A 0 ence length, ξ = 1/ −2τ, respectively. mental consequences. Moreover, the tricritical point is The first-order phase transition can be achieved in the predicted and interpreted in the Standard Model as the Ginzburg-Landau theory by considering quantum correc- absolute stability boundary of the Higgs potential, by tions, which in the Thomas-Fermi approximation [21], analogy with superconductivity. The latest theoretical neglecting fluctuations in ρ, leads to an additional cubic predictions on the meta-stability and instability bound- term in the potential, aries up to the Planck scale of the Standard Model Higgs potential are discussed in the context of the recent results 1 g c e3 V (ρ) = τρ2 + ρ4 − ρ3 , c = . (4) on the observed signal at the LHC. 2 4 3 2π As shown in Fig. 1, the cubic term generates a second 2 QUARTIC INTERACTION AND TRICRITICAL minimum for τ < c /4g, at the minimum, POINT ! c r 4τg ρ˜0 = 1 + 1 − . (5) The Ginzburg-Landau theory of superconductivity is 2g c2 characterized by the following energy density: 2 At the specific point τ1 = 2c /9g, the minimum lies at 1 the same level as the origin for ρ = 2c/3g, where the H(ψ, ∇ψ, A, ∇A) = (∇ + ieA) ψ∗ (∇ − ieA) ψ 1 2 phase transition becomes of first-order (tricritical point). τ g + |ψ|2 + |ψ|4 Therefore, in this point, the coherence length of the ρ- 2 4 field fluctuations becomes, 1 + (∇ × A)2 , (1) 2 1 3rg ξ1 = = , (6) pτ + 3gρ2 − 2cρ c 2 with the order parameter ψ(x) = ρ(x)eiθ(x), where ρ(x) 1 1 and θ(x) are real fields. The vector field is represented by which is the same as the fluctuations around ρ = 0. Fi- A, e is the electric charge of the Cooper pairs4, and the nally, the Ginzburg parameter at the tricritical point is, real constants τ and g give the strength of the quadratic and quartic terms, respectively. If the mass parameter 1r g κ = . (7) 2 e2

2 Their work was done almost simultaneously with [12] on the same ﬂoor at Harvard University. TRICRITICAL POINT IN THE COLEMAN- 3 On a hiking excursion into the mountains near Geneva with Sid WEINBERG MODEL Coleman, H.K. once asked him whether this was really what they proved, he said “yes, but we foolishly did not put it that way”. 4 The Euler number is represented by e, and shall not be confused The intriguing question now is how this result changes with the electric charge e. in the four-dimensional version of the Ginzburg-Landau

2 Ginzburg parameter becomes, r 1 1/ehφ i 1 λ κ = √ c = √ . (12) 2 1/m(φ) 2α e2

The result has the same form as the previously obtained 3-dimensional result, and becomes the same with an appropriate choice of the renormalization scale. Even though these results were computed using only the stability boundary of the corrected quartic potential, without making any use of the dual disorder field theory [13, 21], FIG. 1: Potential for the order parameter ρ with cubic term. the position of the tricritical point does not change from 3 A new minimum develops around ρ1 causing a first-order transition for τ = τ1. to 3+1 dimensions, thus justifying the applicability of the Thomas-Fermi approximation in the tricritical regime. For the Standard Model Higgs potential, the relation between the boundary of absolute stability and the tri- theory, the Coleman-Weinberg model. The effective po- critical point will be discussed in the next section. tential of the Coleman-Weinberg model at one-loop level is [7], HIGGS BOSON MASS AND VACUUM STABILITY 1 λ 3e4 φ2 25 V (φ ) = m2φ2 + φ4 + φ4 log c − , (8) c 2 c 4 c 64π2 c M 2 6 On 4th July 2012, the CMS and ATLAS experiments where the corrected scalar (spin-0) field is represented by announced the discovery of a new boson, compatible 2 φc(x), with a mass term m . Here λ gives the strength with the SM Higgs boson, with global statistical sig- of the quartic term, and M is the value of φc at which nificances of 5.8 sigma (CMS) and 5.9 sigma (ATLAS). the renormalizations are done. Note that we assumed The observed signal currently lies at 125.3 ± 0.4 (stat.) ± λ to be of the same order of e4, and therefore, in the 0.5 (sys.) GeV (CMS) and 126±0.4 (stat.)±0.4 (sys.) GeV one-loop approximation, the scalar loop diagrams were (ATLAS), and no significant deviations from the pre- neglected, since they are of the same order of magnitude dicted SM Higgs boson properties were observed to the as the diagrams with two photon loops. For convenience, present date. a new variable µ can be defined as, Assuming the Standard Model to be valid up to the Planck scale, the Higgs potential develops a new local λ 3e4 M 2 11 = log + , (9) minimum for a positive value of the running quartic cou- 4 64π2 µ2 3 pling with the renormalization scale. However, for a neg- ative quartic coupling, the potential becomes unbounded which turns the effective potential into, from below and, therefore, unstable. Thus the abso- 4 2 lute stability of the electroweak vacuum has its bound- 1 2 2 3e 4 φc 1 V (φc) = m φ + φ log − . (10) ary where the quartic coupling flips sign. This feature 2 c 64π2 c µ2 2 of the SM Higgs potential corresponds precisely to the As described in [7], for a positive m2, the effec- previously discussed tricritical point in a quartic interac- tive potential has a maximum and minimum, for tion, which separates first and second order phase tran- m2 < 3e4µ2e−1/16π2. In particular, the minimum of sitions in superconductors. The phenomenology associ- the potential lies at the same level as the origin if ated with the two physical situations is, of course, quite 2 4 2 −1/2 2 2 2 −1/2 different. While in superconductivity, the spontaneous m = 3e µ e /32π , for hφci = µ e . The mass of the scalar field in the tricritical point is, therefore, symmetry breaking appears as a result of the radiative corrections overcoming the effect of a positive mass term 2 2 ∂ V in the Coleman-Weinberg model, the SM Higgs potential m (φc) = ∂φ2 is characterized by the existence of two non-zero vacua. c φc=hφci Nevertheless, it is clear that the vanishing of the quartic e4hφ i2 hφ i2 2 c c coupling and the degeneracy of the vacuum states corre- = m + 2 6 + 9 log 2 16π µ spond to a tricritical behavior in the two scenarios. 3e4µ2 λ = e−1/2 = hφ i2, (11) The determination of the SM vacuum stability has 16π2 α c been studied in detail in the past two decades [22]. The latest and most precise next-to-next-to-leading-order M 2 11 where α = log µ2 + 3 gives the size of the renormal- (NNLO) prediction of the absolute stability boundary ization scale. Consequently, at the tricritical point, the was established by Degrassi et al. [23], using two-loop

3 renormalization-group equations, one-loop threshold cor- effect of such operators has been studied in the past [27] rections at the electroweak scale (possibly improved with and shown to have a significant influence on the stabil- two-loop terms in the case of pure QCD corrections), and ity and triviality of the Higgs potential. For instance, one-loop improved effective potential. Assuming a top new physics contributions at an energy scale of a few quark mass of mt = 173.1 ± 0.7 GeV [24], and the strong TeV could be enough to ensure the stability of the elec- coupling constant at αs(MZ ) = 0.1184 ± 0.0007 [25], the troweak vacuum. Perhaps the quest for anomalous con- absolute stability boundary up to the Planck scale was tributions to the top quark and Higgs boson SM cou- predicted for a Higgs boson mass, plings at the LHC may bring us interesting surprises in the years ahead [28]. mH [GeV] = 129.4 m [GeV] − 173.1 ± 1.4 t 0.7 SUMMARY α (M ) − 0.1184 ± 0.5 s Z 0.0007 In this Letter, we argue that the tricritical point, obtained for the three-dimensional Ginzburg-Landau the- ± 1.0 (theoretical) . (13) ory with the help of a duality transformation to a disorder By combining in quadrature the theoretical and experi- version, is not expected to significantly change when an- alyzed in the context of the (3+1)-dimensional Coleman- mental uncertainties, the result becomes mH = 129.4 ± 1.8 GeV, distanced by roughly 2 sigma from the LHC re- Weinberg model. This leads us to conclude that the ab- sults. Therefore, one cannot state that the Higgs boson solute stability boundary of the Higgs potential is a tri- lies precisely on the tricritical point of the electroweak critical point of the electroweak interaction, by analogy interactions, nor exclude that possibility. The allowed with superconductivity. The recently obtained result on regions, up to 3 sigma, on the top quark and Higgs bo- the NNLO prediction of the absolute stability bound- son masses measurements, seem to indicate a significant ary, up to the Planck scale, at a Higgs boson mass of preference for meta-stability of the SM potential when mH ≈ 129.4 ± 1.8 GeV, compatible with the observed compared with the latest experimental results from the signals at LHC in the 124-126 GeV mass region, sug- Tevatron and LHC. This tells us there is a non-zero prob- gests that the electroweak interactions make use of the ability of quantum tunneling into the global minimum, tricritical behavior as its natural working point. To val- lying deeper than the electroweak vacuum. As the new idate this statement, we must wait for a greater preci- vacuum appears at a very high-energy scale, the prob- sion of the experimental measurements and theoretical ability of tunneling is very small, with a mean lifetime predictions. Finally, and more strategically, this inter- larger than the age of the Universe. Nonetheless, the ab- pretation may enhance the bridge between the physics solute stability boundary strongly depends on the Higgs of elementary particles and superconductivity, that has boson and top quark masses: slight variations may have led to many important insights since Nambu’s pioneering dramatic implications. 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