Naturalness of effective theories in Wilsonian approach
NNaturalness of effective theories in Wilsonianapproach
T. Krajewski ∗ University of WarsawE-mail:
Z. Lalak
University of WarsawE-mail:
We have computed Wilsonian effective action in a simple model containing scalar field withquartic self-coupling which interacts via Yukawa coupling with a Dirac fermion. The model isinvariant under a chiral parity operation, which can be spontaneously broken by a vev of thescalar field. We have computed explicitly Wilsonian running of relevant parameters which makesit possible to discuss in a consistent manner the issue of fine-tuning and stability of the scalarpotential. This has been compared with the typical picture based on Gell-Mann–Low running.Since Wilsonian running includes automatically integration out of heavy degrees of freedom, therunning differs markedly from the Gell-Mann–Low version. However, similar behavior can beobserved: scalar mass squared parameter and the quartic coupling can change sign from a positiveto a negative one due to running which causes spontaneous symmetry breaking or an instability inthe renormalizable part of the potential for a given range of scales. As for the issue of fine-tuning,since in the Wilsonian approach power-law terms are not subtracted, one can clearly observe thequadratic sensitivity of fine-tuning measure to the change of the cut-off scale. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ a r X i v : . [ h e p - ph ] O c t aturalness in Wilsonian approach T. Krajewski
1. Introduction
The recent discovery of the Higgs boson at the Large Hadron Collider [1, 2], promotes thequestion about the protection of the electroweak breaking scale to one of the most puzzling prob-lems of fundamental physics. The observed compatibility of properties of the newly observed par-ticle, with predictions coming from the Standard Model, additionally strengthens tension betweenthe standard theoretical reasoning which results in a prediction of new physics near the electroweakscale and reality. This situation strengthens the need of revisiting the naturalness principle.Numerous authors [3, 4, 5, 6, 7] propose new definitions of the naturalness. Our goal is lessambitious. We shall try to state clearly a treatment of the fine-tuning based on the Wilsonianeffective action and the corresponding Wilsonian renormalization group. Idea of the Wilsonianeffective action is close to the intuitive understanding of the cut-off regularization. In a standarddiscussion based on quadratic divergences, the artificial meaning of the scale of an effective theoryis given to the regularization parameter Λ . This effects in the regularization dependence of thiskind of analysis. On the contrary, in the Wilsonian method high energy modes are integrated outin a self-consistent, regularization-independent way, and an effective theory has a well-definedeffective action. Moreover, this treatment is universal and depends very weakly on a preferred UVcompletion. The main impact on the effective action from states with masses greater than the scaleof the effective theory can be parametrized by the values of couplings of the Wilsonian effectiveaction. Further corrections are highly suppressed as far as heavy masses are separated from thescale of the effective theory.Given a model where the vacuum expectation value of a scalar field can be generated withquantum corrections, we can also show how the stability of the effective action looks like from thepoint of view of the Wilsonian running. This has been compared with the standard picture basedon the Gell-Mann–Low running. Since the Wilsonian running automatically includes integration ofheavy degrees of freedom, the running differs markedly from the Gell-Mann–Low version. Nev-ertheless, similar behaviour can be observed: the scalar mass-squared parameter and the quarticcoupling can change sign from a positive to a negative one due to running. This causes the spon-taneous symmetry breaking or the instability in the renormalizable part of the potential for a givenrange of scales. However, care must be taken when drawing conclusions, because of the truncationof higher dimension operators. The Gell-Mann–Low running allows one to resume relatively easilya class of operators corresponding to large logarithms to form the RGE improved effective potentialvalid over a huge range of scales. In the Wilsonian approach this would correspond to followingthe running of a large number of irrelevant operators, which is technically problematic.While the simple cut-off analysis of scalar field models has been performed earlier, the goalof the present note is to consistently use the Wilsonian approach, and to make a clear comparisonwith the discussion based on the Gell-Mann-Low running.
2. Basic features of the model
For the sake of clarity we consider a simple model that exhibits certain interesting features ofthe SM. The model consists of a massless Dirac fermion ψ which couples via a Yukawa interaction2 aturalness in Wilsonian approach T. Krajewski to a real scalar field φ with a quartic self-coupling. This Lagrangian takes the form: L = i ψ / ∂ ψ + ∂ µ φ ∂ µ φ − M φ − Y φ ψψ − λ φ . (2.1)The above Lagrangian is symmetric under the (chiral) Z which acts on φ as φ → − φ and on ψ as ψ → i γ ψ . We consider the case of the non-zero vacuum expectation value for the field φ , whichbreaks this symmetry spontaneously. In the broken symmetry phase the Lagrangian density (2.1)expanded around the nontrivial minimum v ( φ = v + ϕ ) will take the form: L = i ψ / ∂ ψ − m ψψ + ∂ µ ϕ∂ µ ϕ − M (cid:48) ϕ − Y ϕψψ − g ϕ − λ ϕ . (2.2)The fermion ψ allows one to model the top quark coupling to the Higgs boson, which is knownto give the main contribution to quadratic divergencies in the mass of the SM scalar and to thehigh-scale instability of the quartic coupling.This model was previously investigated with methods of the FRG in [8, 9, 10], in order toestimate the non-perturbative bound on the Higgs boson mass. The same issue was discussed in[11], with a slightly different Lagrangian. In [12] the stability of potential was discussed with thehelp of the naive cut-off procedure. We have calculated the Wilsonian renormalization group equations at the lowest non-trivialorder. The Wilsonian action can include an infinite number of non-renormalizable operators, how-ever they are suppressed at the low cut-off scale. Hence our truncation in the symmetric phasecontains the following operators: L Λ = i ψ Λ / ∂ ψ Λ + ∂ µ φ Λ ∂ µ φ Λ − M Λ φ Λ − Y Λ φ Λ ψ Λ ψ Λ − λ Λ φ Λ . (2.3)In the ordered phase it is convenient to use fluctuations ϕ Λ around the vacuum expectation value v Λ . In such a case terms generated by expansion φ Λ = v Λ + ϕ Λ must be included and our truncationtakes the form: L Λ = i ψ Λ / ∂ ψ Λ − m Λ ψ Λ ψ Λ + ∂ µ ϕ Λ ∂ µ ϕ Λ − M Λ ϕ Λ − Y Λ ϕ Λ ψ Λ ψ Λ − g Λ ϕ Λ − λ Λ ϕ Λ . (2.4)Wilson coefficients M Λ , Y Λ , λ Λ as well as m Λ and g Λ are defined in [13].Chosen truncation corresponds to the lowest non-trivial order of the Wilsonian running (in thesimilar way as a 1-loop RGEs corresponds to lowest non-trivial order of the Gell-Mann–Low typerunning).
3. Flow equations
It is convenient to express the Wilsonian RGE in terms of dimensionless parameters, becausethen Wilsonian RGEs are a dynamical system of differential equations. We use the dimensionless3 aturalness in Wilsonian approach
T. Krajewski parameters ν Λ : = v Λ Λ , Ω Λ : = M Λ Λ , ω Λ : = m Λ Λ and γ Λ : = g Λ Λ . We define v Λ by the requirement thatthe shift ϕ Λ (cid:55)→ φ Λ − v Λ gives the effective action without terms with odd powers of φ Λ .Wilsonian β -functions for Ω Λ and λ Λ in the ordered phase read as follows: Λ d Ω Λ d Λ = − Ω Λ + Y Λ ( π ) (cid:34) (cid:0) − ω Λ (cid:1) (cid:0) + ω Λ (cid:1) ( ω Λ + ) Ω Λ − ω Λ − ( ω Λ + ) (cid:35) + λ Λ ( π ) ( Ω Λ + ) + γ Λ ( π ) (cid:20)
23 11 + Ω Λ −
23 1 ( + Ω Λ ) + Ω Λ ( + Ω Λ ) (cid:21) − γ Λ (cid:20) Y Λ ( π ) ω Λ ( + ω Λ ) − γ Λ ( π ) ( + Ω Λ ) ) (cid:21) , (3.1) Λ d λ Λ d Λ = λ Λ ( π ) ( + Ω Λ ) − Y Λ ( π ) (cid:34) ( + ω Λ ) − ω Λ ( + ω Λ ) (cid:35) + γ Λ ( π ) ( Ω Λ + ) + λ Λ Y Λ ( π ) (cid:0) − ω Λ (cid:1) ( + ω Λ )( + ω Λ ) + γ Λ λ Λ ( π ) (cid:20)
23 1 ( Ω Λ + ) − ( Ω Λ + ) (cid:21) . (3.2)We obtained Wilsonian RGEs for discussed model using methods discussed in [14].
4. Numerical solutions of RGE
The RGEs from Sec. 3 have been solved numerically. We used the 1-loop matching conditionsin order to compute initial conditions for RGE at Λ =
100 (we use units of GeV through the paper),in terms of the measurable quantities (for definition see [13]).The example solution with the values: m ph = M ph = λ ph = . v ph = g ph = . Y ph = Λ . The dotted line represents the Yukawacoupling Y Λ which runs typically rather slowly. The quartic coupling λ Λ runs faster, because ofthe contribution from the fermionic loop. Couplings ω Λ , Ω Λ and γ Λ for low values of Λ runlike relevant couplings due to the rescaling, but after reaching scales of the order of the masses,they change their behaviour to a slow running near constant value. The behaviour of the abovecouplings for high values of Λ is caused by quadratic divergences (or more precisely by the samediagrams which generate quadratic divergences). The same behaviour is manifested by the vacuumexpectation value ν Λ plotted as a solid line.Solutions with different initial conditions have the same qualitative behaviour. in Fig. 2 weplotted families of solutions with the initial conditions M ph =
125 (dashed line) and M ph = Λ . Similar behaviour can be observed, if one changes the values of other parameters. Fig. 2 The standard treatment is such that in the ordered phase one expresses the symmetric Lagrangian density (2.3) bythe field shifted by its vev: φ (cid:55)→ ϕ + v . This shift introduces terms with the odd powers of ϕ to the Lagrangian density.If we want to recover from the Lagrangian density expressed by ϕ , the value of vev by which φ was shifted, we needto search for the shift ϕ (cid:55)→ φ − v such that terms with the odd powers of φ will be absent. Note that we expand arounda minimum, which means that the coefficient of the linear term vanishes. aturalness in Wilsonian approach T. Krajewski
100 1000 10 Y ωγλΩ ν Λ (a)
100 1000 10 - - - Y ωγλΩ ν Λ (b) Figure 1: Examples of a numerical solution of (a)—Wilsonian and (b)—Gell-Mann–Low typeRGEs corresponding to: m ph = M ph = λ ph = . v ph = g ph = . Y ph = . Λ are strongly suppressed.
5. Fine-tuning
The standard measure ∆ c i of the fine-tuning with respect to the variable c i is defined as ∆ c i = ∂ log v ∂ log c i , (5.1)where c i is a coupling in the model and v is the vacuum expectation value of the field which breaksa symmetry spontaneously (here—chiral parity). As a measure of the fine-tuning of the wholemodel we take [15, 16, 17]: ∆ = (cid:32) ∑ i ∆ c i (cid:33) . (5.2)We have computed ∆ c i for parameters of the effective action as functions of the scale Λ .5 aturalness in Wilsonian approach T. Krajewski
100 1000 10 Λ (a)
100 1000 10 Λ (b)
100 1000 10 Λ (c)
100 1000 10 Λ (d) Figure 2: Solutions for the changed initial condition i.e. M ph multiplied (dashed) or divided (dotted)by the factor , compared with the original one (solid line). The flow of ω Λ , Ω Λ , γ Λ and λ Λ ispresented respectively in (a), (b), (c) and (d).Unfortunately, the effective action for Λ = v Λ for Λ = v ), by the value at Λ = − i.e. v − . Weused Λ = − , because this turns out to be the lowest scale which gives ν Λ safety from numericalerrors. To sum up, we have computed the fine-tuning measure (5.1) by taking numerical derivativesof ν − with respect to dimensionless parameters ω Λ , Ω Λ , γ Λ , λ Λ , Y Λ over the range of scales10 < Λ < .In Fig. 3 the measure (5.2) as a function of scale Λ is shown. The power function ∝ Λ p which has been fitted to the fine-tuning curve is shown as a dashed line. The fitted power is equal2 . ± .
02. The power-like function has been fitted over the interval 10 ≤ Λ ≤ (that is aboveassumed mass thresholds). The reason is the visible change of behaviour of the flow of parametersbelow 10 . Wilsonian effective action in the limit Λ → aturalness in Wilsonian approach T. Krajewski
10 100 1000 10 - Λ Figure 3: The combined fine-tuning measure(5.2) as a function of the scale Λ of the Wilso-nian effective action. Fitted power law is givenas a dashed line. ´ ´ ´ ´ ´ ´ - Y λΩ Λ Figure 4: The example of a solution in which theradiative symmetry breaking takes place. Theplot corresponds to the values Y Λ = . M Λ = × , λ Λ = . Λ = .
6. Vacuum stability
An interesting issue is the question of the spontaneous symmetry breaking and the stabilityof the potential seen from the point of view of the Wilsonian approach. The change of a sign of M Λ during the flow toward IR indicates that the stable vacuum of the theory must have the non-zero vacuum expectation value of the scalar field φ (as long as the quartic coupling stays positive).Moreover, the quartic scalar coupling λ Λ can run negative for higher Λ which shows the similarbehaviour as the one observed in the Gell-Mann–Low type running (Fig. 1b), known from the Stan-dard Model. In the context of the SM, the zero of the quartic self-coupling is usually consideredas an indication of the instability of the electroweak vacuum. In the Wilsonian approach however,simple analysis based on the quartic coupling alone is insufficient, because higher dimension op-erators with higher powers of the scalar field φ , which we suppressed in our truncation (2.3), maydominate the scalar potential for large values of φ . The impact coming from higher dimensionoperators was recently investigated in [10, 11, 18] and [19]. To draw strong conclusions one needsa procedure of resummation of the possibly large contributions to the scalar potential coming fromoperators with all higher powers of φ . However, the observed instability of the Wilsonian quarticcoupling may be seen as an indication of a crossover behaviour at higher scales.The example of a solution demonstrating such features is plotted in Fig. 4. For this solutionthe scalar mass parameter Ω Λ vanishes at the scale Λ = . × and the quartic coupling λ Λ has a zero at Λ = . × . While investigating features of this solution one can notice a strongdependence of the scale of the symmetry breaking on the value of the Yukawa coupling Y Λ . Thisfine-tuning problem makes one choose very precisely the initial condition for the Yukawa couplingin order to make the symmetry breaking scale low.The issue of the spontaneous symmetry breaking can be studied with the help of the numericalapproximation of the phase space presented in Fig. 5. Regions in Fig. 5 marked as "ordered phase"are points which used as initial conditions at Λ = , produce the spontaneous breaking of chiralsymmetry during the flow to Λ = − . In Fig. 5a one can see that if for any scale Λ the coupling7 aturalness in Wilsonian approach T. Krajewski ordered phasesymmetric phase λΩ Λ (a) The subspace of the phase spacegiven by the Eq. Y = . symmetric phase ordered phase Y Ω Λ (b) The subspace of the phase spacegiven by the Eq. λ = . Figure 5: The numerical approximation of the phase space of the model given by the Lagrangiandensity (2.1). Points for which the flow from Λ = to Λ = − crosses Ω = Ω Λ will be lower than certain critical value Ω cr , then Ω will run negative in the IR and the chiralparity will be spontaneously broken. Moreover as can be seen in Fig. 5b the critical value Ω cr israther sensitive to the Yukawa coupling Y . From the behaviour shown in Fig. 5b one concludes that Ω cr increases when the value of Y increases, and for any value of Ω there should exist a criticalvalue of the Yukawa coupling Y cr . We conclude that once Y cr is exceeded, the radiative spontaneoussymmetry breaking appears. Hence the lower right portion of Fig. 5b gives the direct evidence ofthe Coleman–Weinberg mechanism at work.
7. Extended model
In order to father investigate the decoupling of heavy fields we considered an extended modelwith two scalar fields φ and φ , described by the Lagrangian density of the form: L = i ψ / ∂ ψ + ∂ µ φ ∂ µ φ + ∂ µ φ ∂ µ φ − M φ − M φ − Y φ ψψ − Y φ ψψ − λ φ − λ φ − λ φ φ − λ φ φ − λ φ φ . (7.1)In the limit λ , λ , λ , Y → φ (cid:55)→ φ and couplings: M (cid:55)→ M , λ (cid:55)→ λ , Y (cid:55)→ Y . (7.2)Hence one can supposes that for the heavy φ field (the case in which we are interested in), theLagrangian density (2.1) will describe effective theory for presented extended model.The Lagrangian density (7.1) is symmetric under a transformation φ → − φ , φ → − φ and ψ → i γ φ . Moreover if Y = Y =
0) simultaneously with λ = λ = φ aturalness in Wilsonian approach T. Krajewski ( φ ) is independent from the transformation of remaining fields. Is such a case the Lagrangiandensity has two independent symmetries: φ → − φ with ψ → i γ φ and φ → − φ ( φ → − φ with ψ → i γ φ and φ → − φ ). If any of λ , λ or both Yukawa couplings Y , Y are non-zero thenfields φ and φ must have the same quantum numbers and can mix with each other. For the sakeof clarity let us concentrate on this special case. It is convenient to define the quantity:nd i = (cid:12)(cid:12)(cid:12)(cid:12) c EXTi − c EFFi c EXTi (cid:12)(cid:12)(cid:12)(cid:12) (7.3)where the c EXTi stands for one of the following couplings appearing in (7.1): m , M , λ , Y . The c EFFi is the coupling from the effective theory (2.1) which corresponds to c EXTi following the equation(7.2). The quantities 7.3 are referred to as normalized differences.Using Wilsonian RGEs for two theories, among which one is the effective theory for the other,one can try to find out how precise is the decoupling of heavy states in the running. We haveestimated how Wilson coefficients change with scale in two scenarios. In the first case we useWilsonian RGEs for the extended model to run from UV down to IR. In the second case we useRGEs for extended model only down to an intermediate scale Λ matching , lower then the mass M of heavy scalar φ . Below Λ matching we use RGEs for the effective theory with initial conditionsgiven by Wilson coefficients of extended model computed at the matching scale Λ matching . In Fig. 6quantities (7.3) are plotted in two examples which differ by the choice of the scale Λ matching . Thespikes on both plots 6b and 6a correspond to the zeros i.e. the positions of matching scales Λ matching (because our matching condition is equality of couplings c EXTi and c EFFi at Λ matching ). In Fig. 6bnormalized differences as a functions of the scale Λ in the effective action are plotted for the case ofmatching scale equal to the mass of the heavy scalar Λ matching = M = . In this case differencesbetween Wilson coefficients in both theories in the IR limit are of the order of these couplings. InFig. 6a we plotted normalized differences for the matching scale Λ matching one decade lower (withthe same initial conditions for RGEs in the extended model). One can notice that for the lowerscale Λ matching quantities 7.3 fall down rapidly.
8. Conclusions
In this paper we have used the Wilsonian effective action to investigate the fine-tuning and thevacuum stability in a simple model exhibiting the spontaneous breaking of a discrete symmetry andlarge fermionic radiative corrections which are able to destabilize the quartic scalar self-coupling.The regulator independence of the Wilsonian RG provides a consistent and well-defined procedureto analyze the issue of quadratic divergences. In the simplified model simulating certain featuresof the SM, the Wilsonian renormalization group equations have been studied.Numerical solutions of RGEs have revealed an interesting behaviour, caused by the samediagrams which generate quadratic divergences. An operator relevant near Gaussian fixed point (forexample the mass parameter for the scalar particles) can run like a marginal or even an irrelevantoperator, rather than decrease with growing scale. Furthermore solutions for different physical9 aturalness in Wilsonian approach
T. Krajewski
10 100 1000 10 - Λ (a)
10 100 1000 10 - Λ (b) Figure 6: Normalized difference (7.3) between running in effective theory and extended model formatching at two scales: (a)— Λ matching = M /
10 and (b)— Λ matching = M .quantities flow close to each other with increasing scale. The flow in the direction of some commonvalues indicates the severe fine-tuning. In such a situation small changes of boundary values ofparameters at high scale produce very different vacuum expectation values for the scalar field andother measurable quantities at low energies. We have estimated the fine-tuning as a function ofscale of the effective theory. For all parameters the adopted measure of the fine-tuning growsrapidly.It should be stressed that the Wilsonian RGE, in contrast to the Gell-Mann–Low running, au-tomatically accommodate decoupling of heavy particles. As investigated in Section 7, the contribu-tion to the flow coming from particles with masses M heavy greater than the scale Λ of the effectiveaction is strongly suppressed. The main contributions to the interactions generated by heavy statesare integrated out during calculation of the effective action for Λ (cid:28) M heavy , and are included inthe effective Wilson coefficients. These properties of Wilsonian RGE explain why the fine-tuningof the Wilson parameters is so interesting. Let us imagine a more fundamental theory (say theoryA) in which the SM is embedded. If one calculates in theory A the effective action for the scale Λ below, but not very much below, the lowest mass of the particles from the New Physics sector,one obtains certain values of the Wilson coefficients c A . On the other hand one can extrapolate theflow obtained from the SM to the scale Λ and calculate the Wilson coefficients c SM . Couplingscomputed in both ways should match, that is c A Λ = c SM Λ . If the couplings c A are different from c SM at the level of the fine-tuning ∆ c , that is c A Λ ( ± ∆ c ) = c SM Λ , theory A will produce the IR effectiveaction completely different from the SM.We have studied the issue of the spontaneous symmetry breaking due to radiative correctionsin the Wilsonian framework. We have demonstrated that there exists a critical value Ω cr belowwhich Ω runs negative in the IR and the symmetry becomes spontaneously broken. Moreover, thecritical value Ω cr is sensitive to the Yukawa coupling Y . One can see that Ω cr increases when thevalue of Y increases and for any value of Ω there exists a critical value of the Yukawa coupling Y cr . Once Y cr is exceeded, the radiative spontaneous symmetry breaking appears, which is a directevidence of the Coleman–Weinberg mechanism at work.10 aturalness in Wilsonian approach T. Krajewski
Acknowledgments
This work has been supported by National Science Center under research grantDEC-2012/04/A/ST2/00099.
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