Non-perturbative tests for the Asymptotic Freedom in the \mathcal{PT}-symmetric (-\phi^{4})_{3+1} theory

Non-perturbative tests for the Asymptotic Freedom in the -symmetric theory

Abouzeid Shalaby111email: amshalab@mans.edu.eg    Suleiman S. Al-Thoyaib Physics Department, Faculty of Science, Mansoura University, Egypt
Physics Department, Faculty of Science, Qassim University, Saudi Arabia
Abstract

In the literature, the asymptotic freedom property of the theory is always concluded from real-line calculations while the theory is known to be a non-real-line one. In this article, we test the existence of the asymptotic freedom in the theory using mean field approach. In this approach and contrary to the original Hamiltonian, the obtained effective Hamiltonian is rather a real-line one. Accordingly, this work resembles the first reasonable analysis for the existence of the asymptotic freedom property in the -symmetric theory. In this respect, we calculated three different amplitudes of different positive dimensions (in mass units) and find that all of them goes to very small values at high energy scales (small coupling) in agreement with the spirit of the asymptotic freedom property of the theory. To test the validity of our calculations, we obtained the asymptotic behavior of the vacuum condensate in terms of the coupling, analytically, and found that the controlling factor has the value compared to the result from the literature which was obtained via numerical predictions. We assert that the non-blow up of the massive quantities at high energy scales predicted in this work strongly suggests the possibility of the solution of the famous hierarchy puzzle in a standard model with -symmetric Higgs mechanism.

Effective potential, Hierarchy problem, Cosmological constant, non-Hermitian models, -symmetric theories, Quasi-Hermitian field theory.
pacs:
12.60.Cn, 98.80.Cq, 12.90.+b

One of the biggest puzzles in the theory of particle interactions is the Hierarchy problem DJOUADI (). Due to this conceptual problem, all quantities of positive mass dimensions in the scalar sector of the Standard model (e.g. the Higgs mass) blow up to unacceptable values at high energy scales. The worst manifestation of the Hierarchy problem is in the vacuum energy as it behaves like with is a unit mass. This leads to the most unacceptable discrepancy between theory and experiment in the prediction of the cosmological constant (vacuum energy). Indeed, the root of the Hierarchy problem stems from the fact that the scalar Higgs mechanism played by the Hermitian theory has a positive Beta function which up to second order in the coupling is given by;

One can easily show that the positiveness of the Beta function of the theory  will lead to a huge Higgs mass at high energy scales abohir (). By catching the main reason for the Hierarchy problem, one may wonder if the existence of a scalar theory with rather negative Beta function (i.e. asymptotically free) will help in solving the Hierarchy problem in the standard model of particle interactions?. In Ref. abohir (), we argued that such reasoning becomes legitimate since the discovery of the physical acceptability of non-Hermitian and -symmetric theories bendh1 (); abohir (); abophi41p1 (); abophi61p1 (); zadah (); bendr (); bend2005 (); bendg1 ().

In spite of the beauty of the idea of employing a now physically acceptable theory with bounded from above potential to play the role of Higgs mechanism, in the literature, all the claims about the asymptotic freedom property of the -symmetric () scalar field theory were built on a real-line calculation (inaccurate) while the theory is well known to be a non-real line one Symanzik (); bendf (); Frieder (). In view of this realization, our aim in this work is to test the existence of the more than important asymptotic freedom property for -symmetric () scalar field theory but this time using algorithms which proved to be reliable for the study of non-real line problems.

To shed light on how important to employ an asymptotically free -symmetric () scalar field theory to play the role of the Higgs mechanism in the standard model, we mention some possible problems existing in the standard model and its extensions. In the standard model, the spontaneous symmetry breaking adds a large shift to the vacuum energy of the form , where  is the Hamiltonian operator, is dimensionless and is the vacuum condensate Peskin (). This shift is finite but still large. As we will see in this work, contrary to the corresponding Hermitian theory, the vacuum energy of the -symmetric () scalar field theory is tiny for all energy scales and thus gives a clue to benefits that may be drawn by its employment in the standard model. This discrepancy between the features of the two theories (Hermitian and the non-Hermitian theories) can be understood from the fact that for a theory with negative Beta function, the coupling will be dragged to the origin at high energy scales kaku () which in turn drag all the dimnesionfull quantities to small values while the reverse is correct for a theory with positive Beta function. This is the origin of the gauge Hierarchy problem. For the solution of this problem, different algorithms have been introduced. For instance, in the SUSY regime there exists natural cancellation in the dimensionful parameters that turned those parameters protected against perturbations even for very high energy scales susy (). However, SUSY introduces an upper limit to the Higgs mass by 130 and some of its mass spectra are of one which expose this theory direct to the fire of the LHC experiments. Another algorithm for the solution of the Hierarchy problem is to consider the Higgs particle as a composite state bounded by a new set of interactions (Technicolors) tecni (). However, the technicolor model is strongly constrained from precision tests of electroweak theory at LEP and SLC experiments tecni1 () . Also, this algorithm has mass spectra of about one and it is under the direct test of the LHC experiments. Once more, a recent algorithm is suggested for which the symmetry is broken via the compactification of an extra dimension extra (). In fact, particles in this model attain masses through the expectation value of the fifth ( for instance) component of the gauge field. To some physicists, the digestion of extra dimension is not that easy and can be accepted to them at most as a mathematical modeling to the problem.

The introduction of a new scenario that may overcome the Hierarchy problem as well as does not introduce extra problems may be possible. Indeed, the scalar field in the standard model is the source of the hierarchy problem and thus it would be very important to have a scalar field with unproblematic features. In this work we, study the flow of different quantities of positive mass dimension in the -symmetric theory and show that they do not blow up at high energy scales which is the reverse of the behavior of the Hermitian at those scales. Although our calculations are carried out for a one component field while the one used in the standard model has a higher group structure, it is not expected that the group structure will change the amazing asymptotic freedom property of the theory which means that the results in our work strongly recommends the replacement of the conventional Higgs mechanism by a -symmetric one which then is expected to overcome the Hierarchy problem.

The attractive idea of a possible safe employment of the -symmetric Higgs mechanism is in fact confronted by two main technical problems. The first problem is the believe of the lack of a reliable calculational algorithm to follow for -symmetric () field theory. Among the well known non-perturbative methods that works for -symmetric theories is the complex contour treatment applied successfully for the space-time dimensions (quantum mechanics) bendr (). However, this method is not willing to be applicable for higher space-time dimensions. Regarding this problem, in a previous work  aboeff (), we discovered (for the first time) that the mean field approach which is famous in field theory calculations works well for -symmetric () theory. While its applicability in higher space-time dimensions is not questionable, in Ref. aboeff (), we exposed the effective field approach to a quantitative test and showed that it is accurate even at the level of first order in the coupling which means that the first problem has been solved. The second problem that is confronting the progress in the study of -symmetric () theory in the real world of space time dimensions is that the metric operator for this theory is very hard to be obtained. However, in the mean field regime in Ref.aboeff (), we realized that the propagator of the -symmetric () theory has the correct sign. Recently, Jones et.al conjectured that mean field theory may know about the metric. In fact, Jones et.al. showed that the Greens functions in the mean field approach are taking into account the employment of the metric operator jonesgr (); jonesgr1 (). In view of these explanations, we think that a non-problematic -symmetric Higgs mechanism is now possible and it is just a matter of known calculations. In fact, the version of mean field approach used by one of us in Ref. aboeff () mimics the way of breaking the symmetry in the standard model and thus we assert that it is the most known plausible method to use for the study of the -symmetric Higgs mechanism. However, as a first step toward the employment of the -symmetric Higgs mechanism we need to check the existence of the asymptotic freedom property in the prototype example of the one component -symmetric theory using the mean field approach.

The motivation behind the application of the mean field approach for the -symmetric theory is that, in the literature, all the claims about the asymptotic freedom property of the -symmetric () scalar field theory were built on a real-line calculation while the theory is well known to be a non-real line one. In view of this realization, our main target in this work is to assure the existence of the asymptotic freedom property for -symmetric () scalar field theory but this time using the reliable mean field approach (as used by us in Ref. aboeff ()) which implements the use of the metric as well. Although our work stresses the one component -symmetric () scalar field theory, its extension to charged -symmetric () scalar field theory is direct.

Since the calculational algorithm we will follow in this work is the mean field approach as presented in our work in Ref.aboeff (), as a reminder, we summarize its results. To do that, we consider the Lagrangian density of the massless -symmetric () scalar field theory;

where is the field variable, is the canonical conjugate momentum field and is the coupling constant. The effective field approach uses the application of a canonical transformations of the form;

with is a vacuum condensate and is a fluctuating field. Thus one obtains the form

where

(1)

and is the mass of the field .

We used the known relations of the effective potential of the form;

(2)

where and is the mass of the field. Up to first order in the coupling and in dimensions, we were able to obtain the equations;

(3)

For , one can get the parametrization

(4)

Since is imaginary, the effective Hamiltonian in Eq.(1) is non-Hermitian and -symmetric. Moreover, while the original theory is a non-real line one, the effective form is a real-line theory quart (); realline (). In fact, this is a very important realization since the real-line effective field calculations can be extended to higher dimensional cases (field theory) for which non-real line problems can not be treated using the complex contour method.

Note that the results presented in Ref.jonesgr () coincides with our (older) above results. Besides, the conditions , we used in Ref.aboeff () are coincident with the variational conditions , . In fact, this method not only gives accurate results for the energy spectra and the condensate but also results in the correct sign of the propagator (no ghosts) which is assured by the positiveness of the parameter. This result is a clue for the possibility of the disappearance of the metric operator from the calculation of the Greens functions in the effective field approach. In fact, the assertion that mean field approach knows about the metric has been clearly proved by Jones et.al in Refs jonesgr (); jonesgr1 ().

According to the above results, the effective field approach applied by one of us for the first time in Ref.aboeff () is a satisfactory algorithm for the calculations in the -symmetric () scalar field theory which does not need the difficult calculation of the metric operator or the may be impossible complex contour integrations followed in the quantum mechanical case.

Unlike the quantum mechanical case, in quantum field theory one always confronted by infinities in the amplitude calculations. Since we restrict ourself to first order calculations, normal ordering is a valuable tool to exclude infinities at this order of calculations. In this tool, the first order vacuum energy can be obtained by normal ordering the operators in the effective Hamiltonian with respect to two different mass scales Coleman (); efbook (). The benefit of this method is that it accounts not only for first order diagrams but also for all the higher order cactus diagrams wen-Fa (); changcac (). Moreover, in Ref.mag (), it has been shown that the method is equivalent to the first order calculations with the regularization carried out via the addition of a counter term.

To start the algorithm, consider a normal-ordered Hamiltonian density with respect to a mass parameter of the form;

(5)

We can use the relation Coleman (); normal2 ()

(6)

with

(7)

to rewrite the Hamiltonian normal ordered with respect to a new mass parameter . In eq.(6), expanding both sides and equating the coefficients of the same power in yields the result;

(8)

Also, it is easy to obtain the result Coleman ()

(9)

where

with

(10)
(11)

Here, is the gamma function and is the spatial dimension. Accordingly,

(12)

where , , is the Euler number given by

and

Here and are unit masses chosen to make the argument of the logarithm dimensionless. Also, the relation has been employed to fix the renormalization scheme collinbook (). As , can be simplified as;

The mass shift should be accompanied by the canonical transformation; efbook ()

(13)

The field  has mass , is a constant, the field condensate, and is the conjugate momentum (). Therefore, the Hamiltonian in Eq.(5) can be written in the form;


(14)

where

can be found as

(15)

and the field independent terms can be regrouped as;

(16)

Taking , we get

Substituting for and in Eq.(16)we get;

(17)

Or,

where we used the dimensionless parameter such that . Also, in using dimensionless quantities of the form and , we have the dimensionless vacuum energy of the form;

(18)

To understand well the features of this result, let us note that the the effective potential is the generating functional of the one-particle irreducible (1PI) amplitudes Peskin (). This fact results in the relation;

(19)

where is related to the n-point Greens function. For instance, the two-point function can be generated from the effective potential via the second derivative of the effective potential with respect to the condensate, i.e,

(20)

where is the propagator. Since does not depend on the position (zero momentum) we have . Thus, . Since in our work is constrained to be positive, this shows that the propagator has the correct sign (no ghosts). This unexpected result has been explained by Jones et.al as the mean field approach in -symmetric theories knows about the metric.

To analyze our results, we note that the stability condition enforces to be zero. Accordingly, we get the results;

(21)

From these equations one can get the results;

(22)
(23)

Note that these results shows that the vacuum condensate predicted from Eq.(22) is imaginary and thus the Hamiltonian in Eq.(14) is non-Hermitian but -symmetric.

One can solve Eq(23) for as a function of and thus can obtain the dependence of the vacuum energy and the vacuum condensate on the coupling . In Figs.1, 2 and 3 we show the calculations of the vacuum energy, Higgs mass squared and the vacuum condensate , respectively, as a function of the coupling . Before w go on, let us check the accuracy of our calculations. In Ref. bendg1 (), Bender et.al obtained the behavior of the one point function as (numerically). They showed that goes to zero as , where is called the controlling factor and there is related to the coupling. In space-time dimensions they obtained the numerical value . Now, Eq(23) can be rewritten as

where is given by: . Accordingly, as the parameter the coupling . Now writing Eq.(23) in the form;

and since as we have then at this limit one can approximate this equation by

which can be solved to give

with is the Lambert function defined by . Note that, for small arguments and thus as , the parameter can be approximated by;

and in using Eq.(22), we obtain the asymptotic behavior of the one point function as in the form

This shows that we were able to obtain the exponential behavior for the condensate (analytically and for the first time) predicted numerically in Ref. bendg1 (). Moreover, in accounting for the different coupling used in our work from that in Ref.bendg1 (), we find that the controlling factor is given by compared to the numerical prediction of in Ref. bendg1 () as . This result assures the reliability of our analytic calculations.

The result as obtained above seems to be strange as one expects that (no quantum corrections) at this limit. To explain this result, we mention that, up to the first order correction, here is the normalized coupling and if the theory is asymptotically free it means that at the scales . Accordingly, at this limit all quantities of positive mass dimensions goes to zero as well. Accordingly, the result as coincides with the spirit of the asymptotic freedom property concluded from the real-line perturbative calculations. In other words, Eq.(23) agrees with the renormalization group flow of the coupling, small values of the coupling correspond to high energy scales and vice versa. While quantities that have positive dimension in terms of mass unit for the Hermitian theory blow up at high energy scales (large coupling in this case), the corresponding quantities in -symmetric theory tend to tiny values at high energy scales (small coupling in this case). This realization pushes us to believe that the employment of the -symmetric Higgs mechanism in the standard model may solve the famous Hierarchy problem. These interesting features are appearing in our calculations presented in Figs.1, 2 and 3. For instance, in Fig. 1, we plotted the vacuum energy which has a mass dimension of and it is clear that the vacuum energy goes to zero as . This is a very important result because in the corresponding Hermitian theory, vacuum energy represents the worst case of the Hierarchy problem which introduces the cosmological constant problem. In view of our analysis, we strongly believe that the employment of the -symmetric Higgs mechanism will solve the cosmological constant problem too. Note that, finding it as a fully acceptable calculational algorithm for the -symmetric field theory aboeff (); jonesgr (); jonesgr1 (), the mean field approach is a systematic setup to handle the employment of the -symmetric Higgs mechanism in a complete manner.

In Fig. 2  a quantity of mass dimension is represented and also assures the existence of the asymptotic freedom in the-symmetric field theory.

In Fig. 3, the vacuum condensate which has a mass dimension of has been plotted and assures the asymptotic freedom property too.

Let us now speculate about the actual case in the standard model where the Higgs mass dominant contribution has the form DJOUADI ();

(24)

with the different parameters are defined in Ref.DJOUADI () ( here represents a momentum cut off) and since all the species in the standard model attain their masses from the vacuum condensate, thus the masses have even sharper than an exponential decrease in 222The coupling is decreasing as a function of the energy scale as shown in Ref.abohir ().. Hence, one expect that all the particles in the standard model (including Higgs) have finite masses at high energy scales 333The different group structure between theory investigated in this work and the one suitable to break the symmetry in the standard model will affect the numerical values but not the main features of the theory and thus conclusions are supposed to be the same in both cases..

To conclude, we have calculated the vacuum energy for the non-Hermitian and -symmetric scalar field theory in dimensions. We find that the vacuum energy is small for the whole range of energy scales which is a very interesting sign that this theory is a very good candidate to play the role of the Higgs mechanism in the standard model of particle interactions. We assert that the conventional Hermitian has a vacuum energy which blows up at high energy scales as a manifestation of the famous Hierarchy puzzle. In fact, this adds a large value to the cosmological constant and thus enhancing the unacceptable discrepancy between theoretical and experimental predications of the cosmological constant. Accordingly, the discussions here supports the fruitfulness of replacing the Hermitian Higgs mechanism by the -symmetric one.

Since in the literature the asymptotic freedom is always assumed for the theory under consideration by the aid of real-line calculations, we generated the diagrams in Figs.1, 2 and 3 using our effective field calculations. The effective Hamiltonian is a real-line theory and thus conclusions from our calculations are then reliable. In fact, the quantities generated in the graphs have positive mass dimensions and according to asymptotic freedom they have to vanish at very high energy scales (small coupling), which is very clear from the figures. While the naive perturbation analysis in Ref.abohir () shows the same result, the Higgs mass blows up at small energy scales (large coupling). However, at large couplings perturbations are meaningless and conclusions have to be drawn from non-perturbative treatments of the theory. In fact, the non-perturbative effective field approach we used cured this problem as we can realize from Fig.2 that the mass parameter is finite for the whole energy range (or the whole range of the coupling ).

The effective field approach used in this work also knows about the metric as one can realize that the propagator has its correct sign. This prediction has recently proved by Jones et.al in Refs.jonesgr (); jonesgr1 (). Accordingly, there exist no need to take care about the so far unattained metric operator. Thus, the algorithm we followed here is sufficient to tackle the theory and can be employed easily to the realistic case of the standard model. Moreover, the accuracy of the algorithm has been tested in a quantitative manner in Ref.aboeff () and it was found that the effective field approach is reliable for the study of -symmetric theories.

To check the validity of our calculations, we obtained the asymptotic behavior of the vacuum condensate as a function of the coupling when . At this limit, we found that the condensate behaves like a result which was conjectured by bender et.al in Ref.bendg1 (). Moreover, we obtained the controlling factor which is very close to the numerical prediction obtained by bender et.al in Ref.bendg1 ().

Acknowledgements.
We would like to thank Dr. S.A. Elwakil for his support.

References

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Figure 1: The vacuum energy ( has a mass dimensions of 4) of the scalar field theory. For either small (high energy scales) or large (IR energy scales) the vacuum energy is finite.
Figure 2: The mass squared ( of the scalar field theory which has a mass dimensions of 2. In this case also for either small (high energy scales) or large (IR energy scales) is finite.
Figure 3: The absolute value of the vacuum condensate ( has a mass dimensions of 1) of the scalar field theory. For either small (high energy scales) or large (IR energy scales) the vacuum condensate is finite too.
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