A New Solution to the Strong CP Problem

We suggest a new solution to the strong CP problem. The solution is based on the proper use of the boundary conditions for the QCD generating functional integral. We expand the perturbative boundary conditions to both perturbative and nonperturbative fields integrated in the QCD generating functional integral. It allows to nullify the CP odd term in the QCD Lagrangian and thus to solve the strong CP problem. The presently popular solution to the strong CP problem with introducing axions violates the principle of renormalizability of Quantum Field Theory, which is very successful phenomenologically. Our solution obeys the principle of renormalizability of Quantum Field Theory and does not involve new exotic particles like axions.


Introduction
The strong CP problem for a long time is considered as in fact an unsolved, or at least not completely satisfactorily solved, outstanding problem of Quantum Field Theory and Elementary Particle Physics.For an excellent review of the subject see [1], where one can find various aspects of this problem.The most popular presently solution [2], [3] to the strong CP problem introduces new particles -axions [4], [5].Axions became so popular that they are even considered as real candidates for dark matter of the Universe.But presently only restrictions on their possible properties are established in spite of the numerous experimental efforts to discover such exotic particles, see e.g.[6], [7], [8].Besides, the axion solution of the strong CP problem violates the fundamental principle of renormalizability of Quantum Field Theory.This basic principle presently is one of the most phenomenologically successful principles of Elementary Particles Theory.For example, it ensured in Quantum Electrodynamics the agreement between the theory and the experiment for anomalous magnetic moment of the electron within ten decimal points.This impressive agreement convinces us that renormalizable Quantum Field Theory is a correct physical theory.Therefore, in our opinion, it seems to be interesting to find a solution to the strong CP problem which also obeys the principle of renormalizability of Quantum Field Theory.Besides it is desirable to find a solution which does not introduce new exotic particles like axions.This is the goal of the present paper.
To find such a solution we will use in a proper way (a proper way in our opinion) the boundary conditions in the generating functional of Green functions of Quantum Chromodynamics.It is well established what kind of boundary conditions are imposed on the fields of the theory in the functional integral within perturbative approach.These are known boundary conditions which produce the correct form of the perturbative propagators of the fields of the Lagrangian in the considered theory.The derivation of the perturbative propagators using the boundary conditions of the generating functional of Green functions of Quantum Chromodynamics can be found in [9].We will assume that the same boundary conditions are valid for all fields of the theory which are integrated in the functional integral.That is we will suppose that perturbation theory calibrates the whole nonperturbative functional integral.
Our solution will obey the principle of renormalizability of Quantum Field Theory and will not involve new exotic particles like axions.
The CP problem is the question of why the strong interaction does not violate the charge-parity (CP) symmetry, which is the combination of charge conjugation (C) and parity (P) symmetries.The CP symmetry states that the laws of physics should be the same if a particle is replaced by its antiparticle and its spatial coordinates are inverted (P).However, the weak interaction is known to violate the CP symmetry, and there is no fundamental reason why the strong interaction should not do the same.
The CP problem is also related to the origin of the matter-antimatter asymmetry in the universe, which is another unsolved mystery in physics.The CP violation involves scalar fields that couple to the quarks and induce a complex phase in the quark mass matrix.This phase could affect the properties of the neutron stars and black holes in X-ray binaries, such as their mass, radius, magnetic field, and spin, see e.g.[10].Another possible connection is that some models of CP violation involve new particles that have spin1/2 and interact with the standard model particles via a new force [11].These particles could affect the X-ray spectrum or the gravitational waves emitted by the system.
The CP violation in the early universe could have generated primordial magnetic fields that were amplified by the collapse of stars into neutron stars.These fields could then explain the existence of magnetars powered by extremely strong magnetic fields, see e.g.[12].However, this scenario is highly speculative and requires more theoretical and observational support.

Materials and Methods
In the present work we will deal with the Quantum Chromodynamics (QCD) generating functional of Green functions, which will be the basic object of our considerations where dΦ denotes the integration measure of the functional integral Z(J) over all fields Φ k of the theory, gluons and quarks.J k are sources of the fields.The symbol J in Z(J) denotes the full set of sources J k of the fields.Within perturbation theory the QCD Lagrangian L QCD is invariant, in particular, under the combined symmetry transformations CP, where C is the charge conjugation operator and P is the space reflection.More precisely the QCD Lagrangian within perturbation theory is invariant under both the charge conjugation C and the space reflection C.
The essence of the CP problem is that in full nonperturbative QCD one can add to the QCD Lagrangian the CP odd gauge invariant term which seems to be not forbidden from the first principles It is invariant under the charge conjugation C and is not invariant under the space reflection P, hence it is also non invariant under the combined CP transformation.But this term is forbidden by experiments with the rather high precision as we will see it below.The dual field strength tensor Ga µν in (2) is defined in the standard way The θ-term in (2) is purely nonperturbative since it is invisible in perturbation theory because it can be rewritten as a total derivative Here K µ is the known Chern-Simons current The θ-term can be discarded within perturbation theory.It can be easily seen in the Euclidean space since the fields of the theory decrease in the Euclidean space at the time infinities and the total derivative (4) does not contribute to the QCD action.But with the discovery of instantons [13] it was realized that the field configurations with the instanton boundary conditions give nonzero nonperturbative contributions to the action.In particular, the one instanton contribution looks like The key notion here is the famous topological charge which has the following form where K is the Pontryagin number.The topological charge is zero for perturbative fields, i.e. in perturbation theory.But instanton fields, for example in the A 0 = 0 gauge, interpolate between the zero gluon fields Here the matrix U is the Polyakov hedgehog For this instanton configuration one has that the Pontryagin number and correspondingly the topological charge are equal to unity Thus the θ-term gives the nonzero nonperturbative contribution to the QCD action.
In the full QCD, with quarks, there is also contributions to the CP odd part of the QCD Lagrangian from the imaginary phases of the quark mass matrix.The phases can be rotated away by the chiral transformations of quark fields.But there is the famous axial anomaly [14], [15].It generates non invariance of the measure of the Feynman functional integral under chiral transformations [16].Therefore the phases of the quark mass matrix arise before the G G term in the Lagrangian.Hence the parameter which determines the value of the CP violation is in fact where M is the quark mass matrix.Below we shall use the same symbol θ for this parameter to simplify notations assuming that it already includes the effects of the quark mass matrix.
Probably the most essential phenomenological effect of the θ-term is a nonzero electric dipole moment of the neutron d n .The electric dipole moment is given by the effective interaction Lagrangian where is the photon field strength tensor, n stands for the neutron field , σ µν = 1 21 [γ mu γ nu ] is the standard antisymmetric product of Dirac gamma matrices.
The θ-term generates the following electric dipole moment of the neutron Here J µ is the quark electromagnetic current.k µ = p µ f − p µ i , where p µ i is the incoming momentum of the neutron and p µ f is the outgoing momentum of the neutron.ǫ µ (k) is the photon polarization.
The matrix element in the left hand side of eq.( 12) is zero in perturbative theory and is calculated within purely nonperturbative QCD.There are several nonperturbative methods by which the electric dipole moment of the neutron d n was estimated, for the detailed overview see [1] and references therein.Here we give the short summary of the results of the corresponding nonperturbative approaches.The performed bag model calculations produced the following result d n ≈ θ2.7 • 10 −16 e•cm.Shortly after this result the chiral logarithms approach was used to obtain the following estimate d n ≈ θ5.2 • 10 −16 e•cm.The approach of chiral perturbation theory was further developed to produce the slightly less result d n ≈ θ3.3 • 10 −16 e•cm.At last but not at least the calculations based on the QCD sum rules method gave the following again slightly less estimate d n ≈ θ1.2•10 −16 e•cm.All these results have considerable uncertainties of the order of 50 per cent because of essential difficulties of nonperturbative QCD calculations.
But anyway the average theoretical value for d n can be confidently estimated within the same of the order of 50 per cent uncertainty as This should be compared with the most recent experimental value [17] for the electric dipole moment of the neutron d n which is Thus one gets an extremely strong restriction on the value of the θ coupling The explanation of this practically zero value of the coupling θ is the essence of a solution to the strong CP problem.The presently popular solution to the problem is the famous axion solution.It assumes the addition to the QCD Lagrangian the term with the new axion field a(x), which in fact reduces to the shift of the coupling θ in the QCD Lagrangian θ → a(x)/f a + θ.So the corresponding term ∆L θ of eq.( 2) in the QCD Lagrangian becomes as follows After the spontaneous symmetry breaking of the global Peccei Quinn symmetry [2], [3] one calculates the effective potential for the axion field a(x).
Then one finds that when the axion rests at the minimum of this potential, the CP violating term ( 14) nullifies.This is the known axion solution to the strong CP problem.
There are different types of axions suggested in the literature.Let us consider some of them.As the scalar Higgs fields produce vacuum expectation values the electroweak local symmetry group is broken spontaneously.This develops masses of the gauge W and Z intermediate vector bosons.At the same time the global Peccei Quinn U(1) group is also spontaneously broken.This spontaneous breaking of the U(1) global symmetry leads to the appearance of the massless Goldstone boson, which is called the Weinberg-Wilczek (WW) axion in this case [4], [5].In the Standard Model including two Higgs doublets this axion is presented as the following superposition a = 1/v(v φ Imφ 0 − v χ Imχ 0 ), here φ 0 and χ 0 are neutral components of the two Higgs doublets.Besides, v = v 2 φ + v 2 χ ≈ 250 GeV, where v φ and v χ denote vacuum expectation values of the fields φ and χ correspondingly.Within the considered approximation the WW axion is massless.But as it was mentioned above the nonperturbative effects of Quantum Chromodynamics (for example instantons) can generate the potential for the WW axion.In this way the WW axion obtains the nonzero mass value which is estimated according to [4], [5] as following Besides, the decay constant of the WW axion is 1/v.Hence it is clear that the mass and the decay constant of the WW axion is connected to the breaking scale v of electroweak symmetry.This constraint turns out to be too strong and correspondingly the WW axion turns out to be excluded by the existing experimental data.
If the scale of the breaking of the Peccei Quinn symmetry is much more than the electroweak scale v, then according to the above formula the axion is essentially lighter and the decay constant of the axion is much smaller.This type of the light axion could be in agreement with the existing experimental data.
A solution with the light 'invisible' axion was first suggested in [18], [19] (the so called KSVZ axion).In the Ref. [19] this type of the axion is named the phantom axion.It should be underlined that in order to uncouple the 'phantom' from the electroweak scale it is necessary to decouple the proper scalar fields from the standard quarks and couple these fields to very heavy hypothetical fermions carrying color.
To be more precise one should introduce a complex scalar Φ which is coupled to the hypothetical quark field Q, the electroweak singlet in the fundamental representation of the SU(3) color group.
Then the modulus of the scalar Φ is supposed to produce the large vacuum expectation value f / √ 2 and the argument of the field Φ is just the axion field a up to normalization: Further the low energy coupling of this axion to the gluons is as follows In this way the Lagrangian of QCD depends on the expression θ + α(x).More generally, it is possible to introduce more than one quark field Q or to introduce these fields in a higher representation of the SU(3) color group.In this case the coupling of the axion to gluons obtains an integer number N: This multiplier N (should not be confused with the color number N c ) is usually called as the axion index.Hence, in general, the Lagrangian of Quantum Chromodynamics depends on the sum θ + Nα(x).It can be assumed that the nonperturbative QCD effects produce the potential for θ + Nα(x).The latter sum is minimized at the point θ + Nα vac = 0. Hence the strong CP problem is solved in this way.One more way to produce an 'invisible' axion is suggested in [20], [21] (the so called ZDFS axion).In this case one keeps the Peccei Quinn symmetry of the two doublet Standard Model but splits the scales of the Peccei Quinn and electroweak breaking.For this purpose the Lagrangian of the Standard Model is extended, one adds the scalar Standard Model singlet field Σ Then one note that this Lagrangian is in variant under the axial transformations q L → e iα q L , q R → e iα q R , φ → e 2iα φ, χ → e −2iα χ, Σ → e 2iα Σ.
After spontaneous breaking of this axial symmetry the Goldstone particle, the axion, appears as the superposition Here and v Σ represent vacuum expectation values of the fields φ, χ and Σ correspondingly.The vacuum expectation value of the field Σ is not necessarily connected to the scale of the electroweak symmetry breaking.Actually it can be chosen as large as the Grand Unification scale.In this case the considered axion is very light and its decay constant is small.
But first of all, the term with the axion field a(x) in ( 14) has the dynamical dimension (that is the dimensions of the fields plus the dimensions of the derivatives of the fields) five instead of four necessary for renormalizability.Hence the term with the axion field a(x) violates renormalizability of the Lagrangian.As we already have mentioned in the introduction, renormalizability of the Lagrangian is a rather important principle of Quantum Field Theory.This principle turned out to be very successful phenomenologically as it is demonstrated for example by the famous case of the anomalous magnetic moment of the electron.Therefore it is quite important to preserve the principle of renormalizability when solving the strong CP problem.Secondly, the axion is not found experimentally in spite of numerous experimental attempts, as we also have underlined in the introduction.
Therefore we find it necessary to suggest a new solution to the strong CP problem which preserves renormalizability of the theory and does not involve new exotic particles like axions.
Let us now again consider the QCD generating functional (1).It is well known that this integral is not defined yet completely if only the QCD Lagrangian is defined with the corresponding gauge condition.At least within perturbation theory one should impose on the Lagrangian fields the proper boundary conditions.
In perturbation theory one has the well known boundary conditions.For example for the gluon fields one has the following conditions Here the incoming asymptotic gluon fields A a µ,in (x) contain only the positive frequency part and the outgoing gluon fields A a,out µ, (x) contain only the negative frequency part: A a,out µ, where ω = k 2 and v i µ (k) are polarization vectors of the gluons.Here the sums over the gluon polarizations i = 1, 2 are assumed.
These are the known Feynman boundary conditions.They are necessary to obtain the correct form of the perturbative propagators of the fields of the type 1/(k 2 + iǫ) with the correct plus iǫ prescription.Thus with this boundary conditions the gluon fields (the quark fields also) oscillate at the time infinities.After transition to the Euclidean space by means of the Wick rotation t → ix 4 the fields decrease at the time infinities, so in the Euclidean space it is easy to see that in perturbation theory total derivatives in the Lagrangian are zero.
Hence one can write perturbative boundary conditions (15) for all fields Φ i of the QCD Lagrangian symbolically as follows This helpful notation will be used below to formulate the QCD generating functional integral as a compact formula.

Results
Let us now exactly formulate our solution to the strong CP problem.As it is was already underlined above the solution is based on the proper use of the boundary conditions for all Lagrangian fields in the QCD generating functional integral.So let us now again consider the QCD generating functional integral (1. In our opinion, it is necessary and natural to generalize the boundary conditions (17) for perturbative fields to all fields Φ i of the QCD Lagrangian which are integrated in the generating functional integral (1).Then all Lagrangian fields will decrease in the Euclidean space at the time infinities.Hence such a definition of the boundary conditions nullifies all total derivatives in the Lagrangian both for the perturbative contributions and the nonperturbative contributions.Thus the CP odd term in the QCD Lagrangian will be nullified and it solves the strong CP problem.Besides this definition allows to formulate exactly the complete (the perturbative part plus the nonperturbative part) QCD generating functional integral as one compact mathematical formula:

Discussions
The strong CP problem for a long time is in fact considered as the still unsolved (or not completely adequately solved) prominent problem of Quantum Field Theory and Elementary Particle Physics.For an excellent review of the CP problem and related topics see the fist reference of the present paper, where one can discover different aspects of the subject.The presently most popular axion solution [2], [3] to the strong CP problem introduces new exotic elementary particles -axions.Axions are now so popular that they are presently considered as the real candidates for Dark Matter of the Universe.A lot of advanced experiments are performed to discover some sorts of these axions.But presently only some kinds of restrictions on their possible properties are obtained in spite of the numerous huge experimental efforts to find such exotic elementary particles, see for example [6], [7], [8].Besides, the axion solution of the strong CP problem is in contradiction with the fundamental, in our opinion, principle of renormalizability of Quantum Field Theory.This basic, in our opinion, principle presently is one of the most experimentally successful principles of Quantum Field Theory and Elementary Particles Physics.For example, this principle produced for the famous anomalous magnetic moment of the electron within renormalized Quantum Electrodynamics the outstanding agreement between the theoretical value and the experimental value within ten decimal points.This prominent agreement between the theory and the experiment convinces us that renormalizable Quantum Electrodynamics and more generally renormalizable Quantum Field Theory are the proper physical theories.
Therefore, in our opinion, it seems to be important to have a solution to the strong CP problem which is in agreement with the principle of renormalizability of Quantum Field Theory.Besides we suppose that it is desirable to find a solution to the strong CP problem which does not involve new exotic elementary particles like axions or something similar.Therefor the goal of the present paper is to find the solution which satisfies these two conditions (renormalizability and the absence of new exotic particles).
To find such a solution we have used in this paper in a proper way the boundary conditions for the Lagrangian fields in the generating functional of the Green functions of Quantum Chromodynamics (a proper way in our opinion).It is well understood what kind of boundary conditions should be imposed on the fields of the theory in the generating functional integral within perturbation approach.These are the known boundary conditions which generate the necessary form of the perturbative propagators of the fields of the Lagrangian of Quantum Chromodynamics.These boundary conditions produce the correct '+iǫ' prescription for the perturbative propagators of the fields.We have suggested that the same boundary conditions should be valid for all Lagrangian fields integrated in the QCD generating functional integral, i.e. for both perturbative and nonperturbative contributions.That is we have supposed that perturbation theory calibrates the whole nonperturbative functional integral, i.e. it calibrates both the perturbative and the nonperturbative parts of the generating functional integral.
Our solution satisfies two important criteria.The solution does obey the principle of renormalizability of Quantum Field Theory and does not involve new exotic particles like axions.
Let us make now necessary here remarks concerning the famous U(1) problem.The essence of this problem is that the mass of the flavour singlet pseudo-scalar η ′ meson m η ′ ≈ 958MeV is surprisingly heavier than the masses of the flavour octet pseudo-scalar mesons.One can argue that it is not possible to extend the discussed above perturbatve boundary conditions for the fields, which are well established within perturbation theory, to all fields which are integrated in the QCD generating functional integral.Such an extension excludes from the generating functional integral for example the instanton contributions not obeying the perturbative boundary conditions.In particular there is the well known statement [22], [23] that instantons solve the U(1) problem.Hence it seems that they should not be excluded from the theory.But one can note that there is also the well known solution [24], [25] to the U(1) problem using the axial anomaly which was suggested before the discovery of instantons.Therefore one can argue that the U(1) problem can be solved without involving the instantons.Thus the extension of the perturbative boundary conditions to all fields of the generating functional integral is well allowed.

Conclusions
We have suggested a new solution to the strong CP problem.To find such a solution we use in a new way the boundary conditions in the Quantum Chromodynamics generating functional of Green functions.It is well established what kind of boundary conditions are imposed on the fields of the QCD Lagrangian in the functional integral within perturbation theory.We assume that the same boundary conditions are valid for all Lagrangian fields of the functional integral, i.e. for both perturbative and nonperturbative fields.This allows to nullify total derivatives in the QCD action, in particular the CP odd term which can be presented as the total derivative.Hence it solves the strong CP problem.Thus we suppose that perturbation theory calibrate the complete nonperturbative functional integral.
Maybe it is worthwhile also to mention that our solution does not violate any symmetries of Quantum Chromodynamics.We would like to underline once more that our solution to the strong CP problem obeys the principle of renormalizability of Quantum Field Theory and does not involve new exotic particles like axions.