CP and T violation in non-perturbative chiral gauge theories

We give a completely general derivation revealing the precise origin and the quantitative effects of CP and T violations in chiral gauge theories on the lattice.

Obviously the form of the chiral projections based on (1.1) represents a rather special case. Thus firstly the question arises whether the observed symmetry violations really persist in general. Secondly instead of only tracing the violations back to a parameter singularity to reveal the precise reason for them is preferable. Thirdly then to get quantitative hold of the violations is desirable.
In a more general approach [8] it has been shown that the symmetric situation of contiuum theory in the CP case is not admitted due to certain operator properties. The operators G,Ḡ and D there have been functions of a basic unitary operator. Though this formulation includes all chiral operators discussed so far as special cases [9], relying on the mentioned unitary operator introduces unnecessary restrictions and forms an obstacle for a more thorough investigation of the indicated symmetries.
To investigate CP, T and CPT symmetries in a general way, we here first analyze the possible properties of the chiral projections starting from the Dirac operator and imposing only minimal conditions. We find that due to a contribution which inevitably comes with opposite sign in G andḠ one generally getsḠ = G. Furthermore, since the overall sign of the respective contribution remains open, it becomes obvious that in the construction of the chiral projections one is confronted with two distinct possibilities, of which one must be chosen to describe physics.
We next show that CP transformations as well as T transformations interchange the rôles of G andḠ. This together with the fact that one generally hasḠ = G then is seen to constitute the origin of the symmetry violations. With respect to the need of choosing one of the mentioned two possibilities in the construction the interchange under CP and under T transformations means to violate the original choice. On the other hand, CPT symmetry is seen to be generally there and not to be affected byḠ = G.
Finally, considering correlation functions for any value of the index, we point out that the symmetry violation effects enter them via the bases involved. To get quantitative hold of such effects we note that if the related interchange of G andḠ would be supplemented by a change of the respective sign in the construction one would get symmetry. Thus the effect of the violation turns out to be given by the difference of the results for the two sign choices and is seen to become manifest in entirely different subsets of bases contributing to the correlation functions.
In Section 2 we introduce basic relations and analyze the possibilities for the chiral projections. In Section 3 we derive the properties for CP, T and CPT transformations. In Section 4 we consider the effects caused in correlation functions. Section 5 contains our conclusions.

Basic relations
Introducing chiral projections P ± andP ± with P + + P − =P + +P − = 1l and requiring that they satisfyP we get the decomposition of the Dirac operator into Weyl operators Considering P − andP + in the following, we write them in the form which because of P † − = P − = P 2 − andP † + =P + = P 2 + implies unitarity and γ 5 -Hermiticity, According to (2.1) the operators G andḠ are subject to

Spectral representations
We consider a finite lattice and require the Dirac operator to be normal, [D † , D] = 0, and γ 5 -Hermitian, D † = γ 5 Dγ 5 . It then has the spectral representation where the eigenvalues are all different and satisfy Imλ j = 0 and Im λ k > 0. For the orthogonal projections the relations γ 5 P j = P j γ 5 and γ 5 P I k = P II k γ 5 hold and we have where we associate P 0 toλ 0 = 0, i.e. j = 0 to the zero modes of D.
We require that G andḠ are functions of D, which means that their eigenvalues are functions of those of D. Using this and imposing (2.5) we obtain for G andḠ the spectral representations with η j andη 0 taking the values ±1 and phases ϕ k ,φ k being subject to At this point we make the important observation that because of the opposite signs of the j-sums in (2.8) one generally hasḠ = G.

Chiral features
Since γ 5 P j = P j γ 5 we get the decomposition P j = P + j + P − j with γ 5 P ± j = P ± j γ 5 = ±P ± j . Furthermore γ 5 P I k = P II k γ 5 implies Tr(γ 5 P I k ) = Tr(γ 5 P II k ) = 0. For N ± j = Tr P ± j then according to Tr(γ 5 1l) = 0 and (2.7) follows. The index of D is given by For the numbers of the Weyl and anti-Weyl degrees of freedom N = Tr P − andN = TrP + we thus obtain These relations allow us to discuss (2.8) in more detail. Firstly we see that (2.11) gives N − N = 1 2 (η 0 + η 0 )I. Therefore in order to haveN − N = I we must putη 0 = η 0 = 1. Next we note thatN + N = 2d + K. If there is only one term in the j-sums of (2.8), as has been the case for all operators discussed so far [9], using (2.10) we obtain K = −η 1 I. For I = 0 this quite reasonably impliesN + N = 2d. In the more general case admitted here putting η j = η for j = 0 we get K = −ηI and thus alsoN + N = 2d for I = 0. Finally we see that to minimize the differences between G andḠ the k-sums in (2.8) can readily be made equal by requiringφ = ϕ. We thus arrive at the form and remain with the two possible choices η = 1 or η = −1. The important observation here is that to describe physics we are forced to decide for one of such choices.

Transformation properties 3.1 CP transformations
With the charge conjugation matrix C and with R P n ′ n = δ 4 n ′ n P , U CP 4n = U * 4n P and U CP kn = U T k,n P −k for k = 1, 2, 3, where n P = (− n, n 4 ), we have for D the CP transformation in which T denotes transposition and where W CP † = W CP−1 holds. 1 Because G andḠ are functions of D they inherit its transformation properties so that Using (3.2) it becomes obvious that the forms (2.3), The result (3.4) obviously differs from the untransformed relation (3.3) by an interchange of G andḠ. This together with fact that, due to the opposite signs of the j-sums in (2.8), one generally hasḠ = G means violation of the symmetry.
With respect to the need that to describe physics one has to choose a definite value of η in (2.13), the interchange of G andḠ is seen to cause a change violating the original choice of η.

CPT transformations
With R CPT n ′ n = δ 4 n ′ ,−n and U CPT µn = U † µ,−n−μ we get for D the CPT transformation 8) and the analogous relations for G andḠ. Using them we obtain  The basesū σ ′ j and u σi in (4.2) satisfy

Transformations
We now address the case of CP transformations, noting that analogous relations hold for T transformations. With conditions (4.3) and (4.4) being satisfied by u,ū, S,S as well as by u CP ,ū CP , S CP ,S CP , the (equivalence classes of pairs of) bases transform as where the additional unitary operators S ζ andS ζ have been introduced for full generality. Inserting (4.5) into (4.1) gives for the correlation functions where e iϑ CP = detwS ζ · det wS † ζ . Since repetition of the transformation must lead back, S ζ andS ζ are restricted to choices for which ϑ CP is a universal constant. Accordingly the factor e iϑ CP gets irrelevant in full correlation functions and, without restricting generality, we can put ϑ CP = 0.
Though (4.6) then superficially looks "CP covariant", it is affected by the missing CP symmetry of the chiral projections. Indeed, while the pair u,ū is related to one choice of η in (2.13), the pair u CP ,ū CP is related to the other one.

Symmetry-violation effects
To get quantitative hold of the symmetry violations we note that for CP transformations as well as for T transformations an additional change of the value of η in (2.13) would lead to symmetry. Thus the symmetry-violation effect is given by the difference of the results for the two choices of η.
To study this in detail we note that using (2.13) we obtain where . From the representation (4.7) it becomes obvious that for the two choices of η either is involved in the chiral projections. The crucial point now is that this implies the occurrence of the entirely different con- 3) (where since L − − L + = I even the numbers of terms can differ). In the multilinear forms (4.2) then the subsets u + l and u − l of bases, being related to entirely different projections, clearly lead to different results. This in turn causes differences in the correlation functions which give the effects of the symmetry violations.
There remains obviously the question which one of the two choices for η is the appropriate one for the description of physics. However, at present no theoretical principle is in sight to decide about this.

Conclusions
We have investigated CP, T and CPT symmetries in a general way, imposing only minimal conditions, namely normality and γ 5 -Hermiticity of the Dirac operator and that it has a general decomposition into Weyl operators.
We first have analyzed the possible properties of the chiral projections starting from the Dirac operator. It has turned out that due to a contribution in the spectral representations which inevitably comes with opposite sign in the operators G andḠ, which enter the chiral projections, one generally getsḠ = G. Furthermore, because the overall sign of the respective contribution remains open, it has become obvious that in the construction of the chiral projections one is confronted with two distinct possibilities, of which one must be chosen to describe physics.
We next have shown that CP transformations as well as T transformations cause an interchange of the rôles of G andḠ. This together with the observation that one generally getsḠ = G has been seen to constitute the origin of the symmetry violations. With respect to the need of choosing one of the mentioned two possibilities in the construction it has become obvious that the interchange of G andḠ under CP and under T transformations means violation of the original choice. On the other hand, CPT symmmetry has been seen to be generally there and not to be affected byḠ = G.
Finally, using a form of the correlation functions which applies also in the presence of zero modes and for any value of the index, we have pointed out that the symmetryviolation effects enter them via the bases involved. To get quantitative hold of such effects we have noted that if the related interchange of G andḠ would be supplemented by a change of the respective sign in the construction one would get symmetry. Thus the effects of the violations have turned out to be given by the difference of the results for the two choices in the construction of the chiral projections. This has been seen to become manifest in entirely different subsets of bases appearing in the correlation functions.