Revisiting the Okubo-Marshak argument

Modular localization and the theory of string-localized fields have revolutionized several key aspects of quantum field theory. They reinforce the contention that local symmetry emerges directly from quantum theory, but global gauge invariance remains in general an unwarranted assumption, to be examined case by case. Armed with those modern tools, we reconsider here the classical Okubo-Marshak argument on the non-existence of a"strong CP problem"in quantum chromodynamics.


Introduction: string-localized fields
In this paper the case by Okubo and Marshak [ ] against existence of the "strong CP problem" in quantum chromodynamics, which was based on the covariant approach to Yang-Mills theory by Kugo and Ojima [ ], is reassessed from a different theoretical standpoint. For the purpose we bring to bear the theory of string-localized quantum fields (SLF).
Those are not a recent invention. The paper by Dirac [ ] should be regarded as a precedent. Charged SLF, recognizably similar to the modern version, were a brainchild of Mandelstam [ ]. In a different vein, they were treated by Buchholz and Fredenhagen [ ]. In the eighties they were further developed by Steinmann [ -]. They resurfaced over years ago in still pending. The applications of SLF so far concern mostly questions of principle, badly dealt with within conventional QFT. Of which there are plenty. This paper addresses one of them, that can be handled by means of tree graphs.

. Massless bosonic SLF: general theory
The term "string" in the present context precisely denotes a ray starting at a point in Minkowski space M 4 that extends to infinity in a spacelike or lightlike direction. This is the natural limit of the spacelike cones in the intrinsic localization procedure of [ ]. The set of such strings can be parametrized by the one-sheet hyperboloid 3 M 4 ("de Sitter space") of neck radius equal to one.
The denominator ( ) in ( , ) is shorthand for the distribution ( ) + 0 −1 , to be smeared in the string variable . The following key relations, respectively integral and differential, are effortlessly derived: so that ( , ) is a bona fide potential for ( ). For the first identity: ( ) It ought to be clear that the vector potential ( , ) fulfils the equations ( , ) = ( , ) = 0. These "transversality" properties are necessary for the free field ( , ) Not to be confused with the strings of string theory. Use of the limiting case of lightlike strings, parametrized by the celestial spheres 2 ∪ 2 M 4 , simplifies some formulae. They are a little troublesome from the functional-analytic viewpoint, however. acting on the physical Hilbert space [ , Sect. ]. Their role is to reduce the number of degrees of freedom, as required for on-shell photons -in much the same way as the six components of the electromagnetic field reduced by the Maxwell equations propagate two degrees of freedom.
The reader should appreciate fully the deep differences between ( , ) and the ( ) potentials of standard QFT. In particular, there are no "pure gauge" configurations in QED or QCD, when working in SL field theory. The field ( , ) lives on the same Fock space as . Thus the second equation in ( ) is an operator relation; which is not the case for the similar one in the usual framework. All this makes easier the physical interpretation of the present one.
In contrast with the standard formalism, ( , ) is perfectly covariant: for Λ a Lorentz transformation, a translation, and the second quantization of the mentioned unirrep pair of the Poincaré group: It is important to realize that the string-differential of the photon field is a gradient: Naturally, satisfies the wave equation and ( ) 2 = ( , ) = 0. Last, but not least, locality: holds whenever the strings + R +˜ and ′ + R + ′ are spacelike separated for all˜ in some open neighbourhood of : "causally disjoint". A proof is found in [ , App. C].
fields and a test function ℎ ∈ D( 3 ) with integral 1 that averages over the string directions. S[ ; ℎ] is submitted to the customary conditions of causality, unitarity and covariance. One looks for it as a formal power series in , where is the measure on 3 . Only the first-order vertex coupling 1 = 1 ( , ), a Wick polynomial in the free fields, is postulated -already under severe restrictions. It depends on an array = ( 1 , . . . , ) of string coordinates, with the maximum number of SLF appearing in a sub-monomial of 1 . For ≥ 2, the are time-ordered products that need to be constructed.
is a set of measure zero that includes the diagonal = ′ . A similar statement holds for > 2. Only that part of contributes to S[ ; ℎ] which is symmetric under permuting the string coordinates, which are smeared with the same test function ℎ. The extension of the -products across D is the renormalization problem in a nutshell [ ].
The natural and essential hypothesis of interacting SLF theory is simple enough: physical observables and quantities closely related to them, particularly the S-matrix, cannot depend on the string coordinates. This is the intrinsically quantum string independence principle: colloquially, the strings "ought not to be seen". In this paper it will replace the "gauge principle" with advantage.
For the physics of the model described by 1 to be string-independent, one must require that a vector field ( , ) exist such that, after appropriate symmetrization in the string variables so that on applying integration by parts in the "adiabatic limit", as goes to a set of constants, the contribution from the divergence vanishes. Then the covariant S[ ; ℎ] approaches the invariant physical scattering matrix S, therefore ( , Λ)S † ( , Λ) = S, all dependence on the strings disappearing. On the face of it, existence of the adiabatic limit is the property that the be integrable distributions. Due to severe infrared problems, the latter does not hold in QCD, which involves us here. However, a recent breakthrough [ ] rigorously establishes existence of a suitable weaker adiabatic limit (WAL) in QCD, and so the above reasoning can proceed. This symmetry will be heavily exploited in the following derivations. The cited work is concerned with point-local fields. It can be generalized to the string-local setting in low orders. A proof of WAL at all orders in the SLF context is still awaiting.

. The Aste-Scharf argument recast in SLF theory
Proposition . Suppose that we are given massless fields , ( = 1, . . . , ). For their mutual cubic coupling modulo divergences, string independence at first order enables the Wick product combination: where the are completely skewsymmetric coefficients. (Subindices that appear twice are summed over.) Before proceeding, we note that this vertex promulgates a renormalizable theory by power counting. Note also that 1 is intrinsically symmetric in the string coordinates, and that Proof of Prop. . We abbreviate ≡ ( , ) for the SLF and make the Ansatz: for the cubic coupling of the fields, the coefficients 1 being a priori unknown. We shall show that string independence forces this to be of the form 1 as in ( ), where those numerical coefficients are completely skewsymmetric. We first split 1 =: + 2 into a symmetric ( = ) and a skewsymmetric part ( 2 = − 2 ) under exchange of the second and third indices. After symmetrizing ( ) in 2 ↔ 3 , the contribution of yields a divergence: We next split the coefficients 2 = + + − into symmetric and skewsymmetric parts under exchange of the first two indices, ± = ± ± . The skewsymmetric part − does enter into ( ), whereas the symmetric part + contributes ( ) By our basic postulate, this must be a divergence. Since the operators 2 3 1 are linearly independent, that can happen iff the symmetric part + is identically zero.
This means complete skewsymmetry of the 2 ≡ − =: . That is to say, the string independence principle constrains ′′ 1 in ( ) to the form so that the dependence on 3 is trivial and formula ( ) with the stated skewsymmetry condition is established.
It is worth pointing out that the above reasoning for the form of 1 becomes simpler in our SLF context than in the quantum gauge invariance approach [ , Sect. . ], the inference there being in terms of the customary fields and their ungainly retinue of unphysical ones.

. Dealing with string independence at second order: preliminaries
Perturbative string independence should hold at every order in the couplings, surviving renormalization. Now, the do not yet a Lie algebra make; for that one needs to prove a Jacobi identity. This is going to be obtained from string independence in constructing the functional S-matrix at second order in the couplings.
Outside the exceptional set D 2 from ( ), time-ordered products of string-local fields reduce to ordinary products where the order of terms is determined by the geometric time-ordering of the string segments. There, string variation and derivatives commute with time ordering and we have Above, T denotes a generic time-ordering recipe, that is, an extension of the time ordering across D 2 . It does not automatically follow that ( ) holds over the whole ( , 1 , 2 ; ′ , ′ 1 , ′ 2 ) set. The challenge is to manufacture a time-ordered product 2 so that this property holds everywhere after appropriate symmetrization in the string variables. The construction of 2 by solving the obstructions to such an identity will impose the couplings of "non-Abelian gauge theory". The vector quantity will play a central role in our development. Candidate extensions across D 2 are restricted by the requirement that the Wick expansion hold everywhere. We are concerned only with the tree graph for gluon-gluon scattering. Its corresponding amplitude is of the general form where − denotes a vacuum expectation value, the sum in the brackets goes over all the free fields entering the monomials ( ), ( ′ ), and we employ formal derivation within the Wick polynomials.
For time-ordered products of the fields entering 2 in our model, we naturally consider in the first place: for the appropriate spacetime indices * , •, in terms of the respective intertwiners. We need: to be found for instance in our [ ]. Note that in all generality and similarly for ′ . The problem of resolving the obstructions to Eq. ( ) will be reduced to an extension problem for numerical distributions by carefully constructing the contractions ( ). At present we have, with completely skewsymmetric coefficients : Clearly 1 is symmetric under exchange of the string variables. We want 2 ( , 1 , 2 ; ′ , ′ 1 , ′ 2 ) to be symmetric under exchange of ( , 1 , 2 ) ↔ ( ′ , ′ 1 , ′ 2 ). Therefore, resolving all obstructions to Eq. ( ) is equivalent to inspecting the obstruction to investigate how it can be made to vanish after appropriate symmetrization. To the purpose, with the delta-function along the string defined as one obtains, for similarly coloured gluons: These formulae are easily derived by use of Eqs. ( ), ( a) and ( b), respectively.

. The Jacobi identity emerges
We compute, with an eye on ( ) and another on ( ): , and similarly. Employing ( a) and ( b), this obstruction equals as well as skewsymmetry of and Maxwell's equations for ′ , we see that terms of the type form a divergence of an expression supported at the exceptional set D 2 . Integrating by parts in the first line ( ), the obstruction reads, up to a divergence supported at D 2 , We have used that 1 = 1 1 . On symmetrizing in the variables 2 , ′ 1 , ′ 2 , the line ( a) becomes proportional to It follows that the Jacobi identity: is a necessary condition for the obstruction to string independence to vanish.

. The quartic term
We still have to deal with the term of the type ∼ 1 in ( b). Using the skewsymmetry of the , symmetrization of ( b) in the variables ′ 1 , ′ 2 yields the identical symmetrization of whose string-derivative 1 cancels the expression ( ), must be added to 2 . This is the fourgluon coupling, usually attributed to the "covariant derivative" present in the "kinetic" QCD Lagrangian. Now, since in the present dispensation an term is also renormalizable to begin with, we could have introduced it from the outset. Then a discussion parallel to the above leads again to Eq. ( ) with precisely the same second-order coefficient in the coupling constant.
In summary: string independence of the S-matrix at second order holds if and only if the Jacobi identity with completely skewsymmetric for the cubic coupling ( ) holds and the above quartic term ( ) is present at that order in the S-matrix.

Discussion
The outcome of the previous arguments, together with the lessons on QFD in [ ], is that Lie algebra structures of the compact type of necessity govern interactions in QFT. Compactness is related to complete skewsymmetry: a finite-dimensional Lie algebra structure with generators defined by [ , ] = does require = − and the Jacobi identity, but not = − in general. The extra requirement imposed by string independence leads to a negative definite Killing form and thus semisimple compactness -see for instance [ , Sect. . ].
The authors presently know of five different arguments within perturbative QFT for the reductive Lie algebra structure of the interactions: the already classical one by Cornwall, Levin and Tiktopoulos from unitarity bounds at high energy [ ]; the Aste-Scharf analysis referred to in subsection . ; the one in this paper, and two more found in the book [ ]: one in its Chapter , by elaborating on the spinor-helicity formalism for gluon scattering calculations, and another -most charming of them all -in its Chapter , by a variant of Weinberg's "soft limit" reasoning, long ago applied to link helicity 1 particles with charge conservation and helicity 2 particles with universality of gravity. The irrelevance of the gauge condition in the old direct construction of scattering amplitudes by Zwanziger [ ] also comes to mind. This is a good place to take stock of the lessons from our [ ]. Our argument there was also motivated, in a somewhat contrarian way, by Marshak's thoughts on the "neutrino paradigm" in his posthumous book Conceptual Foundations of Modern Particle Physics [ , Chaps. , ] -see [ ] as well. The sole inputs of our treatment of flavourdynamics in it are the physical particle types, masses and charges: "spontaneous symmetry breaking", which comes in succour of the conventional gauge models, is not required, since the SLF models are renormalizable to begin with, irrespectively of mass. String independence at first and second order rules the bosonic couplings, just as in gluodynamics; the non-vanishing masses just bring minor complications. Couplings between bosons and fermions a priori are just asked to respect electric charge conservation and Lorentz invariance. Chirality of the couplings is the outcome of general string independence. The proof requires the presence of the scalar particle and is done with Dirac fermions: from the standpoint of SLF it does not make sense to say that lepton or quark currents are "chiral" in QFD: their couplings are.
The cited book by Marshak is still a very good guide for the discussion that followswitness perhaps to the lack of progress and "deeply disturbing features" [ ] affecting the present theoretical apparatus, contrasting with tremendous progress in the experimental realm during recent years.

. Story of two principles
The "gauge principle", a top-down, classical-geometrical principle which has ruled particle theory for over sixty years, is foreign to QFT. Looking back, a defect -the unavailability in conventional QFT of a Hilbert space framework for massless particles -was elevated into a doctrine. By now bottom-up, inherently quantum principles for the construction of interactions deserve their place in the sun.
If only for sound epistemological reasons, one should ascertain whither the bottom-up approach leads, in present-day particle physics. A Lie algebra and a local Lie group amount to the same thing. Now, the mathematical beauty and power of spin-offs like extended field configurations in classical Yang-Mills geometry is not to be denied -it is enough to recall [ ]. And in turn, it is perfectly legitimate to look for global Lie group features in modern, bottom-up QFT. However, we contend that those need to be constructed as quantum field theory entities -and, as anyone who has had to grapple with (say) the rigorous definition of Wilson loops for interacting fields in SLF formulations well knows, this is more intricate than presuming classical geometry entities to possess non-perturbative quantum counterparts.
Consider, for instance, Dirac's ad-hoc non-integrable phase and magnetic monopole (and their progeny of 't Hooft-Polyakov monopoles). The basic idea was seductive, and led directly and elegantly to electric charge discreteness. Nevertheless, humbler, essentially perturbative arguments on cancellation of anomalies -up to and including the mixed gravitational-gauge anomaly [ ] -recalled by Marshak -see [ , Chap. ] and [ ], among others [ ] -are known to provide an explanation for that discreteness [ , Sect. . ]. Whereas magnetic monopoles of any kind have stubbornly refused to show up.
In the current parlance, fermions in the Standard Model are schizophrenic: non-chiral in QCD, chiral in QFD. To be sure, from the SLF treatment one can reengineer the GWS model and its warts, hypercharges and all, in reverse. This was done in [ , App. D]. This is perhaps the place to comment on the Aharonov-Bohm and Aharonov-Casher effects being held as "proofs" of the "fundamental" character of the standard electromagnetic gauge potential -since the calculation via the electromagnetic field depends on a region where the test particle is not allowed. This is merely a misunderstanding: these effects can be computed by means of the SL -field, which contains the same information as the -field. The deep reasons lie in the mind-boggling entanglement properties of QFT, as compared to ordinary quantum mechanics -concretely in the failure of Haag duality for all quantum massless fields with helicity ≥ 1. This was shown over forty years ago [ ]. Consult [ ] in this respect as well.

. Reassessing the Okubo-Marshak argument
Marshak was very open to topological and differential-geometric constructions in QFT, and actually Chapter of his book [ ], particularly Section . and Subsection . .c, is still a very good place to learn about instantons and "vacuum tunneling" into topologically inequivalent vacua, apparently leading to the degenerate -vacuum, the "strong problem" -since the neutron's electric dipole moment is vanishingly small -and, according to some, to the "axion".
The steps to the claim of existence of such a problem are well known. In the Euclidean setting first, finiteness of the classical action functional of course requires lim | |↑∞ ( ) = 0. For which it is enough to demand that lim | |↑∞ ( ) → † ( ) ( ). It is then said that is "pure gauge". By using an 0 gauge, a winding number is related directly to the Euclidean action -in fact, one is dealing with the homotopy group of the 3-dimensional sphere, which is the group of integer numbers. The next step is to define the vacuum states in Minkowski space in such a way that instantons become "tunneling events" between two different Minkowski vacua | , | ′ with respective winding numbers , ′ satisfying ′ − = . Then it is argued that the "true vacuum" is of the form | = − | with ′ | = 2 ( − ′ ); and the value of is anyone's guess. This is aptly described by Marshak in [ , Subsect. . .c].
Nevertheless, he came to regard the axion hypothesis as a bridge too far. In [ ] it was rigorously proved that the BRS charge "kills" the physical vacuum, which if cyclic must be unique. But that charge and the antiunitary operator for CPT invariance commute, and this obviously demands the zero (or ) value for the -parameter of the instanton makeup [ ]. One could contend that the -vacua are non-normalizable (a sign of trouble in itself) and that the physical vacuum is a superposition of them. However, by means of a simple procedure, Okubo and Marshak showed how in that case CPT invariance still guarantees the experimentally measurable value of to be zero.
The existence of the original "visible" axion had been already disallowed by experiment and observation [ ] by the time of the writing of [ ], leading to a succession of "invisible" axions, that must be "super-light", but still face experimental limitations. Marshak concludes: "It does seem that the odds of finding the 'invisible' axion are rapidly diminishing and that the incentive to carry on the ingenious searches for the 'invisible' axions is fueled more by astrophysical and cosmological interest than by any hope of salvaging the Peccei-Quinn-type solution of the 'strong CP problem' in QCD." This rings even truer years later.
Making a contention to the same purpose in SLF theory is straightforward. The stringlocalized vector potentials live on Hilbert space and act cyclically on the vacuum. In fact, every local subalgebra of operators enjoys this property (this is part of the Reeh-Schlieder property) [ ]). Therefore -vacua are not allowed.
In plainer language: ( ) ↓ 0 at spatial infinity implies ( , ) ↓ 0. Thus only the = 0 vacuum (so to speak) occurs: there are no instantons, and no vacua, and the so-called strong problem is solved. The moral of this part of the story: quantum field theory should stand on its own feet, rather than on classical crutches.

. Coda
The "strong CP problem" and the axion ideas have undergone mutations from the time of their inception to the present day. Relations of the present-day "invisible" axion with the (1) anomaly remain murky -for a recent review of the latter consult [ ]. The contemporary main selling point is still the "axiverse"; in other words, the search for the axion is essentially model-free nowadays. Absence of evidence is not evidence of absence, to be sure -and so hunting for ALPs is bound to go on [ ].
A weak point of most analyses on the present subject is that confinement is not taken into account. However, it does appear to be inimical to violation of CP invariance [ ]. of the Universidad de Costa Rica, MINECO/FEDER project PGC --B-I , and the COST action CA . He still treasures a reprint of [ ] handed to him by Robert E. Marshak during the latter's only visit to Spain, in the fall of , shortly before his untimely death. We are grateful to E. Alvarez, P. Duch, A. Herdegen, K.-H. Rehren, B. Schroer, I. Todorov and J. C. Várilly for comments, discussions and helpful suggestions.

A Time ordering outside the string diagonal
For a vertex coupling 1 of the form ( ), we show 2 := T 1 ( , 1 , 2 ) 1 ( ′ , ′ 1 , ′ 2 ) is fixed outside the string diagonal D 2 from ( ) by the causal factorization property depending on the time ordering of the respective localization regions.
In [ ] the corresponding statement has been shown for the case of first degree stringlocalized fields by "chopping" the strings, and it was indicated why the construction runs into difficulties for higher-degree Wick polynomials. In the present case, we have the lucky circumstance that the relevant Wick products are just ordinary products, see ( ), and we are back to the first-degree case. We give the argument in detail.
The localization region of 1 ( , 1 , 2 ) are the two strings + R + , each of which we chop into a small compact "head" R containing the vertex and an infinite "tail" S : ( ) Accordingly, the string-localized potential decomposes as ( , 1 ) = H ( , 1 ) + T ( , 1 ), where H is localized in the compact "head" R 1 and T in the "tail" S 1 . Same with ( , 2 ). Then the vertex coupling 1 is a linear combination of four terms: : .
( ) The terms TH 1 and TT 1 can be written as and similarly for HT 1 . We have taken T ( 1 ) and T ( 2 ) out of the Wick product; this can be done because all contractions between , and are zero since the indices , , are distinct by skewsymmetry of . The localization regions of these terms are as follows. The operator product in TT 1 is the ordinary product of three linear fields: one localized in the string S 1 , another in S 2 and a third at . TH 1 is the product of a linear field localized in the string S 1 and a Wick product localized in the compact interval R 2 that can be made arbitrarily small around . Similarly for HT 1 . The term HH 1 is a Wick product localized in R 1 ∪ R 2 which also can be made arbitrarily small around .
We decompose 1 ( ′ , ′ 1 , ′ 2 ) in like manner. Then the second order 2 is a sum of terms of the form T KL 1 ( 1 , 2 ) K ′ L ′ 1 ( ′ 1 , ′ 2 ) , where the operator in brackets is the product of string-localized linear fields and (almost) point-localized Wick products. By our hypothesis that 1 ≠ 2 and ′ 1 ≠ ′ 2 , their localization regions are mutually disjoint. By [ , Prop. . ], such regions can be chronologically ordered, eventually after chopping them into segments, and the corresponding operators can be time-ordered according to ( ). As in the proof of in [ , Prop. . ], one sees that the result is Wick's expansion, where the time-ordering appears only within two-point functions. (In contrast to the case considered in [ ], here there are products of time-ordered two-point functions, but these are well-defined since they have disjoint arguments.) Again by skewsymmetry of , only contractions between fields localized on -and ′ -strings occur. Therefore, the restriction 1 ≠ 2 and ′ 1 ≠ ′ 2 can be removed. The proof is complete. [ ] Gaß, C. "Renormalization in string-localized field theories: a microlocal analysis", arXiv , arXiv: . [math-ph]. [ ] Y. Nakamura and G. Schierholz, "Does confinement imply CP invariance of the strong interactions?", Proceedings of Science (LATTICE ), ( ), ; arXiv: . .