Gauge Boson Mass Dependence and Chiral Anomalies in Generalized Massless Schwinger models

I bosonize the position-space correlators of flavor-diagonal scalar fermion bilinears in arbitrary generalizations of the Schwinger model with $n_F$ massless fermions coupled to $n_A$ gauge bosons for $n_F\geq n_A$. For $n_A=n_F$, the fermion bilinears can be bosonized in terms of $n_F$ scalars with masses proportional to the gauge couplings. As in the Schwinger model, bosonization can be used to find all correlators, including those that are forbidden in perturbation theory by anomalous chiral symmetries, but there are subtleties when there is more than one gauge boson. The new result here is the general treatment of the dependence on gauge boson masses in models with more than one gauge symmetry. For $n_A<n_F$, there are fermion bilinears with nontrivial anomalous dimensions and there are unbroken chiral symmetries so some correlators vanish while others are non-zero due to chiral anomlies. Taking careful account of the dependence on the masses, I show how the $n_A<n_F$ models emerge from $n_A=n_F$ as gauge couplings (and thus gauge boson masses) go to zero. When this is done properly, the limit of zero gauge coupling is smooth. Our consistent treatment of gauge boson masses guarantees that anomalous symmetries are broken while unbroken chiral symmetries are preserved because correlators that break the non-anomalous symmetries go to zero in the limit of zero gauge coupling.


Introduction
In this note, I analyze the position-space correlators of flavor-diagonal scalar fermion bilinears in arbitrary generalizations of the massless Schwinger model and focus on the behavior of gauge anomalies in the limit in which some gauge couplings go to zero.We will see that when the dimensional gauge couplings are properly accounted for (which I believe is done here for the first time) the limit is smooth.
We consider a general Schwinger model in 1+1 dimensions with n F massless Dirac fermions, ψ α for α = 1 to n F , and n A ≤ n F vector bosons, A µ j for j = 1 to n A .The Lagrangian is (using summation convention where it does not cause confusion) Note that we have assumed that the gauge couplings are diagonal in the fermion space.This is important to ensure that the model is exactly solvable.We could have non-diagonal couplings as long as the couplings to different gauge bosons commute with one another.But then we can simultaneously diagonalize them with a unitary transformation on the fermion fields, so we will assume that the gauge bosons only couple to diagonal fermion currents.
The gauge couplings are an n F × n A matrix, e in which e αj is the coupling of the jth vector to the αth diagonal fermion current. ( If n F < n A , there are linear combinations of the gauge bosons that do not couple to anything and can be safely ignored, so we will always assume that n F ≥ n A .Each of the massless fermions generates a contribution to the vector boson mass matrix [1] and we will use our freedom to redefine the vector fields to diagonalize the physical vector boson mass so we can write Then the m j are the physical gauge boson masses.The basis of the fermions is fixed up to unbroken flavor symmetries because the currents must be diagonal.And unless some gauge boson masses are equal, the basis of the gauge boson fields is also fixed by (3).We work in Lorenz gauge, Then a generalization of the arguments of [1] 2 shows that the gauge invariant correlation functions to all orders in perturbation theory can be found using the free-field Lagrangian describing massless fermions, Ψ α , massive bosons, B j , and massless scalar ghosts, using the replacements The massive and the massless free scalar propagators, respectively, are3 ξ ≡ e γ E /2 where γ E is Euler's constant (10) The arbitrary dimensional constant in the logarithmic ghost propagators has been fixed so that the B j and C j propagators exactly cancel when the gauge couplings vanish, m j = 0. We will focus on the position-space correlators of the flavor-diagonal fermion-bilinear scalar operators While (11) gives a complete description of the O α± correlators to all orders in perturbation theory, the nonperturbative effects of gauge anomalies are much more transparent if we bosonize.For massless free fermions in 1+1, any non-zero correlator of diagonal fermion bilinears can be calculated by replacing the bilinears with exponentials of free massless scalar fields according to The M α s are arbitrary and correlated with the propagators of the bosonization fields, D α .
There is no mass in the position space correlators of the fermion bilinears and the mass M α is introduced by the bosonization procedure which requires a mass to get the dimensions right.( 12) is a straightforward consequence of Fermi-Dirac statistics.Then The key to using ( 14) most effectively will be to cancel the ghost fields with linear combinations of the bosonization fields.This will be easier if we change notation slightly.
First, we define an orthogonal n F ×n F matrix η αj where the α index labels the fermion as usual but the j index is extended to make the matrix square and a subset b of the j indices are associated with the gauge fields.
we can write Now we can define linear combinations of the bosonization fields and write In this form, it looks like we can cancel the D j fields for j ∈ b with the ghosts.That is precisely what happens in the original Schwinger model and in the Schwinger model with flavors.But in general, this cancellation is not exact because of the different masses associated with different gauge couplings.
The D j propagator is Thus for the D j for j = 1 to n A to exactly cancel the ghosts C j we must have and ( 18) would be most useful if the D j propagator were diagonal for all j.In general, this is impossible and the most useful thing we can do is to choose a common bosonization mass M α = m for the D α s: We could always cancel the ghost for a single j by choosing m = m j .This is the way the issue is typically handled in models with a single gauge boson but it is not adequate for n A > 1.
If we adopt (22), we can almost cancel the ghosts in general.
where b ′ is the complement of b and the E j are "constant" fields with propagators The E j fields keep the engineering dimensions right while eliminating the dependence on the arbitrary bosonization mass m. (23-25) can be used to calculate any matrix element that is non-zero in perturbation theory and we can simplify (23) further by eliminating the E j operators. 4ook at a general perturbatively non-zero correlator involving only bilinears (no higher powers).If there are n α O α+ s there must also be n α O α− s or the correlator would vanish in perturbation theory.We can look separately at the contributions from each E j because the propagators don't mix.So we are interested in We have put arguments (x r ) and (y r ) on the E j from O α+ (x r ) and O α− (y r ) for notational convenience, but the E j are constant operators so the Wick expansion of (27) is independent of the coordinates.The contribution from contractions of the E j (x r )s with the E j (y r )s is The contribution from contractions of the E j (x r )s with as is the contribution from contractions of the E j (y r )s with This means that we can associate a factor with each operator in each O α+ -O α− pair and completely capture the effects of the E j s for perturbatively non-zero correlators.Now we can eliminate the E j s and write We can use (32) to calculate any correlator that is non-zero in perturbation theory.A particularly nice feature of (32) is the way it encodes the diagonal chiral symmetries.The theory is invariant under global translations of the massless D j fields for j ∈ b ′ .These translations generate the n F − n A unbroken chiral symmetries on the fermion bilinears.
O α± has j chiral charge q j = ±η αj for δD j = θ j with j ∈ b ′ . (33) There are no unbroken chiral transformations from translations of D j for j ∈ b, because these D j s have been eaten by the ghosts.The corresponding chiral transformation from translations of the B j for j ∈ b are softly broken by the B j mass terms generated by the gauge anomalies.

Cluster Decomposition
While we only calculate to all orders in perturbation theory, in some situations, cluster decomposition gives us nonperturbative information.For any combination of n o fermion bilinears in some region of space-time, we can look at a correlator that also includes all the conjugate fields in a region far away in space as in 0 where the s u are ±1, as in (11) and the x a and y a are clustered in some region of size ℓ (−(x a − x b ) 2 < ℓ 2 ) around the (arbitrary) origin and z is a large space-like 2-vector.This correlator is calculable in perturbation theory.Using (32), we can write it as where where We can now study the correlator as −z 2 goes to infinity.If b ′ = ∅, that is for n F > n A the correlator will in general go to zero, with the Z factor falling off like 1 where Q j is the total j chiral charge of the operators in the X region, Thus j∈b ′ Q 2 j is the anomalous dimension of the leading operators in the operator product expansion of the set {O αusu (x u )} and I will refer to this as the anomalous dimension of the set and we will have more to say about it below.
But we will begin by assuming that n A = n F so there are no massless fields (b ′ = ∅) and Then for sufficiently large z, all the Bessel functions go to 0 and Z → 1.If the correlator is non-zero as the regions move infinitely far apart, it must factor into a product of contributions in the two regions. 0 This gives non-perturbative information about anomalies.Conversely, if the correlators in the two regions are forbidden by unbroken symmetries, the correlator (34) must vanish as z → ∞.
The above argument means that for n A = n F , up to phases [4,5,6] that are arbitrary and that we will set to zero, we can use (34-37) to calculate any correlator using (32) whether or not it is non-zero in perturbation theory and the result is This makes sense because when b ′ = ∅ with no D fields, there are no global chiral symmetries that are not broken by the gauge boson mass terms.So there are no constraints on the use of (34-37).Because the bosonization mass m is arbitrary, it will cancel explicitly in (42), so we can also write (42) entirely in terms of the m j s as 0 where Equations ( 31), ( 42), (44), and (43) are the basic technical results of this note.(31,42) and (44,43) are equivalent, but (31,42) is more convenient for taking m j s to zero which we will do in the next section.The m j dependence in (31) and ( 44) is useful in understanding the structure of the general theory.Now we can use (31,42) to analyze the general theory for b ′ = ∅.While we could do the general analysis directly, constructing the general theory as the limit of (42) as some gauge couplings go to zero automatically includes the nonperturbative effects of the gauge anomalies and will give us insights into their structure.It may seem surprising that this works at all because the corresponding limit in 3+1 dimensions is quite dangerous.But here everything goes smoothly because we have properly included the important dimensional parameters in (31).
In particular, look at the limit m j → 0 for some j in (42).Because of the product structure of the correlator, we can focus just on the factors that depend on j: If m j −(x u − x u ′ ) 2 is much less than 1 for all the coordinate pairs, we can approximate the Bessel functions by logs: This can be written Finally for b = {1}, if the gauge couplings are not commensurate -as in.
there are no anomaly sets.[7] If there is more than gauge boson, more complicated scenarios are possible.If the components of η αj are commensurate for each j ∈ b, there is an independent pair for anomaly sets for each j ∈ b and they may be reducible.For example, with two gauge bosons, b = 1, 2, we could have a "diagonal color" model: [8,9,10,11,12] The irreducible anomaly sets are {O 1± , O 2∓ }, {O 2± , O 3∓ }, and {O 3± , O 1∓ }.But incommensurate charges can further eliminate all the anomaly sets as in (61) or just limit them, as in this example (again with b = 1, 2): Here because the only linear combination of η α1 and η α2 with commensurate components is η α1 + η α2 , the only irreducible anomaly sets are {O 2± , O 3± }.
One might ask5 whether commensurate components of the η αj vectors is associated with quantization of charge.That is obviously the case for n A = 1, as shown in examples (57,58,61).For n A > 1, the situation is more complicated because generically, the mass-eigenstates of the gauge fields will be linear combinations of the fields in the original Lagrangian.But the η αj for j ∈ b ′ are orthogonal to ALL of the η αj for j ∈ b and thus to any linear combination.So if there is any linear combination of the η αj for j ∈ b with commensurate coefficients, it will be associated with an anomaly set AND the corresponding combination of gauge fields will be coupled to a quantized charge.For example, in (63), the combination η α1 + η α2 describes a coupling to a quantized charge.
In general, the independent anomaly sets will be associated with the independent linear combinations coupled to quantized charges.This is exactly what we would expect from Coleman's interpretation of a θ parameter as a background electric field [13].Only the linear combinations corresponding to quantized charges will have θ parameters because a background field coupled to a non-quantized charge can be canceled by pair production of massless fermions.Thus the independent anomaly sets will be associated with the independent θ parameters.This connection can be obscured in models with more than one gauge boson by a nontrivial gauge boson mass matrix, but we have shown how a complete analysis of the mass dependence describes the general case.

Conclusions
While the physics is trivial, the correlators in generalizations of the massless Schwinger model depend in nontrivial ways on the gauge boson masses.I believe that the complete analysis of the mass dependence in this note clarifies the relationship between models with different numbers of gauge bosons and chiral anomalies.