Helicity operators for mesons in flight on the lattice
Motivated by the desire to construct meson-meson operators of definite relative momentum in order to study resonances in lattice QCD, we present a set of single-meson interpolating fields at non-zero momentum that respect the reduced symmetry of a cubic lattice in a finite cubic volume. These operators follow from the subduction of operators of definite helicity into irreducible representations of the appropriate little groups. We show their effectiveness in explicit computations where we find that the spectrum of states interpolated by these operators is close to diagonal in helicity, admitting a description in terms of single-meson states of identified . The variationally determined optimal superpositions of the operators for each state give rapid relaxation in Euclidean time to that state, ideal for the construction of meson-meson operators and for the evaluation of matrix elements at finite momentum.
for the Hadron Spectrum Collaboration
Hadron spectroscopy is concerned with the study of resonances which appear as poles of hadron scattering amplitudes. These scattering amplitudes are ultimately determined by the underlying theory of quarks and gluons known as QCD. Our best current general-purpose technique for computation of quantities in QCD is lattice QCD, which considers the field theory numerically on a grid of Euclidean space-time points of finite extent. Scattering amplitudes are not directly accessible in the Euclidean theory, but computations of the discrete spectrum of states in a finite volume offer a way to access scattering amplitudes through a formalism presented by LüscherLuscher (1991). Within this framework, narrow meson resonances appear through admixtures of localised single-hadron states, which are interpolated well from the vacuum by local -like operators, and meson-meson states having definite relative momentum determined by the finite-volume boundary conditions.
In order to accurately extract a complete finite-volume meson spectrum, it is necessary to include in a basis of interpolating fields not just local operators resembling single-hadron states, but also some which resemble pairs of mesons projected into definite relative momentum. Our desire then is to form operators that can interpolate the meson-meson pair which will be of the form , where efficiently interpolates meson with momentum (and similarly efficiently interpolates meson ). An optimal operator for with momentum follows from variational diagonalisation of a matrix of single-hadron correlators evaluated with a basis of suitable operators. The subsequent reduction of the contribution of excited mesons to the correlators will mean that the relevant energy levels, and hence signals for scattering, can be extracted at earlier Euclidean times where statistical noise is smaller.
This suggests that we need to develop a basis of single-hadron interpolating fields having non-zero momentum. In lattice QCD this cannot be achieved by simply Lorentz boosting operators at rest, as one is required to take into account the reduced symmetry of a discrete grid of finite extent. The lattice discretisation breaks rotational symmetry at small distances, while the finite volume breaks rotational symmetry at large distances. The latter is of fundamental importance for states at non-zero momentum because the allowed momenta are determined by the boundary conditions which implement the symmetry of the “box”. The relationship between the discrete finite-volume energy spectrum and the infinite volume scattering amplitudes is also determined by the symmetry of the boundary. In most lattice QCD calculations, including those presented in this paper, both the lattice discretisation and the box have the same cubic symmetry, but this need not necessarily be so. For mesons in flight, the symmetry is further reduced to the little group of allowed cubic rotations that leave the meson momentum invariant and it is operators transforming irreducibly under this reduced symmetry group that we will derive.
As well as their use in constructing meson-meson operators, in-flight meson interpolators play a significant role in computations of meson matrix elements of certain currents, such as those used to measure electromagnetic and weak form-factors. Including operators which interpolate mesons at non-zero momentum increases the number of kinematic points available, allowing the form factor to be calculated at more values. These quantities can reveal information about the quark-gluon structure of hadrons Dudek et al. (2009a); Collins et al. (2011).
We will build on recent successDudek et al. (2010) in constructing a basis of meson interpolators at rest that transform irreducibly under the cubic symmetry of the lattice. It was found that on the dynamical lattices of Refs. Edwards et al. (2008); Lin et al. (2009), the effect of the finite lattice spacing was relatively small; operators subduced from definite continuum spin into irreducible representations of the octahedral group overlapped with states in a manner compatible with an effectively restored rotational symmetry. The observed spectrum of states was explained in terms of single-meson states of determined with values subduced across cubic irreps, and, as expected for a calculation featuring only fermion-bilinear operators, there was no clear observation of a spectrum of meson-meson states whose distribution across irreps is determined by the symmetry of the finite-volume boundary conditions.
In this article we follow a similar philosophy, constructing fermion-bilinear operators for mesons in flight which, having a clear interpretation in an infinite volume continuum, allow the continuum quantum numbers of single-hadron states to be identified. We then demonstrate the effectiveness of these constructions by using them to extract an extensive spectrum of isovector mesons in flight at various momenta, with the continuum spins and parities reliably identified. With the extracted spectrum viewed in the finest detail we will observe effects that may be compatible with the admixture of meson-meson states. We propose that by including meson-meson operators constructed from products of the in-flight operators derived herein, in future calculations we will be able to resolve this admixture precisely and extract meson-meson scattering information. While Ref. Luscher (1991) considered only the case of the total system at rest, it has been shown that more information can be gained from calculations that work in a moving frameRummukainen and Gottlieb (1995); Feng et al. (2011).
We begin in Section II by considering an infinite volume continuum, constructing helicity operators and discussing operator-state overlaps. In Section III we move to a finite volume, showing how these operators subduce into lattice little group irreducible representations to give subduced helicity operators. After giving a brief description of the lattice data sets used and the computational method in Section IV, in Section V we describe the method for identifying the of extracted states and show some examples of its application. The extracted spin-identified spectra of mesons in flight are given in Section VI and we conclude in Section VIII.
Ii Mesons in flight in an infinite volume
ii.1 Helicity operators for mesons in flight
We begin by describing the construction of helicity operators which enable the energies of mesons in flight to be extracted in an infinite volume continuum, before considering the overlap of these operators with states. In Section III we will show how these operators can be subduced into the irreducible representations relevant for a cubic lattice in a finite cubic volume, and how, because they have a definite infinite volume interpretation, they enable the identification of the spin and parity of states.
As a starting point, consider the fermion-bilinear operators constructed in Ref. Dudek et al. (2010), referring to that reference for more details,
Here is a gauge-covariant derivative, is any product of Dirac gamma matrices, is a quark field (smeared using distillation in our implementation) and spin, flavour and colour indices have been suppressed for clarity. As described in Dudek et al. (2010), the vector-like gamma matrices and derivatives are expressed in a circular basis and then coupled together using standard Clebsch Gordan coefficients (represented by “CGs” in Eq. 1) to form the operator . However, for consistency between the operator constructions below and conventional definitions for helicity states, here we use a slightly different circular basis compared to that reference,
and similarly for the vector-like gamma matrices. Here is the polarisation vector, given in Appendix B, for a spin-one meson at rest with spin -component .
With full continuum rotational symmetry and , these operators have definite spin (), spin -component () and parity, (determined by the choice of and the number of derivatives), i.e. they only overlap with states having these quantum numbers. In addition, if and correspond to quarks of the same flavour, these operators have definite charge conjugation parity (determined by the choice of , the number of derivatives and how these derivatives are coupled together), generalising to -parity as appropriate. In the continuum the overlap of these operators with states is given by
where the conserved parity and charge conjugation parity quantum numbers have been suppressed.
At non-zero momentum, the spin -component is not a good quantum number unless the momentum is directed along the -axis; in general, Eq. 3 is not applicable to . It is more convenient to consider operators with definite helicity, the projection of the spin component along the direction of . In Appendix A we show that, in analogy to the transformation from basis states with definite spin -component to helicity states, helicity operators can be constructed by
where is a helicity operator with helicity , is a Wigner- matrix, and is the (active) transformation that rotates to . Note that in general there are an infinite number of which rotate to .111If represents a rotation around the -axis by followed by a rotation around the -axis by and finally a rotation around the -axis by , one convention is Chung () and another, the Jacob-Wick convention Jacob and Wick (1959), is . Including an additional arbitrary initial rotation will still give a rotation from to . Different conventions will lead to different phases in the definition of states and operators. There is some subtlety in the appropriate choice of when the symmetry is reduced (e.g. on a finite volume lattice with a finite lattice spacing) and we discuss this below in Section III.
An equivalent way to construct these operators would be to choose an initial circular basis for the vector-like gamma matrices and derivatives defined in terms of the component of spin along instead of the spin -component. These vectors could then be coupled together using the standard Clebsch Gordan coefficients as described above. This emphasises that by constructing helicity operators we’ve just effected a basis change. As will be illustrated below, the resulting basis is much more convenient for studying mesons with non-zero momenta because it respects the symmetries of a system with a meson in flight and helicity is a good quantum number (in an infinite volume continuum).
ii.2 Helicity operator overlaps
Here we discuss the overlap of helicity states onto the helicity operators constructed above, , still considering an infinite volume continuum.
The overlaps for operators with have a simple form, Eq. 3, , and . These constraints arise from the rotational symmetry in 3 spatial dimensions of a system containing a particle at rest, and no further constraints are gained by imposing Lorentz symmetry222Although Lorentz symmetry constraints would relate operators containing the temporal and spatial pieces of Lorentz vectors, e.g. and .. This can be seen explicitly by comparing the Lorentz covariant parameterisations given in Appendix A of Ref. Dudek et al. (2008) and the 3-rotation covariant parameterisations given in Appendix D herein. However, at finite momentum, 3-rotation symmetry, a subgroup of full Lorentz symmetry, provides less stringent constraints, something that can again be seen explicitly by comparing those two different parameterisations. Since in the operator constructions we treat space and time asymmetrically, we will consider in the detail the restricted 3-rotational symmetry constraints.
Using only the constraints arising from 3-rotation symmetry, the overlap of a state of definite ( are the parities at rest) and helicity onto a helicity operator is given by
The helicity of the state, , is constrained to be the same of that of the operator, , but an operator constructed to have integer spin at rest can in general overlap onto states of any integer spin when boosted to non-zero momentum. This is because at non-zero momentum the 3-dimensional rotational group is broken to a subgroup, called the little group, made up of the rotations and reflections which leave the momentum, , invariant. In an infinite volume continuum the little group is U(1) Moore and Fleming (2006a, b), the group of rotations and reflections in two dimensions, regardless of the momentum direction. The irreducible representations of this little group are labelled by the magnitude of helicity (and also, for , a ‘parity’, , described below): . Apart from the two one-dimensional irreps with , these irreps are all two-dimensional. Another consequence of this reduced symmetry is that an operator with can overlap with states of both parities, and the overlap factors, , can depend on . In Appendix D we give the most general parameterisations of these operator-state overlaps and these features can be seen explicitly. Note that any flavour quantum number, such as charge conjugation parity or its generalisation, is still a good quantum number if it is so at rest.
Helicity states at non-zero momentum are not eigenstates of parity because a parity transformation, , reverses the direction of the momentum, . A reflection in a plane containing the momentum direction, (a parity transformation followed by a rotation to bring the momentum direction back to the original direction), preserves . However, helicity states are also in general not eigenstates of because under such a transformation . If we consider a state with along the -axis and a reflection in the plane (), , then we have (Appendix A.1)
where are the quantum numbers of the state at rest and . It can be seen that for the helicity states are eigenstates of with eigenvalue and therefore333and because there is no additional phase depending on the choice of reflection plane (Appendix A.1)
For the helicity states are not eigenstates of 444although we can form eigenstates by taking linear combinations, , they are of limited use here..
If we had started with Lorentz 4-vector gamma matrices and derivatives instead of 3-vectors, we could have constructed operators which only overlap onto states with one at non-zero momentum. Performing a Lorentz boost on the state and operator on the left-hand side of Eq. 3 would leave the right-hand side unchanged, or putting it slightly differently, we can always boost back to the rest frame where Eq. 3 is valid. However, our use of only spatial derivatives precludes us from doing this. We do not include temporal derivatives in our operator constructions because in distillation the quark fields in the operators are only smeared in the spatial directions. This means that we can not form a lattice-discretised temporal derivative which is simply a rotation of a lattice-discretised spatial derivative. In addition, we have used anisotropic lattices and smeared differently in the temporal direction in the action, and these lead to similar problems. If we naively included temporal derivatives, these would be on a different footing to the spatial derivatives and so the resulting operators would not be Lorentz covariant and would overlap onto more than one at non-zero momentum.
Although our operators (and hence the operator-state overlaps) are not Lorentz covariant because they have been built from 3-vectors, in a theory with Lorentz symmetry there are still more stringent constraints on the overlaps than those given in Eq. 5. In Appendix D we show the restrictions on state-operator overlaps which arise from the constraints of a Lorentz symmetric theory applied to our helicity operators. The of the states with which the operator has non-zero overlap depends on which gamma matrices and derivatives the operator is constructed from and how these have been coupled together. An operator which at rest overlaps with only one can in general overlap with states of many at non-zero momentum, but the set of allowed is reduced by Lorentz symmetry constraints compared to the infinite set allowed by 3-rotation symmetry, Eq. 5.
To illustrate the above points, consider a simple example where the fermion bilinear operator consists of a spatial gamma matrix in the Cartesian basis and there are no derivatives. Projected onto zero momentum, this operator, , has non-zero overlap with only the vector state, ,
where is the polarisation vector of the state and is its spin -component. Now consider a non-zero momentum along the positive -axis555chosen to be along the -axis so that the helicity state/operator is the same state/operator, . In a theory with only 3-rotation symmetry and not full Lorentz symmetry, the state-operator overlaps are given in the last column of Table 11 in Appendix D, i.e. the operator has non-zero overlap with states of all integer . For states with the overlaps are666note that the overlap onto the piece of the vanishes, a consequence of Eq. 6
However, in a theory with Lorentz symmetry most of these overlaps are constrained to be zero. For example, there are no Lorentz covariant structures that reduce to the  or  terms above. In this case the operator has non-zero overlap with only two states (shown in the column labelled “L1” in the aforementioned table),
If instead we consider the Lorentz covariant operator , the non-zero overlaps are
In this case we can take linear combinations of the four components to project onto just one of the states: taking the inner product with projects onto the state, whereas taking the inner product with projects onto the component of the state only.
In summary, although simplified somewhat by using helicity operators, the pattern of operator-state overlaps at non-zero momentum is more complicated than at zero momentum. In particular, because we construct operators out of 3-vectors rather than Lorentz 4-vectors, we can not in general form operators that have non-zero overlap with states of only one . However, as we shall show in Section V, the remaining constraints prove to be enough for us to be able to identify the states’ spins and parities. In the following section we consider complications arising from a finite cubic volume.
Iii Mesons in flight and helicity operators on the lattice
In general, the symmetry of the lattice discretisation (which breaks rotational symmetry at small distances) need not be the same as the symmetry of the finite volume, the boundary conditions (which break rotational symmetry at large distances). However, in this work we only consider a cubic lattice in a finite cubic box with periodic boundary conditions, and so both the lattice and the boundary have the same symmetry. In this case, the full rotational symmetry of the continuum is broken to the (double cover) of the octahedral group, , equivalent to the (double cover) of the symmetry group of the cube. Spin is no longer a good quantum number and states are not classified by but instead by the irreducible representations, irreps (), of . In Refs. Dudek et al. (2009b, 2010), it was found that on the dynamical lattices of Refs. Edwards et al. (2008); Lin et al. (2009), the effect of the finite lattice spacing was relatively small. Operators with definite continuum spin (Eq. 1) were subduced into octahedral group irreps; the observed spectrum of states and operator-state overlaps were compatible with an effective restoration of rotational symmetry, and the spectrum was explained in terms of single-meson states of determined with values subduced across these irreps.
In going from zero momentum to non-zero momentum the symmetry is broken further. Whereas, in an infinite volume continuum the little group is the same regardless of the momentum direction, , in a finite volume the particular lattice little group depends on the star of Moore and Fleming (2006a). The allowed are determined by the boundary conditions and the star of is the set of all related by allowed lattice rotations; as a shorthand, we refer to these different stars as different momentum types. Momenta are quantised by the periodic boundary conditions on the cubic box and we give all momenta in units of ; and are respectively the spatial lattice spacing and spatial extent of the lattice in lattice units. The relevant little groups are given in Refs. Moore and Fleming (2006a, b). We summarise the allowed lattice momenta and the corresponding little groups and irreps in Table 1.
|Lattice||Little Group||Irreps ( or )|
|Momentum||(double cover)||(for single cover)|
|, , , ,|
|, , , ,|
|, , ,|
In analogy to Ref. Dudek et al. (2010), we consider subduction coefficients, , which specify how the helicity, , subduces into a little group irrep (row ). Using these we can construct a little group operator, a subduced helicity operator, from a helicity operator:
where with and the spin and parity of the operator . The subduced helicity operators are different orthogonal combinations of the two signs of helicity, and . These subduction coefficients can be calculated using the group theoretic projection formula (for example, see Appendix A of Ref. Dudek et al. (2010)).
In Table 2 we give these subduction coefficients777Note that each subduction coefficient could be multiplied by an arbitrary phase and it would still give a subduction from the helicity to the little group irrep. for momenta of the form , and () for . More details of our conventions are given in Appendix C. For all these momenta, = and subduce onto the and irreps respectively. However, the other contained in those irreps (i.e. which can mix with ) depends on the momentum type: () also has ; () has ; () has . Note that, although the pattern of subductions does not depend on conventions, the relative phases in subduction coefficients can be convention dependent, determined by the helicity operator construction, the rotations used (discussed below) and the particular representation matrices chosen for the two-dimensional irreps.
Instead of constructing little group operators via helicity operators which are then subduced into little group irreps, we could have projected directly from our continuum operators (in the basis) into these little group irreps, but the subduction coefficients would then in general depend on the momentum direction and not just the momentum type. Alternatively, we could have worked only in terms of lattice irreps, subducing from octahedral group irreps to lattice little group irreps. However, the method we have described makes it clear how the operators can be physically interpreted in the limit of an infinite volume continuum and, as we show below, enables the continuum quantum numbers of states to be identified.
As noted above, in a finite volume there is some subtlety in the choice of the in Eq. 4 (out of an infinite number of possibilities) which rotates (not necessarily an allowed lattice momentum) to (an allowed lattice momentum). In an infinite volume continuum the choice is not important as long as one convention is chosen because the two directions perpendicular to are interchangeable; they are related by a rotation. However, on a finite lattice the two transverse directions are not in general interchangeable; they are not related by an allowed lattice rotation. Consider, for example, the two (shortest) lattice vectors perpendicular to , namely and – these have different lengths. Therefore, if the ’s are not chosen appropriately there is the possibility of an inconsistency between correlators from different momentum directions within the same momentum type. For example, if used for and used for are not related by a lattice rotation, this could lead to effectively different definitions of the little group irreps and different bases for little group rows in compared to ; e.g. () from could correspond to () from . This would prove problematic if averaging over different momentum directions to increase statistics and when constructing multi-meson operators which contain a sum over momentum directions with various weights. If and are related by an allowed lattice rotation, these inconsistencies do not arise because there is no allowed lattice rotation that rotates from direction to direction , and so the different irreps or irrep rows cannot be mixed up.
We ensure consistency between different momentum directions by breaking down into two stages: . First, , rotate from to , where is a reference direction for momenta of type (i.e. for the star of ). Note that is not in general an allowed lattice rotation, but this is permissible because all we are effecting by this rotation is a basis transformation. For we choose a lattice rotation which rotates from to . There are in general a finite number of possible choices of lattice rotations for ; the particular choice is not important as long as we make the choice consistently for each momentum direction (for example when later constructing multi-meson operators). More details of our implementation are given in Appendix C.
To illustrate this, consider momentum directions with and choose . If we consider then is just a rotation from to . On the other hand, if we consider then where is that just described and is a lattice rotation that rotates to .
Now that we have described the construction of subduced helicity operators, in the next section we briefly describe some of the lattice and computational details, before explaining our method for determining the continuum spin of extracted states and giving some examples in the following section.
Iv Computational details
We use anisotropic dynamical Clover lattices with lattice parameters described in detail in Refs. Edwards et al. (2008); Lin et al. (2009), having a spatial lattice spacing, , and a temporal lattice spacing approximately times smaller, corresponding to . In Table 3 we give details of the data sets used; in all cases we increase statistics by averaging over equivalent momentum directions within a given momentum type. In the main body of results shown here, we use the data set with lattice size in lattice units. This ensemble has three degenerate dynamical quark flavours, i.e. has SU(3) flavour symmetry, and corresponds to MeV. Some comparisons are performed using the lattices of different volumes and quark masses given in Table 3. We consider isovector mesons and so only connected Wick contractions contribute.
Correlators are constructed using the distillation method Peardon et al. (2009) which, in combination with these anisotropic lattices, has proved fruitful in a number of applications Dudek et al. (2009b, 2010); Bulava et al. (2010); Dudek et al. (2011a, b); Edwards et al. (2011). For studying mesons in flight, an important aspect of distillation is that it allows a large basis of operators at the source and the sink, each projected onto a definite momentum.
Our operator basis consists of all possible combinations of gamma matrices and zero, one or two lattice-discretised gauge-covariant derivatives, coupled together and subduced into lattice little group irreps, to form subduced helicity operators as described above. The notation follows that of Ref. Dudek et al. (2010): an operator contains a gamma matrix, , with the naming scheme given in Table 4, and derivatives coupled to spin , altogether coupled to spin ; refers to the helicity component.
The number of operators in each irrep is given in Table 5. For the two-dimensional irreps we increase statistics by averaging the correlators over the two irrep rows. We consider isovector mesons which are eigenstates of charge conjugation (or -parity) and so the irreps are labelled by charge conjugation parity, , as well as little group irrep, . Correlators were analysed using our implementation of the variational methodMichael (1985); Luscher and Wolff (1990) described in Dudek et al. (2010). We use the baryon mass to set the scale and quote all energies as ratios, , where on the lattices with MeV Lin et al. (2009).
|Group||Number of ops|
As a first test of the subduced helicity operator constructions, for each possible momentum with , , and , all possible correlators were computed, including those off-diagonal in irrep and/or irrep row. This allowed the orthogonality of correlators between different irreps, and between different rows in the same irrep, to be verified, along with the positivity of diagonal correlators and, within each irrep, the hermiticity of the correlator matrix and the consistency of different rows in the same irrep. In addition, the consistency of correlators from different momenta directions with the same momentum type was verified, something which would not necessarily be true if the rotation matrices, , had not been chosen appropriately (see Section III).
V Spin determination
For mesons with non-zero momentum, the identification of the continuum spin of a state extracted in a lattice calculation is more complicated than for mesons at rest; even in an infinite volume continuum, an operator in general can have non-zero overlap with states of many , as discussed in Section II.2. The reduced symmetry compared to the zero momentum case means that more states appear in each irrep leading to many degeneracies in the spectrum; some are dynamical degeneracies of physical states in the true spectrum and others are caused by the reduced symmetry. Trying to identify the continuum quantum numbers by matching degenerate levels in different little group irreps (after taking the continuum limit) will be even more problematic than for mesons at rest. It would appear difficult or impossible to disentangle the states shown in Figs. 8 to 11 without using some more information beyond solely their energies.
For mesons stable against hadronic decay, a general idea as to where to expect different energy levels can be obtained from using the mass of the mesons at rest (extracted on the same lattice) and the dispersion relation,
To determine a precise value for on this lattice, we use our measured energies, , at and the mass from Ref. Dudek et al. (2009b), and fit the dispersion relation888Fitting to a more general form of the dispersion relation with and terms gives a consistent result, , and does not improve the goodness of fit. to obtain a good fit with . However, because of the degeneracies (within statistical uncertainties) in the spectrum, some further information is needed to determine the spin of extracted states. Therefore, following a similar procedure to that developed for mesons at rest in Refs. Dudek et al. (2009b, 2010), we consider the state-operator overlaps.
After subducing to lattice little group irreps, and neglecting the effects of a finite volume and finite lattice spacing, Eq. 5 becomes
where and and are respectively the spin and parity of the operator at rest. As discussed above, even in an infinite volume continuum, at non-zero momentum an operator can in general overlap onto states of many different spins and both parities. We use the constraints on operator-state overlaps which arise from Lorentz symmetry applied to our helicity operators to determine the continuum spin and parity of states. These constraints can be read off from columns “L1”, “L2” and “L3” of Tables 13, 14, 15 and 16 in Appendix D.
We observe that mixings due to the reduced symmetry are small; a subduced operator from a helicity operator with helicity magnitude only overlaps significantly onto states of helicity magnitude and not with other helicities in that irrep. In Fig. 1 we show the normalised correlation matrix on time-slice 5 for the irrep with (including zero, one and two-derivative operators) ordered such that those subduced from come before those from . The correlation matrix is observed to be approximately block diagonal, correlators mixing and are small.
This lack of mixing between different is also seen in the values. For example, consider the irrep. In Fig. 2 we show the ’s for the lightest five states in this irrep with , along with the assignment for each state; for simplicity of presentation, we use a smaller operator basis, only including the seven zero and one-derivative operators. The operator naming scheme is described in Section IV. Also shown on the right of the figure is the spectrum of energy levels extracted in this irrep (the spin-identified spectrum is shown in the left hand panel of Fig. 8).
The same operators are shown for each state but the colour coding varies according to the hypothesis: red bars correspond to operators which overlap onto states with that at rest, blue bars correspond to operators which overlap with such states only at non-zero momentum and hatched grey bars correspond to operators which do not overlap with such states (using the constraints arising from Lorentz symmetry). As a concrete example, consider the first two operators shown, and . These contain, respectively, and in spin space, both with no derivatives, and at zero momentum both overlap only with states. Both contain no vector indices (i.e. no or ) and so Table 13 is relevant but, as explained in Appendix D, because these operators overlap with at rest, we must flip all the state parities in that table. The operator is of the form given in the “L1” column and from there we see that, using Lorentz covariance constraints, it only overlaps onto . In the figures it is therefore coloured red for states and grey for other . On the other hand, is of the form given in the “L2” column and we see that at non-zero momentum it can overlap with both and . It is therefore coloured red for states, blue for states and grey for other .
For all the states shown, it can be seen that the red bars are largest, although these overlaps can be small if there is a mismatch in structure between the operator and the state. The blue bars can be significant, these should be proportional to some power of and so are suppressed compared to the red bars at low momentum, and the grey bars are small. From these figures it can be seen that in each case the can be determined unambiguously. For () the continuum helicities which subduce into the irrep are (Table 2). The restricted operator basis used here does not contain helicities greater than two and so we can not test helicity mixing here. In any case, the lightest spin-four state is expected to be significantly higher in energy than the heaviest states we extract in this analysis.
In Fig. 3 we show the ’s for the overlap of the lightest six states in the irrep with onto the ten zero and one-derivative operators in that irrep, along with the spin assignment hypothesis for each state. Again, for each state, the continuum can be unambiguously determined. For () the continuum helicities which subduce into the irrep are (Table 2) and the operator basis contains operators coming from (the first seven operators) and (the last three operators). It can be seen that for each state only or overlaps are statistically significantly non-zero, further supporting the observation at the level of the correlation matrix, Fig. 1. Furthermore, these overlaps are generally smaller than the overlaps onto operators with the correct continuum helicity but which are constrained to be zero by Lorentz symmetry.
A second stage of the spin identification procedure is to compare the ’s for the overlap of a given state onto a given continuum operator when this operator is subduced into two or more different little group irreps. From Eq. 10, neglecting any finite volume or discretisation effects, these overlap factors should be equal, something that is empirically found to be true for mesons at rest extracted on this lattice in Refs. Dudek et al. (2009b, 2010). For mesons in flight the opportunities for such comparisons are more limited. The only directly analogous situation is when the same subduces into two one-dimensional irreps (as opposed to subducing into one two-dimensional irrep); for example, with subduces into and , and with subduces into and . These comparisons can only be used to confirm the continuum helicity rather than the continuum spin (although determining the continuum helicity does give a minimum value of the continuum spin) and so such comparisons are less useful for mesons in flight than for mesons at rest.
As an example, in Fig. 4 we show the ’s for the lowest two states with overlapping onto the three one-derivative operators subduced into the and irreps. The good agreement between values with the same operator subduced into different irreps is apparent. In Fig. 5 we show the ’s for the lowest two states with overlapping onto the eight zero and one-derivative operators subduced into the and irreps. Again, there is good agreement between the ’s in different irreps.