A lattice QCD study of generalized gluelumps
aa r X i v : . [ h e p - l a t ] J a n Lattice QCD study of generalized gluelumps
Kristen Marsh and Randy Lewis
Department of Physics and Astronomy,York University, Toronto, Ontario, M3J 1P3, Canada
Proposals for physics beyond the standard model often include new colored par-ticles at or beyond the scale of electroweak symmetry breaking. Any new particlewith a sufficient lifetime will bind with standard model gluons and quarks to forma spectrum of new hadrons. Here we focus on colored particles in the octet, de-cuplet, 27-plet, 28-plet and 35-plet representations of SU(3) color because thesecan form hadrons without valence quarks. In every case, lattice creation operatorsare constructed for all angular momentum, parity and charge conjugation quantumnumbers. Computations with fully dynamical lattice QCD configurations producenumerical results for mass splittings within this new hadron spectrum. A previousquenched lattice study explored the octet case for certain quantum number choices,and our findings provide a reassessment of those early results.
I. INTRODUCTION
Quantum chromodynamics (QCD) describes the interactions between colored particlessuch as the color-triplet quarks and color-octet gluons of the standard model, but additionalcolored particles are present in many extensions of the standard model. Supersymmetryrequires gluinos and squarks. String theory provides a broader range of possibilities. Newstrong dynamics would generate a spectrum of new composite particles (recall the techni-hadrons of classic technicolor), and if the new elementary particles (akin to techniquarks)carry QCD color, then the new composite particles occur as octets, decuplets, and other mul-tiplets of QCD color. Studies of new colored particles in the context of the Large HadronCollider therefore go far beyond triplets and octets [1–7], continuing several decades of inter-est in the range of color representations that might be realized beyond the standard model[8–25].Lattice QCD is routinely used to obtain quantitative results from the SU(3) gauge theoryof gluons and quarks. The inclusion of additional particles in an octet [26–32], sextet [26, 33–40], or symmetric [41, 42] representation has also been investigated, in some cases appliedto a new strong interaction rather than to QCD itself. Of more direct relevance to our workis a lattice study by Michael and coworkers [43–46], culminating in Ref. [46] where QCDis coupled to a new heavy color-octet particle representing the gluino of supersymmetry.Given that the gluino is significantly heavier than the QCD scale, Foster and Michael [46]were able to treat the gluino as a static particle, where the spin of the gluino is irrelevant sotheir results are applicable more generally to particles of arbitrary spin. If the static particleis sufficiently stable, then it will couple to surrounding gluons and quarks to form hadronicbound states. Foster and Michael used lattice QCD simulations to produce predictionsfor mass splittings within this new spectrum of hadrons. Specifically, Ref. [46] containsnumerical results for two types of hadrons: gluelumps (having one static octet operatorcoupled to gluon fields, but no valence quarks) and adjoint mesons (having one static octetoperator coupled to a quark-antiquark pair).
TABLE I: The smallest gluelump mass splittings relative to the 1 + − state from the original latticesimulation [46] (where errors are statistical only), compared to model calculations published sub-sequently. See Sec. III for a crucial discussion of lattice systematics. See the original publicationsfor detailed discussions about other parameter choices and systematic issues; this table is merelyan introduction. (To display data from Ref. [50] we chose r = 0 . J P C M ( J P C ) − M (1 + − ) [GeV]Lattice [46] Bag [47] String [48] Coulomb gauge [49] Transverse gluons [50]1 −− −− a + − + − ++ ∼ · · · · · · a This entry repairs a simple typo in column 4 of Table III in Ref. [46], as can be seen by comparing withcolumn 3 of that same table and with Fig. 3 in Ref. [46].
According to Ref. [46], the lightest gluelump has J P C = 1 + − . The predicted mass split-tings of the five next-lightest gluelumps are shown in Table I. Four model calculations [47–50]are also shown in Table I for comparison. We display mass differences because these arewhat emerge directly from the lattice simulations, but in fact the absolute mass scale hasbeen determined in Ref. [51] using a combination of effective field theory and related latticeQCD input. After fixing this absolute mass scale, Ref. [51] then takes the gluelump masssplittings directly from Ref. [46]. We point to potential NRQCD [52] as an example of animportant theoretical development that has requested further lattice studies of gluelumps.The authors of Ref. [46] expressed surprise at the heaviness of their 0 ++ state, and alsoat the degeneracy of 2 + − and 3 + − . Lattice simulations use irreducible representations Λof the octahedral group rather than continuum angular momentum J , so, for example, a J = 2 state should appear for both Λ = E and Λ = T , but Ref. [46] points out that E ++ and T ++2 are not degenerate in their lattice data though the discrepancy is consistent withdegeneracy in the continuum limit. Because of the computational expense, Ref. [46] madeuse of quenched lattices so the authors expect at least a 10% systematic error. The work alsorelied exclusively on operators built from square paths on the lattice which allows access toonly half of the possible Λ P C representations (i.e. 10 out of 20), leaving quantum numberssuch as J P C = 0 + − , 0 − + , 0 −− , and 1 ++ unstudied.In the present work, we extend the basis of operators to the complete set of Λ P C options,and we use dynamical (unquenched) lattices. This provides an opportunity to revisit some ofthe surprises revealed by Foster and Michael in their seminal work, and to predict additionalgluelump masses. We also develop operators for generalized gluelumps by replacing the staticoctet source with a static source having a larger color representation. To avoid the expenseof lattice simulations with valence quarks, we choose representations that need only gluonsto produce a color-singlet generalized gluelump. Specifically we choose dimensions 10, 27,28, and 35. We reiterate that our numerical results make use of dynamical lattice simulationsso that virtual quarks and antiquarks are retained.Static propagators are known to produce particularly large statistical uncertainties inlattice simulations, and a static octet particle is noisier than a static triplet [46, 53]. Weexpect that the larger representations included in the present study will be noisier still. Also,
TABLE II: Young tableaux for representations relevant to this work. Labels inside boxes are toaid the discussion of (anti)symmetrization. n D the Casimir scaling hypothesis [54–56] is the notion that the string tension between stronglyinteracting particles should be proportional to the quadratic Casimir, and standard grouptheory [57, 58] shows that the quadratic Casimirs for our representations, normalized suchthat the triplet has C (3) = 4 /
3, are C (8) = 3, C (10) = 6, C (27) = 8, C (28) = 18, and C (35) = 12. Polyakov loops with all of these representations have been tested previouslyfor Casimir scaling: see Table 2 of Ref. [59]. [For other lattice studies of Casimir scaling andvarious representations in four-dimensional SU(3) gauge theory, sometimes in the context of n -ality, see Refs. [60–69]. The present work deals exclusively with zero n -ality.] In the case ofgluelumps, our simulations confirm that signals for representations with larger Casimirs aredamped more rapidly as a function of Euclidean time, as well as being statistically noisy.Despite these substantial difficulties, the numerical results of this project provide usefulinformation about representations beyond the octet, as well as the octet itself. II. CORRELATION FUNCTIONS
Generalized gluelumps do not involve valence quarks, so the heavy static particle mustbe able to form a color singlet by coupling to a collection of octet gauge fields,8 ⊗ ⊗ ⊗ · · · ∈ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ · · · . (1)Representations of dimension n D = 8, 10, 27, 28, and 35 will be considered in this work.The corresponding Young tableaux, derivable using standard group theory methods [57, 58],are displayed in Table II. Notice that the number of boxes in each tableau is a multiple of3, as required for tableaux built exclusively from octet gauge fields. As will be discussedbelow, each generalized gluelump will have a tensor where the number of indices equals thenumber of columns in its Young tableau.As is standard in lattice QCD simulations, mass splittings will be obtained by computinga correlation function and then observing the exponential dependence on Euclidean time. Acorrelation function that creates a gluelump at Euclidean time τ i and then annihilates it attime τ f is C ( τ f − τ i ) = H ( n D ) α † ( τ i ) G ( n D ) αβ ( τ i , τ f ) H ( n D ) β ( τ f ) . (2)Repeated indices α and β are summed from 1 to n D to produce a gauge-invariant correlationfunction. The operators H and H † that, respectively, annihilate and create the requiredgauge field structure will be developed in Sec. II B. The propagator G for the static particleis described presently. A. Static propagator
A static particle propagates purely in the temporal direction (subscript “4”), so for arepresentation of dimension n D we can write G ( n D ) αβ ( τ i , τ f ) = U ( n D ) αγ ( ~x, τ i ) U ( n D ) γδ ( ~x, τ i + a ) U ( n D ) δǫ ( ~x, τ i + 2 a ) · · · U ( n D ) ζβ ( ~x, τ f ) (3)with repeated Greek indices summed from 1 to n D . Each generalized link U ( n D ) is builtfrom elementary links (one per column of the Young tableau) contracted at each end (e.g.Euclidean times τ i and τ i + a ) with a basis tensor T , U (8) αβ = U ik U ∗ jl T αij T βkl , (4) U (10) αβ = U il U jm U kn T αijk T βlmn , (5) U (27) αβ = U im U jn U ∗ ko U ∗ lp T αijkl T βmnop , (6) U (28) αβ = U io U jp U kq U lr U ms U nt T αijklmn T βopqrst , (7) U (35) αβ = U in U jo U kp U lq U ∗ mr T αijklm T βnopqr , (8)where repeated color indices i, j, k, . . . are summed from 1 to 3.An acceptable basis for the octet representation is T α = λ α / √ λ α is a standardGell-Mann matrix as was used in Ref. [46]. Beyond the octet we find it more convenient touse real T tensors, and for consistency we will also use real matrices for the octet itself.For a Young tableau with n B boxes, we begin with an arbitrary tensor having n B indices.Then we symmetrize all indices within a row. Next we antisymmetrize all indices withinany column having two boxes and multiply that pair of indices by a Levi-Civit`a tensor, thusreducing the number of indices by one for each antisymmetrized column. The final step isto select a real basis of T tensors. For example, consider the 27-plet. An arbitrary 6-indextensor is a ijklmn , and after symmetrization of ( i, j, k, l ), symmetrization of ( m, n ) and thenantisymmetrization of ( i, m ) and ( j, n ), we have b ijklmn = a ijklmn − a mjklin − a inklmj + a mnklij + · · · (9)which reduces to a 4-index tensor, T klpq = 14 ǫ imp ǫ jnq b ijklmn . (10)Evaluation of all 3 = 81 elements of this tensor reveals that it contains 36 distinct entriesbut only 27 of them are linearly independent due to the following 9 constraints: T + T + T = 0 ,T + T + T = 0 ,T + T + T = 0 ,T + T + T = 0 ,T + T + T = 0 ,T + T + T = 0 ,T + T + T = 0 , T + T + T = 0 ,T + T + T = 0 . (11)Our choice for the basis of 27 tensors is given explicitly in Appendix A together with theother representations: octet, decuplet, 28-plet, and 35-plet.As a useful check of these expressions, we calculate a completeness relation for each case:Appendix A verifies that the quantity n D X α =1 T α T α (12)comprises a simple Kronecker delta structure. This is important for the gauge invariance ofour correlation functions.Notice also that our decuplet representation agrees with Appendix B of Ref. [70]. Finally,we mention that we have verified numerically that our real basis of octet T tensors producescorrelation functions that are identical to those obtained in the Gell-Mann basis. B. Creation/annihilation operators
The remaining ingredient needed for the computation of correlation functions is the setof operators, H of Eq. (2), coupling to the generalized gluelumps. An H operator is builtfrom products of gauge links that join to the static particle propagator via a T tensor(from Sec. II A). Planar squares were used for H operators in Ref. [46], but this providesaccess to only half of the possible quantum numbers. Our most basic building block willbe a “chair,” i.e. a 1 × ◦ angle, which provides access to all quantumnumbers. For extra confirmation of numerics, we also ran simulations with the planar squareoperators used by Foster and Michael, and we verified that results are consistent with thecorresponding chair-based operators defined here.Figure 1 displays a pair of chairs touching each other at one lattice site and rotatedinto all of the 24 orientations that are possible on a cubic lattice. Notice that each chairhas a particular direction because a “backward link” U − µ ( x + µ ) = U † µ ( x ) is not equal tothe “forward link” U µ ( x ). Within each pair of chairs in Fig. 1, A + B is a positive parityoperator and A − B is a negative parity operator. Because U µ ( x ) → U † µ ( x ) under chargeconjugation, a “forward” chair plus a “backwards” chair has positive charge conjugation andthe difference between these two chairs has negative charge conjugation.The five bosonic irreducible representations of the octahedral group are Λ = A , A , T , T , and E , and their smallest continuum angular momenta are J = 0, 3, 1, 2, and 2,respectively. For octet gluelumps, the corresponding operators are obtained from specificlinear combinations of the chair-shaped paths in Fig. 1. The steps of a derivation areprovided in Appendix B, and the results are given here: H (8) α ( A ) = X a =1 L (8) a ! ij T αij ,H (8) α ( A ) = X a =1 ( − a L (8) a − X a =13 ( − a L (8) a ! ij T αij , B L =L =L =L = AB AA B BL =L =L =
L =L =L =L = L =L =L =L =
L =L =L =L = L =L =L =L =
21 22 23 2418 19 20(8)(8) (8)(8) (8)(8) (8)(8)(8)(8)(8)(8)(8) (8) (8) (8)(8)(8)(8)(8) (8)(8) (8) (8)
L = B AAAAAAAAAAAAAAA A B BBBBB B BB B BBBB B B BAAA ABBA
FIG. 1: Each chair-shaped path is the product of six gauge links used to build operators for octetgluelumps. Solid lines are the gauge links; dashed lines are just to aid with three-dimensional (3D)visualization. (10)(8)
B A L = B A L =
FIG. 2: Octet chairs and decuplet chairs have the same shape but the product of gauge links differs.Solid lines are the gauge links; dashed lines are just to aid with 3D visualization. A filled circledenotes insertion of a Levi-Civit`a tensor. H (8) α ( T x ) = (cid:16) L (8)6 + L (8)20 + L (8)21 + L (8)11 − L (8)18 − L (8)8 − L (8)9 − L (8)23 (cid:17) ij T αij ,H (8) α ( T y ) = (cid:16) L (8)5 + L (8)19 + L (8)24 + L (8)10 − L (8)17 − L (8)7 − L (8)12 − L (8)22 (cid:17) ij T αij ,H (8) α ( T z ) = (cid:16) L (8)1 + L (8)2 + L (8)3 + L (8)4 − L (8)13 − L (8)14 − L (8)15 − L (8)16 (cid:17) ij T αij ,H (8) α ( T x ) = (cid:16) L (8)6 − L (8)20 + L (8)21 − L (8)11 + L (8)18 − L (8)8 + L (8)9 − L (8)23 (cid:17) ij T αij ,H (8) α ( T y ) = (cid:16) L (8)5 − L (8)19 + L (8)24 − L (8)10 + L (8)17 − L (8)7 + L (8)12 − L (8)22 (cid:17) ij T αij ,H (8) α ( T z ) = (cid:16) L (8)1 − L (8)2 + L (8)3 − L (8)4 + L (8)13 − L (8)14 + L (8)15 − L (8)16 (cid:17) ij T αij ,H (8) α ( E ) = ( v x − v y ) ij T αij ,H (8) α ( E ) = ( v x + v y − v z ) ij T αij ,v x = L (8)6 + L (8)20 + L (8)21 + L (8)11 + L (8)18 + L (8)8 + L (8)9 + L (8)23 ,v y = L (8)5 + L (8)19 + L (8)24 + L (8)10 + L (8)17 + L (8)7 + L (8)12 + L (8)22 ,v z = L (8)1 + L (8)2 + L (8)3 + L (8)4 + L (8)13 + L (8)14 + L (8)15 + L (8)16 . (13)Notice that A and A are one-dimensional representations, T and T are three dimensional,and E is two dimensional.Each decuplet chair contains three paths that begin at a central lattice site (where thetensor T will be placed) and end at a Levi-Civit`a tensor. Each of those three paths is theproduct of three gauge links. The precise definition of L (10)1 is displayed in Fig. 2, and L (10)2 through L (10)24 are defined by applying the same procedure to every chair in Fig. 1. Thedecuplet operators are obtained by making two simple adjustments to Eqs. (13): replaceevery superscript (8) with a superscript (10) and replace every pair of indices ij by the threeindices ijk .The 35-plet is built from a double chair, specifically one octet-type chair and one decuplet-type chair, defined as follows: (cid:16) L (35)1 (cid:17) ijklm = (cid:16) L (8)5 (cid:17) im (cid:16) L (10)9 (cid:17) jkl , (cid:16) L (35)13 (cid:17) ijklm = (cid:16) L (8)21 (cid:17) im (cid:16) L (10)17 (cid:17) jkl , (cid:16) L (35)2 (cid:17) ijklm = (cid:16) L (8)6 (cid:17) im (cid:16) L (10)10 (cid:17) jkl , (cid:16) L (35)14 (cid:17) ijklm = (cid:16) L (8)22 (cid:17) im (cid:16) L (10)18 (cid:17) jkl , (cid:16) L (35)3 (cid:17) ijklm = (cid:16) L (8)7 (cid:17) im (cid:16) L (10)11 (cid:17) jkl , (cid:16) L (35)15 (cid:17) ijklm = (cid:16) L (8)23 (cid:17) im (cid:16) L (10)19 (cid:17) jkl , (cid:16) L (35)4 (cid:17) ijklm = (cid:16) L (8)8 (cid:17) im (cid:16) L (10)12 (cid:17) jkl , (cid:16) L (35)16 (cid:17) ijklm = (cid:16) L (8)24 (cid:17) im (cid:16) L (10)20 (cid:17) jkl , (cid:16) L (35)5 (cid:17) ijklm = (cid:16) L (8)9 (cid:17) im (cid:16) L (10)1 (cid:17) jkl , (cid:16) L (35)17 (cid:17) ijklm = (cid:16) L (8)13 (cid:17) im (cid:16) L (10)21 (cid:17) jkl , (cid:16) L (35)6 (cid:17) ijklm = (cid:16) L (8)10 (cid:17) im (cid:16) L (10)2 (cid:17) jkl , (cid:16) L (35)18 (cid:17) ijklm = (cid:16) L (8)14 (cid:17) im (cid:16) L (10)22 (cid:17) jkl , (cid:16) L (35)7 (cid:17) ijklm = (cid:16) L (8)11 (cid:17) im (cid:16) L (10)3 (cid:17) jkl , (cid:16) L (35)19 (cid:17) ijklm = (cid:16) L (8)15 (cid:17) im (cid:16) L (10)23 (cid:17) jkl , (cid:16) L (35)8 (cid:17) ijklm = (cid:16) L (8)12 (cid:17) im (cid:16) L (10)4 (cid:17) jkl , (cid:16) L (35)20 (cid:17) ijklm = (cid:16) L (8)16 (cid:17) im (cid:16) L (10)24 (cid:17) jkl , (cid:16) L (35)9 (cid:17) ijklm = (cid:16) L (8)1 (cid:17) im (cid:16) L (10)5 (cid:17) jkl , (cid:16) L (35)21 (cid:17) ijklm = (cid:16) L (8)17 (cid:17) im (cid:16) L (10)13 (cid:17) jkl , (cid:16) L (35)10 (cid:17) ijklm = (cid:16) L (8)2 (cid:17) im (cid:16) L (10)6 (cid:17) jkl , (cid:16) L (35)22 (cid:17) ijklm = (cid:16) L (8)18 (cid:17) im (cid:16) L (10)14 (cid:17) jkl , (cid:16) L (35)11 (cid:17) ijklm = (cid:16) L (8)3 (cid:17) im (cid:16) L (10)7 (cid:17) jkl , (cid:16) L (35)23 (cid:17) ijklm = (cid:16) L (8)19 (cid:17) im (cid:16) L (10)15 (cid:17) jkl , (cid:16) L (35)12 (cid:17) ijklm = (cid:16) L (8)4 (cid:17) im (cid:16) L (10)8 (cid:17) jkl , (cid:16) L (35)24 (cid:17) ijklm = (cid:16) L (8)20 (cid:17) im (cid:16) L (10)16 (cid:17) jkl . (14)Notice that a diagram of L (35) n would resemble L (8) n of Fig. 1 except that the chairs in eachlattice cell are in the opposite locations (there is a chair where there was not , and there isnot a chair where there was ). There are two options—the octet-type chair could have beento the left or to the right of the decuplet-type chair when viewed from a certain angle—and Eqs. (14) show which of the two options we have selected. The 35-plet operators areobtained by making two simple adjustments to Eqs. (13): replace every superscript (8) witha superscript (35) and replace every pair of indices ij by the set ijklm .The 27-plet is also built from a double chair, but both are octet-type chairs. The definitionis obtained from Eqs. (14) with these replacements: (35) → (27), (10) → (8), ijklm → ijkl , im → ik , jkl → jl . The 27-plet operators are obtained by making two simple adjustmentsto Eqs. (13): replace every superscript (8) with a superscript (27) and replace every pair ofindices ij by the set ijkl .The 28-plet is built from a double chair; both are decuplet-type. The definition is obtainedfrom Eqs. (14) with these replacements: (35) → (28), (8) → (10), ijklm → ijklmn , im → ijk , jkl → lmn . The 28-plet operators are obtained by making two simple adjustments toEqs. (13): replace every superscript (8) with a superscript (28) and replace every pair ofindices ij by the set ijklmn .To complete the discussion of generalized gluelump operators, notice that the octet and27-plet are eigenstates of charge conjugation, whereas the decuplet, 28-plet, and 35-pletare not. This is evident from the Young tableaux representations of the underlying grouptheory as shown in Table II. Representations with twice as many boxes in the top row as thebottom row have the same number of symmetric and antisymmetric indices. They are theirown antirepresentations and are eigenstates of charge conjugation. Other representationsare “charged” and cannot form states with definite charge conjugation. This property canalso be seen in the color flow in Figs. 1 and 2. Any single octet chair has one color and oneanticolor emanating from the central lattice site, but the decuplet has three colors and noanticolors. TABLE III: Input parameters and standard output parameters (separated by a horizontal line)used in this work were obtained from Ref. [71]. For comparison, parameters used in the quenchedstudy by Ref. [46] are also shown.Source [71] [71] [46] [46] [46] β κ ud · · · · · · · · · κ s · · · · · · · · · c SW · · · · · · · · · L × T ×
40 28 ×
56 12 ×
24 16 ×
48 24 × m π /m ρ · · · · · · · · · q m K − m π /m φ · · · · · · · · · -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 c o rr e l a ti on f un c ti on T -T --T - + T (a) -9 -8 -7 -6 -5 -4 -3 -2 -1 c o rr e l a ti on f un c ti on E ++ E- + E--E + - (b) FIG. 3: Sample correlation functions: (a) the T P C channels containing a static octet particle at β = 2 .
05, (b) the E P C channels containing a static 27-plet particle at β = 1 . III. LATTICE SIMULATIONS
The simulations performed for this work use two ensembles of configurations provided bythe CP-PACS and JLQCD Collaborations [71]. These ensembles are O ( a )-improved due tothe use of the clover coefficient, c SW . The lattice spacings are comparable to the smallestvalues used in Ref. [46]. Precise parameter values are displayed in Table III. Notice that thestrange quark mass is essentially its physical value, but the up and down quarks are not:the pion is about 3.5 times heavier than its physical value.Stout link smearing [72] was applied to the operators of Sec. II B with parameters tunedto reduce contamination from excited states. In the notation of Ref. [72], we use ( ρ, n ρ ) =(0 . ,
15) for the octet and decuplet, and we use ( ρ, n ρ ) = (0 . ,
15) for the 27-plet, 28-plet,and 35-plet. Figure 3 gives an indication of the quality of the data by showing the relativelyclean example of the octet T P C as well as the much more challenging example of the 27-plet E P C .0 TABLE IV: The mass spectrum of gluelumps containing a static octet particle, as determined fromdynamical lattice QCD at two lattice spacings. J denotes the continuum angular momentum ofthe light (gauge) degrees of freedom and does not include the spin of the octet particle. The firsterror is statistical and the second is systematic, from Eq. (16).Λ P C
J M (Λ P C ) − M ( T + − ) [GeV] β = 1 . β = 2 . T −− ± ± ± ± E −− ± ± ± ± T −− ± ± ± ± E + − ± ± ± ± T + − ± ± ± ± A ++1 ± ± ± ± A + − ± ± ± ± A −− ± ± ± ± E ++ ± ± ± ± T ++2 ± ± ± ± T ++1 ± ± ± ± T − +1 ± ± ± ± A − +2 ± ± ± ± E − + ± ± ± ± T − +2 ± ± ± ± A ++2 ± ± ± ± A −− ± ± ± ± A − +1 ± ± ± ± A + − ± ± ± ± Mass differences are obtained by the simultaneous fit of a pair of correlation functions: C = f e − M τ and C = f e − ( M + δM ) τ (15)where f , f , M , and δM are the four fit parameters. The mass difference δM is thephysics we wish to extract, and its statistical uncertainty is determined by bootstrapping[73]. The most important systematic uncertainty comes from choosing the range of timesteps, τ i to τ f , to include in each fit. Fits do not depend significantly on τ f because theinclusion of noisy data at large Euclidean times has a negligible influence. We determined therange of τ i options that all produced a common δM value within one statistical standarddeviation, and then used the smallest τ i in that range because it produces the smalleststatistical uncertainty. A one-sigma systematic error was then assigned to be (cid:12)(cid:12)(cid:12)(cid:12) δM ( τ i ) − δM ( τ i − (cid:12)(cid:12)(cid:12)(cid:12) . (16)Table IV and Fig. 4 contain the final results for mass splittings among gluelumps withthe static particle in the color-octet representation. As is true throughout this article,angular momentum J refers to the light degrees of freedom only; all results apply to a heavy1 -- -- + - ++ + - -- ++ ++ - + - + - + ++ -- - + + - J PC M ( Λ P C ) - M ( T + - ) [ G e V ] β =2.05 β =1.90 FIG. 4: The content of Table IV is displayed visually. Statistical and systematic errors were addedlinearly. particle—color octet in this case—with any spin. Although the central value for the massdifference at β = 2 .
05 tends to be larger than the central value at β = 1 .
90, the effect ismarginal relative to the quoted error bars. Since both lattice spacings are less than 0.1 fmand an improved lattice QCD action has been used, it is not surprising that mass splittingsare essentially independent of lattice spacing. It is also reassuring to see that E P C and T P C are consistent with each other for each P C , since they should couple to the same physicalstate ( J = 2) in the continuum limit.The quenched lattice QCD study of Ref. [46] had access to only 10 of the 20 channelslisted in our Table IV. The raw data for those 10 channels are provided in Table II of Ref. [46](here called “[46]-II” for brevity) without systematic errors, but several options for adjacenttime steps are shown in [46]-II and from this a systematic error defined by our Eq. (16) canbe estimated if desired. The raw data from [46]-II are in reasonable agreement with thepresent work, but we wish to point out some concerns about how [46]-II was used to arriveat final mass splittings in MeV, as listed in [46]-III.To begin, we note that [46]-III was obtained from [46]-II by going through the figure [46]-3. The figure [46]-3 is largely obtained from [46]-II by using the first two time steps (called“ t =2:1” in Ref. [46]) and combining errors from the two energy levels in quadrature. Thisnumerically reproduces the data in [46]-3 with two exceptions, both at β = 5 .
7: the A ++1 data point in [46]-3 is not consistent with [46]-II, and neither is the error bar for T ++2 . Afirst concern is that β = 5 . t =2:1 data are used to obtain2 TABLE V: The mass spectrum of gluelumps containing a static decuplet particle, as determinedfrom dynamical lattice QCD at β = 1 . J denotes the continuum angular momentum of the light(gauge) degrees of freedom and does not include the spin of the decuplet particle. The first erroris statistical and the second is systematic, from Eq. (16).Λ P J M (Λ P ) − M ( A − ) [GeV] T − ± ± E − ± ± T − ± ± T +1 ± ± A +2 ± ± E + ± ± T +2 ± ± A +1 ± ± A − ± ± the continuum limit even though [46]-II shows that they produce mass splittings that differsignificantly from later time steps.These concerns should not detract from the valuable comparison between the presentstudy and [46]-II. Our dynamical lattice QCD study uses two lattice spacings that are veryclose to the finer two spacings of the quenched study in Ref. [46], and produces compatibleresults, which indicates that quenching errors are too small to disentangle from the otheruncertainties. The authors of Ref. [46] reported a lack of degeneracy for E ++ and T ++2 atnonzero lattice spacings, with T ++2 heavier than E ++ , and we see a similar tendency though itis not large relative to the error bars in Fig. 4. Moreover, we now have three other channels(+ − , − +, and −− ) where E and T can be compared, and these are all appropriatelydegenerate when systematic uncertainties are taken into account. The authors of Ref. [46]were surprised by the degeneracy of 2 + − with 3 + − , but in the context of our 20-channel studythis pair of operators has no striking degeneracy. The authors of Ref. [46] were surprisedby the heaviness of the 0 ++ , and we agree that it is heavy, although the extrapolation inRef. [46] is noticeably reduced when the β = 5 . β = 1 .
90 ensemble. Simulations of the β = 2 .
05 lattices were computa-tionally expensive and, like the octet results, we do not anticipate a significant dependenceon lattice spacing between these two β values, so our results at β = 1 .
90 represent predictionsfor the continuum physics spectrum. Mass splittings for gluelumps with the static particlein the color-decuplet representation are shown in Table V. Notice that the A − , which inthe continuum is 0 − , appears to be the lightest state in this spectrum modulo systematicuncertainties.Mass splittings for gluelumps containing a 27-plet static particle are shown in Table VI.All 20 Λ P C channels were attempted, but those omitted from the table produced no usablesignal. Although mass differences are tabulated relative to T ++2 , the data do not ensure thatthis is the lightest state. For both the decuplet and the 27-plet, the E and T channels areconsistent with one another.Correlation functions for the 28-plet and 35-plet contained too few usable time steps togive a meaningful systematic error, so we refrain from presenting numerical results. Never-3 TABLE VI: The resolvable mass spectrum of gluelumps containing a static 27-plet particle, asdetermined from dynamical lattice QCD at β = 1 . J denotes the continuum angular momentumof the light (gauge) degrees of freedom and does not include the spin of the 27-plet particle. Thefirst error is statistical and the second is systematic, from Eq. (16).Λ P C
J M (Λ P C ) − M ( T ++2 ) [GeV] E ++ ± ± A ++1 ± ± T − +2 ± ± E − + ± ± T − +1 ± ± A − +2 ± ± A ++2 ± ± T + − ± ± T ++1 ± ± T + − ± ± E + − ± ± T −− ± ± theless, the writing and running of this code helped us to confirm the operator definitionspresented in Secs. II A and II B and Appendixes A and B—for example, we tested gaugeinvariance through explicit computations with a single configuration in every case.Although none of the operators used in the present work contain valence quarks, physicalstates with valence quarks could have the same quantum numbers as gluelumps. Examples ofsuch states include the adjoint mesons in Ref. [46] that were explored on quenched lattices byusing operators that contain explicit valence quarks. As exemplified by Fig. 8 of Ref. [46],the mass difference between gluelumps and adjoint mesons is difficult to ascertain. Ouruse of dynamical configurations in principle allows adjoint mesons to mix with the gluelumpsignals, but our exclusive use of quark-free operators likely produces only a feeble coupling toadjoint mesons. A combined study of adjoint mesons and gluelumps would require operatorsof both types to be analyzed simultaneously in a matrix that permits mixing between them. IV. CONCLUSIONS
Any extension of the standard model with a long-lived colored heavy particle will containnew hadrons that are QCD bound states of the heavy particle together with gluons andquarks. The lattice QCD study of this new hadron spectrum was pioneered by Michael andcollaborators [43–46], motivated by the color-octet gluino of supersymmetry.The present study has revisited the gluelump spectrum in greater detail. This is the firstlattice simulation to explore the complete set of gluelump quantum numbers, J P C , where J represents the angular momentum of the light degrees of freedom. The heavy particle istreated as static, so its spin decouples. The lightest new state not studied previously is 0 −− ,which is found to be as light as some of the states that were studied in Ref. [46]. Comparisonof E and T representations, both of which couple to J = 2 in the continuum limit, providesa cross-check on systematic errors. A leading systematic error was identified as arising from4the choice of a fitting window in Euclidean time. Comparison of the quenched results fromRef. [46] with the present dynamical results does not reveal any large quenching artifacts.In addition, the present study provides the first results for generalized gluelumps, wherethe heavy particle is not color octet but rather decuplet or 27-plet. The machinery for28-plet and 35-plet computations was also established and tested, so future studies will bestraightforward in those cases as well.Final numerical results are presented in Tables IV, V, and VI. The two β values for octetresults represent two different lattice spacings that agree within uncertainties. Comparisonof Table IV with the previous studies tabulated in Table I shows a general agreement, andindicates that systematic errors cannot be neglected: lattice results are presently limited bysystematics rather than statistics. Future studies can directly use the operators developedhere to perform larger-scale simulations and improve the precision for this spectrum ofgeneralized gluelumps. Acknowledgments
We thank the CP-PACS and JLQCD Collaborations for making their dynamical gaugefield configurations available. This work was supported in part by the Natural Sciences andEngineering Research Council (NSERC) of Canada and by Compute Canada through theShared Hierarchical Academic Research Computing Network (SHARCNET).
Appendix A: BASIS TENSORS FOR EACH REPRESENTATION
To reduce notational clutter, define generalized Kronecker delta functions where indicesin parenthesis are to be permuted through all distinct orderings. A few examples are thefollowing: δ { ij }{ kk } = δ ik δ jk ,δ { ij }{ kl } = δ ik δ jl + δ il δ jk ,δ { ijk }{ lll } = δ il δ jl δ kl ,δ { ijk }{ llm } = δ il δ jl δ km + δ il δ jm δ kl + δ im δ jl δ kl ,δ { ijk }{ lmn } = δ il δ jm δ kn + δ il δ jn δ km + δ im δ jl δ kn + δ im δ jn δ kl + δ in δ jl δ km + δ in δ jm δ kl ,δ { ijkl }{ pppp } = δ ip δ jp δ kp δ lp ,δ { ijkl }{ pppq } = δ ip δ jp δ kp δ lq + δ ip δ jp δ kq δ lp + δ ip δ jq δ kp δ lp + δ iq δ jp δ kp δ lp ,δ { ijkl }{ ppqq } = δ ip δ jp δ kq δ lq + δ ip δ jq δ kp δ lq + δ iq δ jp δ kp δ lq + δ ip δ jq δ kq δ lp + δ iq δ jp δ kq δ lp + δ iq δ jq δ kp δ lp ,δ { ijkl }{ ppqr } = δ ip δ jp δ kq δ lr + δ ip δ jq δ kp δ lr + δ iq δ jp δ kp δ lr + δ ip δ jq δ kr δ lp + δ iq δ jp δ kr δ lp + δ iq δ jr δ kp δ lp + δ ip δ jp δ kr δ lq + δ ip δ jr δ kp δ lq + δ ir δ jp δ kp δ lq + δ ip δ jr δ kq δ lp + δ ir δ jp δ kq δ lp + δ ir δ jq δ kp δ lp . (A1)5The basis used for the octet representation is T ij = δ i δ j , T ij = δ i δ j , T ij = δ i δ j ,T ij = δ i δ j , T ij = δ i δ j , T ij = δ i δ j ,T ij = ( δ i δ j − δ i δ j ) , T ij = √ ( δ i δ j + δ i δ j − δ i δ j ) . (A2)The octet tensors obey the relation X µ =1 T µij T µkl = δ ik δ jl − δ ij δ kl . (A3)The basis used for the decuplet representation is T ijk = δ { ijk }{ } , T ijk = δ { ijk }{ } , T ijk = δ { ijk }{ } ,T ijk = √ δ { ijk }{ } , T ijk = √ δ { ijk }{ } , T ijk = √ δ { ijk }{ } ,T ijk = √ δ { ijk }{ } , T ijk = √ δ { ijk }{ } , T ijk = √ δ { ijk }{ } ,T ijk = √ δ { ijk }{ } . (A4)The decuplet tensors obey the relation X µ =1 T µijk T µlmn = 16 δ { ijk } δ { lmn } . (A5)The basis used for the 27-plet representation is T ijkl = δ { ij }{ } δ { kl }{ } , T ijkl = δ { ij }{ } δ { kl }{ } , T ijkl = δ { ij }{ } δ { kl }{ } ,T ijkl = δ { ij }{ } δ { kl }{ } , T ijkl = δ { ij }{ } δ { kl }{ } , T ijkl = δ { ij }{ } δ { kl }{ } ,T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } (cid:1) , T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } (cid:1) , T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } (cid:1) , T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 12 (cid:0) δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } + δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 12 (cid:0) δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } + δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 12 (cid:0) δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } + δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } (cid:1) , T ijkl = 12 (cid:0) δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } + δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 12 (cid:0) δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } + δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 12 (cid:0) δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } + δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } + δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } + δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } + δ { ij }{ } δ { kl }{ } + δ { ij }{ } δ { kl }{ } (cid:1) ,T ijkl = 1 √ (cid:0) δ { ij }{ } δ { kl }{ } + 2 δ { ij }{ } δ { kl }{ } + 2 δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } − δ { ij }{ } δ { kl }{ } (cid:1) . (A6)The 27-plet tensors obey the relation X µ =1 T µijkl T µmnop = 14 ( δ im δ jn δ ko δ lp + δ im δ jn δ kp δ lo + δ in δ jm δ ko δ lp + δ in δ jm δ kp δ lo ) −
120 ( δ im δ jl δ ko δ np + δ im δ jl δ kp δ no + δ in δ jl δ ko δ mp + δ in δ jl δ kp δ mo + δ im δ lo δ jk δ np + δ im δ lp δ jk δ no + δ in δ lo δ jk δ mp + δ in δ lp δ jk δ mo + δ il δ jm δ ko δ np + δ il δ jm δ kp δ no + δ il δ jn δ ko δ mp + δ il δ jn δ kp δ mo + δ ik δ jm δ lo δ np + δ ik δ jm δ lp δ no + δ ik δ jn δ lo δ mp + δ ik δ jn δ lp δ mo )+ 140 ( δ ik δ jl δ mo δ np + δ ik δ jl δ mp δ no + δ il δ jk δ mo δ np + δ il δ jk δ mp δ no ) . (A7)The basis used for the 28-plet representation is T ijklmn = δ { ijklmn }{ } , T ijklmn = δ { ijklmn }{ } ,T ijklmn = δ { ijklmn }{ } , T ijklmn = 1 √ δ { ijklmn }{ } , T ijklmn = 1 √ δ { ijklmn }{ } , T ijklmn = 1 √ δ { ijklmn }{ } ,T ijklmn = 1 √ δ { ijklmn }{ } , T ijklmn = 1 √ δ { ijklmn }{ } ,T ijklmn = 1 √ δ { ijklmn }{ } , T ijklmn = 1 √ δ { ijklmn }{ } ,T ijklmn = 1 √ δ { ijklmn }{ } , T ijklmn = 1 √ δ { ijklmn }{ } ,T ijklmn = 1 √ δ { ijklmn }{ } , T ijklmn = 1 √ δ { ijklmn }{ } ,T ijklmn = 1 √ δ { ijklmn }{ } , T ijklmn = 1 √ δ { ijklmn }{ } ,T ijklmn = 1 √ δ { ijklmn }{ } , T ijklmn = 1 √ δ { ijklmn }{ } ,T ijklmn = 1 √ δ { ijklmn }{ } , T ijklmn = 1 √ δ { ijklmn }{ } ,T ijklmn = 1 √ δ { ijklmn }{ } , T ijklmn = 1 √ δ { ijklmn }{ } ,T ijklmn = 1 √ δ { ijklmn }{ } , T ijklmn = 1 √ δ { ijklmn }{ } ,T ijklmn = 1 √ δ { ijklmn }{ } , T ijklmn = 1 √ δ { ijklmn }{ } ,T ijklmn = 1 √ δ { ijklmn }{ } , T ijklmn = 1 √ δ { ijklmn }{ } . (A8)The 28-plet tensors obey the relation X µ =1 T µijklmn T µopqrst = 16! δ { ijklmn } δ { opqrst } . (A9)The basis used for the 35-plet representation is T ijklm = δ { ijkl }{ } δ m , T ijklm = δ { ijkl }{ } δ m ,T ijklm = δ { ijkl }{ } δ m , T ijklm = δ { ijkl }{ } δ m ,T ijklm = δ { ijkl }{ } δ m , T ijklm = δ { ijkl }{ } δ m ,T ijklm = 12 δ { ijkl }{ } δ m , T ijklm = 12 δ { ijkl }{ } δ m ,T ijklm = 12 δ { ijkl }{ } δ m , T ijklm = 12 δ { ijkl }{ } δ m ,T ijklm = 12 δ { ijkl }{ } δ m , T ijklm = 12 δ { ijkl }{ } δ m ,T ijklm = 1 √ δ { ijkl }{ } δ m , T ijklm = 1 √ δ { ijkl }{ } δ m ,T ijklm = 1 √ δ { ijkl }{ } δ m , T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m + δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m + δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m + δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m + δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) ,T ijklm = 1 √ (cid:0) δ { ijkl }{ } δ m − δ { ijkl }{ } δ m − δ { ijkl }{ } δ m (cid:1) . (A10)9The 35-plet tensors obey the relation X µ =1 T µijklm T µnopqr = 124 δ { ijlk } δ { nopq } δ mr − (cid:0) δ { jkl } δ { opq } δ im δ nr + δ { jkl } δ { npq } δ im δ or + δ { jkl } δ { noq } δ im δ pr + δ { jkl } δ { nop } δ im δ qr + δ { ikl } δ { opq } δ jm δ nr + δ { ikl } δ { npq } δ jm δ or + δ { ikl } δ { noq } δ jm δ pr + δ { ikl } δ { nop } δ jm δ qr + δ { ijl } δ { opq } δ km δ nr + δ { ijl } δ { npq } δ km δ or + δ { ijl } δ { noq } δ km δ pr + δ { ijl } δ { nop } δ km δ qr + δ { ijk } δ { opq } δ lm δ nr + δ { ijk } δ { npq } δ lm δ or + δ { ijk } δ { noq } δ lm δ pr + δ { ijk } δ { nop } δ lm δ qr (cid:1) . (A11) Appendix B: BUILDING THE OCTET OPERATORS
The construction of operators relies on textbook group theory methods (see, for example,Ref. [74]). Table VII is a reminder of the connection between angular momentum in thecontinuum and on a lattice[75, 76]. To build an octet operator, begin with a single chairand list all possible rotations of it. There are 24 orientations in total, as shown in Fig. 1.The A representation is built from a particular sum, H (8) α ( A ) = X a =1 L (8) a ! ij T αij . (B1)Any octahedral rotation of this sum leaves it invariant, as expected for a J = 0 operator.The A representation is built from a different sum, H (8) α ( A ) = X a =1 ( − a L (8) a − X a =13 ( − a L (8) a ! ij T αij . (B2)Some octahedral rotations of this sum leave it invariant; others return the negative of thesum. The T representation is built from a set of three sums, H (8) α ( T x ) = (cid:16) L (8)6 + L (8)20 + L (8)21 + L (8)11 − L (8)18 − L (8)8 − L (8)9 − L (8)23 (cid:17) ij T αij , (B3) H (8) α ( T y ) = (cid:16) L (8)5 + L (8)19 + L (8)24 + L (8)10 − L (8)17 − L (8)7 − L (8)12 − L (8)22 (cid:17) ij T αij , (B4) H (8) α ( T z ) = (cid:16) L (8)1 + L (8)2 + L (8)3 + L (8)4 − L (8)13 − L (8)14 − L (8)15 − L (8)16 (cid:17) ij T αij . (B5)Any octahedral rotation of one of these sums returns one of the three sums (itself or one ofthe other two) up to ±
1, as expected for a vector with J = 1. The T representation is builtfrom a set of three sums, H (8) α ( T x ) = (cid:16) L (8)6 − L (8)20 + L (8)21 − L (8)11 + L (8)18 − L (8)8 + L (8)9 − L (8)23 (cid:17) ij T αij , (B6)0 H (8) α ( T y ) = (cid:16) L (8)5 − L (8)19 + L (8)24 − L (8)10 + L (8)17 − L (8)7 + L (8)12 − L (8)22 (cid:17) ij T αij , (B7) H (8) α ( T z ) = (cid:16) L (8)1 − L (8)2 + L (8)3 − L (8)4 + L (8)13 − L (8)14 + L (8)15 − L (8)16 (cid:17) ij T αij . (B8)Any octahedral rotation of one of these sums returns one of the three sums (itself or one ofthe other two) up to ±
1. The E representation is built from a set of three differences, v − v , (B9) v − v , (B10) v − v , (B11)where v ≡ (cid:16) L (8)6 + L (8)20 + L (8)21 + L (8)11 + L (8)18 + L (8)8 + L (8)9 + L (8)23 (cid:17) ij T αij , (B12) v ≡ (cid:16) L (8)5 + L (8)19 + L (8)24 + L (8)10 + L (8)17 + L (8)7 + L (8)12 + L (8)22 (cid:17) ij T αij , (B13) v ≡ (cid:16) L (8)1 + L (8)2 + L (8)3 + L (8)4 + L (8)13 + L (8)14 + L (8)15 + L (8)16 (cid:17) ij T αij . (B14)Notice that only two of the three differences are linearly independent. Any octahedralrotation of one of these differences returns one of the three differences (itself or one of theother two) up to ± χ of each representation, which is defined tobe the set of traces of the explicit matrix representation. Since the octahedral group hasfive conjugacy classes, we need to evaluate five traces per representation. To be explicit, wecan use (cid:26) χ ( e ) , χ ( c ( xy )2 ) , χ ( c ( x → y → z → x )3 ) , χ ( c ( z )4 ) , χ (cid:18)(cid:16) c ( z )4 (cid:17) (cid:19)(cid:27) (B15)where c ( xy )2 denotes a 180 ◦ rotation about the line ( x + y = 1 , z = 0) and c ( x → y → z → x )3 denotesa 120 ◦ rotation about the line ( x = y = z ).For the A , A , T , and T representations, it is not necessary to build an explicit matrixrepresentation; we can merely sum the ± ± (itself). For the E representation it is best to build the two-dimensional matrices TABLE VII: The relationship between continuum angular momentum J and octahedral irreduciblerepresentation Λ. Λ J · · · A · · · A · · · E · · · T · · · T · · · (cid:18) H (8) α ( E ) H (8) α ( E ) (cid:19) = √ ( v − v ) − √ ( v + v − v ) ! (B16)and it leads to e = (cid:18) (cid:19) , (B17) c ( z )4 = (cid:18) − (cid:19) . (B18)Under a c ( y )4 rotation, we obtain1 √ v − v ) → √ v − v ) (B19)= 12 (cid:18) √ v − v ) (cid:19) + √ (cid:18) − √ v + v − v ) (cid:19) (B20) − √ v + v − v ) → − √ v + v − v ) (B21)= √ (cid:18) √ v − v ) (cid:19) − (cid:18) − √ v + v − v ) (cid:19) . (B22)Therefore c ( y )4 = √ √ − ! . (B23)All other matrices can be obtained by multiplication of c ( y )4 and c ( z )4 in various orders finding,for example, c ( xy )2 = (cid:18) − (cid:19) , (B24) c ( x → y → z → x )3 = − − √ √ − ! , (B25) (cid:16) c ( z )4 (cid:17) = (cid:18) (cid:19) . (B26)The characters of all representations are collected into Table VIII, and the multiplicities areobtained through standard group theory methods: m ( A ) = 124 (cid:16) χ ( e ) + 6 χ ( c ( xy )2 ) + 8 χ ( c ( x → y → z → x )3 ) + 6 χ ( c ( z )4 ) + 3 χ (( c ( z )4 ) ) (cid:17) = 1 , (B27) m ( A ) = 124 (cid:16) χ ( e ) − χ ( c ( xy )2 ) + 8 χ ( c ( x → y → z → x )3 ) − χ ( c ( z )4 ) + 3 χ (( c ( z )4 ) ) (cid:17) = 1 , (B28) m ( E ) = 124 (cid:16) χ ( e ) − χ ( c ( x → y → z → x )3 ) + 6 χ (( c ( z )4 ) ) (cid:17) = 1 , (B29)2 TABLE VIII: Characters χ of all irreducible representations Λ for the octet operator.Λ χ ( e ) χ ( c ( xy )2 ) χ ( c ( x → y → z → x )3 ) χ ( c ( z )4 ) χ (( c ( z )4 ) ) A A E T T m ( T ) = 124 (cid:16) χ ( e ) − χ ( c ( xy )2 ) + 6 χ ( c ( z )4 ) − χ (( c ( z )4 ) ) (cid:17) = 1 , (B30) m ( T ) = 124 (cid:16) χ ( e ) + 6 χ ( c ( xy )2 ) − χ ( c ( z )4 ) − χ (( c ( z )4 ) ) (cid:17) = 1 . (B31)To summarize, the operators that will be typed into the computer code are those shown inEq. (13). Operators beyond the octet are built from this octet starting point, as describedin Sec. II B. [1] E. Del Nobile, R. Franceschini, D. Pappadopulo, and A. Strumia, Nucl. Phys. B826 , 217(2010) [arXiv:0908.1567 [hep-ph]].[2] F. del Aguila, J. de Blas, and M. Perez-Victoria, J. High Energy Phys. 09 ( ) 033[arXiv:1005.3998 [hep-ph]].[3] T. Han, I. Lewis, and Z. Liu, J. High Energy Phys. 12
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