Three loop DIS and transversity operator anomalous dimensions in the RI' scheme
aa r X i v : . [ h e p - ph ] J un Three loop DIS and transversity operator anomalous dimensionsin the RI ′ scheme ∗ J.A. Gracey,Theoretical Physics Division,Department of Mathematical Sciences,University of Liverpool,P.O. Box 147,Liverpool,L69 3BX,United Kingdom.
Abstract.
We discuss the computation of the three loop anomalous dimensions for variousoperators used in deep inelastic scattering in the MS and RI ′ schemes. In particular the resultsfor the n = 5 and 6 Wilson operators in arbitrary linear covariant gauge in the RI ′ scheme arenew. LTH 747 ∗ Talk presented at 11th International Workshop on Advanced Computing and Analysis Techniques in PhysicsResearch, NIKHEF, Amsterdam, The Netherlands, 23rd-27th April, 2007. Introduction.
The use of symbolic manipulation and computer algebra has been an invaluable tool for provid-ing large loop order results in quantum field theory in general and in quantum chromodynamics(QCD) in particular. The outstanding example is that of the full three loop anomalous dimen-sions for flavour non-singlet and singlet unpolarized operators for deep inelastic scattering as ananalytic function of the operator moment n , [1, 2, 3, 4]. Also the Wilson coefficients have beenprovided to the same precision. This large project, of the order of ten years, required not onlythe extensive use of the symbolic manipulation programme Form , [5], but also its own devel-opment to handle the unforeseen complexity of the computation. An earlier approach, [6, 7],to this problem in deep inelastic scattering was to determine results for fixed (even) momentsusing the
Mincer algorithm, [8], translated into
Form , [9]. For instance, the first few momentswere given in [6, 7] and subsequently those for n = 12 and n = 16 appeared in [10, 11]. At thattime the fixed moment expressions were used to parametrize the full expressions but such anapproach was clearly incomplete lacking the correctness of a full evaluation. However, the re-sults subsequently served as very important independent checks on the final arbitrary n results.Now that the computational algorithm has been established, in principle it can be applied toother operators underlying related phenomenology. For example, the case of polarized Wilsonoperators will be relevant for spin physics. In addition in the spin context there is interestin a similar operator called transversity, [12, 13, 14]. This corresponds to the probability offinding a quark in a transversely polarized nucleon polarized parallel to the nucleon versus thatof the nucleon in the antiparallel polarization. From a theoretical point of view it is similarto the non-singlet unpolarized Wilson operator but experimentally it is not as accessible sincethere is no direct coupling to quarks. Nevertheless there have been proposals to study it atRHIC. Therefore, whilst in principle it is possible to calculate the arbitrary moment three looptransversity operator anomalous dimensions in the MS scheme, it would be important to havestrong independent checks on any future full result. Akin to the 1990’s approach for the Wilsonoperator there is therefore a need for fixed moment calculations. Aside from this motivation,there is a secondary one.One of the ingredients necessary to study the structure functions is the measurement ofthe non-perturbative matrix elements. From the theory point of view, a tool which achievesthis is lattice regularization and various groups, such as QCDSF, have developed a substantialprogramme to determine key matrix elements. (See, for instance, [15, 16, 17].) However, onetechnical aspect of such work is ensuring that the results agree in the continuum with expec-tations from the ultraviolet limit. One approach in this respect is for the lattice results tobe matched onto the perturbative expressions in the chiral limit, where to aid precision onewould prefer the results to as high a loop order as is calculationally feasible. This has beenconsidered in a series of articles, [18, 19, 20], to three loops. One technical issue is that to keeptime (and money) to a minimum, the lattice computations are performed in renormalizationschemes known as regularization invariant (RI) and its modification, RI ′ , [21, 22]. Unlike theMS scheme, they are mass dependent renormalization schemes. Results in this scheme have thento be converted to the standard MS scheme. In the continuum QCD has been renormalized athigh loop order in both RI and RI ′ in [23, 18] and the conversion functions established for variousquantities of interest. Therefore to aid lattice computations of the matrix elements one requiresthe finite part of the analogous Green’s functions but only for low moment since clear signals forhigher moments information are hard to extract from the numerical noise on the lattice. Giventhe need for such accurate results for a specific Green’s function on the lattice, not only for thetransversity operators but also for the Wilson operators, we report the results of recent com-putations in this area, [20, 24]. One consequence of the Green’s function considered is that the2perator anomalous dimensions emerge as a corollary in the RI ′ and MS schemes. In additionto the results already available, [18, 19, 20, 24], we will provide the RI ′ anomalous dimensionsfor the n = 5 and 6 Wilson operators in arbitrary covariant gauge. Although the finite parts arerequired for lattice calculations for low moment, our computations have been extended to highmoment for the transversity operator but only to determine the anomalous dimensions. At theappropriate point we will mention some of the computer algebra aspects of this and previouswork which without the power of Form would have rendered the determination of any resultvirtually impossible.The paper is organised as follows. In section two we introduce our notation and compu-tational strategy in more detail before discussing the appropriate points of the RI ′ scheme insection three. Section four details a simple low moment example, whilst the symbolic manipu-lation issues are recorded in the context of the higher moment calculation in section five. Theexplicit anomalous dimensions are given in section six with a few concluding remarks providedin the final section. The two basic classes of operators we will consider are the non-singlet Wilson operators O ν ...ν n W = S ¯ ψ i γ ν D ν . . . D ν n ψ j (2.1)and the transversity operators O µν ...ν n T = S ¯ ψ i σ µν D ν . . . D ν n ψ j (2.2)where ψ i is the quark field, 1 ≤ i ≤ N f for N f quark flavours, D µ is the covariant derivativeand σ µν = [ γ µ , γ ν ]. The operation of totally symmetrizing with respect to the Lorentz indices { ν i } and ensuring the operator is traceless is denoted by S where the respective but differenttracelessness conditions are given by η ν i ν j O ν ...ν i ...ν j ...ν n W = 0 (2.3)and, [25], η µν i O µν ...ν i ...ν n T = 0 ( i ≥ , η ν i ν j O µν ...ν i ...ν j ...ν n T = 0 . (2.4)For (2.2) the anomalous dimensions are available for all moments n at one and two loops in theMS scheme in [26, 27, 28, 25, 29]. At three loops fixed moment results are available for momentsup to n = 8 for MS and n = 7 for RI ′ , [19, 20, 24]. The specific Green’s functions relevant forthe lattice matching are G ν ...ν n W ( p ) = h ψ ( − p ) O ν ...ν n W (0) ¯ ψ ( p ) i G µν ...ν n T ( p ) = h ψ ( − p ) O µν ...ν n T (0) ¯ ψ ( p ) i (2.5)where the operator is inserted at zero momentum. This allows for the application of the Mincer algorithm, [8, 9], which determines the finite part of scalar massless two point functions usingdimensional regularization in d = 4 − ǫ dimensions to three loops. Unlike earlier approachesto extract anomalous dimensions we do not contract the free Lorentz indices with a null vector.Whilst such a contraction has the effect of excluding the trace terms in the operator itself orthe Green’s function decomposition, the main reason why we cannot follow that route here isthat the lattice makes measurements in different directions of the momentum components. This3llows for the extraction of the values of each of the individual amplitudes into which the Green’sfunctions are decomposed.From the form of the operators there will be n n -point Feynman rules for both the Wilsonand transversity operators, each with two quark legs. However, at the three loop order we willwork at, only the Feynman rules up to and including three gluon legs will be necessary. Hence,there are 3 one loop, 37 two loop and 684 three loop Feynman diagrams contributing to eachGreen’s function. These are generated electronically using the Qgraf package, [30], beforebeing converted into
Form input notation to allow for the application of the
Form version ofthe
Mincer package, [8, 9]. As the
Mincer algorithm is only applicable to scalar Feynmanintegrals, for each moment n the Green’s functions, (2.5), need to be decomposed into invariantamplitudes and Lorentz tensors which respect all symmetry structures. Whilst we will discussmore details later, it suffices to note at this point that for Wilson operators there will be twoindependent amplitudes but three for the transversity case.Although the lattice computations are ultimately performed in the Landau gauge, we willcompute in an arbitrary linear covariant gauge. The associated gauge parameter will act as aninternal checking parameter since, for instance, in the MS scheme the anomalous dimension ofgauge invariant operators are independent of the gauge parameter, [31, 32]. As the computationsare clearly automatic we employ the procedure of [33] where all the diagrams are computed forbare coupling, g o , and gauge parameter α o . To extract the anomalous dimension the renormal-ization constant is fixed (either in the MS or RI ′ schemes) by rescaling these variables by theknown coupling constant and gauge parameter renormalization constants g o = Z g g , α o = Z − α Z A α (2.6)in our conventions, where Z A is the gluon wave function renormalization constant. The remain-ing divergence for each of the Green’s function is absorbed into the operator renormalizationconstant together with a specified finite part in the case of the RI ′ scheme to leave the finiteparts for the lattice matching. In practice the results are determined in the MS scheme first,primarily due to more consistency checks being available before extracting the RI ′ expressions. ′ scheme. We briefly review the parts of the RI ′ scheme needed for the computations discussed here.Originally the scheme was invented in the context of the lattice, [21, 22], but it is not restrictedto a discrete spacetime. The continuum analogue has been studied to three and four looporder in [23, 18]. In general terms it is a mass dependent renormalization scheme where therenormalization of the quark field is chosen to be non-minimal in a way which is appropriate forlattice analyses. The coupling constant (and thus vertex) renormalization is performed in an MSway and so in some sense the RI set of schemes sits between MS and MOM type schemes. Toreduce time (and cost), since taking a derivative on the lattice requires significant computation,the quark 2-point function, Σ ψ ( p ), is renormalized according to the RI ′ prescription, [21, 22],lim ǫ → h Z ′ ψ Σ ψ ( p ) i(cid:12)(cid:12)(cid:12)(cid:12) p = µ = p/ (3.1)where µ is the renormalization point. In other words there are no O ( g ) corrections to Σ ψ ( p )after renormalization at p = µ as these finite parts are absorbed into the quark wave functionrenormalization constant, Z ′ ψ . We use the notation throughout that a ′ on a quantity indicatesthat renormalization has been performed in the RI ′ scheme. Otherwise the scheme is MS. For4ompleteness the RI scheme, which is not of interest to us here, involves taking a momentumderivative of Σ ψ ( p ) first before choosing the result to be the tree value at the renormalizationpoint, [21, 22]. As an extension of the RI ′ scheme in the continuum, the gluon and Faddeev-Popov 2-point functions are renormalized analogous to (3.1). However, as most interest ingeneral is in quark 2-point Green’s functions, there is no real need to pursue this route, unlessone was perhaps intending to consider supersymmetric theories. Similar to the lattice we areultimately interested in converting results from RI ′ to MS and therefore the variables in eachscheme need to be related. Using the standard conversion definitions α ′ = Z ′ A Z A α , a ′ = Z ′ g Z g a (3.2)where a = g / (16 π ), we have the one loop relations, [23, 18], a ′ = a + O ( a ) α ′ = (cid:20) (cid:16)(cid:16) − α − α − (cid:17) C A + 80 T F N f (cid:17) a (cid:21) α + O ( a ) . (3.3)The explicit expressions to three loops are available in [18]. Though it is worth noting that theLandau gauge is preserved in changing between RI ′ and MS. To illustrate the effect the schemeshave on the basic anomalous dimensions, we note γ ψ ( a ) = αC F a + 14 h ( α + 8 α + 25) C A C F − C F − C F T F N f i a γ ′ ψ ( a ) = αC F a + h(cid:16) α + 45 α + 223 α + 225 (cid:17) C A − C F − (80 α + 72) T F N f ] C F a
36 (3.4)where the group theoretic quantities are defined byTr (cid:16) T a T b (cid:17) = T F δ ab , T a T a = C F I , f acd f bcd = C A δ ab (3.5)for a colour group with generators T a and structure functions f abc . Clearly the difference in thenumerical structure in (3.4) only appears at two and higher loops.For the flavour non-singlet operators we are interested in here, we follow a similar route to(3.1) for defining the operator renormalization constant in the RI ′ scheme. Writing Σ ( T ) O ( p ) asthe amplitude in the Lorentz decomposition of (2.5) which contains the tree, ( T ), part of theoperator, we set lim ǫ → h Z ′ ψ Z ′O Σ ( T ) O ( p ) i(cid:12)(cid:12)(cid:12)(cid:12) p = µ = T (3.6)where T is the value of the tree term of amplitude, which may not necessarily be unity giventhe specific (non-unique) way of carrying out the decomposition. We now illustrate the preceeding remarks by discussing the case of the n = 2 transversityoperator in more detail, [19]. First, given the symmetry properties (2.4) the explicit tracelesssymmetrized operator is S ¯ ψσ µν D ρ ψ = ¯ ψσ µν D ρ ψ + ¯ ψσ µρ D ν ψ − d − η νρ ¯ ψσ µλ D λ ψ + 1( d − (cid:16) η µν ¯ ψσ ρλ D λ ψ + η µρ ¯ ψσ νλ D λ ψ (cid:17) (4.1)5n d -dimensions. Inserting (4.1) into the Green’s function G µνρT ( p ), it is decomposed into thethree invariant amplitudes as G µνρT ( p ) = Σ (1) T ( p ) (cid:18) σ µν p ρ + σ µρ p ν − ( d + 2) p σ µλ p ν p ρ p λ + η νρ σ µλ p λ (cid:19) + Σ (2) T ( p ) (cid:16) η µν σ ρλ p λ + η µρ σ νλ p λ − ( d + 1) η νρ σ µλ p λ + ( d − d + 2) p σ µλ p ν p ρ p λ (cid:19) + Σ (3) T ( p ) (cid:16) σ νλ p µ p ρ p λ + σ ρλ p µ p ν p λ + dσ µλ p ν p ρ p λ − η νρ σ µλ p λ p (cid:17) (4.2)in d -dimensions. It is worth noting that this and other decompositions are not unique since onecan always take a linear combination of the three (independent) tensor structures consistent withthe symmetry and traceless properties to form another set of independent amplitudes. However,with this choice one can algebraically form a scalar object which is computed via Mincer . Forinstance, [19],Σ (1) T ( p ) = − d − d −
2) tr (cid:20)(cid:18) σ µν p ρ + σ µρ p ν − ( d + 2) p σ µλ p ν p ρ p λ + η νρ σ µλ p λ (cid:17) G µνρT ( p ) i − d − d − p tr h(cid:16) σ νλ p µ p ρ p λ + σ ρλ p µ p ν p λ + dσ µλ p ν p ρ p λ − η νρ σ µλ p λ p (cid:17) G µνρT ( p ) i . (4.3)This together with Σ ( i ) T ( p ), i = 2 and 3, are the objects of interest for the lattice matching andhave been determined to O ( a ), [19]. For this specific example, we note that the construction ofthe tensor basis as well as the amplitude decomposition can easily be carried out by hand. Thisis primarily due to the small number of free Lorentz indices present. Clearly for the extractionof the anomalous dimensions and amplitudes for the higher moment operators, such a procedurewould be unacceptably time consuming by hand. Moreover, it would be prone to elementaryalgebraic errors. To extract the anomalous dimensions for the higher moment operators, it is clear that onehas to proceed with a computer algebra construction to determine the basis for the independentamplitudes and hence the projections. We discuss the issues in relation to the n = 8 transversityoperator as an example, [24]. For this case there are initially seventeen potential tensors intowhich the Green’s function (2.5) can be decomposed. These are built from the relevant vectorsand tensors of the operator in question, which for transversity are p µ , η µν and σ µν . The onlyconstraint being that the Green’s function has nine independent indices. Given these seventeentensors then within Form it is straightforward to construct the seventeen tensors which havethe correct symmetry, but not traceless, properties. Taking a linear combination of these newobjects with as yet unrelated coefficients, the relationship between these are fixed by imposingthe remaining traceless criterion. In practical terms we take successive pairs of free indices andcontract them. The coefficients of the resulting tensors produce constraints on the seventeeninitial coefficients which can be solved. Whilst there are more contractions than coefficientsthere is redundancy in the system of linear equations which determine the coefficients. Thisis due to the symmetry of the operator itself. However, there is no unique solution and three6oefficients remain unrelated producing three independent amplitudes. (For the Wilson operatorthe corresponding number is two.) For the higher moments, as we are ultimately interested inthe anomalous dimensions, the specific linear combination one uses is not a major issue. Theonly constraint is to choose that projector of the three which leads to the lowest computationtime when
Mincer is applied. The test for this is to compare the run times for each projectorto do the full two loop calculation in an arbitrary linear covariant gauge before generating theresults for the three loop diagrams.The other main computer algebra issue is the construction of the Feynman rules for eachoperator. Given that the Green’s function has free indices one in principle has to constructthe full symmetrized and traceless operator before applying the
Form routine to generate theexplicit Feynman rules for the operator. However, given that the Green’s function will bemultiplied by a projector which is traceless, that part of the operator containing η µν tensorswill automatically give zero upon contracting with the projector. Therefore there is no need tohave an operator which is traceless; only an operator which is symmetrized will suffice. Thiswill reduce computation time since otherwise with a traceless operator there will be an internalintermediate expression swell which will be sorted by Form to produce the equivalent scalarexpression as ignoring the traceless part. For instance, for the n = 8 transversity operator theexpression swell would have been substantial. Finally, in relation to the Feynman rules, only thepart of the operator up to and including two quark and three gluon leg insertions are requiredfor the full three loop computation. To illustrate the size of our higher moment calculation the Form module containing the operator Feynman rule was 36 Mbytes for n = 7 transversity and300 Mbytes for n = 8 transversity, [24]. Indeed the latter calculation could only be performedin the Feynman gauge rather than the full linear covariant or Landau gauges. Even then it tookof the order of 40 days on a dual opteron 64 bit SMP 2GHz machine. Hence, only the MS resultwas determined with the RI ′ scheme anomalous dimension yet to be established.As with all large computations carried out symbolically, it is worth detailing the variouschecks we used in order to be confident that our results are credible. First, for the case ofthe Wilson operators the three loop MS anomalous dimensions are known, [1, 2, 3, 4], andour anomalous dimensions must therefore agree before extracting any finite parts for latticematching. Moreover, for both Wilson and transversity operators the MS expressions have beenshown to be independent of the linear covariant gauge fixing parameter. For the transversity casewe have the checks that the two loop anomalous dimensions must agree with [26, 27, 28, 25, 29]for the various n we consider. At three loops the only substantial check is that the residues ofthe poles in 1 /ǫ and 1 /ǫ have to agree with the renormalization group consistency check. Inother words these are predicted from the one and two loop parts of the anomalous dimensions.In addition for the RI ′ scheme, one can compute the anomalous dimensions either directly fromthe renormalization constants deduced from the Green’s function itself, or from the conversionfunctions, C O ( a, α ), based on the renormalization group. This is defined by C O ( a, α ) = Z ′O Z O (5.1)where the renormalization constants are both expressed in terms of the variables defined in the same scheme. Then the RI ′ anomalous dimension is given by γ ′O (cid:0) a ′ (cid:1) = γ O ( a ) − β ( a ) ∂∂a ln C O ( a, α ) − αγ α ( a ) ∂∂α ln C O ( a, α ) . (5.2)(See, for example, [34].) Therefore, the expression on the left side will agree with the directrenormalization. For all the results presented in the next section we note that they all pass thechecks discussed here. 7 Results.
First, we record the explicit values for the anomalous dimensions of the Wilson operators n = 5and 6 in RI ′ for arbitrary α , which are new. The notation is that the numerical subscript denotesthe moment whilst the superscript, W or T , corresponds to either the Wilson or transversityoperator respectively. We find γ ′ W ( a ) = 9115 C F a + h(cid:16) α + 100575 α + 1729270 (cid:17) C A − C F − T F N f ] C F a h(cid:16) α + 289359000 α − ζ α + 1409428125 α − ζ α + 4758071625 α − ζ + 52067172425) C A + (cid:16) α + 30956400 α − α + 102384000 ζ − C A C F + 6023484800 T F N f − (cid:16) α − ζ α + 1582173000 α + 2514240000 ζ + 36792205400) C A T F N f + (107259600 α + 3680640000 ζ − C F T F N f + (1832544000 ζ − C F i C F a O ( a ) (6.1)and γ ′ W ( a ) = 709105 C F a + h(cid:16) α + 36349425 α + 670295290 (cid:17) C A − C F − T F N f ] C F a h(cid:16) α + 5228103069000 α − ζ α + 25439835416625 α − ζ α + 86004002776125 α − ζ + 988839358918775) C A + (cid:16) α + 158333464800 α − α − ζ − C A C F − (cid:16) α − ζ α + 28468726629000 α + 49102562880000 ζ + 704961641573000) C A T F N f + (2419404145200 α + 73311557760000 ζ − C F T F N f + (31055615136000 ζ − C F + 117065906115200 T F N f i C F a O ( a ) (6.2)where ζ n is the Riemann zeta function. As the Landau gauge is of particular interest, we recordthat the previous two expressions when α = 0 are γ ′ W ( a ) (cid:12)(cid:12)(cid:12) α =0 = 91 C F a + [864635 C A − C F − T F N f ] C F a h (52067172425 − ζ ) C A + (102384000 ζ − C A C F (2514240000 ζ + 36792205400) C A T F N f + (1832544000 ζ − C F + (3680640000 ζ − C F T F N f + 6023484800 T F N f i C F a O ( a ) (6.3)and γ ′ W ( a ) (cid:12)(cid:12)(cid:12) α =0 = 709 C F a + [335147645 C A − C F − T F N f ] C F a h (988839358918775 − ζ ) C A − (5509035504000 ζ − C A C F − (49102562880000 ζ + 704961641573000) C A T F N f + (31055615136000 ζ − C F + (73311557760000 ζ − C F T F N f + 117065906115200 T F N f i C F a O ( a ) . (6.4)For further comparison between the schemes the MS and RI ′ expressions for n = 5 transversityare, [20], γ T ( a ) = 9215 C F a + [189515 C A − C F − T F N f ] C F a h (190836000 ζ + 1975309075) C A − (572508000 ζ + 325464235) C A C F − (1192320000 ζ + 511395100) C A T F N f + (381672000 ζ − C F + (1192320000 ζ − C F T F N f − T F N f i C F a O ( a ) (6.5)and γ ′ T ( a ) = 9215 C F a + h(cid:16) α + 92475 α + 1740690 (cid:17) C A − C F − T F N f ] C F a h(cid:16) α + 1854279000 α − ζ α + 8993896875 α − ζ α + 30074295375 α − ζ + 356401468700) C A + (cid:16) α − α − α + 1076976000 ζ − C A C F − (cid:16) α − ζ α + 10041363000 α + 17858880000 ζ + 253330505600) C A T F N f + 41629683200 T F N f + (1289803200 α + 27164160000 ζ − C F T F N f + (10686816000 ζ − C F i C F a O ( a ) . (6.6)9or completeness we record the next MS anomalous dimensions in the sequence are, [20], γ T ( a ) = 345 C F a + [204770 C A − C F − T F N f ] C F a h (707616000 ζ + 7527909825) C A − (2122848000 ζ + 1373507730) C A C F − (4626720000 ζ + 1841332000) C A T F N f + (1415232000 ζ − C F + (4626720000 ζ − C F T F N f − T F N f i C F a O ( a ) (6.7)and γ T ( a ) = 25835 C F a + [75266555 C A − C F − T F N f ] C F a h (3517994592000 ζ + 38365845513450) C A − (10553983776000 ζ + 5978407701105) C A C F − (24084527040000 ζ + 9039144860900) C A T F N f + (7035989184000 ζ − C F + (24084527040000 ζ − C F T F N f − T F N f i C F a O ( a ) . (6.8)The complete set of three loop transversity anomalous dimensions in MS and RI ′ are given in[18, 19, 20, 24]. We conclude with a few brief remarks. First, the three loop anomalous dimensions are availablefor the transversity operator for each moment up to n = 8 in the MS scheme and n = 7 forthe lattice motivated RI ′ scheme. The former in particular will provide important independentchecks for future explicit arbitrary moment evaluations of the three loop anomalous dimensions.A by-product of the overall project, [18, 19, 20], has been the provision of the finite partsof a Green’s function which are necessary for lattice measurements of matrix elements. Thethree loop perturbative information is essential to obtaining more precise numerical estimates.In addition we have given the (new) RI ′ anomalous dimensions for the n = 5 and 6 Wilsonoperators at three loops. Whilst it is in principle possible to continue with the computation of thetransversity higher moments to n = 9 and beyond, the present method has become too tedious.This is primarily due to the increase in the number of free Lorentz indices on the operatorwhich was originally required for the lattice comparison. Moreover, the actual computationtime as indicated for n = 8 in the Feynman gauge has already become unacceptably long. Anexplicit arbitrary n calculation exploiting the algorithm of [1, 2, 3, 4] would achieve all momentinformation during one run, at possibly a computation time which is not too dissimilar fromthat for one high moment. Acknowledgements.
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