Three loop anomalous dimension of non-singlet quark currents in the RI' scheme

J.A. Gracey

REVIEW

Three loop anomalous dimension of non-singlet quark currents in the RI' scheme

J.A. Gracey

High Energy Physics - Phenomenology · Submitted 11 Apr 2003 · 37 references

Pith's verdict: A careful, well-checked three-loop RI' renormalization with a genuinely new tensor-current anomalous dimension; the only real soft spot is that the tensor finite parts have no external anchor.

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arXiv IDhep-ph/0304113 (version 1)

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CitationJ.A. Gracey. "Three loop anomalous dimension of non-singlet quark currents in the RI' scheme." arXiv:hep-ph/0304113 (2003). https://pith.science/paper/LIGBHHZ7

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Editor's note

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Gracey does what the title says, and does it carefully. The new material is the full three-loop RI' renormalization constants for gluon, ghost, quark and gauge parameter in arbitrary covariant gauge (2.10)-(2.13), the three-loop RI' tensor-current anomalous dimension (4.6), and the four-loop RI' mass anomalous dimension (3.10). The last is a straightforward conversion of the published four-loop MS mass anomalous dimension through the three-loop conversion function computed here. That is standard scheme conversion, not circular, and it is useful to have written down for arbitrary gauge and any Lie group. The claim that the four-loop RI' beta function equals the MS one follows from the three-loop term of the coupling conversion being absent, which is the right argument, and the paper shows it explicitly in (2.24). The checks that exist are real. The MS constants are reproduced, the scalar-current result matches Chetyrkin-Retey in Landau gauge, the vector-current anomalous dimension vanishes as it must, and the coupling renormalization comes out the same from the quark-gluon and ghost-gluon vertices. The paper also records the finite part of the vector Green's function and shows it matches the finite part of the quark two-point function, which is a genuine Slavnov-Taylor check on the projection machinery. On the stress-test note: it is right that the conversion-function agreement (4.11) is not an independent check of the finite parts, since C = Z^{RI'}/Z^{MS} by construction and feeding it into (3.8) reproduces (4.5) up to algebra. But it is not a pure tautology either, because C has to be re-expressed in MS variables using the coupling and gauge-parameter conversion (2.24)-(2.25), so the agreement does test that step. What it does not test is the finite parts of Z^{RI'} themselves. Those rest on the same zero-momentum insertion and projection (4.2) that produce correct finite parts for the scalar operator, which is externally anchored. So the risk is real but small, and I would call it a modest gap rather than a load-bearing one. The scheme-definition caveat is disclosed and defended: this RI' differs from [12] away from Landau gauge and agrees in Landau gauge, which is what lattice groups actually use. No code shipped, which limits independent reproduction, but that is normal for the period. This is a reference-table paper for lattice people doing RI/MOM matching of tensor currents and quark masses, not a conceptual one.

Referee report

Significance

If correct, this is a useful and technically demanding calculation. Lattice QCD determinations of quark masses and of the lowest moment of the transversity operator are quoted in RI' at Landau gauge and must be converted to MS; the conversion functions given here, particularly the three-loop tensor-current result (4.6) and the four-loop mass anomalous dimension (3.10), are exactly the inputs that matching requires. The paper is careful about cross-checks: the MS constants are reproduced before the RI' finite parts are extracted, the scalar channel is anchored to the published Chetyrkin-Retey result, the vector channel to the Ward identity, and the full three-loop renormalization of the Lagrangian is verified through two independent vertices. The arbitrary-gauge treatment, while not needed for lattice practice, supplies genuine internal consistency conditions. The result is a solid, self-contained piece of multiloop technology with a clear phenomenological target.

Major objections (2)

[§4, Eqs. (4.4)-(4.6) and (4.11)] The new content of the tensor-current result is the finite (non-pole) part of Z^{RI'}_{psi sigma psi} in Eq. (4.4), which feeds Eq. (4.6). The two validations offered are (i) that the MS constant reproduces the published Eq. (4.7) and (ii) that the conversion function (4.11) reproduces (4.6) through Eq. (3.8). Check (i) constrains only the pole and ln(4 pi e^{-gamma}) structure, since MS subtraction removes nothing else, and check (ii) is not independent: C = Z^{RI'}/Z^{MS} by construction, so substituting into (3.8) reproduces (4.5) identically. By contrast the scalar result is anchored externally to [12] in Landau gauge and the vector result to the Slavnov-Taylor identity. This is not a claim of error, since the same machinery passes both of those anchors, but the manuscript should state explicitly which parts of (4.6) are externally checked and which are not, and should add one independent handle on the finite parts if one is available. A natural candidate is the alpha-dependence of the three-loop coefficient of (4.6): the alpha^4 term can only arise from the alpha-dependence of (4.4) combined with gamma_alpha from (2.28), and exhibiting that combination explicitly would show that the finite parts and the gauge-parameter anomalous dimension are mutually consistent rather than merely algebraically rearranged.
[§2, Eqs. (2.8)-(2.9), (2.15)-(2.16), (2.24)-(2.25)] The scheme used here is a hybrid: RI' conditions for the fields and the gauge parameter, but an MS-like subtraction of the coupling at the three-point vertices. The author acknowledges in §2 that this differs from the Chetyrkin-Retey definition used in [12] for non-zero alpha, and that the two agree only in Landau gauge. Since the abstract and the section headings advertise results 'in arbitrary covariant gauge', and since the alpha-dependent results are scheme-dependent, the abstract and introduction should state the scheme definition explicitly, including the MS-like coupling subtraction, and should say that the arbitrary-alpha results refer to this definition rather than to the one used in [12]. A short paragraph telling a lattice user which definition to use, and what to do if a published number was obtained with the [12] definition, would materially increase the paper's usefulness and would prevent the arbitrary-gauge results from being cited as if they were scheme-independent.

Minor objections (7)

[Eq. (4.8) versus Eq. (4.6)] I was unable to reproduce Eq. (4.8) from Eq. (4.6). Substituting alpha = 0, C_A = 3, C_F = 4/3 and T_F = 1/2 into (4.6) I obtain (2/243)[572 N_f^2 + (1152 zeta(3) - 29730) N_f + 239529 - 57672 zeta(3)], whereas (4.8) quotes 269259 - 58824 zeta(3) for the constant part. The difference, 29730 - 1152 zeta(3), is exactly minus the N_f bracket, which suggests a transcription slip in (4.8) rather than in (4.6). The two-loop coefficient of (4.8) does follow correctly from (4.6), so the discrepancy is confined to the three-loop constant. Please check which equation is correct.
[Eq. (2.24)] The statement a_RI' = a_MS + O(a_MS^5) 'for all gauges' should be qualified, since the gauge parameters of the two schemes differ from one loop onwards, Eq. (2.25). The equality holds when each coupling is expressed in its own scheme's gauge parameter; as written the phrase could be read as implying that the gauge parameters also coincide.
[§3, paragraph following Eq. (3.4)] The sentence stating that (3.4) 'agrees with the quark mass renormalization constant Z_m deduced at four loops in [12] in the Landau gauge' should be reworded: (3.4) is a three-loop expression, so the comparison with the four-loop result of [12] is made at three loops. As written it suggests a four-loop comparison that is not being performed.
[§1, introduction] The phrase 'the MOM and MOM schemes' appears to be missing a diacritic; the second should presumably be MOM-tilde. Please correct.
[Eq. (3.10)] The quartic Casimir combinations d^{abcd}_A d^{abcd}_F / N_F and d^{abcd}_F d^{abcd}_F / N_F appear in (3.10) without a definition or a reference at the point of use. A one-line definition, or a pointer to [4, 6], would help readers who are not familiar with the light-by-light colour structures.
[Eq. (2.11)] It would be worth stating explicitly that Z_alpha = 1 + O(a^4) is a consequence of the definition (2.9) rather than an accidental result, and that Eq. (2.27) follows from it. The text implies this but does not say it directly.
[§4, after Eq. (4.11)] For lattice users it would be helpful to record a compact Landau-gauge SU(3) expression for the tensor-current conversion function C_{psi sigma psi}, analogous to Eq. (4.8) for the anomalous dimension, so that the matching can be performed without reconstructing it from (4.11).

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The paper

The paper claims

Renormalizing QCD in the RI' scheme at three loops gives lattice practitioners the conversion factors that turn simulated tensor-current and quark-mass matrix elements into MS-bar numbers.

The reading

Lattice QCD produces matrix elements in a scheme called RI', defined by forcing the renormalized quark, ghost and gluon two-point functions to their tree values at a fixed momentum, but the multi-loop anomalous dimensions that continuum predictions are usually quoted in belong to MS-bar. Gracey renormalizes QCD itself at three loops in RI' for arbitrary covariant gauge, checks that the quark-gluon and ghost-gluon vertices give the same coupling-constant renormalization, and finds the RI' beta function coincides with the MS-bar one through four loops. He then extracts the three-loop anomalous dimensions of the scalar, vector and tensor currents by inserting each operator at zero momentum into a massless quark two-point function, recovering the known quark-mass result as a check and producing the tensor-current result as the new one. Because the four-loop MS-bar mass anomalous dimension is already known, the same machinery yields the four-loop RI' mass anomalous dimension in any gauge and for any Lie group. The payoff is a bridge between lattice simulations of transversity and deep-inelastic moments and the perturbative predictions they must be compared against.

What carries the argument

The load-bearing construction is the RI' scheme definition itself: the renormalized quark two-point function is set equal to $p/$, the ghost two-point function to $1/p^2$, and the transverse and longitudinal parts of the gluon polarization to unity, all at $p^2 = mu^2$, with only the finite part absorbed; the coupling renormalization is deliberately kept MS-like, subtracting only poles and $ln(4 pi e^{-gamma})$, so that the Slavnov-Taylor identities survive. The anomalous dimensions of composite operators are then read off from $Z^{RI'}_{psi} Z^{RI'}_{O} langle psi(p) O(0) bar{psi}(-p) rangle = 1$ at $p^2 = mu^2$, with the Green's function decomposed into the two allowed Lorentz structures and projected out. Three-loop massless two-point integrals are evaluated with standard symbolic tools, and the scheme conversion functions $C_i = Z^{RI'}_i / Z^{MS}_i$, together with the derived relation between the RI' and MS-bar gauge parameters, convert MS-bar anomalous dimensions into RI' ones and provide the independent cross-check on every result.

Extended reading notes

Core claim

The paper establishes that QCD can be consistently renormalized at three loops in the RI' scheme in an arbitrary covariant gauge, with the coupling-constant renormalization coming out the same from the quark-gluon and ghost-gluon three-point functions, and that the resulting beta function agrees with the MS-bar one to the order where a difference could first appear, which pushes the agreement to four loops. On that foundation the author computes the RI' anomalous dimensions of the scalar, vector and tensor flavour non-singlet quark currents at three loops from the insertion of the operator at zero momentum in a massless quark two-point function. The scalar case reproduces the quark-mass anomalous dimension, confirming that the operator method matches the older massive-propagator route, and the tensor case, corresponding to the lowest moment of the transversity operator, gives a new gauge-dependent renormalization constant and anomalous dimension. Combining the two- and three-loop conversion functions with the known four-loop MS-bar mass anomalous dimension then yields the four-loop RI' mass anomalous dimension for arbitrary gauge parameter and any Lie group, and the vector current is shown to stay unrenormalized, as a conserved current should.

Load-bearing premise

The load-bearing premise is that the RI' scheme is the one fixed by this paper's own two-point-function conditions together with an MS-style coupling renormalization; the author states that a widely used earlier definition differs from it whenever the gauge parameter is non-zero and agrees only in Landau gauge, so away from Landau gauge these conversion factors belong to a different scheme than the earlier ones.

Editorial extensions

If this is right

Lattice groups gain three-loop conversion factors for the tensor current, so transversity matrix elements can be matched to MS-bar with the same precision already available for quark masses.
The four-loop RI' mass anomalous dimension, valid for any gauge parameter and any colour group, extends the earlier Landau-gauge-only conversion to non-zero gauge fixing.
Because the three-loop RI' beta function equals the MS-bar one, the four-loop RI' beta function is also equal to the MS-bar result, so coupling-constant running needs no separate conversion.
The same zero-momentum operator-insertion method can be applied directly to the twist-2 Wilson operators whose moments appear in the operator product expansion for deep inelastic scattering.
The explicit three-loop relation between the RI' and MS-bar gauge parameters gives a reusable dictionary for any future RI' calculation performed away from Landau gauge.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's insistence that the RI' coupling be renormalized MS-like, rather than by absorbing finite vertex pieces, is a design choice that keeps Slavnov-Taylor identities intact; a reader could reasonably expect that choice to become the standard, since otherwise the scheme cannot be defined consistently beyond two-point functions.
Because the vector current is protected, its finite Green's function doubles as a free consistency check on any future RI' operator calculation, and the author effectively uses it that way.
The gauge-parameter conversion relation is itself a substantial by-product: any RI' result quoted at non-zero $alpha$ can be translated, so non-Landau-gauge lattice simulations become usable for matching rather than being discarded.
Extending the method to operators with gluonic content would require the gluon and ghost field renormalizations computed here, which the author explicitly prepares for but does not use.

Original abstract

We renormalize QCD at three loops in the modified regularization invariant, RI', scheme in arbitrary covariant gauge and deduce that the four loop beta-function is equivalent to the MSbar result. The anomalous dimensions of the scalar, vector and tensor currents are then determined in the RI' scheme at three loops by considering the insertion of the operator in a quark two-point function. The expression for the scalar current agrees with the quark mass anomalous dimension and we deduce an expression for the four loop RI' mass anomalous dimension in arbitrary covariant gauge and for any Lie group.

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We renormalize QCD at three loops in the modified regularization invariant, RI', scheme in arbitrary covariant gauge and deduce that the four loop beta-function is equivalent to the MSbar result. The anomalous dimensions of the scalar, vector and tensor currents are then determined in the RI' scheme at three loops by considering the insertion of the operator in a quark two-point function. The expression for the scalar current agrees with the quark mass anomalous dimension and we deduce an expression for the four loop RI' mass anomalous dimension in arbitrary covariant gauge and for any Lie group.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We renormalize QCD at three loops in the modified regularization invariant, RI', scheme in arbitrary covariant gauge and deduce that the four loop beta-function is equivalent to the MSbar result.	contribution 0.59	0.94	0.37	asserted 1.00
s1	The anomalous dimensions of the scalar, vector and tensor currents are then determined in the RI' scheme at three loops by considering the insertion of the operator in a quark two-point function.	contribution 0.48	0.93	0.20	asserted 1.00
s2	The expression for the scalar current agrees with the quark mass anomalous dimension and we deduce an expression for the four loop RI' mass anomalous dimension in arbitrary covariant gauge and for any Lie group.	result 0.97	0.94	0.25	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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References

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Meson masses and decay constants from unquenched lattice QCD
hep-lat 2009-06 conditional novelty 5.0 of 10

Two-flavour twisted-mass lattice QCD yields the a0, b1 and rho meson masses, the rho leptonic and transverse decay constants, and a direct lattice estimate of the rho-pi-pi coupling.
Renormalization of local quark-bilinear operators for Nf=3 flavors of SLiNC fermions
hep-lat 2014-08 conditional novelty 4.0 of 10

Non-perturbative RI'-MOM renormalization factors for scalar, pseudoscalar, vector, axial-vector and tensor quark bilinears are computed for Nf=3 SLiNC fermions at a=0.074 fm and converted to MS-bar at 2 GeV.
The Theory of Deeply Inelastic Scattering
hep-ph 2012-08 accept novelty 1.0 of 10

A comprehensive review of perturbative QCD calculations for unpolarized and polarized deeply inelastic scattering, covering coefficient functions, anomalous dimensions, heavy-flavor and target-mass corrections, and th...

How the literature uses it

49 arXiv papers cite this in 77 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 45 sentences without a confident label

Contests it 1 paper

We agree with the moments given in [ref] and note a typo in the 15$^th$ moment of [ref] .

The 3-Loop Non-Singlet Heavy Flavor Contributions and Anomalous Dimensions for the Structure Function $F_2(x,Q^2)$ and Transversity · Transversity · contest 0.88

Uses its method or tool 9 papers

This is then converted to the $MS$ scheme using continuum perturbation theory [ref] and evolved to a scale of 2 GeV that allows comparison with phenomenological estimates.

The nucleon isovector tensor charge from lattice QCD using chiral fermions · Renormalization · 0.94
For the exceptional kinematic configuration we use the projectors of [ref] and the perturbative running and matching calculated in [ref] for mass and [ref] for the tensor current.

Step Scaling with off-shell renormalisation · Vertex functions with twisted boundary conditions at a single lattice spacing · 0.93
Since lattice perturbation theory is slowly convergent and high-loop results are unavailable, we first match the local lattice QCD operators of interest non-perturbatively to the regulator independent RI'-MOM scheme [ref] and subsequently translate the result at three-loop order [ref] to the $MS$ scheme.

Magnetic susceptibility of QCD matter and its decomposition from the lattice · Multiplicative QCD renormalization · 1.00

Uses its data or values 7 papers

In deriving the conversion factors via Eqs.([ref], [ref]), we take NLO coefficients of $C_q^→MS$ and $C_q^γ_μ→MS$ from [ref] and the NNLO coefficients from [ref] (setting $C_A=N_c$, $T_F=1/2$), [ref] (setting $w=1$, $r=1$) and [ref] .

RI-(S)MOM to $\overline{\rm MS}$ conversion for $B_K$ at two-loop order · $C_B_K^(S)MOM→MS$ up to NNLO · 0.98
Restricting ourselves in these proceedings to two-loop order in Landau gauge and to quenched approximation (compare e.g. [ref] ), we have [equation] The coefficient $z^RI'_1,0=13.8257$ has been calculated in [ref] , a first rough estimate of $z^RI'_2,0$ has been presented by us at Lattice08 [ref] .

The lattice ghost propagator in Landau gauge up to three loops using Numerical Stochastic Perturbation Theory · The ghost propagator in standard LPT · 0.88
From Ref. [ref] we get for $^S$ and $^P$ [equation] With the help of Ref. [ref] we find for $^T_μν$ [equation] For the vector and axial-vector currents $^V_μ$ and $^A_μ$ one finds from Ref. [ref] [equation] with $R = p_μ^2/p^2$.

Perturbative and Nonperturbative Renormalization in Lattice QCD · Results from continuum perturbation theory · 0.99

Compares with it 7 papers

They are compared to the results in Refs. [ref] and are obtained in a first independent calculation for $γ_qg(N)$, $γ_qq^PS(N)$, $γ_qq^NS,TR(N)$, in Section [ref].

The O(\alpha_s^3) Massive Operator Matrix Elements of O(n_f) for the Structure Function F_2(x,Q^2) and Transversity · Introduction · 1.00
All results agree with the foregoing literature [ref] .

Higher Order Heavy Quark Corrections to Deep-Inelastic Scattering · 3-Loop anomalous dimensions · 1.00
We agree with the results given in [ref] and confirm the $T_F$-contributions for the moments $N=1$ to 8 given in Refs. [ref] .

2- and 3-Loop Heavy Flavor Corrections to Transversity · Anomalous Dimensions · 1.00

Cites it as background 3 papers

It was also shown that the former is equivalent to the $$ scheme and is thus known to three-loop order [ref] , whereas the latter is known to two-loop order [ref] .

Matching factors for Delta S=1 four-quark operators in RI/SMOM schemes · Introduction · 0.83
The corresponding perturbative calculation is relatively simple and those for the quark bilinear currents with all 16 gamma matrices have been obtained at 3 loops or more [ref] .

RI/MOM and RI/SMOM renormalization of quark bilinear operators using overlap fermions · Introduction · 0.96
The anomalous dimensions for transversity are known to NLO [ref] and for a series of moments to 3–loop order [ref] .

The O(\alpha_s^3) Massive Operator Matrix Elements of O(n_f) for the Structure Function F_2(x,Q^2) and Transversity · The Operator Matrix Elements for Transversity · 0.94

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