On the Schroedinger functional in QCD

Stefan Sint

REVIEW

On the Schroedinger functional in QCD

Stefan Sint

High Energy Physics - Lattice · Submitted 20 Dec 1993

Pith's verdict: The paper that puts Wilson fermions into Lüscher's Schrödinger functional, and the derivation holds up; the soft spots are small and mostly acknowledged.

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arXiv IDhep-lat/9312079 (version 1)

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CitationStefan Sint. "On the Schroedinger functional in QCD." arXiv:hep-lat/9312079 (1993). https://pith.science/paper/NOSKCPGT

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Short version: this is the paper that extends Lüscher's Schrödinger functional from pure gauge theory to Wilson fermions, and the construction works. If you care about running-coupling determinations with dynamical quarks, this is the reference you need, and it is done carefully enough to trust the machinery even where the physics is left for later work [abstract:s1, abstract:s2]. The real content is §4: the transfer matrix for Wilson fermions is Lüscher's [4], but the Grassmann coherent-state representation of its kernel is worked out here in detail, and the payoff is that the fermionic boundary conditions fall out rather than being guessed. Only half the spinor components get fixed at each time boundary, which is exactly what a first-order Dirac equation should allow, and the paper makes that look inevitable rather than clever. §6 is the other solid piece: the free Dirac operator is shown to have a unique self-adjoint extension, discrete spectrum, and a zero-mode condition (6.21) that only fires in the infinite-negative-mass limit. That is a clean, checkable result and it is what makes the free propagator well defined for perturbation theory [abstract:s5]. On the soft spots, I part company with the reader on §5. The parity requirement that fixes R_0 and R_T is flagged as the weakest assumption, but the stress-test is right that it is smaller than that: substituting (5.14) into the boundary term of (5.5) and using the boundary conditions (5.4), the X_0 and X_T pieces collapse to constants, so parity only pins down the normalization of the action, not the dynamics. The 'no lattice peculiarities' claim survives [abstract:s4]. Two genuinely minor things: (6.9) looks like a sign typo against (6.13), harmless because (6.19)-(6.20) follow from (6.13); and the renormalizability claim in §7.1 rests on a one-loop calculation cited as [15] 'work in progress', which is the one place the paper asks you to take something on faith. It says so plainly, so I would not hold it against the paper, but it is the open item. The continuum limit is explicitly the naive one, with no O(a) discussion, which is fine for the stated scope. Who should read it: anyone building lattice SF calculations with fermions, and anyone who wants to see how boundary conditions in a regulated theory should be derived rather than postulated. Yes to a serious referee, and yes, I would cite it.

Referee report

Significance

The paper is a foundational contribution: it supplies the missing fermionic half of the SF formulation and does so with a clean, self-contained transfer-matrix/coherent-state derivation that does not rely on unstated lattice folklore. The free Dirac operator analysis in §6 is rigorous and yields a concrete, checkable prediction — the boundary-induced gap (π/2T)² in the massless case, Eq. (6.20) — which is exactly the feature that makes small-volume simulations of light quarks feasible and which underpins the later SF-based determinations of the running coupling. The paper is also honest about scope: renormalizability is presented as an expectation with a one-loop calculation explicitly listed as work in progress [15], and the interacting zero-mode question is flagged in §7.2. These are strengths, not weaknesses.

Major objections (1)

[§5, Eqs. (5.13)–(5.14)] The uniqueness of the continuum boundary terms is stated to require parity invariance, which is what excludes the matrices X_0 and X_T anti-commuting with γ_0 in Eq. (5.14). As the paper's own boundary conditions (5.4) imply, however, the X_0 and X_T contributions reduce to boundary-source-dependent constants (ψ̄ζ_- X_0 ζ_+ at x0=0 and ψ̄ζ'_+ X_T ζ'_- at x0=T), so the parity requirement fixes the normalization of the action rather than the bulk dynamics. The paper should say this explicitly, because as written a reader may over-read the parity requirement as load-bearing for the 'no lattice peculiarities are left over' claim of the abstract. This is a clarification of the central claim's interpretation, not an error in the derivation.

Minor objections (5)

[Eq. (6.9)] The boundary conditions as printed, (-∂_0 - m)P_- w|_{x0=0} = 0 and (-∂_0 + m)P_+ w|_{x0=T} = 0, appear to have the opposite sign on ∂_0 relative to the conditions s'(0) - m s(0) = 0 and s'(T) + m s(T) = 0 stated in Eq. (6.13). Since the eigenfunctions (6.19) and eigenvalues (6.20) are derived from (6.13) and are self-consistent, this looks like a typo in (6.9); please verify and correct.
[§7.1] The list of two gauge-invariant boundary counterterms of dimension three or less, ψ̄ψ and ψ̄γ_0ψ, implicitly assumes invariance under spatial rotations to exclude ψ̄γ_kψ. The paper invokes only parity; a sentence stating that spatial isotropy is also required would make the argument complete.
[§4.1] The positivity of B is stated to require all hopping parameters smaller than 1/6, and this is assumed throughout. A brief remark on what this implies for the range of accessible bare quark masses (in particular that the light-quark region is comfortably inside the bound) would help the reader assess the practical reach of the construction.
[§7.1] The renormalizability statement is presented as an expectation, with the one-loop calculation listed as work in progress [15]. The abstract does not claim renormalizability, so this is a scope note rather than a flaw, but the text could state more sharply that renormalizability is not established in this paper.
[Notation, §3–§4] The symbol K is used both for the hopping-parameter matrix in the fermion action and, as K_0, for the pure-gauge transfer kernel in Eq. (4.6). A notational distinction (e.g. κ for the hopping matrix) would reduce the risk of confusion in §4.3–4.4.

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The paper

The paper claims

The Schrödinger functional — a QCD state prepared on one time slice and read off on another — can be defined with quarks, provided only half of each quark spinor is pinned at each boundary.

The reading

Lüscher and collaborators had shown that the Schrödinger functional, the amplitude for a gauge field to evolve from one prescribed configuration to another over a fixed euclidean time, gives a clean, non-perturbative definition of a running coupling in pure SU(2) and SU(3) gauge theory. Adding quarks is not straightforward: the Dirac equation is first order, so one may fix only half of the spinor components on a boundary, and it is not obvious which half, or whether the resulting lattice action leaves artifacts behind. Sint answers both questions by building the functional from the transfer matrix of Wilson lattice QCD rather than guessing at boundary conditions. The boundary conditions then fall out of the Grassmann integral kernel, and the naive continuum limit produces an action whose boundary terms are shown to be the only ones compatible with smooth classical solutions and parity. If the construction holds, the same small-volume strategy used to extract the running gauge coupling without quarks can be carried over to real QCD, including at light or zero quark mass, because the boundaries themselves supply a minimum frequency and keep the Dirac operator away from zero.

What carries the argument

The transfer matrix of Wilson lattice QCD, converted into an integral operator on Grassmann coherent states. The coherent-state representation turns each factor of the transfer matrix into an explicit kernel $exp(-S[U,V,U'])$ times a bilinear quark exponential, and iterating it $N$ times with inserted completeness relations produces the path integral. The two-spinor decomposition with projectors $P_pm$ is what makes the boundary conditions visible: the kernel can only depend on the crossed Grassmann variables, so exactly half of each spinor appears at each time slice. The second piece of machinery is the variational argument of section 5, which asks that the classical action admit smooth stationary points and be parity invariant; solving the resulting conditions on the boundary matrices $R_0$ and $R_T$ leaves only terms proportional to $P_- gamma_0$ and $P_+ gamma_0$, matching the action derived from the lattice.

Extended reading notes

Core claim

The paper establishes that the Schrödinger functional of pure gauge theory extends to lattice QCD with Wilson fermions, and that the extension is fixed rather than chosen. Starting from Lüscher's transfer matrix for Wilson fermions and representing it as an integral operator on a Grassmann algebra via coherent states, Sint obtains the kernel of the $N$-th power of the transfer matrix as a path integral over euclidean gauge and quark fields. The kernel depends on the Grassmann variables only through the combinations selected by the projectors $P_+ = (1+gamma_0)/2$ and $P_- = (1-gamma_0)/2$: at $x_0=0$ the plus components of the quark field and the minus components of the antiquark field are fixed, and at $x_0=T$ the opposite halves are fixed. Taking $a to 0$ yields the continuum action with explicit boundary terms, and a variational argument shows that these terms are exactly the ones required for the action to possess smooth stationary points while remaining parity invariant, so they are not lattice remnants. Finally, the free Dirac operator built from this action, viewed as an elliptic two-term boundary value problem on $H_+ oplus H_-$, is essentially self-adjoint with a purely discrete spectrum $p_0^2 + p^2 + m^2$, a lowest massless eigenvalue $(pi/2T)^2$, and no zero modes for any finite mass.

Load-bearing premise

The claim that the continuum boundary terms are physical rather than lattice artifacts rests on requiring the classical action to have smooth stationary points and to be parity invariant, and parity is what excludes the leftover boundary matrices that anticommute with $gamma_0$.

Editorial extensions

If this is right

The running coupling of QCD can be computed in small volumes with the same boundary-field technique already used successfully in the pure SU(3) theory, with quark boundary values as an additional dial.
Because the boundaries induce a minimum frequency $(pi/2T)^2$ in the massless case, simulations at light or vanishing quark mass become feasible in small volumes, where the condition number of the lattice Dirac matrix would otherwise blow up.
Setting the quark boundary fields to zero is consistent with renormalizability, since the two allowed boundary counterterms $\bar\psi\psi$ and $\bar\psi\gamma_0\psi$ then vanish, leaving boundary effects that are suppressed by a power of the lattice spacing.
The free propagator is well defined as $D^{-1} = D^\dagger (DD^\dagger)^{-1}$ for all masses, so a perturbative expansion of the QCD Schrödinger functional has a sound starting point.
The construction supplies a criterion for choosing the gauge background: it must make gauge configurations that produce Dirac zero modes extremely unlikely.

Reading between the lines

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

The argument that the continuum boundary terms are not artifacts is a uniqueness argument conditional on parity, not a proof that the regulated lattice theory preserves parity at the boundary; an explicit one-loop check, which the author states is in progress, is the natural place for this to be settled.
If the boundary conditions really do control the low end of the Dirac spectrum, the same mechanism could be used as a spectroscopic tool: the gap $(pi/2T)^2$ is a boundary-induced infrared cutoff whose value is known exactly, so deviations from it in interacting configurations measure how the gauge field distorts the would-be zero-mode sector.
The requirement that smooth stationary points exist is doing real work; in a theory where the classical boundary value problem fails to have smooth solutions, the same lattice derivation would produce boundary terms that no continuum variational principle selects, which suggests the criterion is worth testing in other fermionic boundary constructions.

Original abstract

In a series of publications [\ref{LNWW},\ref{Schroedinger}], L\"uscher et al. have demonstrated the usefulness of the Schr\"odinger functional in pure SU(2) and SU(3) gauge theory. In this paper, it is shown how their formalism can be extended to include fermions. In the framework of Wilson's lattice QCD, we define the Schr\"odinger functional by making use of the transfer matrix formalism. Boundary conditions for the fermions arise naturally. We then take the naive continuum limit of the action and show that no lattice peculiarities are left over. The corresponding free Dirac operator has a unique self-adjoint extension with purely discrete spectrum and no zero modes.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In a series of publications [\ref{LNWW},\ref{Schroedinger}], L\"uscher et al. have demonstrated the usefulness of the Schr\"odinger functional in pure SU(2) and SU(3) gauge theory. In this paper, it is shown how their formalism can be extended to include fermions. In the framework of Wilson's lattice QCD, we define the Schr\"odinger functional by making use of the transfer matrix formalism. Boundary conditions for the fermions arise naturally. We then take the naive continuum limit of the action and show that no lattice peculiarities are left over. The corresponding free Dirac operator has a unique self-adjoint extension with purely discrete spectrum and no zero modes.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In a series of publications [\ref{LNWW},\ref{Schroedinger}], L\"uscher et al. have demonstrated the usefulness of the Schr\"odinger functional in pure SU(2) and SU(3) gauge theory.	background 1.00	0.04	0.04	asserted 1.00
s1	In this paper, it is shown how their formalism can be extended to include fermions.	contribution 0.99	0.89	0.49	asserted 1.00
s2	In the framework of Wilson's lattice QCD, we define the Schr\"odinger functional by making use of the transfer matrix formalism.	method 0.92	0.93	0.07	asserted 1.00
s3	Boundary conditions for the fermions arise naturally.	result 0.53	0.70	0.05	asserted 1.00
s4	We then take the naive continuum limit of the action and show that no lattice peculiarities are left over.	result 0.63	0.94	0.14	asserted 1.00
s5	The corresponding free Dirac operator has a unique self-adjoint extension with purely discrete spectrum and no zero modes.	result 1.00	0.86	0.16	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.65 · evidence from proof or derivation 1.00.

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References

Forward citations

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How the literature uses it

52 arXiv papers cite this in 71 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 · 30 sentences without a confident label

Uses its method or tool 27 papers

We evaluate this identity on a lattice with Schrödinger functional boundary conditions (periodic in space, Dirichlet in time) [ref] with vanishing background field.

Non-perturbative improvement and renormalization of the axial current in N_f=3 lattice QCD · Renormalization Condition · 0.99
As will be discussed in detail in the present article, this problem can be tackled in a systematic way by means of the Symanzik improvement programme [ref] and defining the theory in the Schrödinger functional (SF) scheme [ref] according to the framework presented in Refs. [ref] ; this allows one to cancel the leading, $O(a)$, discretisation artifacts and to achieve $O(a^2)$ scaling towards the continuum limit [ref] .

Non-perturbative renormalisation and improvement of non-singlet tensor currents in $N_\mathrm{f}=3$ QCD · Introduction · 0.92
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These conditions can be compared to those in ferm-bc and be seen to agree – albeit by construction [ref] .

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In the following, we assume that the reader is familiar with the SF in QCD [ref] .

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There are many such schemes for the nonperturbative renormalisation (NPR) of bare lattice quantities, such as RI/MOM [ref] , RI/SMOM [ref] and the Schrödinger functional method [ref] .

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