REVIEW
Parton Degrees of Freedom from the Path-Integral Formalism
High Energy Physics - Phenomenology · Submitted 12 Oct 1999 · 27 references
Pith's verdict: A clean path-integral route to the connected/disconnected sea split, but the load-bearing claim that the connected sea evolves like valence rests on an asserted, not derived, allocation of gluon mixing.
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Referee report
The paper
The paper claims
Liu argues that the proton's sea is not one thing but two, split by the topology of quark paths in the path-integral formulation of deep inelastic scattering, and that the split is visible in how each piece evolves and in how large $x$ is.
The reading
What carries the argument
The load-bearing device is the topological classification of quark skeleton diagrams in the Euclidean four-point function of Eq. (3), combined with a short-distance expansion performed on functions rather than operators. In Fig. 1(a) and 1(b) the quark fields from the nucleon interpolator contract with the currents, so a single quark line runs from $t=0$ to $t=t$; in Fig. 1(c) the quark fields from $J_mu$ and $J_nu$ self-contract and the loop is joined to the proton only through gluons. Because all propagators are sewn together in a colour trace, Liu argues the three classes are gauge invariant. The OPE then follows from substituting the free massless propagator between the currents (Eq. 11), Taylor-expanding the propagator from the later current back to the source (Eq. 12), Fourier transforming in space and Laplace transforming in $tau$ (Eq. 5), and reading off forward matrix elements $langle p | bar Psi O^f_n Psi | p rangle_{C.I.}$ and $langle p | bar Psi O^f_n Psi | p rangle_{D.I.}$ of the twist-two operators $O^f_n = i gamma_{mu_1} (-i/2)^{n-1} overleftrightarrow{D}_{mu_2} cdots overleftrightarrow{D}_{mu_n}$. The even-$n$ moments fix the $q + bar q$ sum rules and the odd-$n$ moments the $q - bar q$ sum rules, which is how the valence number sum rules $int dx, u_v = 2$ and $int dx, d_v = 1$ emerge while the disconnected sea carries zero net charge. Operator mixing then does the decisive work: the connected-insertion moments renormalize with $a^n_{qq}$ alone (Eq. 30), the disconnected-insertion moments pick up the gluon mixing term $(1+(-)^n)/2, a^n_{qG} M^n_G$ (Eq. 31), and this asymmetry is what becomes the claim that connected-sea evolution is decoupled.
Extended reading notes
Core claim
Liu's central claim is that the path-integral formulation of the deep-inelastic hadronic tensor decomposes it into three gauge-invariant contributions distinguished by the topology of quark paths: the valence-plus-connected-sea quarks of Fig. 1(a), the connected-sea antipartons of Fig. 1(b), and the disconnected-sea quarks and antiquarks of Fig. 1(c). Carrying out the operator product expansion inside this formalism, by replacing the propagator between the two currents with the free massless propagator and Taylor-expanding the remaining propagator in the short distance, reproduces the ordinary OPE when the three pieces are summed, but keeps the connected and disconnected insertions separate. Feeding that separation through operator rescaling and mixing gives the paper's main result: the connected-sea moments obey the flavour non-singlet renormalization group equation with the valence anomalous dimension, while only the disconnected-sea moments mix with the gluon operator. Translated into Altarelli-Parisi form, the connected-sea antiparton density evolves exactly like the valence density, with no gluon feeding. From there the paper draws the consequences it is really after: the connected-sea antiparton number is $Q^2$-independent at leading logarithm, the Gottfried sum rule violation $S_G = 1/3 + (2/3)(n_{bar u}^{cs} - n_{bar d}^{cs})$ is a connected-sea effect, and $bar s$ has no connected-sea component at all because a strange quark cannot be contracted with a valence-only nucleon interpolation field.
Load-bearing premise
The argument requires that splitting the path-integral diagrams into connected and disconnected insertions yields a separation that is not only gauge invariant but physically meaningful and separately evolvable at every $Q^2$, and the paper itself concedes that the split depends on the momentum frame of the nucleon even though the summed hadronic tensor does not.
Editorial extensions
If this is right
- The connected-sea antiparton number is conserved, $d/d ln Q^2 int dx, bar q_{cs}(x,Q^2) = 0$, in leading logarithmic approximation, in contrast with the disconnected sea whose number changes as gluons split.
- The Gottfried sum rule violation becomes a sum rule in the connected-sea antiparton numbers, $int_0^1 dx (F_2^p - F_2^n)/x = 1/3 + (2/3)(n_{bar u}^{cs} - n_{bar d}^{cs})$, with the NMC value $0.235 pm 0.026$ implying $n_{bar u}^{cs} - n_{bar d}^{cs} = -0.147 pm 0.039$ at $Q^2 = 4$ GeV$^2$.
- The two seas have different small-$x$ behaviour, $bar u_{cs}, bar d_{cs} sim x^{-1/2}$ from Reggeon exchange and $bar u_{ds}, bar d_{ds}, bar s sim x^{-1}$ from Pomeron exchange, so $bar u - bar d sim x^{-1/2}$ while $(bar u + bar d)/2$ is a mixture of two power laws.
- Global fits should parametrize $(bar u + bar d)/2 = A, bar s + (1/2)(bar u_{cs} + bar d_{cs})$ rather than assuming $(bar u + bar d)/2 = 2 bar s$ as CTEQ4 and MRST do at their input scales, which would change predictions for LHC-era parton distributions at high $Q^2$.
- Summing the connected and disconnected pieces recovers the standard DGLAP equations for the total sea and the valence distribution, so the formalism contains the conventional result as a special case rather than contradicting it.
Reading between the lines
- The claim that the connected sea is a separately evolvable degree of freedom is testable with existing data: a re-fit of DIS, Drell-Yan and dimuon constraints with an explicit connected-sea component would either find a nonzero $bar u_{cs} + bar d_{cs}$ at the few-percent level or push it to zero, and the fit quality difference would be the evidence.
- If connected-sea evolution is truly gluon-decoupled at leading order but the split is frame-dependent as the paper concedes, the natural place to look for the breakdown is next-to-leading order, where the gluon mixing term the paper drops for the connected insertion would have to reappear in some scheme-dependent form.
- The prediction that $bar s$ has no connected component suggests a clean lattice check: computing the disconnected-insertion strangeness loop and the connected-insertion light-sea loop separately should show the light sea carrying a component with valence-like scaling in $Q^2$ and the strange loop not.
- Because the connected-sea antiparton number is conserved only at leading logarithm while the vector-current valence number is protected to all orders, a precise measurement of the $Q^2$ dependence of $n_{bar u} - n_{bar d}$ between, say, 4 and 20 GeV$^2$ would discriminate the connected-sea picture from a purely nonperturbative pion-cloud origin of the asymmetry.
Original abstract
We formulate the hadronic tensor $W_{\mu\nu}$ of deep inelastic scattering in the path-integral formalism. It is shown that there are 3 gauge invariant and topologically distinct contributions. Besides the valence contribution, there are two sources for the sea -- one in the connected insertion and the other in the disconnected insertion. The operator product expansion is carried out in this formalism. The operator rescaling and mixing reveal that the connected sea partons evolve the same way as the valence, i.e. their evolution is decoupled from the disconnected sea and the gluon distribution functions. We explore the phenomenological consequences of this classification in terms of the small x behavior, Gottfried sum rule violation, and the flavor dependence. In particular, we point out that in the nucleon $\bar{u}$ and $\bar{d}$ partons have both the connected and disconnected sea contributions; whereas, $\bar{s}$ parton has only the disconnected sea contribution. This difference between $\bar{u} + \bar{d}$ and $\bar{s}$, as far as we know, has not been taken into account in the fitting of parton distribution functions to experiments.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We formulate the hadronic tensor $W_{\mu\nu}$ of deep inelastic scattering in the path-integral formalism. It is shown that there are 3 gauge invariant and topologically distinct contributions. Besides the valence contribution, there are two sources for the sea -- one in the connected insertion and the other in the disconnected insertion. The operator product expansion is carried out in this formalism. The operator rescaling and mixing reveal that the connected sea partons evolve the same way as the valence, i.e. their evolution is decoupled from the disconnected sea and the gluon distribution functions. We explore the phenomenological consequences of this classification in terms of the small x behavior, Gottfried sum rule violation, and the flavor dependence. In particular, we point out that in the nucleon $\bar{u}$ and $\bar{d}$ partons have both the connected and disconnected sea contributions; whereas, $\bar{s}$ parton has only the disconnected sea contribution. This difference between $\bar{u} + \bar{d}$ and $\bar{s}$, as far as we know, has not been taken into account in the fitting of parton distribution functions to experiments.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We formulate the hadronic tensor $W_{\mu\nu}$ of deep inelastic scattering in the path-integral formalism. | method 0.59 | 0.94 | 0.04 | asserted 0.99 |
| s1 | It is shown that there are 3 gauge invariant and topologically distinct contributions. | result 0.91 | 0.86 | 0.16 | asserted 1.00 |
| s2 | Besides the valence contribution, there are two sources for the sea -- one in the connected insertion and the other in the disconnected insertion. | result 0.91 | 0.76 | 0.10 | asserted 1.00 |
| s3 | The operator product expansion is carried out in this formalism. | method 0.90 | 0.84 | 0.03 | asserted 0.99 |
| s4 | The operator rescaling and mixing reveal that the connected sea partons evolve the same way as the valence, i.e. their evolution is decoupled from the disconnected sea and the gluon distribution functions. | result 0.98 | 0.88 | 0.48 | asserted 1.00 |
| s5 | We explore the phenomenological consequences of this classification in terms of the small x behavior, Gottfried sum rule violation, and the flavor dependence. | contribution 0.91 | 0.89 | 0.08 | asserted 0.83 |
| s6 | In particular, we point out that in the nucleon $\bar{u}$ and $\bar{d}$ partons have both the connected and disconnected sea contributions; whereas, $\bar{s}$ parton has only the disconnected sea contribution. | result 0.96 | 0.75 | 0.18 | asserted 1.00 |
| s7 | This difference between $\bar{u} + \bar{d}$ and $\bar{s}$, as far as we know, has not been taken into account in the fitting of parton distribution functions to experiments. | gap 0.97 | 0.19 | 0.13 | asserted 0.77 |
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Figures
Figures reproduced from arXiv:hep-ph/9910306 by the authors.
References
How the literature uses it
37 arXiv papers cite this in 62 sentences
Papers citing it · 29 sentences without a confident label
Contests it 1 paper
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However, the conventional ansatz $s̅(x) ∼u̅(x) + d̅(x)$ used in recent fittings [ref] is obviously inadequate [ref] .
Moments of Nucleon's Parton Distribution for the Sea and Valence Quarks from Lattice QCD · Conclusion and Discussion · contest 0.71
Uses its method or tool 1 paper
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Instead, we construct the following 4-point correlation function with only a 3-dimensional Fourier transform [equation] and the normal nucleon 2-point function as [equation] Then, the Euclidean hadronic tensor $W^E_μν(p⃗,q⃗,τ)$ is defined by the ratio of the 4-point function to the 2-point function [ref] [equation] where $E_p$ and $m_p$ are the energy and mass of the nucleon.
Towards the nucleon hadronic tensor from lattice QCD · Lattice Formalism of hadronic tensor · 0.92
Cites it as background 26 papers
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While the proposal for a lattice QCD based evaluation has been put forward more than 20 years ago [ref] , it has only recently become feasible to compute the necessary 4-point correlation functions, thanks to the increases in computing power [ref] .
Towards the nucleon hadronic tensor from lattice QCD · Introduction · 0.93
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Several approaches have been pursued to circumvent this difficulty, including analytic continuation [ref] , computing the hadronic tensor [ref] , Compton amplitudes [ref] , and quasi-PDFs [ref] or pseudo-PDFs [ref] , each of which carries unique technical complications.
Parton Physics on a Quantum Computer · Introduction · 0.86
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Recently several methods have been introduced to go beyond the calculations of the first few moments of PDFs on the lattice, such as the path-integral formulation of the deep-inelastic scattering hadronic tensor [ref] , the inversion method [ref] , quasi-PDFs [ref] , and pseudo-PDFs [ref] to obtain the $x$-dependent hadron structure functions.
Pion Valence Quark Distribution from Matrix Element Calculated in Lattice QCD · Introduction · 0.81
Discussion
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