Worldline formalism in a gravitational background

Andrea Zirotti; Fiorenzo Bastianelli

REVIEW

Worldline formalism in a gravitational background

Fiorenzo Bastianelli, Andrea Zirotti

High Energy Physics - Theory · Submitted 17 May 2002 · 20 references

Pith's verdict: A competent method paper: worldline DR plus the -R/8 counterterm and PBC zero-mode factorization make the formalism work for external gravity, though the transverse form factors rest on an unreported Feynman-diagram comparison.

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arXiv IDhep-th/0205182 (version 2)

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CitationAndrea Zirotti, Fiorenzo Bastianelli. "Worldline formalism in a gravitational background." arXiv:hep-th/0205182 (2002). https://pith.science/paper/OZWZGZVY

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Bottom line: this does what it claims. The worldline path integral for a scalar in a gravitational background, regularized with worldline dimensional regularization and the -R/8 counterterm, reproduces the graviton tadpole (27) and the self-energy (42) for arbitrary non-minimal coupling. The ingredients are not new — DR, the ghost measure, and the counterterm all come from [8,9,13]. What is new is the assembly: applying them to gravity and showing the machinery survives intact. That is a real enabler for the subfield, not a breakthrough, and the paper is honest about the distinction (footnote 4 concedes the two-point function may exist elsewhere). What it does well: the tadpole cancellation in (26) is a clean demonstration that the ghost measure does its job. Eq. (42) is a compact and genuinely useful form of the self-energy, and the S1/S2 decomposition makes transversality manifest. Section 5 is the most interesting part — the DBC/PBC comparison, with the non-covariant total derivative (58) appearing at leading order in PBC, is a structural result that explains why Riemann normal coordinates are useless there. Worth having in the literature. Soft spots, in proportion. The stress-test note is right that the Ward identity (40) is a weaker check than it looks: since p_mu S1 = p_mu S2 = 0 by construction, the identity only fixes the coefficient of R1-R2 against the tadpole, i.e. the relative normalization of the one- and two-point computations. It says nothing about the transverse form factors, and it cannot test the -1/8 because it holds for arbitrary xi-bar. So the only check on the physically meaningful part is the Feynman-diagram comparison in section 4, which is asserted without displaying the field-theory result. I would ask for that comparison to be shown, at least in outline. Related: the DR recipe's 'send d to 0 when no ambiguity is left' is quoted from [13] and asserted for I1-I5; a line showing the limit is unambiguous for I3 and I5 would close the loop, since that is exactly where an ambiguity would shift S1/S2. Both are addressable in revision and neither undermines the central claim. The 'infrared effect' reading of (58) in section 5 is offered without derivation — minor, and flagged as interpretation. Who it is for: people working in worldline methods, heat-kernel and Seeley-DeWitt calculations, and anyone wanting a first-quantized handle on gravitational backgrounds. Narrow audience, but a real one. Recommendation: send it out. Careful, honest method paper, real result, clear scope. The revision requests are modest.

Referee report

Significance

If correct, the paper establishes that the worldline formalism is a usable computational scheme for external gravity, not merely for flat-space or gauge backgrounds, and it supplies explicit one- and two-point results that are useful for later work. Two features deserve credit. First, the DBC computation of Section 5 reproduces the standard DeWitt coefficient: Eq. (55) gives Z_DBC = integral sqrt(g)[1 - T(1/12 + xi-bar)R + ...], which with xi-bar = xi - 1/4 is the textbook (1/6 - xi)R, an independent external benchmark on the covariant structure. Second, the explicit total derivative term (Eq. 58) is a concrete, checkable statement about the difference between the PBC and DBC schemes. The paper is also candid about its limitations: abstract:s6 acknowledges the zero-mode factorization problems, Section 6 acknowledges the PBC total-derivative obstruction, and footnote 4 acknowledges that the two-point function was not located in the literature.

Major objections (2)

[Section 4 and footnote 4] The comparison with standard Feynman diagrams, which is the paper's stated evidence for abstract:s5 ('The results obtained by usual field theoretical Feynman diagrams are reproduced in an efficient way'), is asserted but not displayed. The Ward identity (Eq. 40) checks only the longitudinal R1 - R2 structure tied to the tadpole; the transverse form factors S1 and S2 that carry the physical content of Eq. (42) are not constrained by it, and the identity holds for arbitrary xi-bar, so it cannot test the -1/8 in V_DR either. Footnote 4 concedes that the two-point result was not found in the literature, which makes the unreported Feynman-diagram comparison the only external check on the transverse structure. Please display at least the field-theory expression for the two-point function, or a table of the compared form factors, so the reader can verify the agreement rather than take it on trust.
[Section 2, Eqs. (32)-(33)] The values of the integrals I1 through I5 in Eq. (33) are quoted without derivation, and the DR recipe of Ref. [13] is applied in the form 'send d to 0 whenever no ambiguity is left over' without showing that no ambiguity survives for these particular integrals. Since the finite parts of Eq. (42) depend on these values, a line or two of the integration-by-parts identities that reduce I2 through I5 to I1, or a pointer to a worked example in Ref. [13], would make the computation reproducible and would substantiate the 'no ambiguity' claim.

Minor objections (9)

[Eq. (31) and the line below Eq. (41)] Please check the printed form of R3. Contracting the definition as printed with p_mu gives p_mu R3^{mu nu alpha beta} = 2 p^nu p^alpha p^beta / p^2 + p^nu p^beta + p^nu p^alpha, which yields p_mu S2^{mu nu alpha beta} = p^alpha delta^{nu beta} + p^beta delta^{nu alpha} - p^nu p^beta - p^nu p^alpha rather than zero. This looks like a typographical issue, but since the transversality of S2 is used in the Ward identity check, the corrected form should be stated explicitly.
[Section 4, Eq. (42)] The renormalization of the divergences in Eq. (42) at D = 4 is not discussed. The poles in Gamma(-D/2) and Gamma(1-D/2) require counterterms (cosmological constant, Einstein-Hilbert, and R^2 terms); a brief statement of the counterterm structure would help the reader connect Eq. (42) to a renormalized result.
[Section 5, below Eq. (58)] The interpretation of the total derivative term as 'an infrared effect due to the non-local constraint integral dtau y^mu = 0' is heuristic. Either justify it or soften the wording, since the computation itself (Eq. 58) is the solid result and does not depend on the interpretation.
[Footnote 6] The statement that the same total derivative term is found using mode regularization or time slicing is attributed to an unpublished private computation. Consider stating it explicitly as a private communication, or omitting it, so that the published record is not made to carry an unreferenced claim.
[Section 6] The conclusion that 'the secure path is to use the DBC method' for effective-action calculations is reasonable but should be qualified, since the DBC propagators are more laborious and the paper itself notes that the PBC method is more efficient for correlation functions. A sentence stating the trade-off explicitly would be clearer.
[Section 1 and Section 6] The paper mentions Refs. [6] and [18] as the string-inspired rules but does not compare quantitatively with them. A brief remark on whether the results of Eq. (42) agree with, or can be mapped onto, those rules would strengthen the connection the introduction promises.
[Eq. (16)] The statement that the formula is 'exact for xi-bar = 0' should be clarified: presumably it means that for xi-bar = 0 there are no contact terms from the xi-bar R vertex, so the displayed product of single-graviton vertices is complete. A short parenthetical would remove the ambiguity.
[Eq. (26) and Eq. (33)] The coincidence limit in Eq. (26) and the constant '-2' in the expression for I2 in Eq. (33) are quoted without explanation. A brief indication of how the DR prescription handles the coincidence limit of the propagators would make these steps checkable.
[Eq. (15)] The polarization tensors epsilon^{(i)}_{mu nu} are not specified as symmetric or traceless. Since the tensor structures R1 through R5 in Eq. (31) assume a definite symmetry, stating the properties of epsilon^{(i)}_{mu nu} would avoid ambiguity.

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The paper

The paper claims

The paper argues that a scalar particle's worldline path integral, once its ultraviolet behavior is regularized covariantly, reproduces the standard one-loop graviton tadpole and self-energy and passes the general-coordinate Ward identity.

The reading

The paper asks whether the worldline path integral — the first-quantized description of a particle as a loop — can be used when the particle moves in a curved gravitational background, where the path integral needs a careful ultraviolet regulator. It argues that it can: with worldline dimensional regularization, a covariant counterterm proportional to the scalar curvature, and a consistent treatment of the circle's constant zero mode, the one-loop effective action for a scalar field reproduces the standard graviton tadpole and graviton self-energy. The check is not merely a match of two numbers: the self-energy satisfies the general-coordinate Ward identity for arbitrary non-minimal coupling, which is the symmetry test the method must pass to be credible. The paper also separates two ways of handling the zero mode, showing that periodic boundary conditions are efficient for correlation functions but introduce non-covariant total-derivative terms when one computes the effective action directly.

What carries the argument

The machinery is a regulated nonlinear $\sigma$ model on a circle. The covariant path-integral measure is represented by bosonic and fermionic ghosts, the constant zero mode is split off as $x^V_{DR}(V_{DR})=x_0^V_{DR}+y^V_{DR}(V_{DR})$, and the remaining periodic propagators are translationally invariant. Worldline dimensional regularization supplies the finite counterterm $V_{DR}=-R/8$, while the ghost contributions cancel the worst quantum-mechanical divergences. The load-bearing calculational step is the reduction of the self-energy to five worldline integrals $I_1,I_2,I_3,I_4,I_5$, whose values are then fed into the transverse tensor basis $S_1,S_2$ to verify the Ward identity.

Extended reading notes

Core claim

The central claim is that a scalar particle's worldline path integral in an external metric, regularized by worldline dimensional regularization with the counterterm $V_{DR}=-R/8$, yields the same one-loop graviton tadpole and self-energy as ordinary Feynman diagrams. The tadpole comes out as $V_{DR}=-R/8$ contributing to $V_{DR}$, and the two-point function is written in a compact form whose tensor structures $S_1$ and $S_2$ are transverse, so that the Ward identity in eq. (40) holds for any $V_{DR}$. The authors present this as evidence that the worldline method is a practical first-quantized computational scheme for external gravity, not just for flat-space or gauge backgrounds.

Load-bearing premise

The load-bearing premise is that the dimensional-regularization recipe used here — extend the worldline by $d$ infinite dimensions, integrate by parts, then send $dV_{DR}$ whenever no ambiguity remains — really leaves no ambiguity in the five self-energy integrals, so that the quoted finite parts and the Ward-identity check are meaningful.

Original abstract

We analyze the worldline formalism in the presence of a gravitational background. In the worldline formalism a path integral is used to quantize the worldline coordinates of the particles. Contrary to the simpler cases of scalar and vector backgrounds, external gravity requires a precise definition of the ultraviolet regularization of the path integral. Taking into account the UV regularization, we describe the first quantized representation of the one-loop effective action for a scalar particle. We compute explicitly the contribution to the graviton tadpole and self-energy to test the validity of the method. The results obtained by usual field theoretical Feynman diagrams are reproduced in an efficient way. Finally, we comment on the technical problems related to the factorization of the zero mode from the path integral on the circle.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We analyze the worldline formalism in the presence of a gravitational background. In the worldline formalism a path integral is used to quantize the worldline coordinates of the particles. Contrary to the simpler cases of scalar and vector backgrounds, external gravity requires a precise definition of the ultraviolet regularization of the path integral. Taking into account the UV regularization, we describe the first quantized representation of the one-loop effective action for a scalar particle. We compute explicitly the contribution to the graviton tadpole and self-energy to test the validity of the method. The results obtained by usual field theoretical Feynman diagrams are reproduced in an efficient way. Finally, we comment on the technical problems related to the factorization of the zero mode from the path integral on the circle.

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	We analyze the worldline formalism in the presence of a gravitational background.	contribution 0.88	0.89	0.05	asserted 0.92
s1	In the worldline formalism a path integral is used to quantize the worldline coordinates of the particles.	background 0.89	0.08	0.03	asserted 1.00
s2	Contrary to the simpler cases of scalar and vector backgrounds, external gravity requires a precise definition of the ultraviolet regularization of the path integral.	background 0.76	0.15	0.07	asserted 1.00
s3	Taking into account the UV regularization, we describe the first quantized representation of the one-loop effective action for a scalar particle.	contribution 0.74	0.93	0.11	asserted 1.00
s4	We compute explicitly the contribution to the graviton tadpole and self-energy to test the validity of the method.	contribution 0.74	0.96	0.12	asserted 0.97
s5	The results obtained by usual field theoretical Feynman diagrams are reproduced in an efficient way.	result 0.87	0.88	0.54	asserted 1.00
s6	Finally, we comment on the technical problems related to the factorization of the zero mode from the path integral on the circle.	limitation 0.70	0.56	0.04	unclear 0.78

What kind of paper the classifier reports this is: theoretical result 0.90 · evidence from proof or derivation 0.92.

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full text
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P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Dirichlet boundary conditions at x0 (DBC) x0

**Figure 2.** Periodic boundary conditions without zero mode (PBC) In the following we shall consider the one-loop effective action as the generator of 1PI graphs and we shall evaluate directly the terms obtained after functional differentiation. In such a situation total derivative terms which may be present in the effective lagrangian density are harmless. Thus the simplest way to set up the computations in the worldline scheme is to use the “string inspired” method for separating out the zero mode together with worldline dimensional regularization. This way we will obtain the correct…

**Figure 3.** Graviton tadpole The integration over d Dx0 gives the momentum delta function (2π) Dδ D(p), which for simplicity we factorize together with the polarization tensor ǫµν . Momentum conservation eliminates the exponential and the remaining Wick contractions leave us with3 Γ µν (0) = − δ µν 4 Z ∞ 0 dT T e −m2T 1 (4πT) D 2 Z 1 0 dτ ( •∆ • (τ − τ ) + ∆gh(τ − τ )) . (25) The integrals of the propagators can be treated in DR, but it is immediately clear from eq. (21) that only the term from the ghost zero mode contributes •∆ • (τ − τ ) + ∆gh(τ − τ ) = 1 . (26) He

**Figure 4.** Graviton self-energy Γ˜ǫ1,ǫ2 (p1,p2) = − 1 2 Z ∞ 0 dT T e −m2T 1 (4πT) D 2 Z d Dx0

**Figure 5.** Additional graph for graviton self-energy 10

Figures reproduced from arXiv:hep-th/0205182 by the authors.

References

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Forward citations

Cited by 12 Pith papers

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How the literature uses it

35 arXiv papers cite this in 40 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 · 22 sentences without a confident label

Builds on it 1 paper

We have thus extended the gauge-theory result of [ref] , by showing that also in gravitational theories there is a clear connection between the worldline representation of an asymptotic dressed propagator and the more studied worldline quantization on the circle (i.e. the one-loop effective action), which by definition is constructed with denominators only (see e.g. [ref] ).

Generalized Wilson lines and the gravitational scattering of spinning bodies · A trick with conserved charges and the asymptotic limit · 0.93

Uses its method or tool 4 papers

([ref]) in terms of its worldline-representation [ref]

The Relation Between KMOC and Worldline Formalisms for Classical Gravity · From KMOC to worldline observables · 0.97
First, we employ a Euclidean worldline (proper-time) representation [ref] , which makes the winding interpretation manifest and directly exhibits the exponential suppression arising from the worldline-instanton action in warped space.

Axion Quality in Warped Extra-Dimension · Axion potential in the absence of fixed-point localized operators · 1.00
In the sigma model we deal with the corresponding zero mode in the “string inspired” way by imposing [ref] [equation] This yields a translationally invariant formalism on the world-line that allows us to fix $τ_1=0$.

Causality and Micro-Causality in Curved Spacetime · The World-Line Sigma Model, Penrose Limit and Null Congruences · 0.97

Cites it as background 12 papers

Among the several techniques to evaluate functional determinants, the versatile worldline formalism [ref] has proved adaptable to different types of fields on curved spacetimes: scalars [ref] , spinors [ref] (see also [ref] ), and vectors [ref] (where, more generally, antisymmetric tensor fields of arbitrary rank are considered) have been studied with worldline techniques.

Worldline Images for Yang-Mills Theory within Boundaries · Introduction · 0.81
The emission or absorption of quanta, such as photons [ref] , gluons [ref] or gravitons [ref] , is described by inserting suitable vertex operators along the line, in a way that closely resembles string perturbation theory.

Worldline geometries for scattering amplitudes · Introduction · 0.87
Worldline methods have a long-standing history in the study of quantum field theories, see [ref] for a review and [ref] for the inclusion of gravitational backgrounds.

Worldline path integrals for the graviton · Introduction · 1.00

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [13]

F. Bastianelli, O. Corradini and P. van Nieuwenhuizen, Phys. Let t. B 494 (2000) 161 [hep-th/0008045]

View on Pith Pith review arXiv 2000

[2] [1]

R. P. Feynman, Phys. Rev. 80 (1950) 440; J. S. Schwinger, Phys. Rev. 82 (1951) 664; Y. Nambu, Prog. Theor. Phys. 5, 82 (1950); V. Fock, Phys. Z. Sowjetunion 12 (1937) 404

View on Pith 1950

[3] [2]

Z. Bern and D. A. Kosower, Phys. Rev. Lett. 66 (1991) 1669; Nucl. Phys. B 379 (1992) 451; Z. Bern, L. J. Dixon and D. A. Kosower, Phys. Rev. Lett. 70 (1993) 2677 [hep- ph/9302280]

View on Pith 1991

[4] [3]

A.M. Polyakov, “Gauge Fields and Strings” (Harwood, Chur, Switzerland, 1987)

View on Pith 1987

[5] [4]

M.J. Strassler, Nucl. Phys. B 385 (1992) 145 [hep-th/9205205]; M. G. Schmidt and C. Schubert, Phys. Lett. B 318 (1993) 438 [hep-th/9309055]; Phys. Lett. B 331 (1994) 69 [hep-th/9403158]; E. D’Hoker and D. G. Gagne, Nucl. Phys. B 467 (1996) 272 [hep-th/9508131]; Nucl. Phys. B 467 (1996) 297 [hep-th/9512080]

View on Pith arXiv 1992

[6] [5]

C. Schubert, Phys. Rept. 355 (2001) 73 [hep-th/0101036]

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