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Journal articles on the topic 'Vacuum String Field Theory'

Author: Grafiati

Published: 14 March 2023

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1

Bonora, Loriano, Davide Mamone, and Mario Salizzoni. "Vacuum String Field Theory withBfield." Journal of High Energy Physics 2002, no. 04 (April 13, 2002): 020. http://dx.doi.org/10.1088/1126-6708/2002/04/020.

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2

Hata, H., H. Kogetsu, and S. Teraguchi. "Gauge Structure of Vacuum String Field Theory." Journal of High Energy Physics 2004, no. 02 (February 25, 2004): 045. http://dx.doi.org/10.1088/1126-6708/2004/02/045.

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3

Okuyama, Kazumi. "Siegel Gauge in Vacuum String Field Theory." Journal of High Energy Physics 2002, no. 01 (January 31, 2002): 043. http://dx.doi.org/10.1088/1126-6708/2002/01/043.

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4

Bonora, L., C. Maccaferri, R. J. Scherer Santos, and D. D. Tolla. "Bubbling AdS and vacuum string field theory." Nuclear Physics B 749, no. 1-3 (August 2006): 338–57. http://dx.doi.org/10.1016/j.nuclphysb.2006.05.029.

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5

Zeze, S. "New approach to vacuum string field theory." Theoretical and Mathematical Physics 179, no. 3 (June 2014): 689–94. http://dx.doi.org/10.1007/s11232-014-0171-0.

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6

Rastelli, Leonardo, Ashoke Sen, and Barton Zwiebach. "String field theory around the tachyon vacuum." Advances in Theoretical and Mathematical Physics 5, no. 2 (2001): 353–92. http://dx.doi.org/10.4310/atmp.2001.v5.n2.a5.

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7

Bonora, L., D. Mamone, and M. Salizzoni. "GMS solitons from vacuum string field theory." Fortschritte der Physik 51, no. 78 (July 7, 2003): 678–83. http://dx.doi.org/10.1002/prop.200310082.

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8

KAKU, MICHIO. "STRING FIELD THEORY." International Journal of Modern Physics A 02, no. 01 (February 1987): 1–76. http://dx.doi.org/10.1142/s0217751x87000028.

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String theory has emerged as the leading candidate for a unified field theory of all known forces. However, it is impossible to trust the various phenomenological predictions of superstring theory based on classical solutions alone. It appears that the crucial problem of the theory, breaking ten dimensional space-time down to four dimensions, must be solved nonperturbatively before we can extract reliable predictions. String field theory may be the only formalism in which we can resolve this decisive question. Only a rigorous calculation of the true vacuum of the theory will determine which of the many classical solutions the theory actually predicts. In this review article, we summarize the rapid progress in constructing string field theory actions, such as the development of the covariant BRST theory. We also present the newer geometric formulation of string field theory, from which the BRST theory and the older light cone theory can be derived from first principles. This geometric formulation allows us to derive the complete field theory of strings from two geometric principles, in the same way that general relativity and Yang-Mills theory can be derived from two principles based on global and local symmetry. The geometric formalism therefore reduces string field theory to a problem of finding an invariant under a new local gauge group we call the universal string group (USG). Thus, string field theory is the gauge theory of the universal string group in much the same way that Yang-Mills theory is the gauge theory of SU (N). Thus, the geometric formulation places superstring theory on the same rigorous group theoretical level as general relativity and gauge theory.

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9

Gaiotto, Davide, Leonardo Rastelli, Ashoke Sen, and Barton Zwiebach. "Ghost structure and closed strings in vacuum string field theory." Advances in Theoretical and Mathematical Physics 6, no. 3 (2002): 403–56. http://dx.doi.org/10.4310/atmp.2002.v6.n3.a1.

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10

Hata, Hiroyuki, and Hisashi Kogetsu. "Higher Level Open String States from Vacuum String Field Theory." Journal of High Energy Physics 2002, no. 09 (September 11, 2002): 027. http://dx.doi.org/10.1088/1126-6708/2002/09/027.

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11

ANTONOV, D. V. "FLUCTUATING STRINGS IN THE UNIVERSAL CONFINING STRING THEORY AND GLUODYNAMICS." Modern Physics Letters A 13, no. 08 (March 14, 1998): 581–92. http://dx.doi.org/10.1142/s0217732398000632.

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An effective string theory emerging from the bilocal approximation to the method of vacuum correlators in gluodynamics is shown to be well-described by the 4D theory of the massive Abelian Kalb–Ramond field interacting with the string, which is known to be the low energy limit of the universal confining string theory. This correspondence follows from the agreement of the behavior of the coefficient functions, which parametrize the gauge-invariant correlator of two gluonic field strength tensors, known from the lattice data, with their values obtained from the propagator of the Kalb–Ramond field. We discuss this correspondence in several aspects and demonstrate that the mass of the Kalb–Ramond field in this approach plays the role of the inverse correlation length of the vacuum, so that in the massless limit string picture disappears. Next, we apply the background field method, known in the theory of nonlinear sigma models, to obtain the action, which is quadratic in quantum fluctuations around a given (e.g. minimal) string worldsheet. Several nontrivial types of couplings of these fluctuations with the background worldsheet are obtained and discussed.

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12

Bonora, Loriano, Carlo Maccaferri, and Predrag Prester. "Dressed Sliver solutions in Vacuum String Field Theory." Journal of High Energy Physics 2004, no. 01 (January 19, 2004): 038. http://dx.doi.org/10.1088/1126-6708/2004/01/038.

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13

Drukker, Nadav, and Yuji Okawa. "Vacuum string field theory without matter-ghost factorization." Journal of High Energy Physics 2005, no. 06 (June 15, 2005): 032. http://dx.doi.org/10.1088/1126-6708/2005/06/032.

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14

Okuyama, Kazumi. "Ghost Kinetic Operator of Vacuum String Field Theory." Journal of High Energy Physics 2002, no. 01 (January 21, 2002): 027. http://dx.doi.org/10.1088/1126-6708/2002/01/027.

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15

Okuyama, Kazumi. "Ratio of Tensions from Vacuum String Field Theory." Journal of High Energy Physics 2002, no. 03 (March 26, 2002): 050. http://dx.doi.org/10.1088/1126-6708/2002/03/050.

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16

Igarashi, Y., K. Itoh, F. Katsumata, T. Takahashi, and S. Zeze. "Exploring Vacuum Manifold of Open String Field Theory." Progress of Theoretical Physics 114, no. 6 (December 1, 2005): 1269–93. http://dx.doi.org/10.1143/ptp.114.1269.

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17

Bonora, L., D. Mamone, and M. Salizzoni. "B-field and squeezed states in vacuum string field theory." Nuclear Physics B 630, no. 1-2 (May 2002): 163–77. http://dx.doi.org/10.1016/s0550-3213(02)00177-3.

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18

Hata, Hiroyuki, and Sanefumi Moriyama. "Observables as Twist Anomaly in Vacuum String Field Theory." Journal of High Energy Physics 2002, no. 01 (January 29, 2002): 042. http://dx.doi.org/10.1088/1126-6708/2002/01/042.

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19

Bonora, Loriano, Davide Mamone, and Mario Salizzoni. "Vacuum String Field Theory ancestors of the GMS solitons." Journal of High Energy Physics 2003, no. 01 (January 10, 2003): 013. http://dx.doi.org/10.1088/1126-6708/2003/01/013.

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20

Ellwood, Ian. "Rolling to the tachyon vacuum in string field theory." Journal of High Energy Physics 2007, no. 12 (December 7, 2007): 028. http://dx.doi.org/10.1088/1126-6708/2007/12/028.

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21

Okuda, Takuya. "The equality of solutions in vacuum string field theory." Nuclear Physics B 641, no. 1-2 (October 2002): 393–401. http://dx.doi.org/10.1016/s0550-3213(02)00615-6.

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22

Bonora, L., C. Maccaferri, R. J. Scherer Santos, and D. D. Tolla. "Exact time-localized solutions in vacuum string field theory." Nuclear Physics B 715, no. 1-2 (May 2005): 413–39. http://dx.doi.org/10.1016/j.nuclphysb.2005.02.040.

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23

Hata, Hiroyuki, and Teruhiko Kawano. "Open string states around a classical solution in vacuum string field theory." Journal of High Energy Physics 2001, no. 11 (November 17, 2001): 038. http://dx.doi.org/10.1088/1126-6708/2001/11/038.

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24

Okawa, Yuji. "Open string states and D-brane tension from vacuum string field theory." Journal of High Energy Physics 2002, no. 07 (July 1, 2002): 003. http://dx.doi.org/10.1088/1126-6708/2002/07/003.

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25

Yoneya, Tamiaki. "String-coupling constant and dilaton vacuum expectation value in string field theory." Physics Letters B 197, no. 1-2 (October 1987): 76–80. http://dx.doi.org/10.1016/0370-2693(87)90345-5.

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26

MATSUO, YUTAKA. "PROJECTION OPERATORS AND D-BRANES IN PURELY CUBIC OPEN STRING FIELD THEORY." Modern Physics Letters A 16, no. 28 (September 14, 2001): 1811–21. http://dx.doi.org/10.1142/s0217732301005059.

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We study a matrix version of the purely cubic open string field theory as describing the expansion around the closed string vacuum. Any D-branes in the given closed string background can appear as classical solutions by using the identity projectors. Expansion around this solution gives the correct kinetic term for the open strings on the created D-branes while there are some subtleties in the unwanted degree of freedom.

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27

Kehagias, Alexandros, and Hervé Partouche. "The Casimir effect in string theory." International Journal of Modern Physics A 34, no. 09 (March 30, 2019): 1950049. http://dx.doi.org/10.1142/s0217751x19500490.

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We discuss the Casimir effect in heterotic string theory. This is done by considering a [Formula: see text] twist acting on one external compact direction and three internal coordinates. The hyperplanes fixed by the orbifold generator [Formula: see text] realize the two infinite parallel plates. For the latter to behave as “conducting material,” we implement in a modular invariant way the projection [Formula: see text] on the spectrum running in the vacuum-to-vacuum amplitude at one-loop. Hence, the relevant projector to account for the Casimir effect is orthogonal to that commonly used in string orbifold models which is [Formula: see text]. We find that this setup yields the same net force acting on the plates in the context of quantum field theory and string theory. However, when supersymmetry is not present from the onset, finiteness of the resultant force in field theory is reached by adding formally infinite forces acting on either side of each plate, while in string theory, both contributions are finite. On the contrary, when supersymmetry is spontaneously broken à la Scherk–Schwarz, finiteness of each contribution is fulfilled in field and string theory.

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28

CRANE, LOUIS. "STRING FIELD THEORY FROM QUANTUM GRAVITY." Reviews in Mathematical Physics 25, no. 10 (November 2013): 1343005. http://dx.doi.org/10.1142/s0129055x13430058.

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Recent work on neutrino oscillations suggests that the three generations of fermions in the standard model are related by representations of the finite group A(4), the group of symmetries of the tetrahedron. Motivated by this, we explore models which extend the EPRL model for quantum gravity by coupling it to a bosonic quantum field of representations of A(4). This coupling is possible because the representation category of A(4) is a module category over the representation categories used to construct the EPRL model. The vertex operators which interchange vacua in the resulting quantum field theory reproduce the bosons and fermions of the standard model, up to issues of symmetry breaking which we do not resolve. We are led to the hypothesis that physical particles in nature represent vacuum changing operators on a sea of invisible excitations which are only observable in the A(4) representation labels which govern the horizontal symmetry revealed in neutrino oscillations. The quantum field theory of the A(4) representations is just the dual model on the extended lattice of the Lie group E6, as explained by the quantum McKay correspondence of Frenkel, Jing and Wang. The coupled model can be thought of as string field theory, but propagating on a discretized quantum spacetime rather than a classical manifold.

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29

TORRIELLI, ALESSANDRO. "UNITARITY OF NONCOMMUTATIVE FIELD THEORIES FROM STRING THEORY." Modern Physics Letters A 18, no. 33n35 (November 20, 2003): 2525–32. http://dx.doi.org/10.1142/s0217732303012775.

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We improve the study of the lack of perturbative unitarity of noncommutative space-time quantum field theories derived from open string theory in electric backgrounds, enforcing the universality of the mechanism by which a tachyonic branch cut appears when the Seiberg-Witten limit freezes the string in an unstable vacuum. The main example is realized in the context of the on-shell four-tachyon amplitude of the bosonic string, and the dependence of the phenomenon on the brane-worldvolume dimension is analysed. We discuss the possibility of a proof in superstring theory, and finally mention the NCOS limit in this framework.

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30

Maccaferri, Carlo. "Chan-Paton factors and higgsing from vacuum string field theory." Journal of High Energy Physics 2005, no. 09 (September 7, 2005): 022. http://dx.doi.org/10.1088/1126-6708/2005/09/022.

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31

Kwon, O.-Kab, Bum-Hoon Lee, Chanyong Park, and Sang-Jin Sin. "Fluctuations around the tachyon vacuum in open string field theory." Journal of High Energy Physics 2007, no. 12 (December 11, 2007): 038. http://dx.doi.org/10.1088/1126-6708/2007/12/038.

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32

Rastelli, Leonardo, Ashoke Sen, and Barton Zwiebach. "Classical solutions in string field theory around the tachyon vacuum." Advances in Theoretical and Mathematical Physics 5, no. 2 (2001): 393–428. http://dx.doi.org/10.4310/atmp.2001.v5.n2.a6.

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33

AREF’EVA, I. YA, P. B. MEDVEDEV, and A. P. ZUBAREV. "NON-PERTURBATTVE VACUUM FOR SUPERSTRING FIELD THEORY AND SUPERSYMMETRY BREAKING." Modern Physics Letters A 06, no. 11 (April 10, 1991): 949–58. http://dx.doi.org/10.1142/s0217732391000993.

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We analyze possible ground states for the covariant string field theory action of the open NSR-superstring. A remarkable feature of this action is that it possesses a rich set of the auxiliary local fields. We found that some of low-lying scalar auxiliary fields acquire non-zero vacuum expectation values providing a new mechanism for supersymmetry breaking. The gauge vector field become massive while the physical spinor remains massless, thus the supersymmetry is broken in the non-perturbative vacuum.

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34

GRATS, YU V., and A. A. ROSSIKHIN. "VACUUM POLARIZATION NEAR BRANE WORLD COSMIC STRING." Modern Physics Letters A 17, no. 18 (June 14, 2002): 1207–14. http://dx.doi.org/10.1142/s0217732302007296.

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Gravitational field of cosmic strings in theories with extra spatial dimensions differs significantly from that in the 4D Einstein's theory. This means that all gravity induced properties of cosmic strings need to be revised. Here we consider the effect of vacuum polarization outside a straight infinitely thin cosmic string embedded in RS2 brane world.

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35

NILL, FLORIAN. "A CONSTRUCTIVE QUANTUM FIELD THEORETIC APPROACH TO CHERN-SIMONS THEORY." International Journal of Modern Physics B 06, no. 11n12 (June 1992): 2159–98. http://dx.doi.org/10.1142/s0217979292001080.

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Chern-Simons theory is formulated quantum field theoretically in terms of string operators, which are localized on finite non-selfintersecting paths in the zero-time plane R2. It is shown how the Weyl algebra approach to the abelian Chern-Simons theory leads to a bundle theoretic construction of these ‘Chern-Simons string operators’. To proceed to the non-abelian case topological Chern-Simons theory is considered as a specific model of a more general theory describing the quantum kinematics of coloured framed Plek ton s inR2. This theory allows the construction of string operators as pair-creation operators, mapping n-Plekton states into (n+2)-Plekton states. Link invariants are obtained as vacuum correlation functions of string operators and obey a natural version of reflection positivity. The vacuum sector of such a purely kinematical Plekton theory may be recovered from the link invariants by Osterwalder-Schrader reconstruction. To make the theory work one needs the structures of a ribbon graph category, for which the representations of a quasi-triangular quasi-Hopf algebra may serve as a specific realization.

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36

de Lacroix, Corinne, Harold Erbin, Sitender Pratap Kashyap, Ashoke Sen, and Mritunjay Verma. "Closed superstring field theory and its applications." International Journal of Modern Physics A 32, no. 28n29 (October 19, 2017): 1730021. http://dx.doi.org/10.1142/s0217751x17300216.

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We review recent developments in the construction of heterotic and type II string field theories and their various applications. These include systematic procedures for determining the shifts in the vacuum expectation values of fields under quantum corrections, computing renormalized masses and S-matrix of the theory around the shifted vacuum and a proof of unitarity of the S-matrix. The S-matrix computed this way is free from all divergences when there are more than 4 noncompact space–time dimensions, but suffers from the usual infrared divergences when the number of noncompact space–time dimensions is 4 or less.

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37

Ambjørn, J., and Y. Watabiki. "Noncritical String Field Theory for 2d Quantum Gravity Coupled to (p, q)-Conformal Fields." International Journal of Modern Physics A 12, no. 23 (September 20, 1997): 4257–89. http://dx.doi.org/10.1142/s0217751x97002322.

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We propose a noncritical string field theory for 2d quantum gravity coupled to (p, q) conformal fields. The Hamiltonian is described by the generators of the Wp algebra, and the Schwinger–Dyson equation is equivalent to a vacuum condition imposed on the generators of Wp algebra.

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38

ANTONOV, D. V. "NONPERTURBATIVE FIELD CORRELATOR AND STRING REPRESENTATION FOR THE 't HOOFT LOOP AVERAGE IN THE ABELIAN HIGGS MODEL." Modern Physics Letters A 13, no. 09 (March 21, 1998): 659–71. http://dx.doi.org/10.1142/s0217732398000711.

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Making use of the duality transformation, we derive in the Londons' limit of the Abelian Higgs model string representation for the 't Hooft loop average defined on the string worldsheet, which yields the values of two coefficient functions parametrizing the bilocal correlator of the dual field strength tensors. The asymptotic behaviors of these functions agree with the ones obtained within the method of vacuum correlators in QCD in the lowest order of perturbation theory. We demonstrate that the bilocal approximation to the method of vacuum correlators is an exact result in the Londons' limit, i.e. all the higher cumulants in this limit vanish. We also show that at large distances, apart from the integration over metrics, the obtained string effective theory (which in this case reduces to the nonlinear massive axionic sigma model) coincides with the low energy limit of the dual version of 4D compact QED, the so-called universal confining string theory. We derive string tension of the Nambu–Goto term and the coupling constant of the rigidity term for the obtained string effective theory and demonstrate that the latter is always negative, which means the stability of strings, while the positiveness of the former is confirmed by the present lattice data. These data enable us to find the Higgs boson charge and the vacuum expectation value of the Higgs field, which well-described QCD. We also study dynamics of the weight factor of the obtained string representation for the 't Hooft average in the loop space. In conclusion, we obtain string representation for the partition function of the correlators of an arbitrary number of Higgs currents, by virtue of which we rederive the structure of the bilocal correlator of the dual field strength tensors, which yields the surface term in the string effective action.

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39

Ghoshal, Anish, and Florian Nortier. "Fate of the false vacuum in string-inspired nonlocal field theory." Journal of Cosmology and Astroparticle Physics 2022, no. 08 (August 1, 2022): 047. http://dx.doi.org/10.1088/1475-7516/2022/08/047.

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Abstract In this article, we study Coleman bounce in weakly nonlocal theories which are motivated from string field theory. The kinetic term is extended via an infinite series of high-order derivatives, which comes into play at an energy scale M, without introducing any new states or ghosts in the mass spectrum. We calculate the bubble nucleation in thin-wall approximation, treating the system in semi-classical manner. We find that the effect of nonlocal scale M in the theory is to suppress the vacuum tunneling rate from false to true vacuum compared to the standard local bouncing scenario. Likewise, we show that as we move further away from the bubble wall, the effects of nonlocality gets reduced and this suppression is significant only around the wall of the nucleated bubble. From our investigations, we conclude that the main effect is due to the fact that the nonlocality smears the solution of the local bubble profile. However, the energy of the bubble wall remains unaffected by the microscopic nonlocal behavior of the theory in the thin-wall approximation. We also discuss the cases for Lee-Wick theories and applications of our result to cosmology.

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40

Rastelli, Leonardo, Ashoke Sen, and Barton Zwiebach. "Boundary CFT construction of D-branes in vacuum string field theory." Journal of High Energy Physics 2001, no. 11 (November 19, 2001): 045. http://dx.doi.org/10.1088/1126-6708/2001/11/045.

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41

Potting, Robertus, and Joris Raeymaekers. "Ghost sector of vacuum string field theory and the projection equation." Journal of High Energy Physics 2002, no. 06 (June 3, 2002): 002. http://dx.doi.org/10.1088/1126-6708/2002/06/002.

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42

Imamura, Yosuke. "Gauge transformations on a D-brane in Vacuum String Field Theory." Journal of High Energy Physics 2002, no. 07 (July 18, 2002): 042. http://dx.doi.org/10.1088/1126-6708/2002/07/042.

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43

Drukker, Nadav. "On different actions for the vacuum of bosonic string field theory." Journal of High Energy Physics 2003, no. 08 (August 11, 2003): 017. http://dx.doi.org/10.1088/1126-6708/2003/08/017.

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44

Ishida, Akira, Chanju Kim, Yoonbai Kim, O.-Kab Kwon, and D. D. Tolla. "Tachyon vacuum solution in open string field theory with a constantBfield." Journal of Physics A: Mathematical and Theoretical 42, no. 39 (September 8, 2009): 395402. http://dx.doi.org/10.1088/1751-8113/42/39/395402.

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45

Giusto, S., and C. Imbimbo. "Physical states at the tachyonic vacuum of open string field theory." Nuclear Physics B 677, no. 1-2 (January 2004): 52–86. http://dx.doi.org/10.1016/j.nuclphysb.2003.10.037.

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46

Hata, Hiroyuki, and Sanefumi Moriyama. "Reexamining classical solution and tachyon mode in vacuum string field theory." Nuclear Physics B 651, no. 1-2 (February 2003): 3–25. http://dx.doi.org/10.1016/s0550-3213(02)01094-5.

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47

Hata, Hiroyuki, and Sanefumi Moriyama. "Exact results on equations of motion in vacuum string field theory." Physics Letters B 631, no. 3 (December 2005): 141–49. http://dx.doi.org/10.1016/j.physletb.2005.09.081.

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48

INATOMI, SHOKO. "ONE-LOOP VACUUM ENERGY AT THE TACHYON VACUUM." International Journal of Modern Physics: Conference Series 21 (January 2013): 157–58. http://dx.doi.org/10.1142/s2010194513009598.

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We consider one-loop vacuum energy at the tachyon vacuum in cubic bosonic open string field theory. The BRST operator Ql in the theory around an identity-based solution is believed to represent a kinetic operator at the tachyon vacuum. Using homotopy operators for Ql, we find that one-loop vacuum energy at the tachyon vacuum is independent of moduli such as interbrane distances. This result can be interpreted as support for the annihilation of D-branes at the tachyon vacuum even in the quantum theory.

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49

Mavromatos, Nick E., Joan Solà Peracaula, and Spyros Basilakos. "String-Inspired Running Vacuum—The “Vacuumon”—And the Swampland Criteria." Universe 6, no. 11 (November 20, 2020): 218. http://dx.doi.org/10.3390/universe6110218.

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We elaborate further on the compatibility of the “vacuumon potential” that characterises the inflationary phase of the running vacuum model (RVM) with the swampland criteria. The work is motivated by the fact that, as demonstrated recently by the authors, the RVM framework can be derived as an effective gravitational field theory stemming from underlying microscopic (critical) string theory models with gravitational anomalies, involving condensation of primordial gravitational waves. Although believed to be a classical scalar field description, not representing a fully fledged quantum field, we show here that the vacuumon potential satisfies certain swampland criteria for the relevant regime of parameters and field range. We link the criteria to the Gibbons–Hawking entropy that has been argued to characterise the RVM during the de Sitter phase. These results imply that the vacuumon may, after all, admit under certain conditions, a rôle as a quantum field during the inflationary (almost de Sitter) phase of the running vacuum. The conventional slow-roll interpretation of this field, however, fails just because it satisfies the swampland criteria. The RVM effective theory derived from the low-energy effective action of string theory does, however, successfully describe inflation thanks to the ∼H4 terms induced by the gravitational anomalous condensates. In addition, the stringy version of the RVM involves the Kalb–Ramond (KR) axion field, which, in contrast to the vacuumon, does perfectly satisfy the slow-roll condition. We conclude that the vacuumon description is not fully equivalent to the stringy formulation of the RVM. Our study provides a particularly interesting example of a successful phenomenological theory beyond the ΛCDM, such as the RVM, in which the fulfilment of the swampland criteria by the associated scalar field potential, along with its compatibility with (an appropriate form of) the weak gravity conjecture, prove to be insufficient conditions for warranting consistency of the scalar vacuum field representation as a faithful ultraviolet complete representation of the RVM at the quantum gravity level.

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50

Han, Seung Kee. "Vanishing vacuum amplitude of four-dimensional heterotic string theory compactified onN=2superconformal field theory." Physical Review D 39, no. 8 (April 15, 1989): 2322–26. http://dx.doi.org/10.1103/physrevd.39.2322.

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