Three-particle scattering rates and singularities of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>T</mml:mi></mml:math>matrix. I.

Potapov, V. S.; Taylor, John R.

doi:10.1103/physreva.16.2264

Cited by 14 publications

(21 citation statements)

References 5 publications

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“…The reason for this choice is that, for fixed k ∼ m, F i ( k) is exponentially suppressed as L → ∞, since the summand of the sum-integral difference in (15) is smooth. 10 We use this result repeatedly in the following analysis. The same is not true of F ( k), due to the ρ( k) term in Eq.…”

Section: B Perturbative Expansion Of λ0mentioning

confidence: 99%

Threshold expansion of the three-particle quantization condition

Hansen

Sharpe

2016

Phys. Rev. D

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We recently derived a quantization condition for the energy of three relativistic particles in a cubic box [1,2]. Here we use this condition to study the energy level closest to the three-particle threshold when the total three-momentum vanishes. We expand this energy in powers of 1/L, where L is the linear extent of the finite volume. The expansion begins at O(1/L 3 ), and we determine the coefficients of the terms through O(1/L 6 ). As is also the case for the two-particle threshold energy, the 1/L 3 , 1/L 4 and 1/L 5 coefficients depend only on the two-particle scattering length a. These can be compared to previous results obtained using nonrelativistic quantum mechanics [3-5] and we find complete agreement. The 1/L 6 coefficients depend additionally on the two-particle effective range r (just as in the two-particle case) and on a suitably defined threshold three-particle scattering amplitude (a new feature for three particles). A second new feature in the three particle case is that logarithmic dependence on L appears at O(1/L 6 ). Relativistic effects enter at this order, and the only comparison possible with the nonrelativistic result is for the coefficient of the logarithm, where we again find agreement. For a more thorough check of the 1/L 6 result, and thus of the quantization condition, we also compare to a perturbative calculation of the threshold energy in relativistic λφ 4 theory, which we have recently presented in Ref. [6]. Here all terms can be compared and we find full agreement.

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Section: B Perturbative Expansion Of λ0mentioning

confidence: 99%

Threshold expansion of the three-particle quantization condition

Hansen

Sharpe

2016

Phys. Rev. D

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“…1(a), and scale as a 2 /∆E, a 3 / √ ∆E and a 4 log(∆E), respectively, where a is the scattering length. The existence of such singularities is a general field-theoretic result that was established long ago [8][9][10][11]. Our formalism accommodates these divergences by finding that ∆E th depends on a modified quantity, M 3,th , given by subtracting the divergent terms from M 3 [see Fig.…”

Section: Fig 1 (A)mentioning

confidence: 99%

Erratum: Threshold expansion of the three-particle quantization condition [Phys. Rev. D 93 , 096006 (2016)]

Hansen¹,

Sharpe²

2017

Phys. Rev. D

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We recently derived a quantization condition for the energy of three relativistic particles in a cubic box [1,2]. Here we use this condition to study the energy level closest to the three-particle threshold when the total three-momentum vanishes. We expand this energy in powers of 1/L, where L is the linear extent of the finite volume. The expansion begins at O(1/L 3 ), and we determine the coefficients of the terms through O(1/L 6 ). As is also the case for the two-particle threshold energy, the 1/L 3 , 1/L 4 and 1/L 5 coefficients depend only on the two-particle scattering length a. These can be compared to previous results obtained using nonrelativistic quantum mechanics [3][4][5], and we find complete agreement. The 1/L 6 coefficients depend additionally on the two-particle effective range r (just as in the two-particle case) and on a suitably defined threshold three-particle scattering amplitude (a new feature for three particles). A second new feature in the three-particle case is that logarithmic dependence on L appears at O(1/L 6 ). Relativistic effects enter at this order, and the only comparison possible with the nonrelativistic result is for the coefficient of the logarithm, where we again find agreement. For a more thorough check of the 1/L 6 result, and thus of the quantization condition, we also compare to a perturbative calculation of the threshold energy in relativistic λφ 4 theory, which we have recently presented in Ref. [6]. Here, all terms can be compared, and we find full agreement.

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“…(64) 21 In the definition of PV we are using σ and σ † which are continuous functions of a and k. Since these were originally defined only for discrete finite-volume momenta, this requires a continuation of the original functions. We require only that the continuation is smooth and slowly varying.…”

mentioning

confidence: 99%

Relativistic, model-independent, three-particle quantization condition

2014

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We present a generalization of Lüscher's relation between the finite-volume spectrum and scattering amplitudes to the case of three particles. We consider a relativistic scalar field theory in which the couplings are arbitrary aside from a Z2 symmetry that removes vertices with an odd number of particles. The theory is assumed to have two-particle phase shifts that are bounded by π/2 in the regime of elastic scattering. We determine the spectrum of the finite-volume theory from the poles in the odd-particle-number finite-volume correlator, which we analyze to all orders in perturbation theory. We show that it depends on the infinite-volume two-to-two K-matrix as well as a nonstandard infinite-volume three-to-three K-matrix. A key feature of our result is the need to subtract physical singularities in the three-to-three amplitude and thus deal with a divergence-free quantity. This allows our initial, formal result to be truncated to a finite dimensional determinant equation. At present, the relation of the three-to-three K-matrix to the corresponding scattering amplitude is not known, although previous results in the non-relativistic limit suggest that such a relation exists.

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Three-particle scattering rates and singularities of theTmatrix. I.

Cited by 14 publications

References 5 publications

Threshold expansion of the three-particle quantization condition

Threshold expansion of the three-particle quantization condition

Erratum: Threshold expansion of the three-particle quantization condition [Phys. Rev. D 93 , 096006 (2016)]

Relativistic, model-independent, three-particle quantization condition

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