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Stephen R. Sharpe

Publications and source records attributed to Stephen R. Sharpe.

At least 55 records · Page 3Linked to original sources
arXiv2018

Testing the threshold expansion for three-particle energies at fourth order in $ϕ^4$ theory

A relativistic formalism for relating the the energies of the states of three scalar particles in finite volume to infinite volume scattering amplitudes has recently been developed. This formalism has been used to predict the energy of the state closest to threshold in an expansion in powers of $1/L$, with $L$ the box length. This expansion has been tested previously by a perturbative calculation of the threshold energy in $λϕ^4$ theory, working to third order in $λ$ and up to $\mathcal O(1/L^6)$ in the volume expansion. However, several aspects of the predicted threshold behavior do not enter until fourth (three-loop) order in perturbation theory. Here I extend the perturbative calculation to fourth order and find agreement with the general prediction. This check also requires a two-loop calculation of the infinite-volume off-shell two-particle scattering amplitude near threshold. As a spin-off, I check the threshold expansion for two particles to the same order, finding agreement with the result that follows from Lüscher's formalism.

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arXiv2018

Progress report on the relativistic three-particle quantization condition

We describe recent work on the relativistic three-particle quantization condition, generalizing and applying the original formalism of Hansen and Sharpe, and of Briceño, Hansen and Sharpe. In particular, we sketch three recent developments: the generalization of the formalism to include K-matrix poles; the numerical implementation of the quantization condition in the isotropic approximation; and ongoing work extending the description of the three-particle divergence-free K matrix beyond the isotropic approximation.

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arXiv2018

Three-particle systems with resonant subprocesses in a finite volume

In previous work, we have developed a relativistic, model-independent three-particle quantization condition, but only under the assumption that no poles are present in the two-particle K matrices that appear as scattering subprocesses. Here we lift this restriction, by deriving the quantization condition for identical scalar particles with a G-parity symmetry, in the case that the two-particle K matrix has a pole in the kinematic regime of interest. As in earlier work, our result involves intermediate infinite-volume quantities with no direct physical interpretation, and we show how these are related to the physical three-to-three scattering amplitude by integral equations. This work opens the door to study processes such as $a_2 \to ρπ\to πππ$, in which the $ρ$ is rigorously treated as a resonance state.

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arXiv2018

Numerical study of the relativistic three-body quantization condition in the isotropic approximation

We present numerical results showing how our recently proposed relativistic three-particle quantization condition can be used in practice. Using the isotropic (generalized $s$-wave) approximation, and keeping only the leading terms in the effective range expansion, we show how the quantization condition can be solved numerically in a straightforward manner. In addition, we show how the integral equations that relate the intermediate three-particle infinite-volume scattering quantity, $\mathcal K_{\text{df},3}$, to the physical scattering amplitude can be solved at and below threshold. We test our methods by reproducing known analytic results for the $1/L$ expansion of the threshold state, the volume dependence of three-particle bound-state energies, and the Bethe-Salpeter wavefunctions for these bound states. We also find that certain values of $\mathcal K_{\text{df},3}$ lead to unphysical finite-volume energies, and give a preliminary analysis of these artifacts.

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arXiv2017

Threshold expansion of the three-particle quantization condition

We recently derived a quantization condition for the energy of three relativistic particles in a cubic box. 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 ${\cal O}(1/L^3)$, and we determine the coefficients of the terms through ${\cal O}(1/L^6)$. 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 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 $\mathcal O(1/L^6)$. 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. Here all terms can be compared and we find full agreement.

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arXiv2017

Relating the finite-volume spectrum and the two-and-three-particle $S$ matrix for relativistic systems of identical scalar particles

Working in relativistic quantum field theory, we derive the quantization condition satisfied by coupled two- and three-particle systems of identical scalar particles confined to a cubic spatial volume with periodicity $L$. This gives the relation between the finite-volume spectrum and the infinite-volume $\textbf 2 \to \textbf 2$, $\textbf 2 \to \textbf 3$ and $\textbf 3 \to \textbf 3$ scattering amplitudes for such theories. The result holds for relativistic systems composed of scalar particles with nonzero mass $m$, whose center of mass energy lies below the four-particle threshold, and for which the two-particle $K$ matrix has no singularities below the three-particle threshold. The quantization condition is exact up to corrections of the order $\mathcal{O}(e^{-mL})$ and holds for any choice of total momenta satisfying the boundary conditions.

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arXiv2016

Progress in three-particle scattering from LQCD

We present the status of our formalism for extracting three-particle scattering observables from lattice QCD (LQCD). The method relies on relating the discrete finite-volume spectrum of a quantum field theory with its scattering amplitudes. As the finite-volume spectrum can be directly determined in LQCD, this provides a method for determining scattering observables, and associated resonance properties, from the underlying theory. In a pair of papers published over the last two years, two of us have extended this approach to apply to relativistic three-particle scattering states. In this talk we summarize recent progress in checking and further extending this result. We describe an extension of the formalism to include systems in which two-to-three transitions can occur. We then present a check of the previously published formalism, in which we reproduce the known finite-volume energy shift of a three-particle bound state.

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arXiv2016

Progress on the three-particle quantization condition

We report progress on extending the relativistic model-independent quantization condition for three particles, derived previously by two of us, to a broader class of theories, as well as progress on checking the formalism. In particular, we discuss the extension to include the possibility of 2->3 and 3->2 transitions and the calculation of the finite-volume energy shift of an Efimov-like three-particle bound state. The latter agrees with the results obtained previously using non-relativistic quantum mechanics.

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arXiv2016

Expressing the three-particle finite-volume spectrum in terms of the three-to-three scattering amplitude

In this article we complete our formalism relating the finite-volume energy spectrum of a scalar quantum field theory to the three-to-three scattering amplitude, ${\cal M}_3$. In previous work we found a quantization condition relating the spectrum to a non-standard infinite-volume quantity, denoted ${\cal K}_{{\rm df},3}$. Here we present the relation between ${\cal K}_{{\rm df},3}$ and ${\cal M}_3$. We then discuss briefly how our now completed formalism can be practically implemented to extract ${\cal M}_3$ from the finite-volume energy spectrum.

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arXiv2015

Kaon BSM B-parameters using improved staggered fermions from $N_f=2+1$ unquenched QCD

We present results for the matrix elements of the additional $ΔS=2$ operators that appear in models of physics beyond the Standard Model (BSM), expressed in terms of four BSM $B$-parameters. Combined with experimental results for $ΔM_K$ and $ε_K$, these constrain the parameters of BSM models. We use improved staggered fermions, with valence HYP-smeared quarks and $N_f=2+1$ flavors of "asqtad" sea quarks. The configurations have been generated by the MILC collaboration. The matching between lattice and continuum four-fermion operators and bilinears is done perturbatively at one-loop order. We use three lattice spacings for the continuum extrapolation: $a\approx 0.09$, $0.06$ and $0.045\;$fm. Valence light-quark masses range down to $\approx m_s^{\rm phys}/13$ while the light sea-quark masses range down to $\approx m_s^{\rm phys}/20$. Compared to our previous published work, we have added four additional lattice ensembles, leading to better controlled extrapolations in the lattice spacing and sea-quark masses. We report final results for two renormalization scales, $μ=2\;\text{GeV}$ and $3\;\text{GeV}$, and compare them to those obtained by other collaborations. Agreement is found for two of the four BSM $B$-parameters ($B_2$ and $B_3^\text{SUSY}$). The other two ($B_4$ and $B_5$) differ significantly from those obtained using RI-MOM renormalization as an intermediate scheme, but are in agreement with recent preliminary results obtained by the RBC-UKQCD collaboration using RI-SMOM intermediate schemes.

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arXiv2015

Phase structure with nonzero $Θ_{\rm QCD}$ and twisted mass fermions

We determine the phase diagram and chiral condensate for lattice QCD with two flavors of twisted-mass fermions in the presence of nondegenerate up and down quarks, discretization errors and a nonzero value of $Θ_{\rm QCD}$. Although such a theory has a complex action and cannot, at present, be simulated, the results are needed to understand how to tune to maximal twist in the presence of electromagnetism, a topic discussed in a companion paper. We find that, in general, the only phase structure is a first-order transition of finite length. Pion masses are nonvanishing throughout the phase plane except at the endpoints of the first-order line. Only for extremal values of the twist angle and $Θ_{\rm QCD}$ ($ω=0$ or $π/2$ and $Θ_{\rm QCD}=0$ or $π$) are there second-order transitions.

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arXiv2015

Perturbative results for two and three particle threshold energies in finite volume

We calculate the energy of the state closest to threshold for two and three identical, spinless particles confined to a cubic spatial volume with periodic boundary conditions and with zero total momentum in the finite-volume frame. The calculation is performed in relativistic quantum field theory with particles coupled via a $λϕ^4$ interaction, and we work through order $λ^3$. The energy shifts begin at ${\cal O}(1/L^3)$, and we keep subleading terms proportional to $1/L^4$, $1/L^5$ and $1/L^6$. These terms allow a non-trivial check of the results obtained from quantization conditions that hold for arbitrary interactions, namely that of Lüscher for two particles and our recently developed formalism for three particles. We also compare to previously obtained results based on non-relativistic quantum mechanics.

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arXiv2015

Impact of electromagnetism on phase structure for Wilson and twisted-mass fermions including isospin breaking

In a recent paper we used chiral perturbation theory to determine the phase diagram and pion spectrum for Wilson and twisted-mass fermions at non-zero lattice spacing with non-degenerate up and down quarks. Here we extend this work to include the effects of electromagnetism, so that it is applicable to recent simulations incorporating all sources of isospin breaking. For Wilson fermions, we find that the phase diagram is unaffected by the inclusion of electromagnetism---the only effect is to raise the charged pion masses. For maximally twisted fermions, we previously took the twist and isospin-breaking directions to be different, in order that the fermion determinant is real and positive. However, this is incompatible with electromagnetic gauge invariance, and so here we take the twist to be in the isospin-breaking direction, following the RM123 collaboration. We map out the phase diagram in this case, which has not previously been studied. The results differ from those obtained with different twist and isospin directions. One practical issue when including electromagnetism is that the critical masses for up and down quarks differ. We show that one of the criteria suggested to determine these critical masses does not work, and propose an alternative.

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arXiv2014

A relativistic, model-independent, three-particle quantization condition

We present a generalization of Luescher'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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arXiv2014

Phase diagram of non-degenerate twisted mass fermions

We determine the phase diagram and pion spectrum for Wilson and twisted-mass fermions in the presence of non-degeneracy between the up and down quark and discretization errors, using Wilson and twisted-mass chiral perturbation theory. We find that the CP-violating phase of the continuum theory (which occurs for sufficiently large non-degeneracy) is continuously connected to the Aoki-phase of the lattice theory with degenerate quarks. We show that discretization effects can, in some cases, push simulations with physical masses closer to either the CP-violating phase or another phase not present in the continuum, so that at sufficiently large lattice spacings physical-point simulations could lie in one of these phases.

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arXiv2014

Calculation of BSM Kaon B-parameters using Staggered Quarks

We present updated results for kaon B-parameters for operators arising in models of new physics. We use HYP-smeared staggered quarks on the $N_f = 2+1$ MILC asqtad lattices. During the last year we have added new ensembles, which has necessitated chiral-continuum fitting with more elaborate fitting functions. We have also corrected an error in a two-loop anomalous dimension used to evolve results between different scales. Our results for the beyond-the-Standard-Model B-parameters have total errors of $5-10$\%. We find that the discrepancy observed last year between our results and those of the RBC/UKQCD and ETM collaborations for some of the B-parameters has been reduced from $4\!-\!5\,σ$ to $2\!-\!3\,σ$.

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arXiv2014

Phase diagram of non-degenerate Wilson and twisted mass fermions

In this talk we determine the phase diagram and pion spectrum for Wilson and twisted-mass fermions in the presence of non-degeneracy between the up and down quark and discretization errors. We find that the CP-violating phase of the continuum theory, which occurs for sufficiently large non-degeneracy, is continuously connected to the Aoki phase found in the lattice theory with degenerate quarks. Both for the Aoki and first-order scenarios, this results in a critical surface along which at least one of the pions is massless. In the pion spectrum, we focus mainly on the untwisted case, where there is competition between the effects of non-degeneracy and discretization errors. A more extensive analysis can be found in our recent paper [11].

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