Cutkosky Rule Research Articles

Cutkosky rules and unitarity (violation) in D-instanton amplitudes

In perturbative amplitudes in quantum field theory and string field theory, Cutkosky rule expresses the anti-hermitian part of a Feynman diagram in terms of sum over all its cut diagrams, and this in turn is used to prove unitarity of the theory. For D-instanton contribution to a string theory amplitude, the cutting rule needed for the proof of unitarity is somewhat different; we need to sum over only those cut diagrams for which all the world-sheet boundaries ending on some particular D-instanton lie on the same side of the cut. By working with the closed string effective action, obtained after integrating out the open string modes, we prove that the D-instanton amplitudes actually satisfy these cutting rules, provided the effective action is real. The violation of unitarity in the closed string sector of two dimensional string theory can be traced to the failure of this reality condition. In the critical superstring theory, multi-instanton and multi anti-instanton amplitudes satisfy the reality condition. Contribution to the amplitudes from the instanton anti-instanton sector satisfies the reality condition if we make a specific choice of integration cycle over the configuration space of string fields, whereas contribution due to the non-BPS D-instantons will need to either vanish or have an overall real normalization in order for it to give real contribution. We use Picard-Lefschetz theory to argue that these conditions are indeed satisfied in superstring theories.

Nonlocal vertices and analyticity: Landau equations and general Cutkosky rule

We study the analyticity properties of amplitudes in theories with nonlocal vertices of the type occurring in string field theory and a wide class of nonlocal field theory models. Such vertices are given in momentum space by entire functions of rapid decay in certain (including Euclidean) directions ensuring UV finiteness but are necessarily of rapid increase in others. A parametric representation is obtained by integrating out the loop (Euclidean) momenta after the introduction of generalized Schwinger parameters. Either in the original or parametric representation, the well-defined resulting amplitudesare then continued in the complex space of the external momenta invariants. We obtain the alternative forms of the Landau equations determining the singularity surfaces showing that the nonlocal vertices serve as UV regulators but do not affect the local singularity structure. As a result the full set of singularities known to occur in local field theory also occurs here: normal and anomalous thresholds as well as acnodes, crunodes, and cusps that may under certain circumstances appear even on the physical sheet. Singularities of the second type also appear as shown from the parametric representation. We obtain the general Cutkosky discontinuity rule for encircling a singularity by employing contour deformations only in the finite plane. The unitarity condition (optical theorem) is thendiscussed as a special application of the rule across normal thresholds and the hermitian analyticity property of amplitudes.

Causality and unitarity via the tree-loop duality relation

The tree-loop duality relation is used as a starting point to derive the constraints of causality and unitarity. Specifically, the Bogoliubov causality condition is ab initio derived at the individual graph level. It leads to a representation of a graph in terms of lower order cut graphs. Extracting the absorptive part gives then the general unitarity relation (Cutkosky rule). The derivation, being carried out directly in momentum space, holds for any local (polynomial) hermitian interaction vertices. This is in contrast to the technical difficulties arising from contact terms in the spacetime approach based on the largest time equation.

Self-healing of unitarity in Higgs inflation

We reconsider perturbative unitarity violation in the standard model Higgs inflation model. We show that the Cutkosky cutting rule implied by perturbative unitarity is fulfilled at one-loop. This is a strong indication that unitarity is restored order by order in perturbation theory. We then resum certain one-loop diagrams and show that the relevant dressed amplitude fulfills the Cutkosky rule exactly. This is an example of the self-healing mechanism. The original Higgs inflation model is thus consistent and does not require any new physics beyond the standard model at least up to the Planck scale.

The Cutkosky rule of three dimensional noncommutative field theory in Lie algebraic noncommutative spacetime

The Cutkosky rule of three dimensional noncommutative field theory in Lie algebraic noncommutative spacetime

The Cutkosky rule of three dimensional noncommutative field theory in Lie algebraic noncommutative spacetime

We investigate the unitarity of three dimensional noncommutative scalar field theory in the Lie algebraic noncommutative spacetime [x^i,x^j]=2i kappa epsilon^{ijk}x_k. This noncommutative field theory possesses a SL(2,R)/Z_2 group momentum space, which leads to a Hopf algebraic translational symmetry. We check the Cutkosky rule of the one-loop self-energy diagrams in the noncommutative phi^3 theory when we include a braiding, which is necessary for the noncommutative field theory to possess the Hopf algebraic translational symmetry at quantum level. Then, we find that the Cutkosky rule is satisfied if the mass is less than 1/(2^(1/2)kappa).

QCD Approach to B → Dπ Decays and CP Violation

The branching ratios and CP violations of the B → Dπ decays, including both the color-allowed and the color-suppressed modes, are investigated in detail within QCD framework by considering all diagrams that lead to three effective currents of two quarks. An intrinsic mass scale as a dynamical gluon mass is introduced to treat the infrared divergence caused by the soft collinear approximation in the endpoint regions, and the Cutkosky rule is adopted to deal with a physical-region singularity of the on mass-shell quark propagators. When the dynamical gluon mass μg is regarded as a universal scale, it is extracted to be around μg = 440 MeV from one of the well-measured B → Dπ decay modes. The resulting predictions for all branching ratios are in agreement with the current experimental measurements. As these decays have no penguin contributions, there are no direct CP asymmetries. Due to interference between the Cabibbo-suppressed and the Cabibbo-favored amplitudes, mixing-induced CP violations are predicted in the B → D±π∓ decays to be consistent with the experimental data at 1-σ level. More precise measurements will be helpful to extracting weak angle 2β + γ.

Large strong phases and CP violation in the annihilation processes B̄0→K+K-, K*±K∓, K*+K*-

The strong phases and CP violation in the rare (B) over bar (0)-> K+ K-, K*(+/-) K--/+, K*(+) K*(-) decays are investigated. As these decays proceed only via annihilation type diagrams in the standard model (SM), a dynamical gluon mass is introduced to avoid the infrared divergence in the soft endpoint regions. The Cutkosky rule is adopted to deal with a physical-region singularity of the on-mass-shell quark propagators, which leads to a big imaginary part and hence a large strong phase. As a consequence, large CP asymmetries are predicted in those decay modes due to a big interference between the annihilation amplitudes from penguin and tree operators, which may be tested in future more precise experiments.

One-loop unitarity of scalar field theories on Poincaré invariant commutative nonassociative spacetimes

We study scalar field theories on Poincare invariant commutative nonassociative spacetimes. We compute the one-loop self-energy diagrams in the ordinary path integral quantization scheme with Feynman's prescription, and find that the Cutkosky rule is satisfied. This property is in contrast with that of noncommutative field theory, since it is known that noncommutative field theory with space/time noncommutativity violates unitarity in the above standard scheme, and the quantization procedure will necessarily become complicated to obtain a sensible Poincare invariant noncommutative field theory. We point out a peculiar feature of the non-locality in our nonassociative field theories, which may explain the property of the unitarity distinct from noncommutative field theories. Thus commutative nonassociative field theories seem to contain physically interesting field theories on deformed spacetimes.

Chiral3π-exchangeNNpotentials: Results for dominant next-to-leading-order contributions

We calculate in (two-loop) chiral perturbation theory the local $\mathrm{NN}$ potentials generated by the three-pion exchange diagrams with one insertion from the second order chiral effective pion-nucleon Lagrangian proportional to the low-energy constants ${c}_{1,2,3,4}.$ The resulting isoscalar central potential vanishes identically. In most cases these $3\ensuremath{\pi}$-exchange potentials are larger than the ones generated by the diagrams involving only leading order vertices due to the large values of ${c}_{3,4}$ (which mainly represent virtual $\ensuremath{\Delta}$ excitation). A similar feature has been observed for the chiral $2\ensuremath{\pi}$ exchange. We also give suitable (double-integral) representations for the spin-spin and tensor potentials generated by the leading-order diagrams proportional to ${g}_{A}^{6}$ involving four nucleon propagators. In these cases the Cutkosky rule cannot be used to calculate the spectral functions in the infinite nucleon mass limit since the corresponding mass spectra start with a nonvanishing value at the $3\ensuremath{\pi}$ threshold. Altogether, one finds that chiral $3\ensuremath{\pi}$ exchange leads to small corrections in the region $r>~1.4\mathrm{fm}$ where $1\ensuremath{\pi}$ and chiral $2\ensuremath{\pi}$ exchange alone provide a very good strong $\mathrm{NN}$ force as shown in a recent analysis of the low-energy $\mathrm{pp}$-scattering database.

S-Matrix Singularity Structure in the Physical Region. III. General Discussion of Simple Landau Singularities

Drawing on the understanding of unitarity integrals acquired in previous papers [J. Math. Phys. 10, 494, 545 (1969)], we show that unitarity demands that connected amplitudes be singular on the positive-α arcs of all ``simple'' Landau curves in their physical regions. Further, we show that the amplitudes are nonsingular on mixed-α arcs, while on the positive-α arcs their discontinuities are given by the Cutkosky rule. This confirms arguments from perturbation theory and demonstrates how a weak analyticity assumption can generate in an exact way a singularity scheme relating to causality.