One-loop Case Research Articles

Loops in anti de Sitter space

We discuss general one and two-loop banana diagrams and one-loop diagrams with external lines with arbitrary masses on the anti de Sitter spacetime by using methods of AdS quantum field theory in the dimensional regularization approach. The banana diagrams explicitly computed in this paper are indeed the necessary ingredients for the evaluation of the two-loop effective potential of the Standard Model and can be used to extend the flat space results in presence of a negative cosmological constant. In the one-loop case we also compute the effective potential for an O(N) model in d = 4 dimension as an explicit function of the cosmological constant Λ, both exactly and perturbatively up to order Λ. In the two-loop case we show the explicit calculation is possible thanks to a remarkable discrete Källén-Lehmann formula which we found and proved sometimes ago and whose domain of applicability we extend in the present paper.

Loops in de Sitter space

We discuss general one and two-loops banana diagrams with arbitrary masses on the de Sitter spacetime by using direct methods of dS quantum field theory in the dimensional regularization approach. In the one-loop case we also compute the effective potential for an O(N) model in d = 4 dimension as an explicit function of the cosmological constant Λ, both exactly and perturbatively up to order Λ. For the two-loop case we show that the calculation is made easy thanks to a remarkable Källén-Lehmann formula that has been in the literature for a while. We discuss the divergent cases at d = 3 using a contiguity formula for generalized hypergeometric functions and we extract the dominant term at d = 4 proving a general formula to deal with a divergent hypergeometric series.

Cutkosky’s theorem for massive one-loop Feynman integrals: part 1

We formulate and prove Cutkosky’s Theorem regarding the discontinuity of Feynman integrals in the massive one-loop case up to the involved intersection index. This is done by applying the techniques to treat singular integrals developed in Fotiadi et al. (Topology 4(2):159–191, 1965) . We write one-loop integrals as an integral of a holomorphic family of holomorphic forms over a compact cycle. Then, we determine at which points simple pinches occur and explicitly compute a representative of the corresponding vanishing sphere. This also yields an algorithm to compute the Landau surface of a one-loop graph without explicitly solving the Landau equations. We also discuss the bubble, triangle and box graph in detail.

Cuts from residues: the one-loop case

Using the multivariate residue calculus of Leray, we give a precise definition of the notion of a cut Feynman integral in dimensional regularization, as a residue evaluated on the variety where some of the propagators are put on shell. These are naturally associated to Landau singularities of the first type. Focusing on the one-loop case, we give an explicit parametrization to compute such cut integrals, with which we study some of their properties and list explicit results for maximal and next-to-maximal cuts. By analyzing homology groups, we show that cut integrals associated to Landau singularities of the second type are specific combinations of the usual cut integrals, and we obtain linear relations among different cuts of the same integral. We also show that all one-loop Feynman integrals and their cuts belong to the same class of functions, which can be written as parametric integrals.

FDR, an easier way to NNLO calculations: a two-loop case study

In this paper we illustrate the simplifications produced by FDR in NNLO computations. We show with an explicit example that - due to its four-dimensionality - FDR does not require an order-by-order renormalization and that, unlike the one-loop case, FDR and dimensional regularization (DR) generate intermediate two-loop results which are no longer linked by a simple subtraction of the ultraviolet (UV) poles in epsilon. Our case study is the two-loop amplitude for H -> gamma gamma, mediated by an infinitely heavy top loop, in the presence of gluonic corrections. We use this to elucidate how gauge invariance is preserved with no need of introducing counterterms in the Lagrangian. In addition, we discuss a possible four-dimensional approach to the infrared (IR) problem compatible with the FDR treatment of the UV infinities.

Isospin breaking in the phases of the K e4 form factors

Isospin breaking in the K ℓ4 form factors induced by the difference between charged and neutral pion masses is studied. Starting from suitably subtracted dispersion representations, the form factors are constructed in an iterative way up to two loops in the low-energy expansion by implementing analyticity, crossing, and unitarity due to two-meson intermediate states. Analytical expressions for the phases of the two-loop form factors of the K ±→π + π − e ± ν e channel are given, allowing one to connect the difference of form-factor phase shifts measured experimentally (out of the isospin limit) and the difference of S- and P-wave ππ phase shifts studied theoretically (in the isospin limit). The isospin-breaking correction consists of the sum of a universal part, involving only ππ rescattering, and a process-dependent contribution, involving the form factors in the coupled channels. The dependence on the two S-wave scattering lengths $a_{0}^{0}$ and $a_{0}^{2}$ in the isospin limit is worked out in a general way, in contrast to previous analyses based on one-loop chiral perturbation theory. The latter is used only to assess the subtraction constants involved in the dispersive approach. The two-loop universal and process-dependent contributions are estimated and cancel partially to yield an isospin-breaking correction close to the one-loop case. The recent results on the phases of K ±→π + π − e ± ν e form factors obtained by the NA48/2 collaboration at the CERN SPS are reanalysed including this isospin-breaking correction to extract values for the scattering lengths $a_{0}^{0}$ and $a_{0}^{2}$ , as well as for low-energy constants and order parameters of two-flavour χPT.

Direct numerical integration for multi-loop integrals

We present a method to construct a suitable contour deformation in loop momentum space for multi-loop integrals. This contour deformation can be used to perform the integration for multi-loop integrals numerically. The integration can be performed directly in loop momentum space without the introduction of Feynman or Schwinger parameters. The method can be applied to finite multi-loop integrals and to divergent multi-loop integrals with suitable subtraction terms. The algorithm extends techniques from the one-loop case to the multi-loop case. Examples at two and three loops are discussed explicitly.

Relations of loop partial amplitudes in gauge theory by Unitarity cut method

It is well known that under the color-decomposition, one-loop amplitude of gluons contains partial amplitudes of single and double trace structures, and particularly all partial amplitudes of double trace structure can be expressed as a linear combination of partial amplitudes of single trace structure. Using unitarity cut method, we prove that this result is the natural consequence of tree-level Kleiss–Kuijf relation. Generalizing the unitarity cut method to two-loop (triple cut in this case), we show that, unlike the one-loop case, partial amplitudes of double and triple trace structures cannot be expressed as a linear combination of partial amplitudes of leading-color single trace structure. For partial amplitudes of subleading-color single trace structure, we have shown a very non-trivial Kleiss–Kuijf relation for six and seven-point amplitudes, which is one new result of our paper and cannot be obtained by U ( 1 ) -decoupling method. Mysteriously, when we consider the case of eight points, Kleiss–Kuijf relation must be modified for subleading-color single trace partial amplitudes.

On dilatation operator for a renormalizable theory

Given a renormalizable theory we construct the dilatation operator, in the sense of generator of RG flow of composite operators. The generator is found as a differential operator acting on the space of normal symbols of composite operators in the theory. In the spirit of AdS/CFT correspondence, this operator is interpreted as the Hamiltonian of the dual theory. In the case of a field theory with non-abelian gauge symmetry the resulting system is a matrix model. The one-loop case is analyzed in details and it is shown that we reproduce known results from = 4 supersymmetric Yang-Mills theory.

Explicit results for the anomalous three point function and non-renormalization theorems

Two-loop corrections for the <VVA> correlator of the singlet axial and vector currents in QCD are calculated in the chiral limit for arbitrary momenta. Explicit calculations confirm the non-renormalization theorems derived recently by Vainshtein and Knecht et.al. We find that as in the one-loop case also at the two loops the <VVA> correlator has only 3 independent form-factors instead of 4. From the explicit results we observe that the two-loop correction to the correlator is equal to the one-loop result times the constant factor C_2(R) alpha_s/pi in the MSbar scheme. This holds for the full correlator, for the anomalous longitudinal as well as for the non- anomalous thansversal amplitudes. The finite overall alpha_s dependent constant has to be normalized away by renormalizing the axial current according to Witten's algebraic/geometrical constraint on the anomalous Ward identity. Our observations, together with known facts, suggest that in perturbation theory the <VVA> correlator is proportional to the one-loop term to all orders and that the non- renormalization theorem of the Adler-Bell-Jackiw anomaly carries over to the full correlator.

Hard photon production from unsaturated quark–gluon plasma at two-loop level

The hard photon production from bremsstrahlung and annihilation with scattering that arise at two-loop level are estimated for a chemically non-equilibrated quark–gluon plasma in the framework of Hard Thermal Loop (HTL) resummed effective field theory. The rate of photon production is found to be suppressed due to unsaturated phase space compared to equilibrated plasma. For an unsaturated plasma, unlike the effective one-loop case, the reduction in the effective two-loop processes is found to be independent of gluon fugacity, due to an additional collinear enhancement arising from the decrease in thermal quark mass but strongly depends on quark and antiquark fugacities. It is also found that the photon production is dominated by bremsstrahlung mechanism, since the phase space suppression is higher for annihilation with scattering, in contrast to the equilibrated plasma where annihilation with scattering dominates the photon production.

Gauge-independent off-shell fermion self-energies at two loops: The cases of QED and QCD

We use the pinch technique formalism to construct the gauge-independent off-shell two-loop fermion self-energy, both for Abelian (QED) and non-Abelian (QCD) gauge theories. The new key observation is that all contributions originating from the longitudinal parts of gauge boson propagators, by virtue of the elementary tree-level Ward identities they trigger, give rise to effective vertices, which do not exist in the original Lagrangian; all such vertices cancel diagrammatically inside physical quantities, such as current correlation functions or S-matrix elements. We present two different, but complementary derivations: First, we explicitly track down the aforementioned cancellations inside two-loop diagrams, resorting to nothing more than basic algebraic manipulations. Second, we present an absorptive derivation, exploiting the unitarity of the S-matrix, and the Ward identities imposed on tree-level and one-loop physical amplitudes by gauge invariance, in the case of QED, or by the underlying Becchi-Rouet-Stora symmetry, in the case of QCD. The propagator-like sub-amplitude defined by means of this latter construction corresponds precisely to the imaginary parts of the effective self-energy obtained in the former case; the real part may be obtained from a (twice subtracted) dispersion relation. As in the one-loop case, the final two-loop fermion self-energy constructed using either method coincides with the conventional fermion self-energy computed in the Feynman gauge.

NONPERTURBATIVE DISPERSIVE SECTOR IN STRONG (QUASI-)ABELIAN FIELDS

In strong (quasi-)Abelian fields, even at the one-loop level of the coupling constant, quantum fluctuations of fermions induce an effective Lagrangian density whose imaginary (absorptive) part is purely nonperturbative and known to be responsible for the fermion–antifermion pair creation. On the other hand, the induced real (dispersive) part has perturbative and nonperturbative contributions. In the one-loop case, we argue how to separate the two contributions from each other for any strength of the field. We show numerically that the nonperturbative contributions are in general comparable with or larger than the induced perturbative ones. We arrive at qualitatively similar conclusions also for the induced energy density. Further, we investigate numerically the quasianalytic continuation of the perturbative results into the nonperturbative sector, by employing (modified) Borel–Padé. It turns out that in the case at hand, we have to integrate over renormalon singularities, but there is no renormalon ambiguity involved.

Nonperturbative dispersive sector in strong (quasi-)Abelian fields

In strong (quasi-)Abelian fields, even at the one-loop level of the coupling constant, quantum fluctuations of fermions induce an effective Lagrangian density whose imaginary (absorptive) part is purely nonperturbative and known to be responsible for the fermion-antifermion pair creation. On the other hand, the induced real (dispersive) part has perturbative and nonperturbative contributions. In the one-loop case, we argue how to separate the two contributions from each other for any strength of the field. We show numerically that the nonperturbative contributions are in general comparable with or larger than the induced perturbative ones. We arrive at qualitatively similar conclusions also for the induced energy density. Further, we investigate numerically the quasianalytic continuation of the perturbative results into the nonperturbative sector, by employing (modified) Borel-Pade. It turns out that in the case at hand, we have to integrate over renormalon singularities, but there is no renormalon ambiguity involved.

Two-loop Euler–Heisenberg QED pair-production rate

We study the divergence of large-order perturbation theory in the worldline expression for the two-loop Euler–Heisenberg QED effective Lagrangian in a constant magnetic field. The leading rate of divergence is identical, up to an overall factor, to that of the one-loop case. From this we deduce, using Borel summation techniques, that the leading behaviour of the imaginary part of the two-loop effective Lagrangian for a constant E field, giving the pair-production rate, is proportional to the one-loop result. This also serves as a test of the mass renormalization, and confirms the earlier analysis by Ritus.

SCALAR AND VECTOR MIXINGS IN SU(2) × U(1) MODELS

We study the possible mixings between gauge vector fields and scalar fields through their self-energies, arising in models with two Higgs doublets. We derive the relevant set of Schwinger–Dyson equations and the Ward identities that compel the longitudinal parts of the field propagators. A linear Rξ gauge is used and the results are given at all orders in perturbative theory, and some particular aspects of the one-loop case are stressed.

Loop corrections in the spectrum of two-dimensional Hawking radiation

We determine the one- and two-loop backreaction corrections in the spectrum of the Hawking radiation for the Callan - Giddings - Harvey - Strominger (CGHS) model of two-dimensional dilaton gravity by evaluating the Bogoliubov coefficients for a massless scalar field propagating on the corresponding backgrounds. Since the backreaction can induce a small shift in the position of the classical horizon, we find that a positive shift leads to a non-Planckian late-time spectrum, while a null or a negative shift leads to a Planckian late-time spectrum in the leading-order stationary-point approximation. In the one-loop case there are no corrections to the classical Hawking temperature, while in the two-loop case the temperature is three times greater than the classical value. We argue that these results are consistent with the behaviour of the Hawking flux obtained from the operator quantization only for times which are not too late, in accordance with the limits of validity of the semiclassical approximation.

SELF-CONSISTENT SOLUTION OF THE SIMULTANEOUS SCHWINGER-DYSON EQUATION IN STRONG COUPLING QED

In the strong coupling region of QED, we have obtained a numerical solution to the simultaneous Schwinger-Dyson equation for the fermion and the photon propagators. It is shown that there is a critical point separating the spontaneous-chiral-symmetry-breaking (strong coupling) phase and the weak coupling phase. This critical point is consistently interpreted as the second order phase transition point at which the continuum limit of cutoff QED may be taken. The vacuum polarization function obtained in this framework exhibits essentially the same asymptotic uv behavior as that predicted from the one-loop calculation. The scaling behavior is not inconsistent with the mean-field result as predicted from the one-loop case, which favors the triviality of QED.

Schwinger-dyson equation approach to strong coupling QED

In the strong coupling region of QED, we have tried to obtain a solution to the simultaneous Schwinger-Dyson equation for the fermion and the photon propagators. It is shown that there is a critical point separating the spontaneous-chiral-symmetry-breaking (strong coupling) phase and the weak coupling phase. This critical point is consistently interpreted as the second order phase transition point at which the continuum limit of cutoff QED may be taken. The vacuum polarization function obtained numerically in this framework exhibits essentially the same asymptotic ultraviolet behavior as that predicted from the one-loop calculation. The scaling behavior is not inconsistent with the mean-field result as predicted from the one-loop case, which favors the triviality of QED.

GENUS TWO FREE ENERGY FOR HETEROTIC STRINGS

We give an explicit expression for the genus two contribution to the thermal free energy for the heterotic string using Knizhnik's formula for the cosmological constant, and, by analyzing its divergencies we find the critical temperature, which is the same as in the one-loop case.