Following previous work we further explore the possibility that the chirally broken phase of gauge theories admits an infrared fixed point interpretation. The slope of the <math><mrow><mi>β</mi></mrow></math> function, <math><msubsup><mi>β</mi><mo>*</mo><mo>′</mo></msubsup></math>, is found to vanish at the infrared fixed point which has several attractive features such as logarithmic running. We provide a more in-depth analysis of our previous result that the mass anomalous assumes <math><msub><mi>γ</mi><mo>*</mo></msub><mo>=</mo><mn>1</mn></math> at the fixed point. The results are found to be consistent with <math><mi>N</mi><mo>=</mo><mn>1</mn></math> supersymmetric gauge theories. In a second part the specific properties of a dilaton, the (pseudo-) Goldstone, due to spontaneous symmetry breaking are investigated. Dilaton soft theorems indicate that a soft dilaton mass can only originate from an operator of scaling dimension two. In the gauge theory this role is taken on by the <math><mover><mi>q</mi><mo>¯</mo></mover><mi>q</mi></math>-operator. The quantum chromodynamics (QCD) dilaton candidate, the <math><mi>σ</mi><mo>=</mo><msub><mi>f</mi><mn>0</mn></msub><mo>(</mo><mn>500</mn><mo>)</mo></math> meson is investigated and the singlet-octet mixing is found to be relevant. We briefly discuss the dilaton as a candidate for the Higgs boson, which relies on the ratio of dilaton to pion decay constant being close to unity. In QCD this is approximately satisfied but it is remains unclear if this is accidental or whether there is yet to be uncovered principle behind it.

CERN

Roman Zwicky

QCD with an infrared fixed point and a dilaton

Based on explicitly gauge invariant interpolating operators we compute complete next-leading order QED-corrections for leptonic decays. These are sizeable since the helicity-suppression in V-A interactions allows for structure-dependent collinear logs. We have explicitly checked that these logs are absent for helicity-unsuppressed Yukawa-type transitions. Based on B → γ form factors we present the rates for   <math> <msup> <mi>B</mi> <mo>−</mo> </msup> <mo>→</mo> <mfenced> <msup> <mi>μ</mi> <mo>−</mo> </msup> <msup> <mi>τ</mi> <mo>−</mo> </msup> </mfenced> <mover> <mi>ν</mi> <mo>¯</mo> </mover> <mfenced> <mi>γ</mi> </mfenced> </math>  $$ {B}^{-}\to \left({\mu}^{-},{\tau}^{-}\right)\overline{\nu}\left(\gamma \right) $$  in differential and integrated form as a function of the photon energy cut-off   <math> <msubsup> <mi>E</mi> <mi>γ</mi> <mi>cut</mi> </msubsup> </math>  $$ {E}_{\gamma}^{\textrm{cut}} $$ . The effect of the virtual structure-dependent corrections are approximately +5% and +3% for the μ- and τ-channel respectively. The structure dependence of the real radiation exceeds that of the virtual one for   <math> <msub> <mfenced> <msubsup> <mi>E</mi> <mi>γ</mi> <mi>cut</mi> </msubsup> </mfenced> <mi>μ</mi> </msub> </math>  $$ {\left.{E}_{\gamma}^{\textrm{cut}}\right|}_{\mu } $$  &gt; 0.18(3) GeV and is subdominant for the tau channel even when fully inclusive.

United Kingdom

Matthew Rowe

Structure-dependent QED in   <math> <msup> <mi>B</mi> <mo>−</mo> </msup> <mo>→</mo> <msup> <mi>ℓ</mi> <mo>−</mo> </msup> <mover> <mi>ν</mi> <mo>¯</mo> </mover> <mfenced> <mi>γ</mi> </mfenced> </math>  $$ {B}^{-}\to {\ell}^{-}\overline{\nu}\left(\gamma \right) $$

We derive constraints on the four dimensional energy-momentum tensor from gravitational and gauge anomalies. Our work can be considered an extension of Duff’s analysis [1] to include parity-odd terms and explicit symmetry breaking. The constraints imply the absence of the parity-odd   <math> <mi>R</mi> <mover> <mi>R</mi> <mo>~</mo> </mover> </math>  $$ R\overset{\sim }{R} $$  and   <math> <mi>F</mi> <mover> <mi>F</mi> <mo>~</mo> </mover> </math>  $$ F\overset{\sim }{F} $$  terms, for theories whose symmetries are compatible with dimensional regularisation, in a model-independent way. Remarkably, even in the case of explicit symmetry breaking the □R-anomaly is found to be finite and unambiguous after applying the symmetry constraints. We compute mixed gravity-gauge anomalies at leading order and deduce phenomenological consequences for vector bosons associated with global chiral symmetries.

France

Rémy Larue

Jérémie Quevillon

Gravity-gauge anomaly constraints on the energy-momentum tensor

The possibility that gauge theories with chiral symmetry breaking below the conformal window exhibit an infrared fixed point is explored. With this assumption three aspects of pion physics are reproduced if the quark mass anomalous dimension at the infrared fixed point is <math><msub><mi>γ</mi><mo>*</mo></msub><mo>=</mo><mn>1</mn></math>. First, by matching the long-distance scalar adjoint correlation function. Second, by perturbing the fixed point by a small quark mass, the <math><msub><mi>m</mi><mi>q</mi></msub></math>-dependence of the pion mass is reproduced by renormalization group arguments. Third, consistency of the trace anomaly and the Feynman-Hellmann theorem, for small <math><msub><mi>m</mi><mi>q</mi></msub></math>, imply the same result once more. This suggests the following picture for the conformal window; close to its upper boundary <math><msub><mi>γ</mi><mo>*</mo></msub></math> is zero and grows as the number of fermions is reduced until its lower boundary <math><msub><mi>γ</mi><mo>*</mo></msub><mo>=</mo><mn>1</mn></math> is reached, where chiral symmetry breaking sets in. Below, the strongly coupled gauge theory with <math><msub><mi>γ</mi><mo>*</mo></msub><mo>=</mo><mn>1</mn></math> is infrared dual to the free theory of pions. A possible dilaton sector of the scenario will be addressed in a companion paper.

QCD with an infrared fixed point: The pion sector

In this paper we show that <math><msubsup><mi>β</mi><mo>*</mo><mo>′</mo></msubsup></math>, the <math><mi>β</mi></math>-function slopes in the electric and magnetic theories, are equal at the corresponding infrared fixed points. This follows from the scaling of the correlators of the trace of the energy momentum tensors. The slopes <math><msubsup><mi>β</mi><mo>*</mo><mo>′</mo></msubsup></math> determine the scaling dimensions. Our paper can be considered as a commentary to D. Anselmi, M. T. Grisaru, and A. Johansen [Nucl. Phys. B491, 221 (1997)]; it proposes an improved derivation not based on a rather contrived construction by D. Kutasov [Phys. Lett. B 351, 230 (1995)], D. Kutasov and A. Schwimmer [Phys. Lett. B 354, 315 (1995)], and D. Kutasov, A. Schwimmer, and N. Seiberg, [Nucl. Phys. B459, 455 (1996)]. As a byproduct we note that <math><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow><mrow><mo>′</mo></mrow></msubsup></mrow></math>—the slopes of the matter superfield anomalous dimension—vanish at both edges of the conformal window where one of the dual theories is strongly coupled. Finally, we determine the two-coupling magnetic fixed point at weak coupling correcting the result of I. I. Kogan, M. A. Shifman, and A. I. Vainshtein [Phys. Rev. D 53, 4526 (1996); Phys. Rev. D59, 109903(E) (1999)].

United States

Mikhail Shifman

Relating <math><mrow><msubsup><mrow><mi>β</mi></mrow><mrow><mo>*</mo></mrow><mrow><mo>′</mo></mrow></msubsup></mrow></math> and <math><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>Q</mi><mo>*</mo></mrow><mrow><mo>′</mo></mrow></msubsup></mrow></math> in the <math><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow></math> SQCD conformal window

We compute the trace, diffeomorphism and Lorentz anomalies of a free Weyl fermion in a gravitational background field by path integral methods. This is achieved by regularising the variation of the determinant of the Weyl operator building on earlier work by Leutwyler. The trace anomaly is found to be one half of the one of a Dirac fermion. Most importantly we establish that the potential parity-odd curvature term   <math> <mi>R</mi> <mover> <mi>R</mi> <mo>~</mo> </mover> </math>  $$ R\overset{\sim }{R} $$ , corresponding to the Pontryagin density, vanishes. This is to the contrary of some recent findings in the literature which gave rise to a controversy. We verify, that the regularisation does not lead to (spurious) anomalies in the Lorentz and diffeomorphism symmetries. We argue that in d = 2 (mod 4) P- and CP-odd terms cannot appear and that for d = 4 (mod 4) they are absent at least at leading order.

Trace anomaly of weyl fermions via the path integral

We compute the electromagnetic mass difference for the B-, D- and K-mesons using QCD sum rules with double dispersion relations. For the B- and D-mesons we also compute the linear quark mass correction, whereas for the K the standard soft theorems prove more powerful. The mass differences, which have not previously been computed via a double dispersion, are fully consistent with experiment, albeit with large uncertainties.

Isospin mass differences of the B, D and K

Standard interpolating operators for charged mesons, e.g. J  B  =   <math> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ \overline{b} $$  iγ 5 u for B  − , are not gauge invariant in QED and therefore problematic for perturbative methods. We propose a gauge invariant interpolating operator by adding an auxiliary charged scalar Φ B ,   <math> <msubsup> <mi>J</mi> <mi>B</mi> <mfenced> <mn>0</mn> </mfenced> </msubsup> </math>  $$ {\mathcal{J}}_B^{(0)} $$  = J  B  Φ B , which reproduces all the universal soft and collinear logs. The modified LSZ-factor is shown to be infrared finite which is a necessary condition for validating the approach. At   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (α), this is equivalent to a specific Dirac dressing of charged operators. A generalisation thereof, using iterated integrals, establishes the equivalence to all orders and provides a transparent alternative viewpoint. The method is discussed by the example of the leptonic decay B  −  → ℓ  −    <math> <mover> <mi>ν</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\nu} $$  for which a numerical study is to follow. The formalism itself is valid for any spin, flavour and set of final states (e.g. B  −  → π 0 ℓ  −    <math> <mover> <mi>ν</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\nu} $$ ).

Saad Nabeebaccus

Resolving charged hadrons in QED — gauge invariant interpolating operators

Using analytic results obtained in a meson effective theory, that includes all infrared sensitive logs, we build a dedicated Monte Carlo framework to describe QED corrections in   <math> <mover> <mi>B</mi> <mo>¯</mo> </mover> </math>  $$ \overline{B} $$  →   <math> <mover> <mi>K</mi> <mo>¯</mo> </mover> </math>  $$ \overline{K} $$  ℓ + ℓ  −  for a generic form factor. For the neutral mode   <math> <msup> <mover> <mi>B</mi> <mo>¯</mo> </mover> <mn>0</mn> </msup> </math>  $$ {\overline{B}}^0 $$  →   <math> <msup> <mover> <mi>K</mi> <mo>¯</mo> </mover> <mn>0</mn> </msup> </math>  $$ {\overline{K}}^0 $$  ℓ + ℓ  − , we perform a detailed numerical comparison versus those obtained with the general-purpose photon-shower tool PHOTOS. The comparison indicates a good agreement, at the few per-mil level, when focusing on the rare mode only. In addition, our framework allows us to investigate the impact of the charmonium resonances. Interference effects, not described by PHOTOS in the experimental analysis, are found to be small in the dilepton invariant mass region q 2 &lt; 6GeV2, which is used to determine   <math> <msub> <mi>R</mi> <msup> <mi>K</mi> <mfenced> <mo>∗</mo> </mfenced> </msup> </msub> </math>  $$ {R}_{K^{\left(\ast \right)}} $$ . Using a semi-analytic framework we assess the full, rare and resonant, mode. Based thereupon, we discuss strategies to check the subtraction of the resonant mode, which has a sizeable impact at q 2 ≈ 6GeV2 in the electron mode.

Switzerland

Gino Isidori

Davide Lancierini

QED in   <math> <mover> <mi>B</mi> <mo>¯</mo> </mover> </math>  $$ \overline{B} $$  →   <math> <mover> <mi>K</mi> <mo>¯</mo> </mover> </math>  $$ \overline{K} $$  ℓ + ℓ  −  LFU ratios: theory versus experiment, a Monte Carlo study

<math><mrow><mi>C</mi><mi>P</mi></mrow></math> violation in <math><mi>Δ</mi><msub><mi>A</mi><mrow><mi>CP</mi></mrow></msub><mo>=</mo><mo>−</mo><mn>0.154</mn><mo>(</mo><mn>29</mn><mo>)</mo></math>, in the <math><msup><mi>D</mi><mn>0</mn></msup><mo>→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo>/</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>K</mi><mo>−</mo></msup></math> system, is established, and its central value is 1 order of magnitude above the naive Standard Model (SM) estimate. It remains unclear whether this is due to currently incalculable strong interaction matrix elements or genuine new physics, such as a shift in <math><msub><mi>O</mi><mn>8</mn></msub></math> with a weak phase. We show that interference of the long-distance (LD) terms with the <math><msub><mi>O</mi><mn>8</mn></msub></math> matrix element can give rise to <math><msubsup><mi>A</mi><mrow><mi>C</mi><mi>P</mi></mrow><mrow><mi>D</mi><mo>→</mo><mi>V</mi><mi>γ</mi></mrow></msubsup><mo>=</mo><mi>few</mi><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math> (for reference values <math><mi>Im</mi><mo>[</mo><msubsup><mi>C</mi><mn>8</mn><mrow><mi>NP</mi></mrow></msubsup><mo>]</mo><mo>≈</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math>). In addition, it is pointed out that the ratio of left- to right-handed (photon polarization) LD amplitudes is measurable in time-dependent <math><mrow><mi>C</mi><mi>P</mi></mrow></math> asymmetries. We argue that both theory and experimental consideration favor weak annihilation (WA) as the dominant LD contribution. More definite progress could be achieved by either computing the radiative corrections to WA or the measurement of the charged modes <math><msubsup><mi>D</mi><mrow><mo>(</mo><mi>d</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow><mo>+</mo></msubsup><mo>→</mo><mo>(</mo><mi>ρ</mi><mo>,</mo><msup><mi>K</mi><mo>*</mo></msup><msup><mo>)</mo><mo>+</mo></msup><mi>γ</mi></math> and <math><msub><mi>D</mi><mi>s</mi></msub><mo>→</mo><msup><mi>ρ</mi><mo>+</mo></msup><mi>γ</mi></math>.

James Lyon

<math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>CP</mi></mrow></msub><mo>[</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mrow><mn>0</mn><mo>,</mo><mo>+</mo></mrow></msubsup><mo>→</mo><mi>V</mi><mi>γ</mi><mo>]</mo></mrow></math> from a large <math><msub><mi>O</mi><mn>8</mn></msub></math> contribution