Results of a study of the  $$K^+ \rightarrow \pi ^{0} e^{+} \nu \gamma $$  <math> <mrow> <msup> <mi>K</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mn>0</mn> </msup> <msup> <mi>e</mi> <mo>+</mo> </msup> <mi>ν</mi> <mi>γ</mi> </mrow> </math>   decay at OKA setup are presented. More than 32,000 events of this decay are observed. The differential spectra over the photon energy and the photon–electron opening angle in kaon rest frame are presented. The branching ratios, normalized to that of  $$K_{e3}$$  <math> <msub> <mi>K</mi> <mrow> <mi>e</mi> <mn>3</mn> </mrow> </msub> </math>   decay are calculated for different cuts on  $$E^*_\gamma $$  <math> <msubsup> <mi>E</mi> <mi>γ</mi> <mo>∗</mo> </msubsup> </math>   and  $$cos\Theta ^{*}_{e\gamma }$$  <math> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <msubsup> <mi>Θ</mi> <mrow> <mi>e</mi> <mi>γ</mi> </mrow> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msubsup> </mrow> </math>  . In particular, the branching ratio for  $$E^{*}_{\gamma }&gt;30$$  <math> <mrow> <msubsup> <mi>E</mi> <mi>γ</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msubsup> <mo>&gt;</mo> <mn>30</mn> </mrow> </math>   MeV and  $$\Theta ^{*}_{e \gamma }&gt;20^{\circ }$$  <math> <mrow> <msubsup> <mi>Θ</mi> <mrow> <mi>e</mi> <mi>γ</mi> </mrow> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msubsup> <mo>&gt;</mo> <msup> <mn>20</mn> <mo>∘</mo> </msup> </mrow> </math>   is measured R =  $$\frac{Br(K^+ \rightarrow \pi ^{0} e^{+} \nu _{e} \gamma ) }{Br(K^+ \rightarrow \pi ^{0} e^{+} \nu _{e})} $$  <math> <mfrac> <mrow> <mi>B</mi> <mi>r</mi> <mo>(</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mn>0</mn> </msup> <msup> <mi>e</mi> <mo>+</mo> </msup> <msub> <mi>ν</mi> <mi>e</mi> </msub> <mi>γ</mi> <mo>)</mo> </mrow> <mrow> <mi>B</mi> <mi>r</mi> <mo>(</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mn>0</mn> </msup> <msup> <mi>e</mi> <mo>+</mo> </msup> <msub> <mi>ν</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> </mfrac> </math>   = = (0.587±0.010(stat.)±0.015(syst.)) $$\times 10^{-2}$$  <math> <mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math>  , which is in a good agreement with ChPT  $$O(p^{4})$$  <math> <mrow> <mi>O</mi> <mo>(</mo> <msup> <mi>p</mi> <mn>4</mn> </msup> <mo>)</mo> </mrow> </math>   calculations.

Russia

A. Polyarush

S. Akimenko

A. Artamonov

A. Blik

V. Burtovoy

V. Bychkov

S. Donskov

Italy

V. Duk

A. Filin

S. Filippov

A. Gorin

E. Gushchin

A. Inyakin

G. Kekelidze

G. Khaustov

S. Kholodenko

A. Khudyakov

V. Kolosov

V. Kravtsov

Yu. Kudenko

V. Kurshetsov

V. Lishin

V. Lysan

M. Medynsky

Yu. Mikhailov

V. Obraztsov

V. Polyakov

V. Romanovsky

V. Rykalin

A. Sadovsky

V. Samoilenko

M. Shapkin

O. Stenyakin

O. Tchikilev

V. Uvarov

O. Yushchenko

B. Zalikhanov

Study of  $$K^+ \rightarrow \pi ^{0} e^{+} \nu \gamma $$  <math> <mrow> <msup> <mi>K</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mn>0</mn> </msup> <msup> <mi>e</mi> <mo>+</mo> </msup> <mi>ν</mi> <mi>γ</mi> </mrow> </math>   decay with OKA setup

A precise measurement of the vector and axial-vector form factors difference $$F_V-F_A$$ <math><mrow><msub><mi>F</mi><mi>V</mi></msub><mo>-</mo><msub><mi>F</mi><mi>A</mi></msub></mrow></math>  in the $$K^+\rightarrow {\mu ^+}{\nu _{\mu }}{\gamma }$$ <math><mrow><msup><mi>K</mi><mo>+</mo></msup><mo>→</mo><msup><mi>μ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>μ</mi></msub><mi>γ</mi></mrow></math>  decay is presented. About 95K events of $$K^+\rightarrow {\mu ^+}{\nu _{\mu }}{\gamma }$$ <math><mrow><msup><mi>K</mi><mo>+</mo></msup><mo>→</mo><msup><mi>μ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>μ</mi></msub><mi>γ</mi></mrow></math>  are selected in the OKA experiment. The result is $$F_V-F_A=0.134\pm 0.021(stat)\pm 0.027(syst)$$ <math><mrow><msub><mi>F</mi><mi>V</mi></msub><mo>-</mo><msub><mi>F</mi><mi>A</mi></msub><mo>=</mo><mn>0.134</mn><mo>±</mo><mn>0.021</mn><mrow><mo>(</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>t</mi><mo>)</mo></mrow><mo>±</mo><mn>0.027</mn><mrow><mo>(</mo><mi>s</mi><mi>y</mi><mi>s</mi><mi>t</mi><mo>)</mo></mrow></mrow></math> . Both errors are smaller than in the previous $$F_V-F_A$$ <math><mrow><msub><mi>F</mi><mi>V</mi></msub><mo>-</mo><msub><mi>F</mi><mi>A</mi></msub></mrow></math>  measurements.

United Kingdom

A. Popov

V. Semenov

Measurement of the $$K^+\rightarrow {\mu ^+}{\nu _{\mu }}{\gamma }$$ <math><mrow><msup><mi>K</mi><mo>+</mo></msup><mo>→</mo><msup><mi>μ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>μ</mi></msub><mi>γ</mi></mrow></math>  decay form factors in the OKA experiment

A high statistics data sample of the decays of $$K^+$$ <math><msup><mi>K</mi><mo>+</mo></msup></math>  mesons to three charged particles was accumulated by the OKA experiment in 2012 and 2013. This allowed to select a clean sample of about 450 events with $$K^{+}\rightarrow \pi ^{+}\pi ^{-}\pi ^{+}\gamma $$ <math><mrow><msup><mi>K</mi><mo>+</mo></msup><mo>→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>-</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mi>γ</mi></mrow></math>  decays with the energy of the photon in the kaon rest frame greater than 30 MeV. The measured branching fraction of the $$K^{+}\rightarrow \pi ^{+}\pi ^{-}\pi ^{+}\gamma $$ <math><mrow><msup><mi>K</mi><mo>+</mo></msup><mo>→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>-</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mi>γ</mi></mrow></math> , with $$E_{\gamma }^{*} &gt; 30\ \hbox {MeV}$$ <math><mrow><msubsup><mi>E</mi><mrow><mi>γ</mi></mrow><mrow><mrow></mrow><mo>∗</mo></mrow></msubsup><mo>&gt;</mo><mn>30</mn><mspace width="4pt"></mspace><mtext>MeV</mtext></mrow></math>  is equal to $$(0.71 \pm 0.05) \times 10^{-5}$$ <math><mrow><mrow><mo>(</mo><mn>0.71</mn><mo>±</mo><mn>0.05</mn><mo>)</mo></mrow><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>5</mn></mrow></msup></mrow></math> . The measured differential branching fraction over photon energy is compared with the prediction of the chiral perturbation theory to $${\mathcal {O}}(p^{4})$$ <math><mrow><mi>O</mi><mo>(</mo><msup><mi>p</mi><mn>4</mn></msup><mo>)</mo></mrow></math> . A search for an up-down asymmetry of the photon with respect to the hadronic system decay plane is also performed.

A. Konstantinov

Study of the decay $$K^{+}\rightarrow \pi ^{+}\pi ^{-}\pi ^{+}\gamma $$ <math><mrow><msup><mi>K</mi><mo>+</mo></msup><mo>→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>-</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mi>γ</mi></mrow></math>  in the OKA experiment

We study the finite density phase transition in the lattice QCD at real chemical potential. We adopt a canonical approach and the canonical partition function is constructed for N f = 2 QCD. After derivation of the canonical partition function we calculate observables like the pressure, the quark number density, its second cumulant and the chiral condensate as a function of the real chemical potential. We covered a wide range of temperature region starting from the confining low to the deconfining high temperature; 0 . 65 T c ≤ T ≤ 3 . 62 T c . We observe a possible signal of the deconfinement and the chiral restoration phase transition at real chemical potential below T c starting from the confining phase. We give also the convergence range of the fugacity expansion.