Construction of the pion scalar form factor from few poles and zero - PowerPoint PPT Presentation

Construction of the pion scalar form factor from few poles and zero Robert Kami´ nski IFJ PAN, Kraków Stanislav Dubnicka, Andrej Liptaj Institute of Physics, Slovak Academy of Sciences, Bratislava Anna Zuzana Dubnickova Department of Theoretical Physics, Comenius University, Bratislava Meson2016, Kraków

Pion scalar form factor < π i ( p 2 ) | � uu + ¯ dd ) | π j ( p 1 ) > = δ ij Γ π ( t ) m (¯ t = ( p 2 − p 1 ) 2 and m = 1 where � 2 ( m u + m d ) . It posses all known properties of the pion vector electromagnetic FF F π ( t ) like • analyticity in t -plane beside cuts on the positive real axis from two-pion threshold t = 4 m 2 π to + ∞ • elastic unitarity condition Im Γ π ( t ) = Γ π ( t ) e − i δ 0 0 sin δ 0 0 , where δ 0 0 ( t ) is the S -wave isoscalar ππ phase shift • asymptotic behavior Γ π ( t ) | t |→∞ ∼ 1 t • reality condition Γ ∗ π ( t ) = Γ π ( t ∗ ) • normalization, however, now to the pion sigma term value Γ( 0 ) = ( 0 . 99 ± 0 . 02 ) m 2 π to be predicted by the χ PT , ∂ M 2 ∂ M 2 Γ π ( 0 ) = m u ∂ m u + m d π π ∂ m d

Earlier analyses • T. N. Truong and R. S. Willey, Phys. Rev. D40 (1989) 3635, • J. F. Donoghue, J. Gasser and H. Leutwyler, Nucl. Phys. B343, 341 (1990), • B. Moussallam, Eur. Phys. J. C14 (2000) 111, • G. Colangelo, J. Gasser and H. Leutwyler, Nucl. Phys. B603 (2001) 125, • "Scalar form factors of light mesons" ; B. Ananthanaryan, I. Caprini, G. Colangelo, J. Gasser, H. Leutwyler, PLB’2004, • "The Quadratic scalar radius of the pion and the mixed π − K radius" , F. J. Yndurain, Phys. Lett. B578, 99 (2004)

Our (Bratislava-Kraków) analysis -the method � ∞ � t t ′ ( t ′ − t ) dt ′ � δ 0 0 ( t ′ ) Γ π ( t ) = P n ( t ) exp , (1) π 4 m 2 π tan δ Γ ( t ) = A 1 q + A 3 q 3 + A 5 q 5 + A 7 q 7 + ... (2) 1 + A 2 q 2 + A 4 q 4 + A 6 q 6 + ... � q ′ ln ( 1 + A 2 q ′ 2 + A 4 q ′ 4 )+ i ( A 1 q ′ + A 3 q ′ 3 + A 5 q ′ 5 ) � � ∞ ( q 2 + 1 ) ( 1 + A 2 q ′ 2 + A 4 q ′ 4 ) − i ( A 1 q ′ + A 3 q ′ 3 + A 5 q ′ 5 ) dq ′ Γ π ( t ) = P n ( t ) exp × , (3) ( q ′ 2 + 1 )( q ′ 2 − q 2 ) 2 π i −∞ � M � n = 0 a n q n � φ ( q ′ ) dq ′ = 2 π i Γ π ( q ) = , Res n (4) � N i = 1 ( q − q i ) n = 1 explicit form of the pion scalar FF: ( q + q 2 )( q + q 3 )( q + q 4 )( q + q 5 ) × ( i + q 2 )( i + q 3 )( i + q 4 )( i + q 5 ) ( q − q 1 ) Γ π ( t ) = P n ( t ) ( i − q 1 ) where P n ( t ) is any polynomial normalised at t = 0 to one.

First results (2014) PHYSICAL REVIEW D 90, 114003 (2014) "Pion scalar form factor and another confirmation of the existence of the f 0 ( 500 ) meson" Results: PDG: f 0 ( 500 ) : 388 ± 23 − i ( 301 ± 33 ) MeV 400 − 550 − i ( 200 − 350 ) MeV f 0 ( 980 ) : 1066 ± 142 − i ( 110 ± 97 ) MeV 990 ± 20 − i ( 25 − 50 ) MeV

New results (2015/2016); new data for δ ππ 200 200 S-wave isoscalar ππ phase shifts S-wave isoscalar ππ phase shifts 6 6 6 6 150 150 4 4 4 4 | Γ (t)| | Γ (t)| | Γ (t)| | Γ (t)| 100 100 2 2 2 2 50 50 0 0 0 0 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 -3 -3 -3 -3 -2 -2 -2 -2 -1 -1 -1 -1 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 t (GeV 2 ) t (GeV 2 ) t (GeV 2 ) t (GeV 2 ) t (GeV 2 ) t (GeV 2 ) Results: PDG: f 0 ( 500 ) : 467 ± 22 − i ( 257 ± 30 ) MeV 400 − 550 − i ( 200 − 350 ) MeV f 0 ( 980 ) : 983 ± 76 − i ( 48 ± 25 ) MeV 990 ± 20 − i ( 25 − 50 ) MeV

New results (2015/2016); poles tan δ Γ ( t ) = A 1 q + A 3 q 3 + A 5 q 5 + A 7 q 7 + ... 1 + A 2 q 2 + A 4 q 4 + A 6 q 6 + ... fit to the output fit to the input A 1 0 . 220 fixed 0 . 220 fixed 0 . 151 ± 0 . 020 0 . 128 ± 0 . 016 A 3 A 5 − 0 . 0144 ± 0 . 0016 − 0 . 0125 ± 0 . 0013 A 2 − 0 . 055 ± 0 . 038 − 0 . 085 ± 0 . 023 A 4 − 0 . 0089 ± 0 . 0048 − 0 . 0051 ± 0 . 0029 f 0 ( 500 ) 451 ± 14 − i 260 ± 33 467 ± 14 − i 261 ± 29 f 0 ( 980 ) 988 ± 81 − i 52 ± 32 987 ± 76 − i 47 ± 25 ( q + q 2 )( q + q 3 )( q + q 4 )( q + q 5 ) × ( i + q 2 )( i + q 3 )( i + q 4 )( i + q 5 ) ( q − q 1 ) Γ π ( t ) = P n ( t ) ( i − q 1 ) q 1 = 0 . 00 − i 1 . 943 ± 0 . 20 q 2 = 3 . 397 ± 0 . 40 + i 0 . 196 ± 0 . 04 q 3 = − 3 . 397 ± 0 . 40 + i 0 . 196 ± 0 . 04 q 4 = 1 . 385 ± 0 . 10 + i 1 . 085 ± 0 . 12 q 5 = − 1 . 385 ± 0 . 10 + i 1 . 085 ± 0 . 12

New results (2015/2016); analytical structure of the Γ( s ) q 1 = 0 . 00 − i 1 . 943 ± 0 . 20 q 2 = 3 . 397 ± 0 . 40 + i 0 . 196 ± 0 . 04 q 3 = − 3 . 397 ± 0 . 40 + i 0 . 196 ± 0 . 04 q 4 = 1 . 385 ± 0 . 10 + i 1 . 085 ± 0 . 12 q 5 = − 1 . 385 ± 0 . 10 + i 1 . 085 ± 0 . 12

New results (2015/2016) 200 200 S-wave isoscalar ππ phase shifts S-wave isoscalar ππ phase shifts 6 6 6 6 150 150 4 4 4 4 | Γ (t)| | Γ (t)| | Γ (t)| | Γ (t)| 100 100 2 2 2 2 50 50 0 0 0 0 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 -3 -3 -3 -3 -2 -2 -2 -2 -1 -1 -1 -1 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 t (GeV 2 ) t (GeV 2 ) t (GeV 2 ) t (GeV 2 ) t (GeV 2 ) t (GeV 2 ) Results: PDG: f 0 ( 500 ) : 467 ± 14 − i ( 261 ± 29 ) MeV 400 − 550 − i ( 200 − 350 ) MeV f 0 ( 980 ) : 987 ± 76 − i ( 47 ± 25 ) MeV 990 ± 20 − i ( 25 − 50 ) MeV

New results (2015/2016) 200 200 S-wave isoscalar ππ phase shifts S-wave isoscalar ππ phase shifts 6 6 6 6 150 150 4 4 4 4 | Γ (t)| | Γ (t)| | Γ (t)| | Γ (t)| 100 100 2 2 2 2 50 50 0 0 0 0 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 -3 -3 -3 -3 -2 -2 -2 -2 -1 -1 -1 -1 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 t (GeV 2 ) t (GeV 2 ) t (GeV 2 ) t (GeV 2 ) t (GeV 2 ) t (GeV 2 ) Results: GKPY eqs: f 0 ( 500 ) : 467 ± 14 − i ( 261 ± 29 ) MeV 457 ± 14 − i ( 279 ± 11 ) MeV f 0 ( 980 ) : 987 ± 76 − i ( 47 ± 25 ) MeV 996 ± 7 − i ( 25 ± 10 ) MeV

Correction caused by squared scalar radius Scalar radius: χ PT : r 2 = 0 . 61 ± 0 . 04 fm 2 � ∞ δ Γ( s ) < r 2 > = 6 π ds 4 m 2 π s 2 and from 6 < r 2 > π Γ π ( s ) / Γ( π ( 0 ) = 1 + 1 s s + O ( s 2 ) we have: < r 2 > = 6 ∂ Γ π ( s ) at s = 0 ∂ s Our result: → 0 . 76 ± 0 . 12 fm 2 five-parameter (two for f 0 ( 600 ) + two for f 0 ( 980 ) + one for lhc) − Therefore at least two poles (two parameters) more are needed Then explicit form of the corrected pion scalar FF: ( q − q 1 ) Γ π ( t ) = P n ( t ) ( q + q 2 )( q + q 3 )( q + q 4 )( q + q 5 )( q + q 6 )( q + q 7 ) × ( i + q 2 )( i + q 3 )( i + q 4 )( i + q 5 )( i + q 6 )( i + q 7 ) ( i − q 1 )

Final results Additional constrain in the fit: r 2 = 0 . 61 gives: r 2 = 0 . 61 ± 0 . 08 fit to the output fit to the input A 1 0 . 220 fixed 0 . 220 fixed A 3 0 . 150 ± 0 . 018 0 . 123 ± 0 . 014 − 0 . 0123 ± 0 . 0013 − 0 . 0135 ± 0 . 0013 A 5 A 7 0 . 0034 ± 0 . 0012 0 . 0041 ± 0 . 0011 A 2 − 0 . 045 ± 0 . 033 − 0 . 065 ± 0 . 022 A 4 − 0 . 0089 ± 0 . 0048 − 0 . 0051 ± 0 . 0029 A 6 − 0 . 0023 ± 0 . 0018 − 0 . 0031 ± 0 . 0015 f 0 ( 500 ) 461 ± 14 − i 259 ± 32 468 ± 14 − i 261 ± 29 f 0 ( 980 ) 991 ± 79 − i 51 ± 30 990 ± 74 − i 46 ± 25 q 1 = 0 . 00 − i 1 . 824 ± 0 . 18 3 . 452 ± 0 . 40 + i 0 . 186 ± 0 . 04 q 2 = q 3 = − 3 . 452 ± 0 . 40 + i 0 . 186 ± 0 . 04 q 4 = 1 . 376 ± 0 . 10 + i 1 . 091 ± 0 . 12 q 5 = − 1 . 376 ± 0 . 10 + i 1 . 091 ± 0 . 12 q 6 = 3 . 985 ± 0 . 09 − i 0 . 085 ± 0 . 09 q 7 = − 3 . 985 ± 0 . 09 − i 0 . 085 ± 0 . 09

Analytical structure of the Γ( s ) q 1 = 0 . 00 − i 1 . 824 ± 0 . 18 q 2 = 3 . 452 ± 0 . 40 + i 0 . 186 ± 0 . 04 q 3 = − 3 . 452 ± 0 . 40 + i 0 . 186 ± 0 . 04 q 4 = 1 . 376 ± 0 . 10 + i 1 . 091 ± 0 . 12 q 5 = − 1 . 376 ± 0 . 10 + i 1 . 091 ± 0 . 12 q 6 = 3 . 985 ± 0 . 09 − i 0 . 085 ± 0 . 09 q 7 = − 3 . 985 ± 0 . 09 − i 0 . 085 ± 0 . 09

Final results 10 250 250 S-wave isoscalar ππ phase shifts S-wave isoscalar ππ phase shifts 8 200 200 6 150 150 | Γ (s)| 4 100 100 2 50 50 0 0 0 0.0 0.0 0.5 0.5 1.0 1.0 1.5 1.5 0.0 0.5 1.0 1.5 s (GeV 2 ) s (GeV 2 ) s (GeV 2 ) Results: GKPY eqs: f 0 ( 500 ) : 468 ± 14 − i ( 261 ± 29 ) MeV 457 ± 14 − i ( 279 ± 11 ) MeV f 0 ( 980 ) : 990 ± 74 − i ( 46 ± 25 ) MeV 996 ± 7 − i ( 25 ± 10 ) MeV

Final results 5 paramaters 7 parameters 250 250 250 250 S-wave isoscalar ππ phase shifts S-wave isoscalar ππ phase shifts S-wave isoscalar ππ phase shifts S-wave isoscalar ππ phase shifts 200 200 200 200 150 150 150 150 100 100 100 100 50 50 50 50 0 0 0 0 0.0 0.0 0.5 0.5 1.0 1.0 1.5 1.5 0 0 1 1 2 2 3 3 4 4 2 ) 2 ) 2 ) 2 ) s (GeV s (GeV s (GeV s (GeV

Final results 5 paramaters 7 parameters 10 10 8 8 6 6 | Γ (s)| | Γ (s)| 4 4 2 2 0 0 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 s (GeV 2 ) s (GeV 2 )

Conclusions • good fit to the data + DR (GKPY egs): χ 2 = 0 . 88 pdf, • ππ scalar scattering length = 0.22, • two lowest scalar resonances f 0 ( 500 ) and f 0 ( 980 ) with very good parameters are found, • scalar radius r 2 = 0 . 61 ± 0 . 8 fm 2 , • first explicit form of the ππ scalar form factor is found