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

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

Nov 06, 2022 144 likes •333 views

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

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SLIDE 1

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

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SLIDE 2

Pion scalar form factor

< πi(p2) | m(¯ uu + ¯ dd) | πj(p1) >= δijΓπ(t) where t = (p2 − p1)2 and

  • m = 1

2(mu + md).

It posses all known properties of the pion vector electromagnetic FF Fπ(t) like

  • analyticity in t-plane beside cuts on the positive real axis from two-pion threshold

t = 4m2

π to +∞

  • elastic unitarity condition ImΓπ(t) = Γπ(t)e−iδ0

0 sin δ0

0, where δ0 0(t) is the S-wave

isoscalar ππ phase shift

  • asymptotic behavior Γπ(t)|t|→∞ ∼ 1

t

  • reality condition Γ∗

π(t) = Γπ(t∗)

  • normalization, however, now to the pion sigma term value

Γ(0) = (0.99 ± 0.02)m2

π to be predicted by the χPT,

Γπ(0) = mu

∂M2

π

∂mu + md ∂M2

π

∂md

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SLIDE 3

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)
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SLIDE 4

Our (Bratislava-Kraków) analysis -the method

Γπ(t) = Pn(t)exp t π ∞

4m2

π

δ0

0(t′)

t′(t′ − t) dt′ , (1) tan δΓ(t) = A1q + A3q3 + A5q5 + A7q7 + ... 1 + A2q2 + A4q4 + A6q6 + ... (2) Γπ(t) = Pn(t)exp

  • (q2 + 1)

2πi × ∞

−∞

q′ln (1+A2q′2+A4q′4)+i(A1q′+A3q′3+A5q′5)

(1+A2q′2+A4q′4)−i(A1q′+A3q′3+A5q′5)

(q′2 + 1)(q′2 − q2) dq′

  • ,

(3) Γπ(q) = M

n=0 anqn

N

i=1(q − qi)

,

  • φ(q′)dq′ = 2πi
  • n=1

Resn (4) explicit form of the pion scalar FF: Γπ(t) = Pn(t) (q − q1) (q + q2)(q + q3)(q + q4)(q + q5) × (i + q2)(i + q3)(i + q4)(i + q5) (i − q1) where Pn(t) is any polynomial normalised at t = 0 to one.

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SLIDE 5

First results (2014)

PHYSICAL REVIEW D 90, 114003 (2014) "Pion scalar form factor and another confirmation of the existence of the f0(500) meson" Results: PDG: f0(500): 388 ± 23 − i(301 ± 33) MeV 400 − 550 − i(200 − 350) MeV f0(980): 1066 ± 142 − i(110 ± 97) MeV 990 ± 20 − i(25 − 50) MeV

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SLIDE 6

New results (2015/2016); new data for δππ

0.2 0.4 0.6 0.8

50 100 150 200

t (GeV2) S-wave isoscalar ππ phase shifts

0.2 0.4 0.6 0.8

50 100 150 200

t (GeV2) S-wave isoscalar ππ phase shifts

  • 3
  • 2
  • 1

1 2 3 2 4 6 t (GeV2) |Γ(t)|

  • 3
  • 2
  • 1

1 2 3 2 4 6 t (GeV2) |Γ(t)|

  • 3
  • 2
  • 1

1 2 3 2 4 6 t (GeV2) |Γ(t)|

  • 3
  • 2
  • 1

1 2 3 2 4 6 t (GeV2) |Γ(t)|

Results: PDG: f0(500): 467 ± 22 − i(257 ± 30) MeV 400 − 550 − i(200 − 350) MeV f0(980): 983 ± 76 − i(48 ± 25) MeV 990 ± 20 − i(25 − 50) MeV

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SLIDE 7

New results (2015/2016); poles

tan δΓ(t) = A1q + A3q3 + A5q5 + A7q7 + ... 1 + A2q2 + A4q4 + A6q6 + ... fit to the output fit to the input A1 0.220 fixed 0.220 fixed A3 0.151 ± 0.020 0.128 ± 0.016 A5 −0.0144 ± 0.0016 −0.0125 ± 0.0013 A2 −0.055 ± 0.038 −0.085 ± 0.023 A4 −0.0089 ± 0.0048 −0.0051 ± 0.0029 f0(500) 451 ± 14 − i260 ± 33 467 ± 14 − i261 ± 29 f0(980) 988 ± 81 − i52 ± 32 987 ± 76 − i47 ± 25 Γπ(t) = Pn(t) (q − q1) (q + q2)(q + q3)(q + q4)(q + q5) × (i + q2)(i + q3)(i + q4)(i + q5) (i − q1) q1 = 0.00 −i1.943 ± 0.20 q2 = 3.397 ± 0.40 +i0.196 ± 0.04 q3 = −3.397 ± 0.40 +i0.196 ± 0.04 q4 = 1.385 ± 0.10 +i1.085 ± 0.12 q5 = −1.385 ± 0.10 +i1.085 ± 0.12

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SLIDE 8

New results (2015/2016); analytical structure of the Γ(s)

q1 = 0.00 −i1.943 ± 0.20 q2 = 3.397 ± 0.40 +i0.196 ± 0.04 q3 = −3.397 ± 0.40 +i0.196 ± 0.04 q4 = 1.385 ± 0.10 +i1.085 ± 0.12 q5 = −1.385 ± 0.10 +i1.085 ± 0.12

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SLIDE 9

New results (2015/2016)

0.2 0.4 0.6 0.8

50 100 150 200

t (GeV2) S-wave isoscalar ππ phase shifts

0.2 0.4 0.6 0.8

50 100 150 200

t (GeV2) S-wave isoscalar ππ phase shifts

  • 3
  • 2
  • 1

1 2 3 2 4 6 t (GeV2) |Γ(t)|

  • 3
  • 2
  • 1

1 2 3 2 4 6 t (GeV2) |Γ(t)|

  • 3
  • 2
  • 1

1 2 3 2 4 6 t (GeV2) |Γ(t)|

  • 3
  • 2
  • 1

1 2 3 2 4 6 t (GeV2) |Γ(t)|

Results: PDG: f0(500): 467 ± 14 − i(261 ± 29) MeV 400 − 550 − i(200 − 350) MeV f0(980): 987 ± 76 − i(47 ± 25) MeV 990 ± 20 − i(25 − 50) MeV

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SLIDE 10

New results (2015/2016)

0.2 0.4 0.6 0.8

50 100 150 200

t (GeV2) S-wave isoscalar ππ phase shifts

0.2 0.4 0.6 0.8

50 100 150 200

t (GeV2) S-wave isoscalar ππ phase shifts

  • 3
  • 2
  • 1

1 2 3 2 4 6 t (GeV2) |Γ(t)|

  • 3
  • 2
  • 1

1 2 3 2 4 6 t (GeV2) |Γ(t)|

  • 3
  • 2
  • 1

1 2 3 2 4 6 t (GeV2) |Γ(t)|

  • 3
  • 2
  • 1

1 2 3 2 4 6 t (GeV2) |Γ(t)|

Results: GKPY eqs: f0(500): 467 ± 14 − i(261 ± 29) MeV 457 ± 14 − i(279 ± 11) MeV f0(980): 987 ± 76 − i(47 ± 25) MeV 996 ± 7 − i(25 ± 10) MeV

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SLIDE 11

Correction caused by squared scalar radius

Scalar radius: χPT: r2 = 0.61 ± 0.04 fm2 < r 2 >= 6

π

∞

4m2

π ds

δΓ(s) s2

and from Γπ(s)/Γ(π(0) = 1 + 1

6 < r2 >π s s + O(s2)

we have: < r 2 >= 6 ∂Γπ(s)

∂s

at s = 0 Our result: five-parameter (two for f0(600) + two for f0(980) + one for lhc) − → 0.76 ± 0.12 fm2 Therefore at least two poles (two parameters) more are needed Then explicit form of the corrected pion scalar FF: Γπ(t) = Pn(t) (q − q1) (q + q2)(q + q3)(q + q4)(q + q5)(q + q6)(q + q7) × (i + q2)(i + q3)(i + q4)(i + q5)(i + q6)(i + q7) (i − q1)

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SLIDE 12

Final results

Additional constrain in the fit: r2 = 0.61 gives: r 2 = 0.61 ± 0.08 fit to the output fit to the input A1 0.220 fixed 0.220 fixed A3 0.150 ± 0.018 0.123 ± 0.014 A5 −0.0123 ± 0.0013 −0.0135 ± 0.0013 A7 0.0034 ± 0.0012 0.0041 ± 0.0011 A2 −0.045 ± 0.033 −0.065 ± 0.022 A4 −0.0089 ± 0.0048 −0.0051 ± 0.0029 A6 −0.0023 ± 0.0018 −0.0031 ± 0.0015 f0(500) 461 ± 14 − i259 ± 32 468 ± 14 − i261 ± 29 f0(980) 991 ± 79 − i51 ± 30 990 ± 74 − i46 ± 25 q1 = 0.00 − i1.824 ± 0.18 q2 = 3.452 ± 0.40 + i0.186 ± 0.04 q3 = −3.452 ± 0.40 + i0.186 ± 0.04 q4 = 1.376 ± 0.10 + i1.091 ± 0.12 q5 = −1.376 ± 0.10 + i1.091 ± 0.12 q6 = 3.985 ± 0.09 − i0.085 ± 0.09 q7 = −3.985 ± 0.09 − i0.085 ± 0.09

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SLIDE 13

Analytical structure of the Γ(s)

q1 = 0.00 − i1.824 ± 0.18 q2 = 3.452 ± 0.40 + i0.186 ± 0.04 q3 = −3.452 ± 0.40 + i0.186 ± 0.04 q4 = 1.376 ± 0.10 + i1.091 ± 0.12 q5 = −1.376 ± 0.10 + i1.091 ± 0.12 q6 = 3.985 ± 0.09 − i0.085 ± 0.09 q7 = −3.985 ± 0.09 − i0.085 ± 0.09

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SLIDE 14

Final results

0.0 0.5 1.0 1.5

50 100 150 200 250

s (GeV2) S-wave isoscalar ππ phase shifts

0.0 0.5 1.0 1.5

50 100 150 200 250

s (GeV2) S-wave isoscalar ππ phase shifts

0.0 0.5 1.0 1.5 2 4 6 8 10 s (GeV2) |Γ(s)|

Results: GKPY eqs: f0(500): 468 ± 14 − i(261 ± 29) MeV 457 ± 14 − i(279 ± 11) MeV f0(980): 990 ± 74 − i(46 ± 25) MeV 996 ± 7 − i(25 ± 10) MeV

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SLIDE 15

Final results

5 paramaters 7 parameters

0.0 0.5 1.0 1.5

50 100 150 200 250

s (GeV

2)

S-wave isoscalar ππ phase shifts

0.0 0.5 1.0 1.5

50 100 150 200 250

s (GeV

2)

S-wave isoscalar ππ phase shifts

1 2 3 4

50 100 150 200 250

s (GeV

2)

S-wave isoscalar ππ phase shifts

1 2 3 4

50 100 150 200 250

s (GeV

2)

S-wave isoscalar ππ phase shifts

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SLIDE 16

Final results

5 paramaters 7 parameters

0.0 0.5 1.0 1.5 2 4 6 8 10 s (GeV2) |Γ(s)| 0.0 0.5 1.0 1.5 2 4 6 8 10 s (GeV2) |Γ(s)|

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SLIDE 17

Conclusions

  • good fit to the data + DR (GKPY egs): χ2 = 0.88 pdf,
  • ππ scalar scattering length = 0.22,
  • two lowest scalar resonances f0(500) and f0(980) with very good parameters

are found,

  • scalar radius r2 = 0.61 ± 0.8 fm2,
  • first explicit form of the ππ scalar form factor is found