Bled Workshops in Physics Vol. 16, No. 1 p. 40
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015

Partial wave analysis of n photoproduction data with analyticity constraints*
M. Hadzimehmedovica, V. Kashevarovc, K. Nikonovc, R. Omerovica, H. Osmanovica, M. Ostrickc, J. Stahova, A. Svarcb, L. Tiatorc
a University of Tuzla, Faculty of Science, Bosnia and Herzegovina b Rudjer Boskovic Institute, Zagreb, Croatia
c Institut fur Kernphysik, Johannes Gutenberg Universtat Mainz, Germany
Abstract. We perform partial wave analysis of the n photoproduction on data. The obtained multipoles are consistent with the fixed-t analyticity and fixed-s analyticity. A fixed-t analyticity is imposed using Pietarinen expansion method. The invariant amplitudes obey the required crossing symmetry.
1	Introduction
A big problem in partial wave analyses are ambiguities of partial wave solutions. More than one set of partial waves describe equally well the experimental data. A first attempt to solve this problem was to require smoothness of partial waves as a function of energy. It was shown that this criteria was not enough to achieve a unique partial wave solution [1]. Furthermore, it was shown that more stringent constraints, based on the analytic properties of invariant amplitudes from Mandelstam hypothesis, should be taken into consideration. An efficient method for imposing the fixed-t analyticity on invariant amplitudes was proposed by E. Pietarinen [2-5] and was used in Karlsruhe-Helsinki partial wave analysis of nN scattering data KH80 [6-8]. In our partial wave analysis of n-photoproduction data we follow main ideas from Karlsruhe-Helsinki analysis. The method consists of two separate analyses: Fixed-t amplitude analysis (FT AA) and a single energy partial wave analysis (SE PWA). The two analyses are coupled in such a way that results from one are used as a constraint in another in an iterative procedure. The resulting partial waves (multipoles) describe experimental data adequately and are consistent with fixed-t and fixed-s analyticity as well.
2	Preparing experimental data for partial wave analysis
Our data base consists of the following experimental data: — Differential cross sections at 120 energies in the range 710 MeV < Elab < 1395 MeV [9];
* Talk presented by J. Stahov
Partial wave analysis of r| photoproduction data
41
—	Beam asymmetry I at 15 energies in the range 724 MeV — 1472 MeV [10];
—	Target asymmetry T at 12 energies in the range 725 MeV — 1350 MeV [11];
—	Double asymmetry F at 12 energies in the range 725 MeV — 1350 MeV [11].
In SE PWA experimental data are required at a predetermined set of energies. Experimental values of beam asymmetry, target asymmetry and double polarization asymmetry are interpolated to 113 energies, where data on differential cross sections are available. A spline fit method with x2/dp = 0.7 (DP-number of data points) was used. FT AA requires experimental data at predetermined set of t values. Using the same method, data previously prepared for SE PWA were shifted to 40 t values in the range t G [—1.00 GeV2, —0.05 GeV2].
3 Fixed-t amplitude analysis
Following definition in Ref. [12], in description of n-meson photoproduction, we use crossing symmetric invariant amplitudes Bi, B2, B6, and B8/v. For a given value of variable t amplitudes are represented by two Pietarinen expansions in the form
N
Fk(v2,t) = FkN(v2,t) + (1 + zi)Y_ bu ' 4 +(1 + Z2)Y_ b2i 4, (1)
i=1 i=1
where Fk stands for invariant amplitudes Bk. FkN are explicitly known nucleon pole contributions and s,u and v = (s — u)/4m with the proton mass m are Mandelstam variables. The conformal variables z1 and z2 are defined as
a1 — V Vth1 — v2	a2 — y vth.2 — V
zi = - ,	, zt = - ,	.	(2)
a + v vthi — v2	a2 + y vth.2— v2
vth1 and vth2 correspond to the n and n photoproduction thresholds (yp —> n0p and Yp —> np). N and N2 are number of parameters in expansion (1) (in our applications N1, N2 « 15). a1 and a2 are parameters which determine distribution of points on a unit circle (|z11 = |z2| = 1 ). Coefficients b1k) and b2k) in expansion (1) are determined by minimizing a quadratic form
X2 = Xdata + Xpw +	(3)
The term xdata is the standard exspression containing all the data at a fixed-t value
Xdata =	(Dn'P(v2,t'— Dni'(v2,t))2 ,	(4)
D n=1	Dn
where D stands for measurable quantities (a0 = da/dH, ct0 • T, a0 • F, ct0 • I). The sum goes over all N D available experimental values of measured quantities D for a given t value. D^ are predicted values in terms of coefficients in expansion (1).
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J. Stahov et al.
A second term xpW is also a usual x2 expression containing as "data" the helicity amplitudes calculated from the partial wave solution
x2pw = q L z (iReHkiMt'v2) - Re HPW (t,v2)]2 k=1 i=1 I
(£r)2
ki
+
[Im Hkit (t,v2) - Im H?W (t,v2)] (£i)ki
(5)
In the first iteration H{PW are calculated from an initial, already existing solution. In the subsequent iterations H£w are calculated from partial waves obtained in the single energy partial wave analysis (SE PWA) of the same set of experimental data. The weight factor q and errors £ki are unknown. They are adjusted in such a way that xiata ~ XpW. ® is Pietarinen's penalty function in the form
® = + ®2 + ®3 + 04,
(6)
where ®k is defined as

i=1
k = AikZ b^M (i + 1)3 + À2^ b
i=1
/k) J2i
(i + 1)3
A11, A21,..., A"14, À24 are weight factors determined according to the convergence test function method [5]. The final result of the fixed-t amplitude analysis consists of 40 sets of coefficients b1k) and b2k). The invariant amplitudes may be calculated at any c.m. energy W and scattering angle 0 in the physical region. Helicity amplitudes are used as a constraint in a SE PWA. Helicity amplitudes in terms of invariant amplitudes are given in the Appendix.
2
2
3.1 Single energy partial wave analysis
In the single energy partial wave analysis we minimize a quadratic form:
X2 = xiata + Xí=T .
(7)
Xdata is again a standard expression containing all the data at a given energy. For a given observable D, measured at ND angles 0i, contribution to the xdata reads:
(x
2 Ï data/d
= L
i=1
D
exp
(0i) - Dfit (0i)
A
xdata
Di
2.
data D
Dexp (0i) are experimental values of observable D with corresponding experimental errors ADi. Dfit (0i) are values of observable D calculated from partial waves which are parameters in the fit. The second term x2T is also a usual x2
2
Partial wave analysis of r| photoproduction data
41
expression containing as "data" the helicity amplitudes Hk from the fixed-t amplitude analysis. It has the form
Nc
xft =
k=1i=1
Re Hk (0i) - Re H™ (0i)
(£r )
ki
+
Im Hk (0i) - Im Hku (0i)
(£i)
ki
The angles 0t are calculated using the formula
cos 0i =
ti
m^ + 2kq '
cos 0i G [-1.00, +1.00],
(8)
where mn, q, and w are mass, c.m. momentum and c.m. energy of the n meson, and k is the c.m. momentum of the photon. Nc is the number of angles at which constraining amplitudes are given. Errors of real and imaginary parts (eR), (ei) are not determined. They are adjusted in such a way that xiata ~ Xft. After performing SE PWA at predetermined energies, the obtained partial wave values are used as a constraint in the fixed-t amplitude analysis. The "data" in the term xpW of (3) are to be calculated using these partial waves.
Our iterative procedure is shown in Fig 1.
Helicity amplitudes from initial solution
At each i-value perform fixed-i amplitude analysis Minimize:
X Xdata ~^~Xpw
At each of N energies perform single energy partial wave analysis Minimize:
X* = Xdata + Xft
Use results from single energy partial wave analysis to calculate helicity amplitudes which are used as a constraint in fixed-i amplitude analysis
Fig. 1. (Color online) Iterative procedure in a combined single energy partial wave analysis and fixed-t amplitude analysis.
2
2
To make our analysis easier to follow, we give more details about important steps after preparing input data as described in section 2.
1.	Take an initial solution (MAID [13] or Bonn-Gatchina [14,15]) and calculate all four invariant amplitudes Bt(W, t) at all t values and energies where input data are available.
2.	Perform the Pietarinen expansion for all invariant amplitudes using equation (1) with conformal variables defined in formula (2).
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3.	Calculate helicity amplitudes from invariant amplitudes (see Appendix).
4.	For all t values perform a non-linear fit of observables minimizing the quadratic form (3). As starting values of parameters b|k) and bi,k) take coefficients obtained in step 2. Calculate term xpw using initial solution to calculate Hpw. This step completes the FT AA.
5.	At a given energy W calculate helicity amplitudes Hk(W, cos9i), where cos9i are given by formula (8). Use coefficients b|k) and bi,k) from FT AA for corresponding t-values.
6.	Perform a non-linear SE PWA using helicity amplitudes obtained in step 5 as a constraint. As starting values for partial waves (multipoles) use the same initial solution as in step 1.
7.	Use results from step 6 in step 1 and perform next iteration. Our preliminary results show that, depending on the strength of constraints, it is enough to perform 2-3 iterations to get a stable final solution.
In Fig. 2 fits of invariant amplitudes are shown at t = -0.15 GeV2. Multipoles with L < 3, obtained after two iterations, are shown in Figs. 3 and 4. The Eta-Maid2015b solution was chosen as a starting solution in both analyses, FT AA and SE PWA.
Real part
3 2
„ 1
CM
n 0
CD
Imag part 1
0.5 0
-0.5 -1 -1.5 -2 -2.5 -3 -3.5
1500 1600 1700 1800 1900 2000 2100 2200 W[MeV]
1500 1600 1700 1800 1900 2000 2100 2200 W[MeV]
1500 1600 1700 1800 1900 2000 2100 2200 W[MeV]
1500 1600 1700 1800 1900 2000 2100 2200 W[MeV]
Fig. 2. (Color online) Red diamonds and blue circles show initial real and imaginary values of invariant amplitudes. As initial solution invariant amplitudes for t = -0.15 GeV2 from etaMAID2015b [13] are used. The red and blue lines show the Pietarinen fits to real and imaginary parts of invariant amplitudes, respectively
Partial wave analysis of r| photoproduction data	41
Real part
25 20 15 10 5 0 -5
-10 1450
2.5 2 1.5 1
0.5 0
-0.5
1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
Imag part 0.25 0.2 0.15 0.1 0.05 0
-0.05 -0.1 -0.15 -0.2 -0.25
145015001550160016501700175018001850
1450 1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
W[MeV]
3 2.5 2 1.5 1
0.5 0
-0.5 -1 -1.5 -2 -2.5
1450 1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
Real part
0.2 0.15 0.1 0.05 0
-0.05
-0.1
0.5 0.4 0.3
Ê a2 E 0.1
oj 0
S-0.1 -0.2 -0.3 -0.4
1450 1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
Imag part 0.8
0.6
0.4
I 0.2
CVJ 0 LU
-0.2 -0.4 -0.6
1450 1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
0.6 0.4 0.2 I 0
cvj -0.2 M
-0.4 -0.6
1450 1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
-0.8
1450 1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
Fig. 3. (Color online) Real and imaginary parts of multipoles obtained from SE PWA in 2nd iteration are shown as red diamonds and blue circles. The initial solution etaMAID2015b is given as red and blue solid lines.
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J. Stahov et al.
Real part
0.15 0.1 m 0.05
[ 0
h >
^ -0.05 -0.1
-0.15 1450
) 1550 1600 1650 1700 1750 1800 1850 W[MeV]
1500
1550 1600 1650 1700 1750 1800 1850 W[MeV]
Imag part 0.2
0.1 F 0 -0.1 3-0.2 -0.3 -0.4
1450 1500 1550 1600 1650 1700 1750 1 W[MeV]
0.15 0.1 m] 0.05
^ 0 3-
^ -0.05 -0.1 -0.15
1450 1500 1550 1600 1650 1700 1750 1800 1850 W[MeV]
Fig. 4. [Continued from previous page.] Caption as in Fig. 3.
4 Conclusions
A SE PWA with fixed-t constraints has been performed and multipoles, consistent with crossing symmetry and fixed-t analyticity, have been obtained. The helicity amplitudes from fixed-t show good consistency with fixed-s analyticity. It implies that our amplitudes are consistent with both, fixed-t and fixed-s analyticity.
Acknowledgment
This work was supported in part by the Federal Ministry of Education and Science, Bosnia and Herzegovina, Grant No. 05-39-3545-1/14 and by the Deutsche Forschungsgemeinschaft, Collaborative Research Center 1044.
Appendix
A Multipole expansion of invariant amplitudes
In partial wave analysis of pseudoscalar meson photoproduction it is convenient to work with CGLN amplitudes [16] giving simple representations in terms of
Partial wave analysis of r| photoproduction data 47 electric and magnetic multipoles and derivatives of Legendre polynomials
Ft = ¿[(IM1+ + El+)Pl'+1 (x) + ((I + 1 )Ml+ + El_)Pl'_1 (x)], 1=0
CO
F 2 = + 1)Ml+ + 1Ml-]Pl(x), l=i
CO
F3 = ^[(El+ - Ml+)Pl'+1 + (El- + Ml-)Pl-1 (x)],
(A.1)
l=i
F4 =	- Ei+ - Mi- - El-]Pl"(x).
l=2
Another common set of amplitudes are helicity amplitudes, which are linearly related to the CGLN amplitudes
1	9
Hi = : sin 9cos 9(F3 + F4),
H2 = V2cos 2[(F2 - Fi ) + 1 cos 9 (F3 - F4)],
1	9
H3 = — sin 9 sin 9 (F3 - F4),
H4 = V2sin 2[(Fi + F2)+ 1 + cos 9 (F3 + F4)]. The relations between CGLN and invariant amplitudes are given by
(A.2)
/FA		/BA
F2	= M •	B2
F3		B6
VF4/		VBs/
(A.3)
with the matrix M:
M =
1
2W(s - m2)
(s-m2)	(s-m2)	0 (t-m2 )(m-W) 2as7 (t-m^n)	0 (t-mn)(m+W) 2a4 (t-m2)
ai 0 2(m+W)	a2 0 2(m-W)		
ai (m+W)	a2 (m-W)	a3 (s-u)	a4 (s-u)
ai	a2	2a3	2a4
and
ai
a2 =
V(Ei + m)(E2 + m) 8nW
V(Ei - m)(E2 - m) 8nW
(A.4)
V(Ei - m)(E2 - m)(E2 + m)
a3 =-8nw-=a2 •(E2+m)
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J. Stahov et al.
y^Ei + m)(E2 + m)(E2 - m) =-8nW-= ai ^ (E2 - m)
s +1 + u = ^ = 2m2 +
2	s — u
m_, v = —-
4m
where E1 and E2 are c.m. energies of the incoming and outgoing nucleons and W is the total c.m. energy.
References
1.	J. E. Bowcock and H. Burkhardt, Rep. Prog. Phys. 38 1099 (1975).
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5.	E. Pietarinen, Nuovo Cim. 12A 522 (1972).
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13.	V. Kashevarov, Proceedings from Mini-Workshop Bled 2015.
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104 Povzetki v slovenščini
Analiza delnih valov za podatke pri fotoprodukciji mezona n z upoštevanjem omejitev zaradi analiticnosti
M. HadZimehmedovič a, V. Kashevarovc, K. Nikonovc, R. Omerovič a, H. Osman-oviča, M. Ostričkc, J. Stahova, A. Svarcb in L. Tiatorc
a University of Tuzla, Faculty of Science, Bosnia and Herzegovina b Rudjer Boskovič Institute, Zagreb, Croatia
a Institut fuer Kernphysik, Johannes Gutenberg Universtaet Mainz, Germany
Izvedemo analizo delnih valov za podatke pri fotoprodukciji n. Dobljeni multi-poli so v skladu z analiticnostjo pri fiksnem t in pri fiksnem s. Analiticnost pri fiksnem t zagotovimo s Pietarinenovo metodo. Invariantne amplitude ubogajo zahtevano navzkrizno simetrijo.
Napredek pri poznavanju sklopitev nevtrona
W. J. Briscoe in I. Strakovsky
The George Washington University, Washington, DC 20052, USA
Podajamo pregled prizadevanj skupine GW SAID za analizo fotoprodukcije pio-nov na nevtronski tarci. Razlocitev izoskalarnih in izovektorskih elektromagnetnih sklopitev resonanc N* in A* zahteva primerljive in skladne podatke na protonski in na nevtronski tarci. Interakcija v koncnem stanju igra kriticno vlogo pri najsodobnejši analizi in izvrednotenju podatkov za proces yn —» nN pri eksperimentih z devteronsko tarco. Ta je pomemben sestavni del tekocih programov v laboratorijih JLab, MAMI-C, SPring-8, CBELSA in ELPH.
Vzbujanje barionskih resonanc s fotoprodukcijo mezonov
Lothar Tiatora in Alfred Svarcb
a Institut fuer Kernphysik, Johannes Gutenberg Universtaet Mainz, Germany b Rudjer Boskovic Institute, Zagreb, Croatia
Spektroskopija lahkih hadronov je se vedno zivahno podrocje v fiziki jedra in delcev. Celo 50 let po odkritju Roperjeve resonance in vec kot 30 let po pionirskem delu Hoehlerja and Cutkoskyja je se veliko odprtih vprasanjglede barionskih resonanc. Danes je glavni vzbujevalni mehanizem fotoprodukcija in elektropro-dukcija mezonov, merjena na elektronskih pospesevalnikih kot so MAMI, ELSA in JLab. V zdruzenem prizadevanju izvrednotimo lege in jakosti polov iz parcialnih valov, dobljenih z analizo parcialnih valov pri nedavnih meritvah polarizacij ob uporabi analiticnih omejitev iz disperzijskih relacijpri fiksnem t. Poseben poudarek pri barionskih resonancah je na strukturi pola na razlicnih Rieman-novih ploskvah.