Singularities and Essential Singularities in ˇ N ...

The Production Representation Partial Waves N/D Calculations Dispersion Relations, Virtual Poles and Essential Singularities Discussions on 1/2− Octet Baryons

Singularities and Essential Singularities in π N

Scattering Amplitudes

Hanqing Zheng

Sichuan University

�H�Æ, &²§Jul. 28th, 2021

Hanqing Zheng SCU

Singularities and Essential Singularities in π N Scattering Amplitudes


Contents

The Production Representation

Partial Waves

N/D Calculations

Dispersion Relations, Virtual Poles and Essential Singularities

Discussions on 1/2− Octet Baryons

Hanqing Zheng SCU



The Production (PKU) Representation

The factorized S matrix and the separable singularities:

Sphy . =∏i

SRi · Scut . (1)

Scut : no longer contains any pole:

Scut = e2iρf (s) ,

f (s) =s

π

∫L

ImLf (s ′)

s ′(s ′ − s)+

s

π

∫R

ImR f (s ′)

s ′(s ′ − s). (2)

Subtraction constant can be determined!

Mandelstam Analyticity(Polynomial boundedness of scattering

amplitudes) [Z. Y. Zhou and H.Z., NPA, 2006]

f (0) = 0 . (3)

Hanqing Zheng SCU



The background phase contribution

The phase is additive, δ(s) =∑

i δRi+ δb.g ..

δb.g .(s) = ρ(s)f (s) . (4)

ρ(s) =

√(s−(m1+m2)2)(s−(m1−m2)2)

s .

ImL,R f (s) = − 1

2ρ(s)log |Sphy (s)| , S = 1 + 2iρT . (5)

The approximation scheme:

discfL = disc{ 1

2iρ(s)log[S [X ](s)

]} . (6)

Background contributes negatively to scattering phase shift and

scattering lengths. Crucial in stabilizing the pole position!Hanqing Zheng SCU



Hanqing Zheng SCU



Quantum Scattering Theory Correspondence

For a finite range potential:

S(k) = e−2ikR∞∏1

kn + k

kn − k, (7)

where k is the (single) channel momentum and kn pole locations

on the complex k plane. The Eq. (7) automatically predicts a

negative background contribution!

Eq. (7) written down by [Ning Hu, Phys. Rev. 1948] [T. Regge 1958]

Hanqing Zheng SCU



Phase shift components

I PKU representation → conventionally additive phase shift

I Phase shift contributions

I bound states → negative phase shift

I virtual states (usually hidden !) → positive phase shift

I resonances → positive phase shift

I left hand cut → (empirically) negative phase shift (proved in

quantum mechanical potential scatterings)

[T. Regge 1958 Nuovo Cimento]

Hanqing Zheng SCU



Tree level phase shift results in πN

scatterings

L2I 2J convention, W =√s, data: green triangles [SAID: WI 08]

1 . 0 8 1 . 1 0 1 . 1 2 1 . 1 4 1 . 1 6- 2 0

- 1 5

- 1 0

- 5

0

5

1 0

L e f t h a n d c u t P o l e s T o t a l D a t a

Phas

e shif

t (Deg

ree)

W ( G e V )

S 1 1

1 . 0 8 1 . 1 0 1 . 1 2 1 . 1 4 1 . 1 6

- 1 0

- 5

0 L e f t h a n d c u t P o l e s T o t a l D a t a

Phas

e shif

t (Deg

ree)

W ( G e V )

S 3 1

1 . 0 8 1 . 1 0 1 . 1 2 1 . 1 4 1 . 1 6- 3 5

- 3 0

- 2 5

- 2 0

- 1 5

- 1 0

- 5

0

5


Phas

e shif

t (Deg

ree)

W ( G e V )

P 1 1

1 . 0 8 1 . 1 0 1 . 1 2 1 . 1 4 1 . 1 6

- 5

0


Phas

e shif

t (Deg

ree)

W ( G e V )

P 3 1

Hanqing Zheng SCU



L2I 2J convention, W =√s, data: green triangles [SAID: WI 08]

1 . 0 8 1 . 1 0 1 . 1 2 1 . 1 4 1 . 1 6

- 5

0

5


Phas

e shif

t (Deg

ree)

W ( G e V )

P 1 3

1 . 0 8 1 . 1 0 1 . 1 2 1 . 1 4 1 . 1 6- 1 5- 1 0- 505

1 01 52 02 53 03 54 0


Phas

e shif

t (Deg

ree)

W ( G e V )

P 3 3

Hanqing Zheng SCU



Discrepancies in S11 and P11 channels

I Large missing positive contributions

I Possible interpretations

I one loop contributions? numerical uncertainties?

I contributions from other branch cuts?

I hidden poles - virtual states, crazy resonances below threshold,

or some extremely broad states?

I Once subtraction, logarithmic form → not sensitive to chiral

orders and numerical details

Hanqing Zheng SCU



Finding P11 hidden pole

I Initially one resonance → two virtual states → one survives, the

other is nearly absorbed by the point (MN −mπ)2

I sc = −9 GeV2, virtual pole: 980 MeV, χ2P11/d.o.f = 0.201.

I An extra CDD pole is needed in P11 channel [A. Gasparyan and M.F.M. Lutz

2010 NPA]

1 . 0 8 1 . 1 0 1 . 1 2 1 . 1 4 1 . 1 6

- 1 . 0

- 0 . 5

0 . 0 H i d d e n p o l e s f i t D a t a

Phas

e shif

t (Deg

ree)

W ( G e V )

P 1 1

Hanqing Zheng SCU



P11 channel: shadow pole of the nucleon

I Analytical continuation: S II = 1/S I.

Second sheet poles → first sheet zeros.

I Expansion: S I ∼ a/(s −M2N) + b + · · ·

I Arbitrary non-zero b → the virtual state

I Perturbation calculation → virtual state at 976 MeV; fit →980 MeV

Hanqing Zheng SCU



Finding S11 hidden pole

[Y.F. Wang, D. L. Yao, HQZ, EPJC 2018]

I sc = −0.08 GeV2, ΛR = 4.00 GeV.

I Hidden pole → a “crazy resonance” below threshold

(0.861± 0.053)− (0.130± 0.075)i GeV

sc (GeV2) Pole position (GeV) Fit quality χ2/d.o.f

−0.08 0.808− 0.055i 0.109

−1 0.822− 0.139i 0.076

−9 0.883− 0.195i 0.034

∞ 0.914− 0.205i 0.018

Hanqing Zheng SCU



S11 channel: lowest potential-nature

resonance?

I S11 channel → no s-channel intermediate states → potential

nature interaction

I Square-well potential (µ: reduced mass)

U(r) = 2µV (r) =

−2µV0 (r ≤ L),

0 (r > L),

I Phase shift (k ′ = (k2 + 2µV0)1/2)

δsw(k) = arctan

[k tan k ′L− k ′ tan kL

k ′ + k tan (kL) tan (k ′L)

]Hanqing Zheng SCU



I Fit result (20 data): L = 0.829 fm and V0 = 144 MeV,

χ2sw/d.o.f = 0.740

I Pole position: k = −346i MeV → 0.872− 0.316i GeV.

Hidden pole fit (0.861± 0.053)− (0.130± 0.075)i GeV

1 . 0 8 1 . 1 0 1 . 1 2 1 . 1 4 1 . 1 60

2

4

6

8

1 0 S q u a r e W e l l F i t D a t a

Phas

e shif

t (Deg

ree)

W ( G e V )

S 1 1

Hanqing Zheng SCU



O(p3) results· · ·[Y.F. Wang, D. L. Yao, HQZ, Front. Phys.(Beijing)2019] [Y.F. Wang, D. L. Yao, HQZ, CPC 43 2019]

I The same cut-off condition

I Chiral order does not impact on the existence of the S11 and

P11 states

I O(p3) greatly improves the fit quality in other channels that

are impossible to fit the data at O(p2).

1 . 0 8 1 . 1 0 1 . 1 2 1 . 1 4 1 . 1 6

- 1 0

- 5

0

5

1 0

O ( p 1 ) T o t a l O ( p 2 ) T o t a l O ( p 3 ) T o t a l D a t a

Phas

e shif

t (Deg

ree)

W ( G e V )

S 1 1

1 . 0 8 1 . 1 0 1 . 1 2 1 . 1 4 1 . 1 6- 3 0

- 2 5

- 2 0

- 1 5

- 1 0

- 5

0

O ( p 1 ) T o t a l O ( p 2 ) T o t a l O ( p 3 ) T o t a l D a t a

Phas

e shif

t (Deg

ree)

W ( G e V )

P 1 1

Hanqing Zheng SCU



sN∗ = 895(81)− i164(23) MeV

sv = 966(18)MeV.

Hanqing Zheng SCU



N∗(890) pole in N/D method

QuZhi Li et al., 2021, ArXive:2102.00977

T = N/D . (8)

D(s) = 1− s − s0

π

∫R

ρ(s ′)N(s ′)

(s ′ − s)(s ′ − s0)ds ′ ,

N(s) = N(s0) +s − s0

π

∫L

D(s ′)ImL[T (s ′)]

(s ′ − s)(s ′ − s0)ds ′ .

(9)

ImLT as an input

Analytic continuation:

DII(s) = D(s) + 2iρN(s) , NII(s) = N(s) , (10)

Hanqing Zheng SCU



A toy model calculation

ImLT =∑i

γiδ(s − si ) (11)

Case I Case II

s1 0 −m2N

γ1 (GeV2) 0.79 1.34√spole(GeV) 0.810 - 0.125i 0.788 - 0.185i

Table: Subthreshold pole locations using input Eq. (11).

.

Hanqing Zheng SCU



Figure: left: fit to the S11 channel phase shift data, taking Case II as an

example; right: the ‘spectral function’ ImLf (s)/s of Case I and II. Notice

that the singularity at s = 0 in Case II is due to the kinematical

singularity in ρ(s).

Hanqing Zheng SCU



O(p2) calculation

Partial wave projection of χPT amplitudes encounter a severe

problem at s = 0,

T[O(pn)

](s → 0) ∼ Cs−n−1/2 , (12)

Violating Froissart bound. General argument gives instead

T ∼ s−α∆(0) (13)

where α∆(0) is the intercept parameter of the Regge trajectory of

∆(1232). An N/D calculation is nevertheless still doable with

√s = 1.01± 0.19i GeV , (14)

within reasonable range of LECs of O(p2) lagrangian.Hanqing Zheng SCU



A ‘realistic’ model calculation

discT (s) = discT (1)(s) + discT ρ(s) + disc [a + bs√

s] . (15)

Figure: The l .h.c . by t-channel ρ exchange; u-channel N exchange.

√s = 0.90− 0.20iGeV . (16)

Hanqing Zheng SCU



Figure: Phase shift decomposition: only contributions from physical

ingredients are plotted including their summation ‘Total’. It clearly

demonstrates that spurious contributions cancel each other, otherwise

curve ‘Total’ cannot get close to the data.Hanqing Zheng SCU



Figure: Comparison among different “spectral” functions. The singular

behaviors of T (s) at s = 0 are O(s−5/2), O(s−1/2) and O(s0) for O(p2)

χPT , model Eq. (15) and Case II, respectively.

Hanqing Zheng SCU



Virtual poles

u channel nucleon pole exchange contributes a cut ∈ [cL, cR ],

with cL =(m2

N−m2π)2

m2N

and cR = m2N + 2m2

π.

s → cL : T (s)→ − g2m4N

16πF 2 (4m2N −m2

π)ln

s − cLcL − cR

,

s → cR : T (s)→ g2m2N(m2

N + 2m2π)

πF 2(4m2N −m2

π)ln

cR − cLs − cR

, (17)

s → cL, S ' AcL + BcL lns − cLcL − cR

,

s → cR , S ' AcR + BcR lns − cRcR − cL

,(18)

AcL = AcR = 1 +g2mNmπ

8πF 2+ O

(m3π

),

BcL = BcR =g2mNmπ

16πF 2+ O

(m3π

).

(19)

Hanqing Zheng SCU



I S(cL),S(cR)→ −∞ which are exact (correct to any order of

chiral expansions)

I S = +1 at sL and sR by definition.

I S(s) is real when s ∈ (sL, cL) ∪ (cR , sR)

⇒ There have to be two S matrix zeros: one below cL and another

above cR , on the real axis. Their locations (vL and vR) are:

vL = cL − (cR − cL)e−AcL/BcL ,

vR = cR + (cR − cL)e−AcR/BcR .

(20)

These two virtual poles in total give a large contribution to the

phase shift. E.g., roughly 50◦ at√s = 1.16GeV, which seems to

completely destroy the picture presented in Fig. 3.Hanqing Zheng SCU



Dispersion relations for f (s)

Figure: Real (left) and imaginary (right) part of Scut(s) when s lies in

(cL, cR). The dot-dashed line comes from N/D solution, the yellow solid

line is obtained from O(p1) χPT results.

Scut =Sphys∏

p Sp × SvL × SvR

(21)

Hanqing Zheng SCU



On (cL, cR) the imaginary part of T (1) reads:

ImT (1)(s) =g2m2

Ns2

16F 2 (s − sL)2 (s − sR)2

×[m2

N

s(s − cL)

(−m2

N −m2π + s

)− mN√

s(s − cR)

(−m2

N + m2π + s

)],

ImT (1) develops a zero at s = sc ' m2N −

m4π

2m2N

and changes sign

when s crosses sc . Immune of chiral corrections.

Hanqing Zheng SCU



Figure: f (s)��p�ã

Hanqing Zheng SCU



Modify dispersion relations for f

f (s) =ln−Scut

2iρ(s)− π

2ρ(s)(22)

Justified by the fact that both f and f give the same value in the

physical region! ρ = ρ in the physical region but its cut ∈ [sL, sR ].

f no longer contains branch point sc , and its cut ∈ [−∞, cR ].

It is verified that the cut ∈ [sL, cL] exactly cancels the contribution

from the two virtual poles at vL and vR when vL = cL, vR = cR ;

leaves previous calculations unchanged.

Hanqing Zheng SCU



Higher partial waves

©ÅÚ^Ý�Ì�L�ªµ

T I ,J++ = 2mNA

I ,JC (s) + (s −m2

π −m2N)B I ,J

C (s) §

T I ,J+− = − 1√

s[(s −m2

π + m2N)AI ,J

S (s) + mN(s + m2π −m2

N)B I ,JS (s)]

§�UìXe�ª|Üµ

T I ,J± = T I ,J

++ ± T I ,J+−§ (23)

��T I ,J± ©OéA;��Äþl = J ∓ 1/2�P = (−1)J±1/2"

S I ,J± = 1 + 2iρT I ,J

± (24)

Hanqing Zheng SCU



Higher partial waves

I 3«m(cR , sR)S§Ó ^�I = 1/2��Ñ�3��J�¶

I = 3/2��KØ(½"

I 3«m(sL, cL)Sµ3|:cL?§�J = 1/2, 5/2, 9/2, ...�§

T1/2,J+ �T

3/2,J− → +∞¶ �J = 3/2, 7/2, 11/2, ... �§

T1/2,J− �T

3/2,J+ → +∞"Ïd�±íÑµI = 1/2�§

l = 0, 2, 4, 6...(P = −) ��3��J�¶ I = 3/2�§

l = 1, 3, 5, 7...(P = +) ��3��J�"

Hanqing Zheng SCU



Location of Virtual Poles

éu«m(cR , sR)p�J�vJR±§�J�u,��ê�§

vJR±¬�XJO� Åì�CcR§��J →∞�Âñ�cR" é

uv1/2,JL± �v

3/2,JL± K¬ÅìÂñ�cL"

Hanqing Zheng SCU



Essential Singularities for πN scatterings

[Li QZ, HZ, to appear]

T (s, t) = 16π∞∑

J=1/2

(2J + 1)T J(s)dJ1/2,−1/2(cos θ) §

T II+±(s, t) = 8π

∑J=1/2

(2J + 1)

[T J

+(s)

SJ+(s)

±T J−(s)

SJ−(s)

]dJ

1/2,±1/2(cos θ)

s = cL, cR , accumulation of poles on sheet II. So essential

singularities of T (s, t) on sheet II of s.

Hanqing Zheng SCU



Hanqing Zheng SCU



Physics of N∗(890)?

Hanqing Zheng SCU



Physics of N∗(890)

N ′ searched for desperately for 50 years.

Courtesy of Igor Strakovsky. talk given at EHS–2019

Hanqing Zheng SCU



Λ(1330)

Listed in RPP, the lowest lying Λ baryons with JP = 12

−are

Λ(1380)(**) and Λ(1405)(****). There is no recent experimental

information available for the former. However, [Guo Oller 13] found two

poles in the KN,Σπ couple channel UχPT model, with√s = 1388± 9− i114+24

−25 MeV on RS-III, and√s = 1421+3

−2 − i19+8−5 MeV on RS-III too.

Hanqing Zheng SCU



Σ(1475) JP =??

The lowest lying Σ baryons with possible spin parity JP = 12

−

are Σ(1620)(*), Σ(1750)(***). In [Zychor et al. 06] a possible hyperon

state with M = 1480± 15MeV, Γ = 60± 15 MeV is observed with

the ANKE spectrometer at COSY-Julich. In the 2020 edition of

RPP, the “bump” Σ(1480) is however removed. Its parity is not

determined in early edition of RPP as well.

In models, which couple nucleons with kaons and pions,

quasi-bound states can be generated with relatively low masses. In

[Garcia 04], a pole with the quantum numbers of the Σ, which might

be identified with the Σ(1480), is found with a mass of 1446 MeV,

and a large width of 343 MeV.

Hanqing Zheng SCU



Ξ(1620) JP = 12

?

Experimental evidences have been accumulated in [Sumihama et al.

19], [Guo et al. 07], [Briefel 77]. The spin quantum number of Ξ(1620) is not

determined.

In model calculations it was explained as a KΛ molecular

state [Ramos 02], [Chen 19].

Hanqing Zheng SCU



Hanqing Zheng SCU



Conclusions

It is apparent that the discovery of a light 1/2− nucleon state

is crucial for the completion and establishment of the lowest lying

1/2− octet baryons, and it will definitely contribute significantly to

our understanding of the strong interaction physics.

Hanqing Zheng SCU



– Thanks for Patience!Hanqing Zheng SCU



Backups

Hanqing Zheng SCU



pion-nucleon scatterings

I The πN scattering → one of the most fundamental and

important processes in nuclear or hadron physics

I Decades of researching

I Various experiments and phenomena

(L2I 2J convention, W =√s, Sr = 1− η2)[SAID: WI 08]

1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 00

5 0

1 0 0

1 5 0

Phas

e shif

t (Deg

ree)

W ( M e V )

S 1 1

1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0 S 1 1

S r

W ( M e V )

Hanqing Zheng SCU



1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 0- 7 0

- 6 0

- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0

1 0

2 0

Phas

e shif

t (Deg

ree)

W ( M e V )

S 3 1

1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0 S 3 1

S r

W ( M e V )

1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 00

5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

Phas

e shif

t (Deg

ree)

W ( M e V )

P 1 1

1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0 P 1 1

S r

W ( M e V )

Hanqing Zheng SCU



1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 0- 8 0

- 6 0

- 4 0

- 2 0

0

Phas

e shif

t (Deg

ree)

W ( M e V )

P 3 1

1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0 P 3 1

S r

W ( M e V )

1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 0- 3 0

- 2 5

- 2 0

- 1 5

- 1 0

- 5

0

5

1 0

Phas

e shif

t (Deg

ree)

W ( M e V )

P 1 3

1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 0

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8P 1 3

S r

W ( M e V )

Hanqing Zheng SCU



1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 00

5 0

1 0 0

1 5 0

2 0 0

Phas

e shif

t (Deg

ree)

W ( M e V )

P 3 3

1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8P 3 3

S r

W ( M e V )

Hanqing Zheng SCU



Theoretical discussions

I Problems to study

I Low energy properties:

πN σ-term, subthreshold expansions

[C. Ditsche et. al. 2012 JHEP][Hoferichter et. al. 2016 Phys.Rept.]

I Intermediate resonances: ∆(1232), N∗(1440), N∗(1535) · · ·

I Methods

I Perturbative calculation [J.M. Alarcon et. al. 2012] [Y. H. Chen et. al. 2013]

I Couple channel Lippmann-Schwinger Equation

[O. Krehl et. al. 2000 PRC]

I Dispersion technique [A. Gasparyan and M.F.M. Lutz 2010]

I Roy-Steiner equation [C. Ditsche et. al. 2012] [Hoferichter et. al. 2016]

Hanqing Zheng SCU



I S11 channel (L2I 2J convention): N∗(1535)

[N. Kaiser et. al. 1995 PLB][J. Nieves et. al. 2000 PRD]

I lies above the P- wave first resonance N∗(1440)

I large couple channel effects with πN and ηN

I P11 channel: N∗(1440) (Ropper resonance), various puzzles

I low mass, large decay width, coupling to σN channel...

[O. Krehl et. al. 2000 PRC]

I two-pole structure? [R. A. Arndt et. al. 1985 PRD]

I second sheet complex branch cut in P11 channel?

[S. Ceci et. al. 2011 PRC]

I A method is needed to examine the relevant channels carefully

and to exhume more physics behind

I low energy

I model independent

Hanqing Zheng SCU



πN Lagrangian

I Covariant baryon chiral perturbation theory, SU(2) case.

I Lagrangians [N. Fettes et. al. 2000 Ann. Phys. ]

I O(p1)µ

L(1)πN = N

(iD/−M +

1

2gu/γ5

)N

I O(p2) (“〈〉” stands for trace in isospin space)µ

L(2)πN = c1〈χ+〉NN − c2

4M2N

〈uµuν〉(NDµDνN + h.c.)

+c3

2〈uµuµ〉NN − c4

4Nγµγν

[uµ, uν

]N

Hanqing Zheng SCU



Conventions

I Conventions

Dµ = ∂µ + Γµ

Γµ =1

2

[u†(∂µ − irµ)u + u(∂µ − ilµ)u†

]uµ = i

[u†(∂µ − irµ)u − u(∂µ − ilµ)u†

]χ± ≡ u†χu† ± uχ†u

χ = 2B0(s + ip)

h µν =

[Dν , u

µ]

+[Dµ, uν

]In calculation 2B0s → 2B0mq = m2

π, other sources are

switched off

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Isospin decomposition

I Symmetric part vs. anti-symmetric part

T (πa + Ni → πa′

+ Nf) = χ†f

(δaa′T S +

1

2

[τ a′, τ a]TA

)χi

I Isospin channels

T I=1/2 = T S + 2TA

T I=3/2 = T S − TA

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Helicity structureI Lorentz structure (where D = A + (s − u)B/(4MN))

T S,A = u(p′, s′)[AS,A(s, t) +

1

2(q/+ q/′)BS,A(s, t)

]u(p, s)

= u(p′, s′)[DS,A(s, t) +

iσµνqνq′µ

2MBS,A(s, t)

]u(p, s)

I Helicity amplitudes (zs = cos θ)

T++ = (1 + zs

2)

12 [2MNA(s, t) + (s −m2

π −M2N)B(s, t)]

T+− = −(1− zs

2)

12 s−

12 [(s −m2

π + M2N)A(s, t) + MN(s + m2

π −M2N)B(s, t)]

I Partial wave projection

T J++ =

1

32π

∫ 1

−1dzsT++(s, t(s, zs))dJ

−1/2,−1/2(zs)

T J+− =

1

32π

∫ 1

−1dzsT+−(s, t(s, zs))dJ

1/2,−1/2(zs)

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Channels to be analyzed

L2I 2J convention

T (S11) = T++(I = 1/2, J = 1/2) + T+−(I = 1/2, J = 1/2)

T (S31) = T++(I = 3/2, J = 1/2) + T+−(I = 3/2, J = 1/2)

T (P11) = T++(I = 1/2, J = 1/2)− T+−(I = 1/2, J = 1/2)

T (P31) = T++(I = 3/2, J = 1/2)− T+−(I = 3/2, J = 1/2)

T (P13) = T++(I = 1/2, J = 3/2) + T+−(I = 1/2, J = 3/2)

T (P33) = T++(I = 3/2, J = 3/2) + T+−(I = 3/2, J = 3/2)

Each channel satisfies unitarity condition.

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Branch cut structure

[S. W. MacDowell 1959 PR][J. Kennedy and T. D. Spearman 1961 PR]

t channelWhen t ′ takes the value from σt to +∞, the trajectory ⇒

I when t ′ ∈ [4m2π, 4m

2N ], the cut appear as a circle

Res2 + Ims2 = (m2N −m2

π)2, and the endpoint to the right

s = (m2N −m2

π)2 corresponds to t ′ = σt = 4m2π;

I when t ′ ∈ (4m2N ,+∞), s− generates the cut (−∞,m2

π −m2N),

and −∞ corresponds to t ′ → +∞;

I when t ′ ∈ (4m2N ,+∞), s+ generates the cut (m2

π −m2N , 0),

and actually 0 corresponds to t ′ → +∞;

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u channel

The solution is

s1(u′) =(m2

π −m2N)2

u′, s2(u′) = 2(m2

π + m2N)− u′ . (25)

There is a nucleon pole u′ = m2N , giving a segment cut

((m2π −m2

N)2/m2N , 2m

2π + m2

N). When u′ > σu = (mπ + mN)2, s1

gives (0, (mN −mπ)2) and s1 → 0 just when u′ → +∞; while s2

generates (−∞, (mN −mπ)2) with s2 → −∞ when u′ → +∞.

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Tree level left-hand cut

I Tree level left-hand cut of S

I (−∞, (MN −mπ)2]→ From logs and relativistic kinematics!

I [(M2N −m2

π)2/M2N ,M

2N + 2m2

π]→ u channel nucleon exchange

→ very small

I The main contribution of f (s) (with a cut-off sc)

f (s) =s

π

∫ (MN−mπ)2

sc

σ(w)dw

w(w − s)

I The dispersion spectral function

σ(w) = Im{ ln |Stree|

2iρ(w)

}= − ln |1 + 2iρ(w)Ttree|

2ρ(w)

negative definite

I Right-hand inelastic cuts are omitted for the momentHanqing Zheng SCU



Right-hand inelastic cut

I Right-hand inelastic cut contribution →positive definite

fR′(s) =s

π

∫ Λ2R

(2m+M)2

σ(w)dw

w(w − s)

σ(w) = −{ ln[η(w)]

2ρ(w)

}I η: inelasticity, from SAID WI 08 data and extrapolation

I Cut-off: ΛR = 4.00GeV

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Right-hand inelastic cut

1 . 0 8 1 . 1 0 1 . 1 2 1 . 1 4 1 . 1 60

2

4

6

8

Phas

e shif

t (Deg

ree)

W ( G e V )

P 1 1 S 1 1

1 . 0 8 1 . 1 0 1 . 1 2 1 . 1 4 1 . 1 60 . 00 . 20 . 40 . 60 . 81 . 01 . 21 . 4

Phas

e shif

t (Deg

ree)

W ( G e V )

S 3 1 P 3 3 P 1 3 P 3 1

Far from enough!!

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γN → πN process

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[Ma Y. et al., CPC, 2021]

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