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• Veneziano model and application to Dalitz plot analysis

Adam Szczepaniak, Indiana U./JLab

Motivation

Properties

Application to J/ 3 decays

Generalizations (dual models)

• 1

2

3

4

s = (1 + 2)2 t = (1 + 3)2 u = (1 + 4)2

physical domain in the s-channel

1 2

3_

4_

|zs| < 11 3

2_

4_

|zt| < 1

E3 = -E3 > 0_

physical domain in the t-channel

E2 = -E2 > 0_

s + t +u = 4m2

2-to-2 kinematicse.g. 0 0 -> 0 0 : A(s,t,u)

E4 = -E4 > 0_

E4 = -E4 > 0_

• A(s, t) =1X

l=0

(2l + 1)Al(s)Pl(zs)

t-channel1

2

3

4

s-channel

1 + 2 -> 3 + 4 1 + 3 -> 2 + 4 __

A(s, t)

peaks in s from resonances in (12)

peaks in t from resonances in (13)

_

1

2

3

4

_

_

A(s, t) =1X

l=0

(2l + 1)Al(t)Pl(zt)

zs = 1 +2t

s 4m2 zt = 1 +2s

t 4m2

to reproduce peaks in t (or s) need to continue the s (or t)channel p.w. sum outside its domain of convergence

• s

ut s-channel

u-channelt-channel

s=mr2

t=mr2 u=mr2

zeros from Pl speaks from poles

large -zs large +zs

zs = 1 +2t

s 4m2

• Analytical continuation: simple example

t(s, zs) = (1 zs)s 4m2

2

A(s, t) = A(s, t(s, zs)) =1X

l=0

(2l + 1)Al(s)Pl(zs)

well defined for |zs| < 1

-1 +1-3

|zs| < 1 in the s-channel and zs < -1 in the t-channel

A(zs) =1X

l=0

(zs)l = 1 zs + z2s +

A(zs) =1

1 + zs

• for large -zs

= 1

t(s, zs) = (1 zs)s 4m2

2

A(s, t) = A(s, t(s, zs)) =1X

l=0

(2l + 1)Al(s)Pl(zs)

compare with starting point

we derived the large +t behavior !

there is a pole in the physical t region (resonance) if non-integer, tasks as a function of t the amplitude has a

branch point: particle production at lathe t-channel energy the leading term at large t is simple: it must come from some

specific property of the p.w. series

A(zs) =1

1 + zs= (zs)

[1 +O(1/zs)]

t / zs when (t ! +1)

Exercise: find the large-zs behavior using the Sommerfeld-Watson transform.

• (s) =1

2+ s+ i(s) (s)(s) = Im(s)

(s) =p1 4m2/s

A1(s) ~ Breit-Wigner of the rho-meson Re((s)) = 1/2 + s : linear Regge trajectory Im((s)) = is related to resonance widths the relation between (trajectory) and (residue) follows from

unitarity: Im A1(s) = |Al(s)|2 (s) Resonances with different spins in A1, A3, A5, are related by

poles in l of the function Al

Analytical continuation: realistic example

1X

l=0

zls (zs)l

l /1 ei

sin(zs)Exercise:show that

Exercise:show that (*) follows from unitarity

(*)

Exercise:show that (**) has a Breit-Wigner form

(**)

m

A(s, t) =1X

l=0

(2l + 1)Al(s)Pl(zs) =1X

l=0

(s)

l (s)[Pl(zs) Pl(zs)]

2

• Regge theory = origin and properties of singularities of p.w. in the angular momentum

plane

the leading term at large t is simple: it must come from some specific property of the p.w. series -> it comes from right most singularity of partial waves in the angular momentum plane

it is possible (simple) to construct models of partial waves in one channel (e.g. s) which have Regge poles and produce right asymptotic behavior in another (e.g. t). It is not easy to do it simultaneously

1X

l=0

zls (zs)l

l /1 ei

sin(zs)

• s-channel p.w. : s-channel resonances

Q: Should you add t-channel resonances (interference model) ?

LmaxX

l=0

(2l + 1)Al(t)Pl(zt) ! sLmax

A finite number of t-channel resonances will break s-channel analyticity

An infinite number of t-channel resonance -> double counting of A(s,t)

Veneziano model = has simultaneous resonances in s and t-channel and proper asymptotic behavior. To do this requires

number of p.w/resonances (why?)

A(s, t) =1X

l

(2l + 1)Al

(s)Pl

(s), Al

(s) =X

i=Regge poles

i

(s)

l i

(s)

A: No. (resonances in t and s are dual not additive)

• What functions have an infinite number of poles (resonances) to be

used to represent (model) the amplitude A(s,t) ?

The Gamma function !!!

• (z) (1)n

(n+ 1)

1

n+ zz ~ -n

|(z)|

A(s, t) (t)(s)

so we want something like

• A(s, t) (t)(s)

to connect poles at t (or s) = 1,2,3 with physical masses use the Regge trajectory

n,m determine location of first poles, e.g. (-(s)) has, for (s) = 0.5 + s, the first pole at s=-1/2 i.e. particle with imaginary mass. But (1-(s)) has the first pole at s=+1/2 i.e. the rho-meson

simultaneous poles in s and t (overlapping channels) are unexpected. To remove them use

A(s, t) (n (s))(n (t))

A(s, t) (n (s))(n (t))(n+m (s) (t))

n m 1

what is missing is kinematical (spin) and symmetry (e.g. isospin) factors

it has poles and zeros!

• Examples:

->

after isospin decomposition there are three scalar amplitudes A(s,t,u), B(s,t,u), C(s,t,u) -> Veneziano

N -> N

there are two scalar amplitudes A(s,t), B(s,t)

V ->

there is one scalar amplitude

Exercise: verify (*)

(*)

• J/

BESIII, Phys.Lett. B710 (2012) 594-599

standard (isobar)

A(s, t) =(J(s))(J(t))(J(s) J(t))

dual model

J/dual model

PRELIMINARY

! ! 3

• Veneziano amplitude: compact expression for the full amplitude

A(s,t) can be written as sum over resonances in ether channel.

Note: in V-model resonance couplings, , are fixed!

A(s, t) =((s))((t))((s) (t))

A(s, t) =X

k

k(t)

k (s) =X

k

k(s)

k (t)

k(t) / (1 + (t))(2 + (t)) (k + (t))

(s) = a+ bs

Exercise: verify (*)

(*)

• 2

s1 s2 s3 s4 s5 s6 s

l

1

2

3

4

5

leading 1st 2nd 3rd

FIG. 1: Spectrum in the s-channel of the generalizedVeneziano amplitude model of Eq. 1. The leading and daugh-ter Regge trajectories are marked by thin solid lines and res-onances by dots at integer values of spin (l). The dashedand dotted thick lines illustrate resonance contributes to in-dividual amplitudes, A2,1 and A4,3, respectively. All (infinitenumber) of resonances on and to the right of the dashed linecontribute to A2,1, while resonances on and to the right of thedotted line contribute to A4,3.

leading trajectory and from all subsequent daughter tra-jectories. The amplitudes A

n,2 have poles originatingfrom the the 1st daughter and all subsequent daughters,A

n,3 from the 2nd and all higher daughters, etc. Thus wecan use m to label Regge trajectories and define,

(m)(s) (s) (m 1) (6)

so that (1)(s) (s) corresponds to the leading trajec-tory, (2) the 1st daughter and so on. The spectrum isillustrated in Fig. 4. For fixed-t, the asymptotic behaviorof A

n,m

(s, t) at large-s reflects presence of resonances inthe crossed channel. Using Stirlings formula one finds,

An,m

(s ! 1, t) / 1s(n

t

)s(m)(t) (7)

For large-s the tensor factor in Eq. 1 is proportional to sand the full amplitude has the expected Regge limit,

A(s, t, u) / s(t) (8)

arising from the leading, m = 1 trajectory. The signaturefactor will be discussed later.

III. REMOVAL OF POLES

As described in the preceding section, an amplitudeA

n,m

with fixed n and m contains an infinite number ofpoles in a two-body channel it describes. Since produc-tion of resonances is process dependent it is necessary to

find a generalization of the amplitude that allows for theresidues to be process dependent. One possibility is touse a linear combination

An,m

(s, t) ! A(s, t) =X

n1,nm1cn,m

An,m

(s, t) (9)

The coecients cn,m

need to be chosen in such a waythat As only couples to resonances that contribute tothe process in question. For example, in the case of anisoscalar boson strongly coupled to three pions, isospinconservation demands each pair of pions be produced inisospin-1. Bose statistics then eliminates all spin-evenresonances in s t and u channels of this reaction.One way to proceed is to construct linear combinations

of amplitudes An,m

that eliminate all, but selected par-tial waves and then take linear combinations of partialwaves. Alternatively one can attempt data analysis witha finite number of linear combinations of the A

n,m

s andlet the fit to data determine coecients c

n,m

[? ]. Wefind the former more appealing for several reasons. Firstof all, when studying resonance properties one is forcedto work with partial waves. Proper description of reso-nances, however, requires that unitarity is satisfied andRegge trajectories are non-linear, while the Venezianomodel forces Regge trajectories to be real and linear.Even though there are extensions of the Veneziano modelallowing for non-linear trajectories, implementation ofunitarity is much simpler at the level of partial waves.We therefore need to be able to isolate partial waves.Using the Veneziano amplitudes as building blocks, how-ever, we will be able match the low-energy behavior ofpartial waves with the asymptotic high-energy limit de-termined by Regge poles. This is important as it providesa constraint on data analysis that extends beyond whatresonances alone can fix.Since each A

n,n

amplitude contains an infinite numberof poles, in order to cancel all, but a finite number ofpoles an infinite number of coecients c

n,m

s in Eq. 9must be non vanishing. It is not dicult to find a rela-tion between the coecients, which decouples all, but afinite number of poles. Consider, for example, keepingonly the pole at (s) = 1 i.e. at s = s1. This pole isonly present in the amplitude A1,1 since amplitudes withn > 1 have the lowest pole at s

n

> s1. There is onlyone amplitude A1,m = A1,1 so a single coecient c1,1 de-termines coupling to the pole at s = s1. The amplitudeA1,1, however, also has poles at higher masses locatedat

s

= 2, 3, with residues that are polynomials int of the order of 1, 2, , respectively. If we only wantto keep the pole at (s) = 1, these higher mass polesof A1,1 have to be canceled by similar poles present inamplitudes with n > 1.The pole in A1,1 at s = 2 can only by canceled by the

same pole in the two amplitudes A2,m, m = 1, 2 since forn > 2 no other A

n,m

has this pole. The amplitudes A2,1and A2,2 are polynomials in t of the order of O(1) andO(0) respectively. We can therefore uniquely determinetwo coecients, c2,1 and c2,2 in terms of c1,1 so that the

Veneziano Model Motivated Amplitudes for Dalitz Plot Analysis

A.P.Szczepaniak1

1Physics Department Indiana University, Bloomington, IN 47405, USA

Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403, USA

Jeerson Laboratory, 12000 Jeerson Avenue, Newport News, VA 23606, USA

(Dated: December 15, 2013)

Needs to be written

PACS numbers:

I. INTRODUCTION

Unless partial waves can be related to the full ampli-tude, partial waves, beyond the resonance contributionpartial waves are largely unconstrained. The Venezianomodel provides this connections.

II. GENERALIZED VENEZIANO AMPLITUDES

We consider the generalized Veneziano model in thecontext of decays of mesons with heavy quarks to finalstates with light quarks. Specifically we focus on the ap-plication to decays of vector charmonia, e.g. J/ or 0 tothree pions. For simplicity we neglect pion messes and,in units of GeV, use M2 = O(10) for the mass squared ofthe decaying particle. The generalization to other chan-nels is in principle straightforward.

The Veneziano model for the amplitude A describinga decay of a vector meson with momentum p and helicity to three pions, V (p,) ! i(p1)j(p2)k(p3) is givenby

A(s, t, u) = ijk

(p,)p1p

2 p

3

[An,m

(s, t) +An,m

(s, u) +An,m

(t, u)] (1)

with s + t + u = M2 and the scalar amplitudes An,m

sdefined by

An,m

(s, t) (n s)(n t)(n+m

s

t

). (2)

Here n,m are positive integers, 1 m n, and s

is ashorthand for (s), the leading Regge trajectory. We donot use the Veneziano amplitude with n = 0 since with representing the leading meson trajectory it would pro-duce a ghost pole. The lower limit on m, m = 1 guaran-tees that A(s, t, u) has the expected high-energy behavior(see below) and the upper limit, m = n guarantees ab-sence of double poles in overlapping channels.

The Veneziano formula exhibits the behavior of A thatis expected in the large-N

c

limi of QCD. In this limit theQCD boson spectrum is saturated by narrow resonancesand confinement produces linear Regge trajectories, i.e.(s) = 0 + 0s. This spectrum is manifested in thesingularities of A

n,m

(s, t), which has simple poles in sand t-channels and no double poles. For example thes-channel poles are located at s = s

n+k determined by

(sn+k) = n+ k (3)

with k being a nonnegative integer. The contribution ofa pole to the amplitude A

n,m

(s, t) is then given by

A(s sn+k) =

n,m,k

(t)

sn+k s

(4)

with the residue

n,m,k

(t) =(1)k

0k!

(n t

)

(m k t

)=

(1)k

0k!(m k

t

)(m k + 1 t

) (m k + Lmax

1 t

)

being a polynomial in t of the order Lmax

= Lmax

(k) k + n m 0. Thus in each channel s, t, or u, the fullamplitude given by Eq. 1 describes k, degenerate, narrow(zero) width resonances with spins in the range,

1 l Lmax

(k) + 1. (5)

The additional factor of 1 originates from the tensor fac-tor in front of the r.h.s of Eq. 1.

The integers n and m determine which resonances con-tribute poles to the amplitude. It follows form Eqs. 3,5that amplitudes with m = 1, A

n,1 have poles from the

Veneziano Model Motivated Amplitudes for Dalitz Plot Analysis

A.P.Szczepaniak1

1Physics Department Indiana University, Bloomington, IN 47405, USA

Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403, USA

Jeerson Laboratory, 12000 Jeerson Avenue, Newport News, VA 23606, USA

(Dated: December 15, 2013)

Needs to be written

PACS numbers:

I. INTRODUCTION

Unless partial waves can be related to the full ampli-tude, partial waves, beyond the resonance contributionpartial waves are largely unconstrained. The Venezianomodel provides this connections.

II. GENERALIZED VENEZIANO AMPLITUDES

We consider the generalized Veneziano model in thecontext of decays of mesons with heavy quarks to finalstates with light quarks. Specifically we focus on the ap-plication to decays of vector charmonia, e.g. J/ or 0 tothree pions. For simplicity we neglect pion messes and,in units of GeV, use M2 = O(10) for the mass squared ofthe decaying particle. The generalization to other chan-nels is in principle straightforward.

The Veneziano model for the amplitude A describinga decay of a vector meson with momentum p and helicity to three pions, V (p,) ! i(p1)j(p2)k(p3) is givenby

A(s, t, u) = ijk

(p,)p1p

2 p

3

[An,m

(s, t) +An,m

(s, u) +An,m

(t, u)] (1)

with s + t + u = M2 and the scalar amplitudes An,m

sdefined by

An,m

(s, t) (n s)(n t)(n+m

s

t

). (2)

Here n,m are positive integers, 1 m n, and s

is ashorthand for (s), the leading Regge trajectory. We donot use the Veneziano amplitude with n = 0 since with representing the leading meson trajectory it would pro-duce a ghost pole. The lower limit on m, m = 1 guaran-tees that A(s, t, u) has the expected high-energy behavior(see below) and the upper limit, m = n guarantees ab-sence of double poles in overlapping channels.

The Veneziano formula exhibits the behavior of A thatis expected in the large-N

c

limi of QCD. In this limit theQCD boson spectrum is saturated by narrow resonancesand confinement produces linear Regge trajectories, i.e.(s) = 0 + 0s. This spectrum is manifested in thesingularities of A

n,m

(s, t), which has simple poles in sand t-channels and no double poles. For example thes-channel poles are located at s = s

n+k determined by

(sn+k) = n+ k (3)

with k being a nonnegative integer. The contribution ofa pole to the amplitude A

n,m

(s, t) is then given by

A(s sn+k) =

n,m,k

(t)

sn+k s

(4)

with the residue

n,m,k

(t) =(1)k

0k!

(n t

)

(m k t

)=

(1)k

0k!(m k

t

)(m k + 1 t

) (m k + Lmax

1 t

)

being a polynomial in t of the order Lmax

= Lmax

(k) k + n m 0. Thus in each channel s, t, or u, the fullamplitude given by Eq. 1 describes k, degenerate, narrow(zero) width resonances with spins in the range,

1 l Lmax

(k) + 1. (5)

The additional factor of 1 originates from the tensor fac-tor in front of the r.h.s of Eq. 1.

The integers n and m determine which resonances con-tribute poles to the amplitude. It follows form Eqs. 3,5that amplitudes with m = 1, A

n,1 have poles from the

Veneziano Model Motivated Amplitudes for Dalitz Plot Analysis

A.P.Szczepaniak1

1Physics Department Indiana University, Bloomington, IN 47405, USA

Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403, USA

Jeerson Laboratory, 12000 Jeerson Avenue, Newport News, VA 23606, USA

(Dated: December 15, 2013)

Needs to be written

PACS numbers:

I. INTRODUCTION

Unless partial waves can be related to the full ampli-tude, partial waves, beyond the resonance contributionpartial waves are largely unconstrained. The Venezianomodel provides this connections.

II. GENERALIZED VENEZIANO AMPLITUDES

We consider the generalized Veneziano model in thecontext of decays of mesons with heavy quarks to finalstates with light quarks. Specifically we focus on the ap-plication to decays of vector charmonia, e.g. J/ or 0 tothree pions. For simplicity we neglect pion messes and,in units of GeV, use M2 = O(10) for the mass squared ofthe decaying particle. The generalization to other chan-nels is in principle straightforward.

The Veneziano model for the amplitude A describinga decay of a vector meson with momentum p and helicity to three pions, V (p,) ! i(p1)j(p2)k(p3) is givenby

A(s, t, u) = ijk

(p,)p1p

2 p

3

[An,m

(s, t) +An,m

(s, u) +An,m

(t, u)] (1)

with s + t + u = M2 and the scalar amplitudes An,m

sdefined by

An,m

(s, t) (n s)(n t)(n+m

s

t

). (2)

Here n,m are positive integers, 1 m n, and s

is ashorthand for (s), the leading Regge trajectory. We donot use the Veneziano amplitude with n = 0 since with representing the leading meson trajectory it would pro-duce a ghost pole. The lower limit on m, m = 1 guaran-tees that A(s, t, u) has the expected high-energy behavior(see below) and the upper limit, m = n guarantees ab-sence of double poles in overlapping channels.

The Veneziano formula exhibits the behavior of A thatis expected in the large-N

c

limi of QCD. In this limit theQCD boson spectrum is saturated by narrow resonancesand confinement produces linear Regge trajectories, i.e.(s) = 0 + 0s. This spectrum is manifested in thesingularities of A

n,m

(s, t), which has simple poles in sand t-channels and no double poles. For example thes-channel poles are located at s = s

n+k determined by

(sn+k) = n+ k (3)

with k being a nonnegative integer. The contribution ofa pole to the amplitude A

n,m

(s, t) is then given by

A(s sn+k) =

n,m,k

(t)

sn+k s

(4)

with the residue

n,m,k

(t) =(1)k

0k!

(n t

)

(m k t

)=

(1)k

0k!(m k

t

)(m k + 1 t

) (m k + Lmax

1 t

)

being a polynomial in t of the order Lmax

= Lmax

(k) k + n m 0. Thus in each channel s, t, or u, the fullamplitude given by Eq. 1 describes k, degenerate, narrow(zero) width resonances with spins in the range,

1 l Lmax

(k) + 1. (5)

The additional factor of 1 originates from the tensor fac-tor in front of the r.h.s of Eq. 1.

The integers n and m determine which resonances con-tribute poles to the amplitude. It follows form Eqs. 3,5that amplitudes with m = 1, A

n,1 have poles from the

Veneziano Model Motivated Amplitudes for Dalitz Plot Analysis

A.P.Szczepaniak1

1Physics Department Indiana University, Bloomington, IN 47405, USA

Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403, USA

Jeerson Laboratory, 12000 Jeerson Avenue, Newport News, VA 23606, USA

(Dated: December 15, 2013)

Needs to be written

PACS numbers:

I. INTRODUCTION

Unless partial waves can be related to the full ampli-tude, partial waves, beyond the resonance contributionpartial waves are largely unconstrained. The Venezianomodel provides this connections.

II. GENERALIZED VENEZIANO AMPLITUDES

We consider the generalized Veneziano model in thecontext of decays of mesons with heavy quarks to finalstates with light quarks. Specifically we focus on the ap-plication to decays of vector charmonia, e.g. J/ or 0 tothree pions. For simplicity we neglect pion messes and,in units of GeV, use M2 = O(10) for the mass squared ofthe decaying particle. The generalization to other chan-nels is in principle straightforward.

The Veneziano model for the amplitude A describinga decay of a vector meson with momentum p and helicity to three pions, V (p,) ! i(p1)j(p2)k(p3) is givenby

A(s, t, u) = ijk

(p,)p1p

2 p

3

[An,m

(s, t) +An,m

(s, u) +An,m

(t, u)] (1)

with s + t + u = M2 and the scalar amplitudes An,m

sdefined by

An,m

(s, t) (n s)(n t)(n+m

s

t

). (2)

Here n,m are positive integers, 1 m n, and s

is ashorthand for (s), the leading Regge trajectory. We donot use the Veneziano amplitude with n = 0 since with representing the leading meson trajectory it would pro-duce a ghost pole. The lower limit on m, m = 1 guaran-tees that A(s, t, u) has the expected high-energy behavior(see below) and the upper limit, m = n guarantees ab-sence of double poles in overlapping channels.

The Veneziano formula exhibits the behavior of A thatis expected in the large-N

c

limi of QCD. In this limit theQCD boson spectrum is saturated by narrow resonancesand confinement produces linear Regge trajectories, i.e.(s) = 0 + 0s. This spectrum is manifested in thesingularities of A

n,m

(s, t), which has simple poles in sand t-channels and no double poles. For example thes-channel poles are located at s = s

n+k determined by

(sn+k) = n+ k (3)

with k being a nonnegative integer. The contribution ofa pole to the amplitude A

n,m

(s, t) is then given by

A(s sn+k) =

n,m,k

(t)

sn+k s

(4)

with the residue

n,m,k

(t) =(1)k

0k!

(n t

)

(m k t

)=

(1)k

0k!(m k

t

)(m k + 1 t

) (m k + Lmax

1 t

)

being a polynomial in t of the order Lmax

= Lmax

(k) k + n m 0. Thus in each channel s, t, or u, the fullamplitude given by Eq. 1 describes k, degenerate, narrow(zero) width resonances with spins in the range,

1 l Lmax

(k) + 1. (5)

The additional factor of 1 originates from the tensor fac-tor in front of the r.h.s of Eq. 1.

The integers n and m determine which resonances con-tribute poles to the amplitude. It follows form Eqs. 3,5that amplitudes with m = 1, A

n,1 have poles from the

n m 1

no-double poles correct asymptotic limit

(s) =1

2+ s

Exercise: verify (*)

(*)

• Resonances couplings, , should depend on final state particles: a linear superposition of Veneziano amplitudes can

be used to suppress or enhance individual resonances or trajectories

2

s1 s2 s3 s4 s5 s6 s

l

1

2

3

4

5

leading 1st 2nd 3rd

FIG. 1: Spectrum in the s-channel of the generalizedVeneziano amplitude model of Eq. 1. The leading and daugh-ter Regge trajectories are marked by thin solid lines and res-onances by dots at integer values of spin (l). The dashedand dotted thick lines illustrate resonance contributes to in-dividual amplitudes, A2,1 and A4,3, respectively. All (infinitenumber) of resonances on and to the right of the dashed linecontribute to A2,1, while resonances on and to the right of thedotted line contribute to A4,3.

leading trajectory and from all subsequent daughter tra-jectories. The amplitudes A

n,2 have poles originatingfrom the the 1st daughter and all subsequent daughters,A

n,3 from the 2nd and all higher daughters, etc. Thus wecan use m to label Regge trajectories and define,

(m)(s) (s) (m 1) (6)

so that (1)(s) (s) corresponds to the leading trajec-tory, (2) the 1st daughter and so on. The spectrum isillustrated in Fig. 4. For fixed-t, the asymptotic behaviorof A

n,m

(s, t) at large-s reflects presence of resonances inthe crossed channel. Using Stirlings formula one finds,

An,m

(s ! 1, t) / 1s(n

t

)s(m)(t) (7)

For large-s the tensor factor in Eq. 1 is proportional to sand the full amplitude has the expected Regge limit,

A(s, t, u) / s(t) (8)

arising from the leading, m = 1 trajectory. The signaturefactor will be discussed later.

III. REMOVAL OF POLES

As described in the preceding section, an amplitudeA

n,m

with fixed n and m contains an infinite number ofpoles in a two-body channel it describes. Since produc-tion of resonances is process dependent it is necessary to

find a generalization of the amplitude that allows for theresidues to be process dependent. One possibility is touse a linear combination

An,m

(s, t) ! A(s, t) =X

n1,nm1cn,m

An,m

(s, t) (9)

The coecients cn,m

need to be chosen in such a waythat As only couples to resonances that contribute tothe process in question. For example, in the case of anisoscalar boson strongly coupled to three pions, isospinconservation demands each pair of pions be produced inisospin-1. Bose statistics then eliminates all spin-evenresonances in s t and u channels of this reaction.One way to proceed is to construct linear combinations

of amplitudes An,m

that eliminate all, but selected par-tial waves and then take linear combinations of partialwaves. Alternatively one can attempt data analysis witha finite number of linear combinations of the A

n,m

s andlet the fit to data determine coecients c

n,m

[? ]. Wefind the former more appealing for several reasons. Firstof all, when studying resonance properties one is forcedto work with partial waves. Proper description of reso-nances, however, requires that unitarity is satisfied andRegge trajectories are non-linear, while the Venezianomodel forces Regge trajectories to be real and linear.Even though there are extensions of the Veneziano modelallowing for non-linear trajectories, implementation ofunitarity is much simpler at the level of partial waves.We therefore need to be able to isolate partial waves.Using the Veneziano amplitudes as building blocks, how-ever, we will be able match the low-energy behavior ofpartial waves with the asymptotic high-energy limit de-termined by Regge poles. This is important as it providesa constraint on data analysis that extends beyond whatresonances alone can fix.Since each A

n,n

amplitude contains an infinite numberof poles, in order to cancel all, but a finite number ofpoles an infinite number of coecients c

n,m

s in Eq. 9must be non vanishing. It is not dicult to find a rela-tion between the coecients, which decouples all, but afinite number of poles. Consider, for example, keepingonly the pole at (s) = 1 i.e. at s = s1. This pole isonly present in the amplitude A1,1 since amplitudes withn > 1 have the lowest pole at s

n

> s1. There is onlyone amplitude A1,m = A1,1 so a single coecient c1,1 de-termines coupling to the pole at s = s1. The amplitudeA1,1, however, also has poles at higher masses locatedat

s

= 2, 3, with residues that are polynomials int of the order of 1, 2, , respectively. If we only wantto keep the pole at (s) = 1, these higher mass polesof A1,1 have to be canceled by similar poles present inamplitudes with n > 1.The pole in A1,1 at s = 2 can only by canceled by the

same pole in the two amplitudes A2,m, m = 1, 2 since forn > 2 no other A

n,m

has this pole. The amplitudes A2,1and A2,2 are polynomials in t of the order of O(1) andO(0) respectively. We can therefore uniquely determinetwo coecients, c2,1 and c2,2 in terms of c1,1 so that the

s

Re (s)Re (s) = a + b s

(770)

(1450)(1570)

3(1690)

(1900)

3(1990)

(2150)

3 (2250)

5 (2350)

M = p1p

2p

3

A(s, t, u)

A =X

n,m

cn,m

(n (s))(n (t))(n+m (s) (t)) + (s, u) + (t, u)

even-spin s do not couple to and should decouple in J/3

coupling of odd-spin s depend of can depend vary depending on trajectory

• An,m(s, t) =1X

k

(t)

k (s) =1X

k

(s)

k (t)

How to isolate individual poles ?

A1,1 =(1 s)(1 t)(2 s t)

has poles at s=1,2,3,...

A2,1 =(2 s)(2 t)(3 s t)

have poles at s=3,4,5,...

A2,2 =(2 s)(2 t)(4 s t)

has poles at s=2,3,4,...

A3,1, A3,2, A3,3

A4,1, A4,2, A4,3, A4,4

s

t

have poles at s=4,5,6,...

n m 1

Use a linear combination of A2,1 and A2,2 to remove pole at s =2

Use a linear combination of A3,1, A3,2 ,A3,3, to remove pole at s =3,

etc.

has poles at s=2,3,4,...

• 2

s1 s2 s3 s4 s5 s6 s

l

1

2

3

4

5

leading 1st 2nd 3rd

FIG. 1: Spectrum in the s-channel of the generalizedVeneziano amplitude model of Eq. 1. The leading and daugh-ter Regge trajectories are marked by thin solid lines and res-onances by dots at integer values of spin (l). The dashedand dotted thick lines illustrate resonance contributes to in-dividual amplitudes, A2,1 and A4,3, respectively. All (infinitenumber) of resonances on and to the right of the dashed linecontribute to A2,1, while resonances on and to the right of thedotted line contribute to A4,3.

leading trajectory and from all subsequent daughter tra-jectories. The amplitudes A

n,2 have poles originatingfrom the the 1st daughter and all subsequent daughters,A

n,3 from the 2nd and all higher daughters, etc. Thus wecan use m to label Regge trajectories and define,

(m)(s) (s) (m 1) (6)

so that (1)(s) (s) corresponds to the leading trajec-tory, (2) the 1st daughter and so on. The spectrum isillustrated in Fig. 4. For fixed-t, the asymptotic behaviorof A

n,m

(s, t) at large-s reflects presence of resonances inthe crossed channel. Using Stirlings formula one finds,

An,m

(s ! 1, t) / 1s(n

t

)s(m)(t) (7)

For large-s the tensor factor in Eq. 1 is proportional to sand the full amplitude has the expected Regge limit,

A(s, t, u) / s(t) (8)

arising from the leading, m = 1 trajectory. The signaturefactor will be discussed later.

III. REMOVAL OF POLES

As described in the preceding section, an amplitudeA

n,m

with fixed n and m contains an infinite number ofpoles in a two-body channel it describes. Since produc-tion of resonances is process dependent it is necessary to

find a generalization of the amplitude that allows for theresidues to be process dependent. One possibility is touse a linear combination

An,m

(s, t) ! A(s, t) =X

n1,nm1cn,m

An,m

(s, t) (9)

The coecients cn,m

need to be chosen in such a waythat As only couples to resonances that contribute tothe process in question. For example, in the case of anisoscalar boson strongly coupled to three pions, isospinconservation demands each pair of pions be produced inisospin-1. Bose statistics then eliminates all spin-evenresonances in s t and u channels of this reaction.One way to proceed is to construct linear combinations

of amplitudes An,m

that eliminate all, but selected par-tial waves and then take linear combinations of partialwaves. Alternatively one can attempt data analysis witha finite number of linear combinations of the A

n,m

s andlet the fit to data determine coecients c

n,m

[? ]. Wefind the former more appealing for several reasons. Firstof all, when studying resonance properties one is forcedto work with partial waves. Proper description of reso-nances, however, requires that unitarity is satisfied andRegge trajectories are non-linear, while the Venezianomodel forces Regge trajectories to be real and linear.Even though there are extensions of the Veneziano modelallowing for non-linear trajectories, implementation ofunitarity is much simpler at the level of partial waves.We therefore need to be able to isolate partial waves.Using the Veneziano amplitudes as building blocks, how-ever, we will be able match the low-energy behavior ofpartial waves with the asymptotic high-energy limit de-termined by Regge poles. This is important as it providesa constraint on data analysis that extends beyond whatresonances alone can fix.Since each A

n,n

amplitude contains an infinite numberof poles, in order to cancel all, but a finite number ofpoles an infinite number of coecients c

n,m

s in Eq. 9must be non vanishing. It is not dicult to find a rela-tion between the coecients, which decouples all, but afinite number of poles. Consider, for example, keepingonly the pole at (s) = 1 i.e. at s = s1. This pole isonly present in the amplitude A1,1 since amplitudes withn > 1 have the lowest pole at s

n

> s1. There is onlyone amplitude A1,m = A1,1 so a single coecient c1,1 de-termines coupling to the pole at s = s1. The amplitudeA1,1, however, also has poles at higher masses locatedat

s

= 2, 3, with residues that are polynomials int of the order of 1, 2, , respectively. If we only wantto keep the pole at (s) = 1, these higher mass polesof A1,1 have to be canceled by similar poles present inamplitudes with n > 1.The pole in A1,1 at s = 2 can only by canceled by the

same pole in the two amplitudes A2,m, m = 1, 2 since forn > 2 no other A

n,m

has this pole. The amplitudes A2,1and A2,2 are polynomials in t of the order of O(1) andO(0) respectively. We can therefore uniquely determinetwo coecients, c2,1 and c2,2 in terms of c1,1 so that the

3

first order polynomial in t at the s = s2 pole of A1,1matches the polynomial in t at the same pole of A2,1 andA2,2 to produce a vanishing residue. Similarly, at thes

= 3 pole of A1,1, the residue is an O(2) polynomialin t. This pole is present in A2,1 and A2,2 with residuesO(2) and O(1) polynomials, respectively, and it s alsopresent in A3, 1, A3,2 and A3,3 with residues of the orderof O(2), O(1) and O(0), respectively. With c2,1 and c2,2already fixed, c3,1, c3,2 and c3,3 are now uniquely deter-mined by c1,1 and by the requirement that the residueof the

s

= 3 pole, which is an O(2) polynomial in t,vanishes. Continuing in this way all poles in s satisfying(s) > 1 can be decoupled. Specifically we find

cn,1 =

c1,1(n)

, cn,2 =

c1,1(n 1) , cn,m = 0 for m > 2,

(10)so that

A1(s, t) = c1,12

s

t

(1 s

)(1 t

). (11)

This simple result could have been anticipated sinceA(s, t) is symmetric in s and t and completely deter-mined by the poles. It is worth noting, however, thatan infinite sum of the A

n,m

s resulted in an amplitudethat has fixed poles in s and t and not Regge poles. Wewill return to this point in the following subsection. Theprocedure can be generalized to produce amplitudes withisolated poles at any higher, integer value of

s

and t

.For example, to construct an amplitude with a single polein s at (s) = 3, one starts with the three amplitudesA3,m, m = 1, 2, 3 and determines the coecients cn,m forn > 3 in terms of c3,1, c3,2 and c3,3 that remove all polesat (s) > 3. Since the residue of the (s) = 3 pole isa polynomial of O(2) in t having three parameters c3,m,m = 1, 2, 3 determining the amplitude A(s, t) enablesto decompose the residue in terms of an arbitrary linearcombination of partial waves with l = 0, 1, 2. We note,however, that once A is used in place of A

n,m

in the ex-pression for the full amplitude in Eq. 1 the (s) = 3 polewill represent (narrow) resonances with spin increased byone unit, i.e l = 1, 2, 3. The coecients c3,m, m = 1, 2, 3can be therefore be chosen to decouple the l = 2 isobarsin the decay of an iso-scalar vector meson to three pions.The amplitude with the = 3 poles is then given by

A3(s, t) = a3,0(6

s

t

)(a3,1 s t)(a3,2 s t)(3

s

)(3 t

)(12)

where the as are linear combinations of the c3.ms. Thefirst factor in the numerator guarantees that A does nothave the double pole at

s

= t

= 3. It is followed bya product of two monomials in

s

+ t

that all togethergenerate O(2) polynomial in s or t at the pole of

t

= 3or

s

= 3 respectively.

A. Regge poles

After all but a finite number of poles have been re-moved from, say the t-channel the resulting amplitudeat large values of the cross-channel energy, s, behaves assJ where J is the highest spin reached at poles in

s

.For example, for the amplitude in Eqs. 11, 12 J = 0 andJ = 2, respectively. The expected asymptotic behavior,however, is s(t)1 (cf. Eqs. 7,8). The Regge behaviorcan only emerge from an infinite number of poles, there-fore we need to modify the procedure outlined above andallow for infinite number of poles to be present in A.Since the location of the poles increases with energy, theeect of poles, located at n > N is small on amplitudesfor s > N has the desired Regge behavior /s(t)1. As expected, the amplitude is free from polesin the range 1 < (s) N , and the same holds for thet-channel. Furthermore, for N large enough i.e. N >>M2 the contribution of the undesired, high-energy polesat > N on the low-energy region, < M2 is powersuppressed

A1(s, t < M2;N) = c1,12

s

t

(1 s

)(1 t

)

1 +O

M2

N

(14)and can be interpreted as background. The generaliza-tion of Eq. 13 to a pole at = n is

An

(s, t;N) = an,0

2n s

t

(n s

)(n t

)

n1

i=1 (an,i s t)

(N + 1 s)(N + 1 t)(N + 1 n)(N + n+ 1

s

t

)

(15)

In the following we this set of amplitudes to describe J and 0, three pion decays.

IV. APPLICATION TO VECTORCHARMONIUM DECAYS

Decays of vector charmonia have been studied byMARKII, CLEO, BaBar and BES, and more recently byBESIII. One of the original motivation was the perturba-tive QCD prediction. The QCD calculation is based onfactorization of the cc wave function from the perturba-tive production of light quark pairs. The predicted ratioof branching ratios, 0 ! //J/ ! / of 12% ap-pears to be significantly larger then measurements, which

3

first order polynomial in t at the s = s2 pole of A1,1matches the polynomial in t at the same pole of A2,1 andA2,2 to produce a vanishing residue. Similarly, at thes

= 3 pole of A1,1, the residue is an O(2) polynomialin t. This pole is present in A2,1 and A2,2 with residuesO(2) and O(1) polynomials, respectively, and it s alsopresent in A3, 1, A3,2 and A3,3 with residues of the orderof O(2), O(1) and O(0), respectively. With c2,1 and c2,2already fixed, c3,1, c3,2 and c3,3 are now uniquely deter-mined by c1,1 and by the requirement that the residueof the

s

= 3 pole, which is an O(2) polynomial in t,vanishes. Continuing in this way all poles in s satisfying(s) > 1 can be decoupled. Specifically we find

cn,1 =

c1,1(n)

, cn,2 =

c1,1(n 1) , cn,m = 0 for m > 2,

(10)so that

A1(s, t) = c1,12

s

t

(1 s

)(1 t

). (11)

This simple result could have been anticipated sinceA(s, t) is symmetric in s and t and completely deter-mined by the poles. It is worth noting, however, thatan infinite sum of the A

n,m

s resulted in an amplitudethat has fixed poles in s and t and not Regge poles. Wewill return to this point in the following subsection. Theprocedure can be generalized to produce amplitudes withisolated poles at any higher, integer value of

s

and t

.For example, to construct an amplitude with a single polein s at (s) = 3, one starts with the three amplitudesA3,m, m = 1, 2, 3 and determines the coecients cn,m forn > 3 in terms of c3,1, c3,2 and c3,3 that remove all polesat (s) > 3. Since the residue of the (s) = 3 pole isa polynomial of O(2) in t having three parameters c3,m,m = 1, 2, 3 determining the amplitude A(s, t) enablesto decompose the residue in terms of an arbitrary linearcombination of partial waves with l = 0, 1, 2. We note,however, that once A is used in place of A

n,m

in the ex-pression for the full amplitude in Eq. 1 the (s) = 3 polewill represent (narrow) resonances with spin increased byone unit, i.e l = 1, 2, 3. The coecients c3,m, m = 1, 2, 3can be therefore be chosen to decouple the l = 2 isobarsin the decay of an iso-scalar vector meson to three pions.The amplitude with the = 3 poles is then given by

A3(s, t) = a3,0(6

s

t

)(a3,1 s t)(a3,2 s t)(3

s

)(3 t

)(12)

where the as are linear combinations of the c3.ms. Thefirst factor in the numerator guarantees that A does nothave the double pole at

s

= t

= 3. It is followed bya product of two monomials in

s

+ t

that all togethergenerate O(2) polynomial in s or t at the pole of

t

= 3or

s

= 3 respectively.

A. Regge poles

After all but a finite number of poles have been re-moved from, say the t-channel the resulting amplitudeat large values of the cross-channel energy, s, behaves assJ where J is the highest spin reached at poles in

s

.For example, for the amplitude in Eqs. 11, 12 J = 0 andJ = 2, respectively. The expected asymptotic behavior,however, is s(t)1 (cf. Eqs. 7,8). The Regge behaviorcan only emerge from an infinite number of poles, there-fore we need to modify the procedure outlined above andallow for infinite number of poles to be present in A.Since the location of the poles increases with energy, theeect of poles, located at n > N is small on amplitudesfor s > N has the desired Regge behavior /s(t)1. As expected, the amplitude is free from polesin the range 1 < (s) N , and the same holds for thet-channel. Furthermore, for N large enough i.e. N >>M2 the contribution of the undesired, high-energy polesat > N on the low-energy region, < M2 is powersuppressed

A1(s, t < M2;N) = c1,12

s

t

(1 s

)(1 t

)

1 +O

M2

N

(14)and can be interpreted as background. The generaliza-tion of Eq. 13 to a pole at = n is

An

(s, t;N) = an,0

2n s

t

(n s

)(n t

)

n1

i=1 (an,i s t)

(N + 1 s)(N + 1 t)(N + 1 n)(N + n+ 1

s

t

)

(15)

In the following we this set of amplitudes to describe J and 0, three pion decays.

IV. APPLICATION TO VECTORCHARMONIUM DECAYS

Decays of vector charmonia have been studied byMARKII, CLEO, BaBar and BES, and more recently byBESIII. One of the original motivation was the perturba-tive QCD prediction. The QCD calculation is based onfactorization of the cc wave function from the perturba-tive production of light quark pairs. The predicted ratioof branching ratios, 0 ! //J/ ! / of 12% ap-pears to be significantly larger then measurements, which

3

first order polynomial in t at the s = s2 pole of A1,1matches the polynomial in t at the same pole of A2,1 andA2,2 to produce a vanishing residue. Similarly, at thes

= 3 pole of A1,1, the residue is an O(2) polynomialin t. This pole is present in A2,1 and A2,2 with residuesO(2) and O(1) polynomials, respectively, and it s alsopresent in A3, 1, A3,2 and A3,3 with residues of the orderof O(2), O(1) and O(0), respectively. With c2,1 and c2,2already fixed, c3,1, c3,2 and c3,3 are now uniquely deter-mined by c1,1 and by the requirement that the residueof the

s

= 3 pole, which is an O(2) polynomial in t,vanishes. Continuing in this way all poles in s satisfying(s) > 1 can be decoupled. Specifically we find

cn,1 =

c1,1(n)

, cn,2 =

c1,1(n 1) , cn,m = 0 for m > 2,

(10)so that

A1(s, t) = c1,12

s

t

(1 s

)(1 t

). (11)

This simple result could have been anticipated sinceA(s, t) is symmetric in s and t and completely deter-mined by the poles. It is worth noting, however, thatan infinite sum of the A

n,m

s resulted in an amplitudethat has fixed poles in s and t and not Regge poles. Wewill return to this point in the following subsection. Theprocedure can be generalized to produce amplitudes withisolated poles at any higher, integer value of

s

and t

.For example, to construct an amplitude with a single polein s at (s) = 3, one starts with the three amplitudesA3,m, m = 1, 2, 3 and determines the coecients cn,m forn > 3 in terms of c3,1, c3,2 and c3,3 that remove all polesat (s) > 3. Since the residue of the (s) = 3 pole isa polynomial of O(2) in t having three parameters c3,m,m = 1, 2, 3 determining the amplitude A(s, t) enablesto decompose the residue in terms of an arbitrary linearcombination of partial waves with l = 0, 1, 2. We note,however, that once A is used in place of A

n,m

in the ex-pression for the full amplitude in Eq. 1 the (s) = 3 polewill represent (narrow) resonances with spin increased byone unit, i.e l = 1, 2, 3. The coecients c3,m, m = 1, 2, 3can be therefore be chosen to decouple the l = 2 isobarsin the decay of an iso-scalar vector meson to three pions.The amplitude with the = 3 poles is then given by

A3(s, t) = a3,0(6

s

t

)(a3,1 s t)(a3,2 s t)(3

s

)(3 t

)(12)

where the as are linear combinations of the c3.ms. Thefirst factor in the numerator guarantees that A does nothave the double pole at

s

= t

= 3. It is followed bya product of two monomials in

s

+ t

that all togethergenerate O(2) polynomial in s or t at the pole of

t

= 3or

s

= 3 respectively.

A. Regge poles

After all but a finite number of poles have been re-moved from, say the t-channel the resulting amplitudeat large values of the cross-channel energy, s, behaves assJ where J is the highest spin reached at poles in

s

.For example, for the amplitude in Eqs. 11, 12 J = 0 andJ = 2, respectively. The expected asymptotic behavior,however, is s(t)1 (cf. Eqs. 7,8). The Regge behaviorcan only emerge from an infinite number of poles, there-fore we need to modify the procedure outlined above andallow for infinite number of poles to be present in A.Since the location of the poles increases with energy, theeect of poles, located at n > N is small on amplitudesfor s > N has the desired Regge behavior /s(t)1. As expected, the amplitude is free from polesin the range 1 < (s) N , and the same holds for thet-channel. Furthermore, for N large enough i.e. N >>M2 the contribution of the undesired, high-energy polesat > N on the low-energy region, < M2 is powersuppressed

A1(s, t < M2;N) = c1,12

s

t

(1 s

)(1 t

)

1 +O

M2

N

(14)and can be interpreted as background. The generaliza-tion of Eq. 13 to a pole at = n is

An

(s, t;N) = an,0

2n s

t

(n s

)(n t

)

n1

i=1 (an,i s t)

(N + 1 s)(N + 1 t)(N + 1 n)(N + n+ 1

s

t

)

(15)

In the following we this set of amplitudes to describe J and 0, three pion decays.

IV. APPLICATION TO VECTORCHARMONIUM DECAYS

Decays of vector charmonia have been studied byMARKII, CLEO, BaBar and BES, and more recently byBESIII. One of the original motivation was the perturba-tive QCD prediction. The QCD calculation is based onfactorization of the cc wave function from the perturba-tive production of light quark pairs. The predicted ratioof branching ratios, 0 ! //J/ ! / of 12% ap-pears to be significantly larger then measurements, which

remove all poles but the one at =1

... but the Regge limit is now lost !

remove all poles between N 2

3

first order polynomial in t at the s = s2 pole of A1,1matches the polynomial in t at the same pole of A2,1 andA2,2 to produce a vanishing residue. Similarly, at thes

= 3 pole of A1,1, the residue is an O(2) polynomialin t. This pole is present in A2,1 and A2,2 with residuesO(2) and O(1) polynomials, respectively, and it s alsopresent in A3, 1, A3,2 and A3,3 with residues of the orderof O(2), O(1) and O(0), respectively. With c2,1 and c2,2already fixed, c3,1, c3,2 and c3,3 are now uniquely deter-mined by c1,1 and by the requirement that the residueof the

s

= 3 pole, which is an O(2) polynomial in t,vanishes. Continuing in this way all poles in s satisfying(s) > 1 can be decoupled. Specifically we find

cn,1 =

c1,1(n)

, cn,2 =

c1,1(n 1) , cn,m = 0 for m > 2,

(10)so that

A1(s, t) = c1,12

s

t

(1 s

)(1 t

). (11)

This simple result could have been anticipated sinceA(s, t) is symmetric in s and t and completely deter-mined by the poles. It is worth noting, however, thatan infinite sum of the A

n,m

s resulted in an amplitudethat has fixed poles in s and t and not Regge poles. Wewill return to this point in the following subsection. Theprocedure can be generalized to produce amplitudes withisolated poles at any higher, integer value of

s

and t

.For example, to construct an amplitude with a single polein s at (s) = 3, one starts with the three amplitudesA3,m, m = 1, 2, 3 and determines the coecients cn,m forn > 3 in terms of c3,1, c3,2 and c3,3 that remove all polesat (s) > 3. Since the residue of the (s) = 3 pole isa polynomial of O(2) in t having three parameters c3,m,m = 1, 2, 3 determining the amplitude A(s, t) enablesto decompose the residue in terms of an arbitrary linearcombination of partial waves with l = 0, 1, 2. We note,however, that once A is used in place of A

n,m

in the ex-pression for the full amplitude in Eq. 1 the (s) = 3 polewill represent (narrow) resonances with spin increased byone unit, i.e l = 1, 2, 3. The coecients c3,m, m = 1, 2, 3can be therefore be chosen to decouple the l = 2 isobarsin the decay of an iso-scalar vector meson to three pions.The amplitude with the = 3 poles is then given by

A3(s, t) = a3,0(6

s

t

)(a3,1 s t)(a3,2 s t)(3

s

)(3 t

)(12)

where the as are linear combinations of the c3.ms. Thefirst factor in the numerator guarantees that A does nothave the double pole at

s

= t

= 3. It is followed bya product of two monomials in

s

+ t

that all togethergenerate O(2) polynomial in s or t at the pole of

t

= 3or

s

= 3 respectively.

A. Regge poles

After all but a finite number of poles have been re-moved from, say the t-channel the resulting amplitudeat large values of the cross-channel energy, s, behaves assJ where J is the highest spin reached at poles in

s

.For example, for the amplitude in Eqs. 11, 12 J = 0 andJ = 2, respectively. The expected asymptotic behavior,however, is s(t)1 (cf. Eqs. 7,8). The Regge behaviorcan only emerge from an infinite number of poles, there-fore we need to modify the procedure outlined above andallow for infinite number of poles to be present in A.Since the location of the poles increases with energy, theeect of poles, located at n > N is small on amplitudesfor s > N has the desired Regge behavior /s(t)1. As expected, the amplitude is free from polesin the range 1 < (s) N , and the same holds for thet-channel. Furthermore, for N large enough i.e. N >>M2 the contribution of the undesired, high-energy polesat > N on the low-energy region, < M2 is powersuppressed

A1(s, t < M2;N) = c1,12

s

t

(1 s

)(1 t

)

1 +O

M2

N

(14)and can be interpreted as background. The generaliza-tion of Eq. 13 to a pole at = n is

An

(s, t;N) = an,0

2n s

t

(n s

)(n t

)

n1

i=1 (an,i s t)

(N + 1 s)(N + 1 t)(N + 1 n)(N + n+ 1

s

t

)

(15)

In the following we this set of amplitudes to describe J and 0, three pion decays.

IV. APPLICATION TO VECTORCHARMONIUM DECAYS

Decays of vector charmonia have been studied byMARKII, CLEO, BaBar and BES, and more recently byBESIII. One of the original motivation was the perturba-tive QCD prediction. The QCD calculation is based onfactorization of the cc wave function from the perturba-tive production of light quark pairs. The predicted ratioof branching ratios, 0 ! //J/ ! / of 12% ap-pears to be significantly larger then measurements, which

has Regge limit is for s > N

• The new model does this in a systematic way. In addition it allows for imaginary non-linear (and complex) trajectories without introducing ancestors

In the past this was done by choosing an arbitrary set of n,m and fitting c(n,m) to the data (e.g. Lovelace, Phys. Lett. B28, 265 (1968),Altarelli, Rubinstein, Phys. Rev. 183, 1469 (1969))

n: number of Regge trajectoriesan,i: determine resonance couplingsN: determines the onset of Regge behavior(s), (t) = Re + i Im : with Im related to resonance widths

• from A.Celentano

• 3

first order polynomial in t at the s = s2 pole of A1,1matches the polynomial in t at the same pole of A2,1 andA2,2 to produce a vanishing residue. Similarly, at thes

= 3 pole of A1,1, the residue is an O(2) polynomialin t. This pole is present in A2,1 and A2,2 with residuesO(2) and O(1) polynomials, respectively, and it s alsopresent in A3, 1, A3,2 and A3,3 with residues of the orderof O(2), O(1) and O(0), respectively. With c2,1 and c2,2already fixed, c3,1, c3,2 and c3,3 are now uniquely deter-mined by c1,1 and by the requirement that the residueof the

s

= 3 pole, which is an O(2) polynomial in t,vanishes. Continuing in this way all poles in s satisfying(s) > 1 can be decoupled. Specifically we find

cn,1 =

c1,1(n)

, cn,2 =

c1,1(n 1) , cn,m = 0 for m > 2,

(10)so that

A1(s, t) = c1,12

s

t

(1 s

)(1 t

). (11)

This simple result could have been anticipated sinceA(s, t) is symmetric in s and t and completely deter-mined by the poles. It is worth noting, however, thatan infinite sum of the A

n,m

s resulted in an amplitudethat has fixed poles in s and t and not Regge poles. Wewill return to this point in the following subsection. Theprocedure can be generalized to produce amplitudes withisolated poles at any higher, integer value of

s

and t

.For example, to construct an amplitude with a single polein s at (s) = 3, one starts with the three amplitudesA3,m, m = 1, 2, 3 and determines the coecients cn,m forn > 3 in terms of c3,1, c3,2 and c3,3 that remove all polesat (s) > 3. Since the residue of the (s) = 3 pole isa polynomial of O(2) in t having three parameters c3,m,m = 1, 2, 3 determining the amplitude A(s, t) enablesto decompose the residue in terms of an arbitrary linearcombination of partial waves with l = 0, 1, 2. We note,however, that once A is used in place of A

n,m

in the ex-pression for the full amplitude in Eq. 1 the (s) = 3 polewill represent (narrow) resonances with spin increased byone unit, i.e l = 1, 2, 3. The coecients c3,m, m = 1, 2, 3can be therefore be chosen to decouple the l = 2 isobarsin the decay of an iso-scalar vector meson to three pions.The amplitude with the = 3 poles is then given by

A3(s, t) = a3,0(6

s

t

)(a3,1 s t)(a3,2 s t)(3

s

)(3 t

)(12)

where the as are linear combinations of the c3.ms. Thefirst factor in the numerator guarantees that A does nothave the double pole at

s

= t

= 3. It is followed bya product of two monomials in

s

+ t

that all togethergenerate O(2) polynomial in s or t at the pole of

t

= 3or

s

= 3 respectively.

A. Regge poles

After all but a finite number of poles have been re-moved from, say the t-channel the resulting amplitudeat large values of the cross-channel energy, s, behaves assJ where J is the highest spin reached at poles in

s

.For example, for the amplitude in Eqs. 11, 12 J = 0 andJ = 2, respectively. The expected asymptotic behavior,however, is s(t)1 (cf. Eqs. 7,8). The Regge behaviorcan only emerge from an infinite number of poles, there-fore we need to modify the procedure outlined above andallow for infinite number of poles to be present in A.Since the location of the poles increases with energy, theeect of poles, located at n > N is small on amplitudesfor s > N has the desired Regge behavior /s(t)1. As expected, the amplitude is free from polesin the range 1 < (s) N , and the same holds for thet-channel. Furthermore, for N large enough i.e. N >>M2 the contribution of the undesired, high-energy polesat > N on the low-energy region, < M2 is powersuppressed

A1(s, t < M2;N) = c1,12

s

t

(1 s

)(1 t

)

1 +O

M2

N

(14)and can be interpreted as background. The generaliza-tion of Eq. 13 to a pole at = n is

An

(s, t;N) = an,0

2n s

t

(n s

)(n t

)

n1

i=1 (an,i s t)

(N + 1 s)(N + 1 t)(N + 1 n)(N + n+ 1

s

t

)

(15)

In the following we this set of amplitudes to describe J and 0, three pion decays.

IV. APPLICATION TO VECTORCHARMONIUM DECAYS

Decays of vector charmonia have been studied byMARKII, CLEO, BaBar and BES, and more recently byBESIII. One of the original motivation was the perturba-tive QCD prediction. The QCD calculation is based onfactorization of the cc wave function from the perturba-tive production of light quark pairs. The predicted ratioof branching ratios, 0 ! //J/ ! / of 12% ap-pears to be significantly larger then measurements, which

All poles below = N except at = n

s

t

s = n

t =

n

at s=n residue is a polynomial in t of order n-1 (remember to add 1 from the Levi-Civita tensor)

Veneziano Model Motivated Amplitudes for Dalitz Plot Analysis

A.P.Szczepaniak1

1Physics Department Indiana University, Bloomington, IN 47405, USA

Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403, USA

Jeerson Laboratory, 12000 Jeerson Avenue, Newport News, VA 23606, USA

(Dated: December 15, 2013)

Needs to be written

PACS numbers:

I. INTRODUCTION

Unless partial waves can be related to the full ampli-tude, partial waves, beyond the resonance contributionpartial waves are largely unconstrained. The Venezianomodel provides this connections.

II. GENERALIZED VENEZIANO AMPLITUDES

We consider the generalized Veneziano model in thecontext of decays of mesons with heavy quarks to finalstates with light quarks. Specifically we focus on the ap-plication to decays of vector charmonia, e.g. J/ or 0 tothree pions. For simplicity we neglect pion messes and,in units of GeV, use M2 = O(10) for the mass squared ofthe decaying particle. The generalization to other chan-nels is in principle straightforward.

The Veneziano model for the amplitude A describinga decay of a vector meson with momentum p and helicity to three pions, V (p,) ! i(p1)j(p2)k(p3) is givenby

A(s, t, u) = ijk

(p,)p1p

2 p

3

[An,m

(s, t) +An,m

(s, u) +An,m

(t, u)] (1)

with s + t + u = M2 and the scalar amplitudes An,m

sdefined by

An,m

(s, t) (n s)(n t)(n+m

s

t

). (2)

Here n,m are positive integers, 1 m n, and s

is ashorthand for (s), the leading Regge trajectory. We donot use the Veneziano amplitude with n = 0 since with representing the leading meson trajectory it would pro-duce a ghost pole. The lower limit on m, m = 1 guaran-tees that A(s, t, u) has the expected high-energy behavior(see below) and the upper limit, m = n guarantees ab-sence of double poles in overlapping channels.

The Veneziano formula exhibits the behavior of A thatis expected in the large-N

c

limi of QCD. In this limit theQCD boson spectrum is saturated by narrow resonancesand confinement produces linear Regge trajectories, i.e.(s) = 0 + 0s. This spectrum is manifested in thesingularities of A

n,m

(s, t), which has simple poles in sand t-channels and no double poles. For example thes-channel poles are located at s = s

n+k determined by

(sn+k) = n+ k (3)

with k being a nonnegative integer. The contribution ofa pole to the amplitude A

n,m

(s, t) is then given by

A(s sn+k) =

n,m,k

(t)

sn+k s

(4)

with the residue

n,m,k

(t) =(1)k

0k!

(n t

)

(m k t

)=

(1)k

0k!(m k

t

)(m k + 1 t

) (m k + Lmax

1 t

)

being a polynomial in t of the order Lmax

= Lmax

(k) k + n m 0. Thus in each channel s, t, or u, the fullamplitude given by Eq. 1 describes k, degenerate, narrow(zero) width resonances with spins in the range,

1 l Lmax

(k) + 1. (5)

The additional factor of 1 originates from the tensor fac-tor in front of the r.h.s of Eq. 1.

The integers n and m determine which resonances con-tribute poles to the amplitude. It follows form Eqs. 3,5that amplitudes with m = 1, A

n,1 have poles from the

A1 has (770) A3 has (1700), 3(1690) A5 has (2150), 3(2250), 5(2350)

• 2

s1 s2 s3 s4 s5 s6 s

l

1

2

3

4

5

leading 1st 2nd 3rd

FIG. 1: Spectrum in the s-channel of the generalizedVeneziano amplitude model of Eq. 1. The leading and daugh-ter Regge trajectories are marked by thin solid lines and res-onances by dots at integer values of spin (l). The dashedand dotted thick lines illustrate resonance contributes to in-dividual amplitudes, A2,1 and A4,3, respectively. All (infinitenumber) of resonances on and to the right of the dashed linecontribute to A2,1, while resonances on and to the right of thedotted line contribute to A4,3.

leading trajectory and from all subsequent daughter tra-jectories. The amplitudes A

n,2 have poles originatingfrom the the 1st daughter and all subsequent daughters,A

n,3 from the 2nd and all higher daughters, etc. Thus wecan use m to label Regge trajectories and define,

(m)(s) (s) (m 1) (6)

so that (1)(s) (s) corresponds to the leading trajec-tory, (2) the 1st daughter and so on. The spectrum isillustrated in Fig. 4. For fixed-t, the asymptotic behaviorof A

n,m

(s, t) at large-s reflects presence of resonances inthe crossed channel. Using Stirlings formula one finds,

An,m

(s ! 1, t) / 1s(n

t

)s(m)(t) (7)

For large-s the tensor factor in Eq. 1 is proportional to sand the full amplitude has the expected Regge limit,

A(s, t, u) / s(t) (8)

arising from the leading, m = 1 trajectory. The signaturefactor will be discussed later.

III. REMOVAL OF POLES

As described in the preceding section, an amplitudeA

n,m

with fixed n and m contains an infinite number ofpoles in a two-body channel it describes. Since produc-tion of resonances is process dependent it is necessary to

find a generalization of the amplitude that allows for theresidues to be process dependent. One possibility is touse a linear combination

An,m

(s, t) ! A(s, t) =X

n1,nm1cn,m

An,m

(s, t) (9)

The coecients cn,m

need to be chosen in such a waythat As only couples to resonances that contribute tothe process in question. For example, in the case of anisoscalar boson strongly coupled to three pions, isospinconservation demands each pair of pions be produced inisospin-1. Bose statistics then eliminates all spin-evenresonances in s t and u channels of this reaction.One way to proceed is to construct linear combinations

of amplitudes An,m

that eliminate all, but selected par-tial waves and then take linear combinations of partialwaves. Alternatively one can attempt data analysis witha finite number of linear combinations of the A

n,m

s andlet the fit to data determine coecients c

n,m

[? ]. Wefind the former more appealing for several reasons. Firstof all, when studying resonance properties one is forcedto work with partial waves. Proper description of reso-nances, however, requires that unitarity is satisfied andRegge trajectories are non-linear, while the Venezianomodel forces Regge trajectories to be real and linear.Even though there are extensions of the Veneziano modelallowing for non-linear trajectories, implementation ofunitarity is much simpler at the level of partial waves.We therefore need to be able to isolate partial waves.Using the Veneziano amplitudes as building blocks, how-ever, we will be able match the low-energy behavior ofpartial waves with the asymptotic high-energy limit de-termined by Regge poles. This is important as it providesa constraint on data analysis that extends beyond whatresonances alone can fix.Since each A

n,n

amplitude contains an infinite numberof poles, in order to cancel all, but a finite number ofpoles an infinite number of coecients c

n,m

s in Eq. 9must be non vanishing. It is not dicult to find a rela-tion between the coecients, which decouples all, but afinite number of poles. Consider, for example, keepingonly the pole at (s) = 1 i.e. at s = s1. This pole isonly present in the amplitude A1,1 since amplitudes withn > 1 have the lowest pole at s

n

> s1. There is onlyone amplitude A1,m = A1,1 so a single coecient c1,1 de-termines coupling to the pole at s = s1. The amplitudeA1,1, however, also has poles at higher masses locatedat

s

= 2, 3, with residues that are polynomials int of the order of 1, 2, , respectively. If we only wantto keep the pole at (s) = 1, these higher mass polesof A1,1 have to be canceled by similar poles present inamplitudes with n > 1.The pole in A1,1 at s = 2 can only by canceled by the

same pole in the two amplitudes A2,m, m = 1, 2 since forn > 2 no other A

n,m

has this pole. The amplitudes A2,1and A2,2 are polynomials in t of the order of O(1) andO(0) respectively. We can therefore uniquely determinetwo coecients, c2,1 and c2,2 in terms of c1,1 so that the

(1400)

m2 =ps2 =

r2 1

2= 1.23

(1900),3(1990),

m4 =ps4 =

r4 1

2= 1.87

• 4

yield approximately 1% for this ratio. Our objective,however, is not to discuss the origin of the discrepancywith the predicted couplings but instead focus on theanalysis of the decay amplitude.

Both, the J/ and 0 decays show a clear signal of production. In additional there is indication of resonancebeing produced in the third and fifth resonance region,i.e. 3, 5. We will thus attempt to fit the di-pion massdistribution with three amplitudes, A1, A3 and A5, re-spectively. We take N = 20 which is above the availablephase space as as long as N > M2 we find very littlesensitivity to N In terms of the s-channel partial waves,fl

(s), the scalar amplitude in Eq. 1

F (s, t, u) X

n=1,3,5

An

(s, t;N)+An

(s, u;N)+An

(t, u;N)

(16)is given by

F (s, t, u) =X

l=odd

fl

(s)P 0l

(z) (17)

where, ignoring the pion mass, z = (t u)/(M2 s) isthe cosine of s-channel scattering angle and P

l

are theLegendre polynomial. As discussed in the preceding sec-tion, the pole at

s

= 1 contributes only to the l = 1wave. From A1(s, t, ;N) and A1(s, u;N) in F (s, t, u) forthe pole contribution to f1(s), one finds, (A1(t, u;N) doesnot contribute to the pole term

s

)

f (1)1 (s) =g(1)1 (M

2)

1 (s) (18)

where g(1)1 (M2) = a1,0(M2) and an,i are defined by

Eq. 15. At the s

= 3, 5 poles the amplitudesA3(s, t;N), A5(s, u;N) contribute to l = 1, 3 and l =1, 3, 5 partial waves, respectively

f (3)1,3 (s) =g(3)1,3(M

2)

3 (s) , f(5)1,3,5(s) =

g(5)1,3,5(M2)

5 (s) . (19)

The superscripts on f (n)l

denote the position of the polesi.e. n =

s

and the resonance couplings, gn,l

are obtainedfrom,

gn,l

=

Z 1

1

dz

2[P

l1(z)Pl+1(z)][ResAn(n, t;N)+(t ! u)]

(20)where Res refers to the residue of the amplitude of Eq. 15at pole at

s

= n.For each, channel, J/ and 0 we fit nine real pa-

rameters, a1,0, a3,0, a3,1, a3,2, a5,0, a5,4. In addition weallow the trajectories to be imaginary when appearing inthe denominators of A

n

in order to be able to accountfor the finite width of the resonances. The trajectoryis expected to be approximately equal to

(s) = 1 + 0(sm2

) + i0m

0.47 + 0.9s+ 0.1ips 0.07 (21)

1.5 1.7 1.9100

150

200

250

0 1 2 3M20

200

400

600

800

Even

ts 1

03/2

.4M

eV

FIG. 2: Dalitz plot projection of the di-pion mass distributionfrom J/ decay. The solid is the result of the fit with threeamplitudes and the dashed line with the amplitude A1 alone.The insert shows the mass region of the 3 and its contributionfrom the fit with the full set of amplitudes (solid line) ascompared. Absence of the structure at 1.7GeV from the fitwith the A1 amplitude is indicated by the dashed line.

0 1 2 3 4M20

1

2

Even

ts 1

03/ 2

.4M

eV

FIG. 3: Dalitz plot projection of the di-pion mass distributionfrom 0 decay. The solid is the result of the fit with threeamplitudes and the dashed line is the fit with A1 alone.

where we also included the phase space factor,ps 4m2

in the imaginary part. With (s) = 1, 3, 5 the six res-onances corresponding to the amplitudes in Eqs. 18,19can be assigned to (770) ( = 1), 0(1700), 3(1690),( = 3) and 00(2150) 3(2250), 5(2350) for = 5, re-spectively.

The data and results of the fit are shown in Figs. ??.The data is taken from for the resent measurement by theBESIII collaboration. Unfortunately having no access tothe Dalitz plot distribution we were able to analyze the

4

yield approximately 1% for this ratio. Our objective,however, is not to discuss the origin of the discrepancywith the predicted couplings but instead focus on theanalysis of the decay amplitude.

Both, the J/ and 0 decays show a clear signal of production. In additional there is indication of resonancebeing produced in the third and fifth resonance region,i.e. 3, 5. We will thus attempt to fit the di-pion massdistribution with three amplitudes, A1, A3 and A5, re-spectively. We take N = 20 which is above the availablephase space as as long as N > M2 we find very littlesensitivity to N In terms of the s-channel partial waves,fl

(s), the scalar amplitude in Eq. 1

F (s, t, u) X

n=1,3,5

An

(s, t;N)+An

(s, u;N)+An

(t, u;N)

(16)is given by

F (s, t, u) =X

l=odd

fl

(s)P 0l

(z) (17)

where, ignoring the pion mass, z = (t u)/(M2 s) isthe cosine of s-channel scattering angle and P

l

are theLegendre polynomial. As discussed in the preceding sec-tion, the pole at

s

= 1 contributes only to the l = 1wave. From A1(s, t, ;N) and A1(s, u;N) in F (s, t, u) forthe pole contribution to f1(s), one finds, (A1(t, u;N) doesnot contribute to the pole term

s

)

f (1)1 (s) =g(1)1 (M

2)

1 (s) (18)

where g(1)1 (M2) = a1,0(M2) and an,i are defined by

Eq. 15. At the s

= 3, 5 poles the amplitudesA3(s, t;N), A5(s, u;N) contribute to l = 1, 3 and l =1, 3, 5 partial waves, respectively

f (3)1,3 (s) =g(3)1,3(M

2)

3 (s) , f(5)1,3,5(s) =

g(5)1,3,5(M2)

5 (s) . (19)

The superscripts on f (n)l

denote the position of the polesi.e. n =

s

and the resonance couplings, gn,l

are obtainedfrom,

gn,l

=

Z 1

1

dz

2[P

l1(z)Pl+1(z)][ResAn(n, t;N)+(t ! u)]

(20)where Res refers to the residue of the amplitude of Eq. 15at pole at

s

= n.For each, channel, J/ and 0 we fit nine real pa-

rameters, a1,0, a3,0, a3,1, a3,2, a5,0, a5,4. In addition weallow the trajectories to be imaginary when appearing inthe denominators of A

n

in order to be able to accountfor the finite width of the resonances. The trajectoryis expected to be approximately equal to

(s) = 1 + 0(sm2

) + i0m

0.47 + 0.9s+ 0.1ips 0.07 (21)

1.5 1.7 1.9100

150

200

250

0 1 2 3M20

200

400

600

800

Even

ts 1

03/2

.4M

eV

FIG. 2: Dalitz plot projection of the di-pion mass distributionfrom J/ decay. The solid is the result of the fit with threeamplitudes and the dashed line with the amplitude A1 alone.The insert shows the mass region of the 3 and its contributionfrom the fit with the full set of amplitudes (solid line) ascompared. Absence of the structure at 1.7GeV from the fitwith the A1 amplitude is indicated by the dashed line.

0 1 2 3 4M20

1

2

Even

ts 1

03/ 2

.4M

eVFIG. 3: Dalitz plot projection of the di-pion mass distributionfrom 0 decay. The solid is the result of the fit with threeamplitudes and the dashed line is the fit with A1 alone.

where we also included the phase space factor,ps 4m2

in the imaginary part. With (s) = 1, 3, 5 the six res-onances corresponding to the amplitudes in Eqs. 18,19can be assigned to (770) ( = 1), 0(1700), 3(1690),( = 3) and 00(2150) 3(2250), 5(2350) for = 5, re-spectively.

The data and results of the fit are shown in Figs. ??.The data is taken from for the resent measurement by theBESIII collaboration. Unfortunately having no access tothe Dalitz plot distribution we were able to analyze the

• B5 amplitude:

Reggeons/Resonances in all 5 channels