Large-N C resonance relations from partial wave analyses

Large-NC resonance relations from partial-wave analyses

J.J. Sanz-Cillero

Large-NC

resonance relations

from partial wave

analysesJ.J. Sanz-Cillero (IFAE - UAB)

EuroFlavours 07, 14-16 November 2007

Univ. Paris-Sud XI - Orsay

Z.H. Guo, J.J. Sanz-Cillero and H.Q. Zheng [ JHEP 0706 (2007) 030 ];

arXiv:0710.2163 [hep-ph]


J.J. Sanz-Cillero

Organization of the talk:Organization of the talk:

• Dispersive calculation for -scattering at large NC

• Matching PT at low energies

• Resonance coupling relations and LEC predictions

• Testing phenomenological lagrangians and LEC resonance estimates

• Conclusions


J.J. Sanz-Cillero

• Former large-NC resonance analysis have looked at

a) Form factors,

b) 2-point Green-functions

c) 3-point Green-functions

• Scattering amplitudes are the next in the line:

There have been studies on forward scattering

We propose the analysis of the PW scattering amplitudes

• In forward scattering s, t and u-channels have similar asymptotics

In PW amplitudes Each has a clearly distinguishable structure

Motivation


J.J. Sanz-Cillero

• In general, the description in terms of couplings of a lagrangian

usually does not provide an intuitive picture of the 1/NC expansion:

Expansion parameter in the hadronic 1/NC theory ?

Model dependence of a lagrangian realization ?

…

• However, maybe we can reach a better understanding/agreement

if we express resonance couplings

in terms of physical parameters (like masses and widths)

Moreover…Moreover…


J.J. Sanz-Cillero

Dispersive Dispersive

calculation of calculation of --

scatering scatering

at large-Nat large-NCC


J.J. Sanz-Cillero

2

+ 0I I I IJ J J J

-4m

s dt s dtT (s) - T (0) = ImT (t) + ImT (t)

t - s t - s

• T-matrix dispersive relation for : [Guo, Zheng & SC’07]

Resonance inputs:

•Right-hand cut cut (s-channel)(s-channel)

•Left-hand cut (t- and u-channels)(t- and u-channels)


J.J. Sanz-Cillero

• At large-NC s-channel narrow-width resonance exchanges:

For the right-hand cut:For the right-hand cut: TJI partial wave ONLY IJ

resonance

I 2J R R R

R

1ImT (t) = M Γ δ t-M

ρ2

R 2R

4m = 1-

Mρ

Right-hand cut

with0t


J.J. Sanz-Cillero

• We substitute this ImT(t)

in the right-hand side dispersive integral and obtain

I sR R RJ 2

R R

1 s M ΓT (s) =

M - s ρ

which can be identified with the exchange

of a tree-level resonance R

in the s-channel

R

MR , R


J.J. Sanz-Cillero

• Crossing symmetry relations for right and left-hand cut:

I+J

I stJ II'2

J' I'

1+(-1)ImT (s) = (2J'+1)C

s-4m

2

2

4m -sI'

J J' J'2 24m

2t 2s × dt P 1+ P 1+ ImT (t)

s-4m t-4m

0s

24t m

Left-hand cut

(true for any s<0

for large-NC tree-level amplitudes)


J.J. Sanz-Cillero

TJI partial wave almost EVERY narrow-width state RJ’

I’

contributes in the t and u-channels

I+JI 2 2 stJ R II'2

1+(-1)ImT (s) = θ(s-M +4m ) × (2J'+1)C

s-4m

2R

J J' R R2 2 2R R

2M 2s 1 × P 1+ P 1+ × M Γ

s-4m M -4m

ρ


J.J. Sanz-Cillero

which can be identified with the exchange

of a tree-level scalar resonance

in the crossed-channel

• By placing ImT(s) in the left-hand cut disperive integral:

Explicit analytical expression TJI(s)tR

for the contribution from the exchange

of a resonance R in the t (and u) channels.

• For instance, for R=S and the partial wave T11:

2 2 2 2 2 2 2 2I tS S S S S S SJ 2 2 2 2 2 4 2

S S S

2 M Γ s+2M -4m s+M -4m 2m -M s+M -4msT (s) = - + ln + ln

3 2m (s-4m ) (s-4m ) M 8m M

ρ

S

MR , R


J.J. Sanz-Cillero

I I I sR I tRJ J J J

R R

T (s) - T (0) = T (s) + T (s)

• Putting the different contributions together gives

• In our analysis, only the first V and S resonances have been included.

• Problems when higher-spin resonances were included.

Final dispersive expression


J.J. Sanz-Cillero

Matching Matching PT PT

at large-Nat large-NCC


J.J. Sanz-Cillero

• We perform a chiral expansion

of the resonance contributions TsR and TtR

I sR 2 4 6JT (s) = (p ) + (p ) + (p ) + ... O O O

I tR 2 4 6JT (s) = (p ) + (p ) + (p ) + ... O O O

in powers of s and m2

• For T(s) and T(0) , we use the values provided by PT up to (p6)

(amplitudes expressed in terms of s, m

2

and m-independent constants)

MR , R

LECs


J.J. Sanz-Cillero

• This produces a matching equation of the form,

I PT I PT I sR I tRJ J J J

R R

T (s) - T (0) = T (s) + T (s)

where we match left and right-hand side

order by order in (m2)m sn ,

LECs MR , R


J.J. Sanz-Cillero

• We have taken the matching up to (p6):

At (p2) we match the terms



(m2) NOT PRESENT

(s) mod-KSRF relation

(m4) NOT PRESENT

(s m2) Reson. relation

(s2) L2, L3

(m6) NOT PRESENT

(s m4) r2 - 2rf

(s2m2) r3, r4

(s3) r5, r6


J.J. Sanz-Cillero

• Simultaneous analysis of the IJ=11,00,20 channels

Compatible system of 18 equations: rank 9

number of unknowns = 9

• One must take into consideration that MR and R

are the physical large-NC masses and they also depend on m

4

2

2RR

R33R R R

+ (m )mΓ Γ

= 1 + ,M M M

O

…


J.J. Sanz-Cillero

Matching at (p2):

the (s1 m0) term Modified-KSRF relation

(constraint)(constraint)

• To exemplify the matching, we explicitly show this case:

V S

3 32V S

1 3Γ Γ + ... = + + ...

96 f 2M 9M s s

IJ=11 :

IJ=00 :

IJ=20 :

V S

3 32V S

1 9Γ 2Γ + ... = + + ...

16 f M 3M s s

V S

3 32V S

- 1 9Γ Γ + ... = - - + ...

32 f 2M 3M s s


J.J. Sanz-Cillero

• The three channels provide exactly the same constraint

which is a modification of the KSRF relation

that takes into account S resonances and crossed exchanges:

V S

3 32V S

1 9Γ 2Γ = +

16 f M 3M

to be compared to the original result,

V2 3

V

6Γ1 =

16 f M

2

V

g MΓ =

48

with

[Kawarabayashi & Suzuki’66]

[Riazuddin & Fayazuddin’66]

The original KSRF relation is recovered

in our analysis of the IJ=11 channel

if we neglect the impact from S resonances and crossed V exchanges


J.J. Sanz-Cillero

Matching at (p4):

the (s1 m2) term Novel resonance constraintconstraint

• The three channels provide exactly the same constraint:

This new relation provides a constraint between

the m2 corrections to masses and widths.

S V

S V5 5S V

2Γ 9Γ0 = +6 + +6

3M M


J.J. Sanz-Cillero

Matching at (p4):

the (s2 m0) term PredictionPrediction for L2 and L3

• The three channels provide two compatible constraints for the LECs: V4

2 5V

ΓL = 12 f

M

S V43 5 5

S V

2Γ 9ΓL = 4 f -

3M M

where similar results in terms of widths and masses

were also found in previous works [ Bolokhov et al.’93]


J.J. Sanz-Cillero

Matching at (p6):

(s m4), (s2 m

2), (s3 m0) PredictionPrediction for r3,4,5,6 and r2 -

2rf

• The three channels provide compatible constraints for the LECs:

S V6 6S S2 f S S7 7

S V

2 Γ Γ

r - r = 64 f 1+ + + f 7584+ + 3 6M M

with R and R given by the chiral corrections,

...3r = [Guo, Zheng & SC’07]

6

2 4

2 4RR

R R33R R R R

+ (m )m mΓ Γ

= 1 + + M M M M

O

4

2

2RR

R55R R R

+ (m )mΓ Γ

= 1 + M M M

O


J.J. Sanz-Cillero

Origin of the relations

• Good high-energy behaviour

• Good low-energy behaviour

Once subtracted

dispersion relations

PT matching


J.J. Sanz-Cillero

On the consistency On the consistency

of of

phenomenological phenomenological

lagrangianslagrangians


J.J. Sanz-Cillero

• Linear Sigma Model [’60,’70,’80,’90…]

• Gauged Chiral Model [Donoghue et al.’89]

• Resonance Chiral Theory (RT) [Ecker et al.’89]

and extended versions of RT [Cirigliano et al.’06]

We analysed a series of different phenomenological lagrangians:


J.J. Sanz-Cillero

• For sake of lack of time I will not explain

the first two cases in detail

(although they are exhaustively analysed in [Guo, Zheng & SC’07])

Nevertheless, the conclusion was that:

- Our dispersive predictions of the LECs

exactly agreed those obtained through the standard procedure

(integrating out the heavy resonances)

- We extracted constraints between resonance couplings

that were intimately related to the asymptotic high-energy behaviour


J.J. Sanz-Cillero

…Hence, I will focus on the last type of lagrangian.

First we will analyse

the original version of [Ecker et al’89],

the Minimal Resonance Chiral Theorythe Minimal Resonance Chiral Theory


J.J. Sanz-Cillero

• Non-linear realization for the Goldstones

• No assumptions on the vector and scalar nature

• Originally, only linear operators in the resonance fields were considered in the lagrangian:

RL μ μVd μ m + μ

i G = c S u u + c S + V u u +...

2

KinORL L L L

2(p )R T PT = + +

with the linear terms including only (p2 ) tensors,

[Ecker et al.’89]

• Procedure:Procedure: 1) First, we compute MR, R

2) Second, we check our relations


J.J. Sanz-Cillero

1.) We get the widths at LO in 1/NC:

33 22SV dV

V SV SV S 4 4

3 c M G M M =M , M =M , Γ = , Γ =

48 f 16 f

S

2d mm

SS 2 2dS

m 16 c c4 cΓ = Γ 1 + -6 + - +...

c fM

V

22Vd m

VV 2 22V S

m 16 c c MΓ = Γ 1 + -6 - +...

fM M


J.J. Sanz-Cillero

• Integrating out the resonances in the generating functional,

one gets the LECs corresponding to this action: 2V

2 2V

GL =

22dV

3 2 2V S

c3GL = -

2.) We compare the standard results and our LECs predictions:

[Ecker et al.’89]

• And using the dispersive predictions one gets a complete agreement: 2

V4 V2 5 2

VV

GΓL = 12 f =

M

22S V4 dV

3 5 5 2 2V SS V

c3G2Γ 9ΓL = 4 f = -

3M M

(SIMILAR AGREEMENT WAS FOUND IN THE ANALYSIS OF THE OTHER LAGRANGIANS)


J.J. Sanz-Cillero

• From the modified-KSRF constraint we get,

2 2d V

2 2

2c 3G1 = +

ff

• And the S,V constraint yields,

2 2d V

m m2 2

4c 6G1- c = c

ff

But notice that for

both constraints are incompatible

mc 0

…and study the resonance relations:


J.J. Sanz-Cillero

What is the problem in this What is the problem in this case?case?

• If we introduce the operator cm <S +> ,

it must come together with other operators

(if it is introduced alone, wrong results)

• What is special in the cm operator?

It is an operator that couples the scalar to the vacuum

proportionally to mq

• This makes f and the S-, V- vertices m dependent

even at large-NC

• However, we will see that this m dependence

may be produced by other operators not considered

S

SV


J.J. Sanz-Cillero

Extensions to Extensions to

Resonance Chiral Resonance Chiral

TheoryTheory


J.J. Sanz-Cillero

For a clearer understanding we will focus first on the scalar sector:

• Allowing a more general structure in the resonance lagrangian,

the scalar mass and width gain additional (m2)

corrections

from the extra resonance operators [Cirigliano et al.’06],

SL S μ S μ6 + μ 7 μ + = λ S , u u + λ S u u

SSL SS μ SS μ SS1 μ 2 μ 2 + = λ S S u u + λ Su S u + λ SS

SSSL SSS SSS μ0 1 μ = λ S SS + λ S S S

<R(p4)>

<RR(p2)>

<RRR(p2)>


J.J. Sanz-Cillero

• In order to compute the amplitudes free of scalar tadpoles we perform the mq-dependent shift,

M(o)

m 0q 2

S

4 c B S S +

M

• This provides the contribution to the mass and S- vertex,

now free of S tadpoles.

The S- interaction, in the isospin limit shows the structure,

[SC’04]

L M effeff r μ eff r r μ r 2 r rd μ m + 0 q μ S

1 = c S u u + c S ( - 4B ) + S S - M SS +...

2

With the m dependent parameters

2effd d d 2

S

mc =c 1 + δc + ...

M

2S S S SS SS SSSm

d 6 7 1 2 1 md d

4c2Mδc = 2λ +λ + 2λ +λ - 2λ c

c c

and MSeff = MS + O(m

2), cmeff = cm + O(m

2),


J.J. Sanz-Cillero

• Likewise, the pion decay constant gets m corrections at large NC,

12

2- d m

2 2S

m 4c c f =f Z = f 1+ f +... , with f =

fM

• Hence, the ratio /M3 for the scalar becomes,

eff 2 2S d S m

23 4S dS

Γ 3c 4m c = 1+ -1

M 16 f cM

S

2

d 2d

f = 2 c - 4 f + f

c


J.J. Sanz-Cillero

• Following a similar procedure for the vector we would have an effective coupling,

leading to the ratio,

2effV V V 2

V

mG =G 1 + δG + ...

M

eff 2 3V V V

3 4V

Γ G =

M 48 f

V

2V

V 2S

M = 2 G - 4 f

M


J.J. Sanz-Cillero

• Finally, putting everything together one gets the KSRF and S,V constraints:

22d dV V

2 2 2 24 4 2V S S S

2c (2 c -4 f)3G 2 G 4 f 1 2 f - + + = 0

ff fM M M M

22dV

4 4 2

2c3G 1 + =

ff f

which can be easily combined in the single form

effeff 2 2dV

4 4 2

2c3G 1 + =

ff f

But what is the meaning of this?


J.J. Sanz-Cillero

• At high energies the amplitude behaves like

Oeffeff 2 2

1 0dV1 2 2 2

2 c 3 G sT (s) = 1 - - + s

96 ff f [ SIMILAR RESULT FOR

IJ=00,20 ]

It is then clear now that the KSRF and S,V constraints

are equivalent to demanding a good behaviour

at high (and low) energies

Chiral lagrangians


J.J. Sanz-Cillero

ConclusionsConclusions


J.J. Sanz-Cillero

• New dispersive method for the the study

of LECs and resonance constraints at large-NC

• Easy implementation of high & low-energy constraints

independent of the realization of the resonance lagrangian

• Successfully checked for a wide set of different

phenomenological lagrangians

• Useful tool for future studies of other scattering amplitudes


J.J. Sanz-Cillero


J.J. Sanz-Cillero

1.)1.) Linear Sigma Linear Sigma

ModelModel


J.J. Sanz-Cillero

• Only Scalar + Goldstones (no Vectors)

• For our first check we use the LM, where the scalar and the Goldstones are introduced in a linear realization:

• Simple model with useful properties that give a first insight of the meaning of these constraints.

• Procedure: 1) First, we compute MS, S

2) Second, we check our relations

22 2 2 2 2 2 2 2

L M

1 1 1 = + σ + μ +σ - λ +σ +f m σ

2 2 4 L


J.J. Sanz-Cillero

• Renormalizability

• Chiral symmetry

Good high-energy behaviour

Good-low-energy behaviour

No place for further constraints

However, renormalizability is not the keypoint,

as we will see in the next example.

The KSRF and S,V constraints are trivially fulfilled for any value of

and

S

32S

1 2 Γ =

16 f 3 MKSR

F

S,V 0S =

[ T(s) ~ O(s0) when s∞ ]


J.J. Sanz-Cillero

2.)2.) Gauged Chiral Gauged Chiral

ModelModel


J.J. Sanz-Cillero

• Only Vector + Goldstones (no Scalars)

• The and a1 are introduced as gauge bosons

in the (p2) PT lagrangian :

2 2 2μ † †o

G M μ

f m f = D U D U + U+ U

2 2

L

μ μ 2 μ μμ μ o μ μ

1- L L + R R + M L L + R R 4

However, due to the -a1 mixing,

one finds a highly non-trivial interaction,

which makes the calculation of the -scattering

rather involved

[Donoghue et al.’89]


J.J. Sanz-Cillero

• Integrating out the resonances in the lagrangian,

one gets the corresponding LECs at large-NC : 2 4

2 4V

g fL =

2 4

3 4V

3 g fL = -

• If we now use the dispersive predictions we get exactly the same:

2V

V V

g MΓ =

2 4V4

2 5 4VV

g fΓL = 12 f =

M

2 4

V43 5 4

VV

3 g f9ΓL = 4 f = -

M

with

(p(p44) LECs :) LECs :


J.J. Sanz-Cillero

• The S,V constraint is trivially obeyed since we find

0

2

5V V

V 2 3V V m

M Γ d = = -6

Γ dm M

• This is not so for the KSRF constraint, which gives

V

32V

1 9Γ =

16 f M

2

2 2

V

3 g f 1 =

M

Origin of these constraints? Observe the -scattering amplitude at s∞ :

2

2 21 01 2

V

3 g fsT (s) = 1 - + s

96 f M

O [ SIMILAR RESULT FOR

IJ=00,20 ]

Resonance Resonance constraints :constraints :

V + 6

2V

V V

g MΓ =

= (m0) + 0 x (m

2)

TRIVIAL S,V relationKSRF relation


J.J. Sanz-Cillero

• Through the explicit integration of the heavy resonances in the generating functional one gets the LECs corresponding to this action:

2V

2 2V

GL =

22dV

3 2 2V S

c3GL = -

• We also get the widths at LO in 1/NC,

33 22SV dV

V SV SV S 4 4

3 c M G M M =M , M =M , Γ = , Γ =

48 f 16 f

S

2d mm

SS 2 2dS

m 16 c c4 cΓ = Γ 1 + -6 + - +...

c fM

V

22Vd m

VV 2 22V S

m 16 c c MΓ = Γ 1 + -6 - +...

fM M


J.J. Sanz-Cillero

• If we now use the dispersive predictions we get exactly the right results for the LECs:

2V4 V

2 5 2VV

GΓL = 12 f =

M

22S V4 dV

3 5 5 2 2V SS V

c3G2Γ 9ΓL = 4 f = -

3M M

In complete agreement

with the original lagrangian calculation [Ecker et al.’89]

Large-N C resonance relations from partial wave analyses

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