Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st...

Structure of NLO corrections in 1/NC J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21st 2006

Structure of

next-to-leading order

corrections in 1/NC

J.J. Sanz Cillero, IPN-Orsay

Hadrons & Strings, Trento, July 21st 2006


BottomBottom

UpUp

Very bottomVery bottom

Just general QCD properties:

4D-QFT description of hadrons


Just general QCD properties:

•4D-QFT description with hadronic d.o.f.

•Chiral symmetry invariance (nf light flavours)

•1/NC expansion around the ‘t Hooft large-NC limit:

NC ∞, NC s fixed

Pole structure of amplitudes at large NC (tree-level)

•Analiticity + matching QCD short-distance behaviour

(parton logs + s logs + OPE)

V,V’,…

[ ‘t Hooft 74 ]

[ Callam et al.’69]

[Colleman et al.’69]

[Bando et al.’85][Ecker et al.’89]

…, [ Peris et al.’98][Catà et al.’05]

[SC’05]


Why going up to NLO in 1/NWhy going up to NLO in 1/NCC??

• To validate the large-NC limit: NLO under control

• To show the phenomenological stability of the 1/NC series

• To increase the accuracy of the predictions

• To make real QFT in 1/NC, not just narrow-width ansate

• To understand sub-leading effects (widths, exotica,…)

• Because we already have it there (even we don’t know it)


Large-NLarge-NCC QCD, QCD,

NLO in 1/NNLO in 1/NCC

andand

NNCC=3 QCD=3 QCD


QFT description of amplitudes at large NC

Infinite number of hadronic states

+Goldstones from the SSB (special)

• Infinite set of hadronic operators in = i i

(but don’t panic yet; this already happens in large-NC QCD)

• Chiral symmetry invariance

• Tree-level description of the amplitudes:

strengths Zk (residues) and masses Mk (pole positions)


2q……

aa11(1260)(1260)(770)(770)

LO in 1/NC

(tree-level)

Im 2(q )

Re{q2}Im{q2}

[ ‘t Hooft 74 ][ Witten 79]


…… 2q

aa11(1260)(1260)(770)(770)

LO + NLO in 1/NC

(tree-level + one-loop)

Re{q2}Im{q2}


2q

aa11(1260)(1260)(770)(770)

LO + SLO in 1/NC

+ Dyson-Schwinger summation

(tree-level + one-loop widths)

Re{q2}Im{q2}

Unphysical Riemann-sheets


Truncation of Truncation of

the large-Nthe large-NCC spectrum spectrum


Minimal Hadronical Approximation

[Knecht & de [Knecht & de Rafael’98]Rafael’98]

Large-NC (infinite # of d.o.f.)

• Lack of precise knowledge on the high-lying spectrum

• Relative good knowledge of low-lying states


Minimal Hadronical Approximation

[Knecht & de [Knecht & de Rafael’98]Rafael’98]

Large-NC (infinite # of d.o.f.)

Approximate large-NC (finite # of d.o.f. –lightest ones-)

• Lack of precise knowledge on the high-lying spectrum

• Relative good knowledge of low-lying states


Ingredients of a Ingredients of a Resonance Chiral Theory Resonance Chiral Theory

(R(RT)T)

• Large NC U(nf) multiplets

• Goldstones from SSB

• MHA: First resonance multiplets (R=V,A,S,P)


(2) L L L L L1 1 2 1 2 3

1 1 2 1 2 3

R T PT R R ,R R ,R ,RR R ,R R ,R ,R

= + + + + ...

[ Ecker et al.’89]


L2

+(2)PT

F= u u +

4

[ ]+V V

V

i= V f + V u ,

F G[ , u

2 2V]

2 2

L

LS[ , ,S] LA[ , ,A] LP[ ,P]

L1

1

RR

(2)L PT

……… [Moussallam’95], [Knecht &

Nyffeler’01][ Cirigliano et al.’06]

[Pich,Rosell & SC, forthcoming]


[ Weinberg’79]

couplings iRR, i

RRR

1 2 1 2 3

1 2 1 2 3

R ,R R ,R ,RR ,R R ,R ,R

+ L L

[ Gasser & Leutwyler’84]



We must build the RWe must build the RTT

that best mimics QCD at large-Nthat best mimics QCD at large-NCC

• Chiral symmetry invariance:

Ensures the right low-energy QCD structure (PT),

even at the loop level!

• At short-distances:

Demand to the theory the high-energy power behaviour prescribed by QCD (OPE)[Shifman et al ’79]

[ Weinberg’79]

[ Gasser & Leutwyler’84,’85]

[ Catà & Peris’02]

[Harada & Yamawaki’03]

[ Rosell, Pich & SC’04, forthcoming’06]


C

MHA

N Π(s) (2 )OPE < >

Π(s) = ( - s)åO l

ll

s -∞

• Constraints among the couplings i and masses MR at NC∞

e.g., Weinberg sum-rules

[Weinberg’67]

1

2 2 2aF - F - F = 0 r p

1 1

2 2 2 2a aF M - F M = 0r r

LR(2)< > =0OLR(4)< > =0O


It is possible to develop the RT up to NLO in 1/NC

One Loop Diagrams

NLO Contributions

However, Loops=UV Divergences!!

New NLO pieces (NLO couplings)?

Removable through EoM

if proper short-distance

RT at LO

[Rosell, Pich & SC’04]

[Ecker et al.’89], …

[Catà & Peris’02]

[Rosell, Pich & SC, forthcoming’06]

[Rosell et al.’05]


MHA

LO+NLO Π(s) (2 )OPE < >

Π(s) = ( - s)åO l

ll

s -∞

• Constraints among i and masses MR : LO + NLO contribution

e.g., WSR, 1

2 2( )aN O2

2 LF - F - F + = 0 δr p

1 1

2 2 2 2a a

NLO(4)F M - F M + 0 δ = r r

Again, one must build the RAgain, one must build the RT that best T that best mimics QCD,mimics QCD,

but now up to NLO in 1/NC :

- Natural recovering of one-loop PT at low energies

- Demanding QCD short-distance power behaviour



However,… plenty of problems

• The # of different operators is ~102 (NOW YOU CAN PANIC!!!)

• Even with just the lightest resonances one needs

~30 form-factors k(s) to describe all the possible

intermediate two-meson states in LR(s)

• Systematic uncertainty due to the MHA

• Eventually, inconsistences between constraints when more and more amplitudes under analysis

• Need for higher resonance multiplets

• Even knowing the high-lying states,

serious problems to manage the whole large-NC

spectrum

[Rosell et al.’05]


[SC’05]

[Bijnens et al.’03]


General General

properties at properties at

NLO in 1/NNLO in 1/NCC


Interesting set of QCD matrix elements

• QCD amplitudes depending on a single kinematic variable q2

• Paradigm: two-point Green-functions,

e.g., left-right correllator LR(q2), scalar correllator SS(q2), …

also two-meson form factors <M1 M2||0> ~ (q2)

• We consider amplitudes determined by their physical right-hand cut.

For instance, partial-wave projections into TIJ(s)

transform poles in t and u variables into continuous left-hand cut in s variable.

2 4 -iqx Π(q ) = i dx e T (x) (0) †O O

[SC, forthcoming’06]


Essentially, we consider amplitude

with an absorptive part of the form

2

k kk=1

1 ImΠ(t) = Z δ t-M

This information determines

the QCD content of

the two-point Green-functions


• Exhaustive analysis of the different cases:

1. Unsubtracted dispersive relations

Infinite resonance large-NC spectrum

2. m-subtracted dispersive relations

Straight-forward generalization


Unsubtracted dispersion Unsubtracted dispersion relationsrelations

• This is the case when (s)0 for |s|∞

• In this case one may use the analyticity of (s) and

consider the complex integral

• Providing at LO in 1/NC the correlator expresion

+

1 dt dt 1 Π(s) = Π(t) = ImΠ(t)

2 i t-s t-sR

C

k2N

k=1 k

Z Π(s) =

M - sR1, R2,

…


• Up to NLO in 1/NC one has tree-level + one-loop topologies

• The finite (renormalized) amplitudes contain up to doble

poles

2LO+NLO 2k

1 Π(s)

M - s

…so the dispersive relation

must be performed a bit more carefully…


ZOOM


ssMk,r

2

ZOOMZOOM

2

(1) (2)k k

2 2LO+NLOk=1 k,r k,r

D D= (s) + +

M -s M -s(s)


ssMk,r

2

ZOOMZOOM

2

(1) (2)k k

2 2LO+NLOk=1 k,r k,r

D D= (s) + +

M -s M -s(s)

with the finite contribution

2

1 1Im Im

k=1

B-

r 2R

k=12k,r

t (M )

M -tdt 2 = lim - lim (s) (t) (t

t-s t )

- s→0 2k,r→ M

R


…where, in addition to the spectral function (finite),

one needs to specify the value of:

• Each residue

• Each double-pole coefficient

• Each renormalized mass

22k

(1)k ,r

2k,rt=M

d= - Re MD -t

dt(t)

22k,r

(2)k

2k,rt=M

= - Re M -tD (t)

2k,rM

2...

(1) (2)k k

2LO+NLOk

2k,r

=1 k,r

= + + -s

D D

M -sM(s)


What’s the meaning of all What’s the meaning of all this is in QFT language?this is in QFT language?

• Consider separately the one-loop contributions (s)1-loop

• Absorptive behaviour of (s)1-loop = (s)OPE at |s|∞

• Possible non-absorptive in (s)1-loop ≠ (s)OPE at |s|∞

(but no physical effect at the end of the day)

• Counterterms in (s)tree = behaviour as (s)OPE at |s|∞


If one drops appart the any “nasty” non-absorptive contribution in

(s)1-loop

(s)1-loop fulfills the same dispersion relations as (s)LO+NLO

2

(1),1-loop (2),1-loopk k

2 21-loopk=1 k,r k,r

D D= (s) + +

M -s M -s(s)

Same Same finite finite

functionfunctionUV UV

divergencedivergencess

+ …


But, the LO operators are precisely those needed

for the renormalization of these UV-divergences

Renormalization of the Zk and Mk2 up to NLO in 1/NC:

2

2

k,r k,r k,r k 2k k,r2 2 2tree

k=1 k,r k,r k,r

Z Z Z M = + - + M Z

M - s M - s M - sΠ(s) O

Finite Finite renormalized renormalized

couplingscouplings

Counter-Counter-termsterms

k k,r kZ = Z + Z

2 2 2k k,r kM = M + M

NNLO in 1/NC


…leading to the renormalization conditions,

(1),1-loop (1)k k kD + Z = c

(2),1-loop 2 (2)k k,r k kD - Z M = c

with ck(1) and ck

(2) setting the renormalization scheme

(for instance, ck(1)=ck

(2)=0 for on-shell scheme )

• Hence, the amplitude becomes finally finite:

2

(1) (2)

k,r k k2 2LO+NLO

k=1 k,r k,r

Z + c c= + + (s)

M -s M -s(s)


…leading to the renormalization conditions,

(1),1-loop (1)k k kD + Z = c

(2),1-loop 2 (2)k k,r k kD - Z M = c

with ck(1) and ck

(2) setting the renormalization scheme

(for instance, ck(1)=ck

(2)=0 for on-shell scheme )

• Hence, the amplitude becomes finally finite:

k,r2LO+NLO

k=1 k,r

Z= + (s)

M -s(s) On-shell On-shell

schemescheme


And what about those “nasty” non-And what about those “nasty” non-absorptive terms?absorptive terms?

• This terms are not linked to any ln(-s) dependence Purely analytical contributions

• They would require the introduction of local counter-terms

• Nevertheless, when summing up, they both must vanish (so (s)0 for |s|∞)

m(m)

non-abs.m

= a s (s) UV

divergences

m(m)

local-ct.m

= s (s) C

NLO local couplings

m(m) (m)

non-abs. + local-ct.m

= +a s = 0(s) C


m-subtracted dispersion m-subtracted dispersion relations relations

• Other Green-functions shows a non-vanishing behaviour

(s)sm-1 when |s|∞

• In that situations, one need to consider not (s) but

some m-subtracted quantity like the moment of order

m:

• This contains now the physical QCD information, and

can be obtained from the spectral function: +

m(m)

m+1

(-s) dt 1 (s) = ImΠ(t)

(t - s)R

A

m(m) 1 d

(s) -s Π(s) m! ds

A


To recover the whole (s) one needs to

specify m subtraction constants

at some reference energy s=sO

(0) (m-1)O O (s ), ..., (s ) A A

• These subtractions are not fixed by QCD

(e.g., in the SM, VV(sO) is fixed by the photon wave-function renormalization)


• Providing at LO in 1/NC the pole structures

C

m(m) k

m+1N 2k=1

k

(-s) Z (s) =

M - sA R1, R2,

…


…but at the end of the day, at NLO

one reaches the same kind of renormalization conditions (1),1-loop (1)

k k kD + Z = c

(2),1-loop 2 (2)k k,r k kD - Z M = c

…and an analogous structure for the renormalized moment:

(1) m (2) m

(m)k,r k(m) km+1 m+2LO+NLO 2 2

k=1k,r k,r

Z + c (-s) c (-s)(s) = + + (s)

M -s M -s A A

Finite (from the spectral function)Renormalized tree-level


…but at the end of the day, up to NLO

one reaches exactly the same renormalization conditions (1),1-loop (1)

k k kD + Z = c

(2),1-loop 2 (2)k k,r k kD - Z M = c

…and an analogous structure for the renormalized moment:

m

(m)k,r(m)m+1LO+NLO 2

k=1k,r

Z (-s)(s) = + (s)

M -s A A

Finite (from the spectral function)Renormalized tree-level

On-shell On-shell

schemescheme


Renormalizability?Renormalizability?


• RT descriptions of (s) inherites the “good renormalizable

properties” from QCD, through the matching in the UV (short-

distances)

• Caution on the term “renormalizability”: Infinite # of renormalizations

• The LO operators cover the whole space of possible UV divergences

(for this kind of (s) matrix elements)

• Inner structure of the underlying theory:

The infinity of renormalizations are all related

and given in terms of a few “hidden” parameters (NC and NCs in

our case)

(see, for instance, the example of QED5 [Álvarez & Faedo’06])


• General “renormalizable” structures in other matrix elements?

Appealing!!

Larger complexity (s1,s2,…)

Multi-variable dispersion relations, crossing symmetry,…

Next step: three-point GF and scattering amplitudes


ConclusionsConclusions


• General QCD properties + 1/NC expansion:

Already valuable information

• Decreasing systematic errors

• Increasing accuracy

• Proving that QCDNC=3 has to do with QCDNC∞

• MHA:

Relevance of

NLO in 1/NC

-Introduces systematic uncertainties

-Makes calculation feasible

Nevertheless, at some point the 4D-QFT becomes unbearably

complex


• AdS dual representations of QCD are really welcome:

They provide nice/compact/alternative description of QCD

Extremely powerful technology

• However, there are several underlying QCD features

that must be incorporated:

- Chiral Symmetry and Goldstones from SSB

- Short-distance QCD (parton logs + s logs + OPE)

- “Renormalizable” structure for (s) amplitudes at NLO in 1/NC

in terms of a few AdS parameters


• Chiral order parameter: No pQCD contribution

• Isolates the effective PT coupling L8 (quark mass <-> pGoldstone

mass )

• Less trivial case than the J=1 correlators

• Two-point Green functions:

• We focus the attention on the SS-PP with I=1

Interest of this Interest of this correlatorcorrelator

2 4 iqx Π(q ) = i dx e T q q(x) q q(0) †


• Resonance Chiral Theory framework (RT):

Construction of the lagrangian

PROGRAM:




• 2-body form-factors at LO in 1/NC:

QCD short-distance constraints on the FF at LO in 1/NC

PROGRAM:

Tree-level






• Derivation of S-P (dispersive relations):

QCD short-distance constraints on S-P up to NLO in 1/NC

PROGRAM:

Tree-level

1-loop






• Derivation of S-P (dispersive relations):

QCD short-distance constraints on S-P up to NLO in 1/NC

• Recovering PT at low energies:

Low energy constants up to NLO in 1/NC: L8

PROGRAM:

Tree-level

1-loop

1-loop


RRT lagrangianT lagrangian


Ingredients of RIngredients of RTT

• Large NC U(nf) multiplets

• Goldstones from SSB (,K,8,0)

• MHA: First resonance multiplets (V,A,S,P)


• Just (p2) operators

• Chiral limit

(2) L L L L L1 1 2 1 2 3

1 1 2 1 2 3

R T PT R R ,R R ,R ,RR R ,R R ,R ,R

= + + + + ...


L2

+(2)PT

F= u u +

4

[ ]+V V

V

i= V f + V u ,

F G[ , u

2 2V]

2 2

L

LS[ , ,S] LA[ , ,A] LP[ ,P]

L1

1

RR

(2)L PT

……… [Moussallam’95], [Knecht &

Nyffeler’01][ Cirigliano et al.’06]

[Pich,Rosell & SC, forthcoming]


[ Weinberg’79]

couplings iRR, i

RRR

1 2 1 2 3

1 2 1 2 3

R ,R R ,R ,RR ,R R ,R ,R

+ L L




2-body form-factors2-body form-factors


Optical theorem and the 1/NOptical theorem and the 1/NCC expansionexpansion

• At LO in 1/NC, t is given by tree-level (1-particle intermediate states)

1Im 2 2

i i1P-cut

(t) F (t-M )

22

• 1-P cuts: asymptotic behaviour

0

O2i21P-cuti

F 1(t)

M -t t

t


• 2-P cuts: asymptotic behaviour??

• At NLO in 1/NC, 2-particle intermediate states:

1Im F

2

2P-cut

(t) (t) (t)

22


1Im

2P-cut2P-cut

(t) (t)

O1P-cut

1(t)

t

1Im

F O2

2

1(t) (t)

t

<

[Brodsky & Lepage’79]

1Im

2P-cut2P-cut

(t) , (t) 0

1 1Im Im pQCD

2P-cut

(t) < (t) const. V,A V,A

t

t

ARGUMENTS:


General FF analysis:

VV

, V, A, …, V, A, …

, V, , V, ,,……

AA

V, A, S, …V, A, S, …

, V, , V, ,,……

SS

, V, A, …, V, A, …

, V, , V, ,,……

PP

V, A, S, …V, A, S, …

, V, , V, ,,……


SS-PP correlatorSS-PP correlator

at one loopat one loop


The example of LThe example of L88::


• At LO in 1/NC one has the resonance exchange

2 2 2 2 2 22 0 0 m 0 m

S-P 2 2 2 2 2S P

2B F 16 B c 16 B d(q ) = + -

q M -q M -q


The example of LThe example of L88::


• At LO in 1/NC one has the resonance exchange

2 2 2 2 2 22 0 0 m 0 m

S-P 2 2 2 2 2S P

2B F 16 B c 16 B d(q ) = + -

q M -q M -q

which at low energies becomes,

O2 2m m

2 2S

2 22 2 20

S-PP

02

c d-

2B F(q ) =

2+ 32 B +

M 2

M(q )

q

CN8 L


• Matching OPE for S-P:

2 2 2 20 m m-2 B F - 8 c + 8 d = 0

O2 2 2 2 2 S-P0 m S m P (4)2 B - 8 c M + 8 d M = 0

C

222 Pm 2 2N

P S

M Fc

8 M -M

C

222 Sm 2 2N

P S

M Fd

8 M -M

[ Golterman & peris’00 ]



one gets at low energies,

O2 22 2

2 2 20S- 2P 2

P2

S0

F F+

16 M2B F

(q ) = + 32 B 16

+ (qq M

)

2 2 2 20 m m-2 B F - 8 c + 8 d = 0

O2 2 2 2 2 S-P0 m S m P (4)2 B - 8 c M + 8 d M = 0

C

222 Pm 2 2N

P S

M Fc

8 M -M

C

222 Sm 2 2N

P S

M Fd

8 M -M

CN -38 L 0.7 10




one gets at low energies,

O2 22 2

2 2 20S- 2P 2

P2

S0

F F+

16 M2B F

(q ) = + 32 B 16

+ (qq M

)

2 2 2 20 m m-2 B F - 8 c + 8 d = 0

O2 2 2 2 2 S-P0 m S m P (4)2 B - 8 c M + 8 d M = 0

C

222 Pm 2 2N

P S

M Fc

8 M -M

C

222 Sm 2 2N

P S

M Fd

8 M -M

CN -38 L 0.7 10 ??????

-30.5 10



1 2

1 2

2 2 2 r 2 2 r 22 20 0 m 0 m

S-P S-P2 r 2 2 r 2 2 m ,mm ,mS P

2B F 16 B c 16 B d(q )= + - + (q )

q M -q M -q

• Up to NLO in 1/NC S-P shows the general structure

with the 2-P contributions from dispersion relations

2

1Im

iR M

1 21 2

2S-P S-P2m ,m

m ,m

dt(q ) (t)

t-

q

depending on the correponding couplings i,

fixed before at LO in 1/NC in the FF analysis


• Exact definition of the integral:

2 1Im

1 2

r 2R S-P

tm ,m

2 - lim M -t (t)

2R→M

1 1Im Im

0

1 21 2 1 2

2S-P S-P S-P2 2m ,m

m ,m m ,m

dt dt (q ) = lim (t) + (t)

t-q t-q

2R

2R

M

→0M +

tMR

2


• Example: contribution



F d m0 2 2

S

4 c c t(t) = 2B 1 +

F M -t Tree-level Tree-level SFFSFF



F d m0 2 2

S

4 c c t(t) = 2B 1 +


F2S

0 2S

M(t) = 2B

M -t Short-distance SFF Short-distance SFF (correlator)(correlator)



F d m0 2 2

S

4 c c t(t) = 2B 1 +


F2S

0 2S

M(t) = 2B


1Im

F22 2

2 f 0 SfS-P 2

S

n B Mn(t) t (t) t

M -t

Optical Optical theoremtheorem



F d m0 2 2

S

4 c c t(t) = 2B 1 +

F M -t

22 2 2 22 f 0 S

S-P 2 2 2 2S S S

n B M q -q(q ) -1+ +ln

M -q M M

Tree-level Tree-level SFFSFF

F2S

0 2S

M(t) = 2B


1Im

F22 2

2 f 0 SfS-P 2

S

n B Mn(t) t (t) t

M -t

Optical Optical theoremtheorem

Dispersion Dispersion relationsrelations


2-particle channels:2-particle channels:

•Goldstone-Goldstone

()

•Resonance-Goldstone

(R)

•Resonance-Resonance

Suppressed

Neglected


Full recovering of Full recovering of PTPT

at one loopat one loop


• Result in PT within U(nf):

Low energy expansion Low energy expansion at one loop at one loop

O2 2 2

2 2 20 0fS-P 02

2r8 2

2B F B n(q ) = + 32 B + 1- + (q )

-qL ( ) l

2 8n

q

TO NOTICE:

•Exact cancellation of dependence

•Presence of the massless ln(-q2) from loop

•Analytical part (L8 coupling constant)


• Tree level:

O2 2 r 2 r 2

2 2 20 m mS-P 02 r 2 r 2tree

S P

2B F c d(q ) = + 32 B - + (q )

q 2 M 2 M

Analytical Analytical LO +NLOLO +NLO


• Tree level:

O2 2 r 2 r 2

2 2 20 m mS-P 02 r 2 r 2tree

S P

2B F c d(q ) = + 32 B - + (q )

q 2 M 2 M

• Intermediate state

O2

2 20fS-

2

2S

P

B n(q ) = -1-

-

qln + (q )

2 8

Chiral log Chiral log

NLONLO



• Tree level:

O2 2 r 2 r 2

2 2 20 m mS-P 02 r 2 r 2tree

S P

2B F c d(q ) = + 32 B - + (q )

q 2 M 2 M


O2

2 20fS-

2

2S

P

B n(q ) = -1-

-

qln + (q )

2 8

• Intermediate state R

[ ic O2 2fS-P R

n(q ) = ] + (q )

2

constant

Chiral log Chiral log NLONLO

Analytical Analytical NLONLO



• Tree level:

O2 2 r 2 r 2

2 2 20 m mS-P 02 r 2 r 2tree

S P

2B F c d(q ) = + 32 B - + (q )

q 2 M 2 M


O2

2 20fS-

2

2S

P

B n(q ) = -1-

-

qln + (q )

2 8

• Intermediate state R

[ ic O2 2fS-P R

n(q ) = ] + (q )

2

• Intermediate state RR NEGLECTED NEGLECTED

constant

Chiral log Chiral log NLONLO

Analytical Analytical NLONLO



Matching OPE for S-P(q2) ~

1/q4 up to NLO in 1/NC

2 r 2 r 2m m

2NLO- F + 8 c - 8 d - = F 0

2Sf

2NLO 2

Mn = + ...

3 F

2

with

( <4> not considered, competition <4> vs.

NLO)




2 r 2 r 2m m

2NLO- F + 8 c - 8 d - = F 0

2Sf

2NLO 2

Mn = + ...

3 F

2

with

NL

2r 2 r 2m Om

Fc = d + 1+

8


NLO)




2 r 2 r 2m m

2NLO- F + 8 c - 8 d - = F 0

2Sf

2NLO 2

Mn = + ...

3 F

2

with

NL

2r 2 r 2m Om

Fc = d + 1+

8

NLOr 2 r

8 tree

2 r 2 r 2r 2P Sm

r 2 r 2 r 2S P S

2NLOm m

r 2 r 2S P

F 1+ M -Md =

c d-

2 M 2= +

16 M 2 M MML

Δ


NLO)


InputsInputs

VM = 776 16 MeV Parameters Parameters needed needed at LO in 1/Nat LO in 1/NCC

(appearing only NLO in S-P)

AM = 1.23 0.04 GeVr

S SM M = 0.98 -1.3 GeVr

P PM M = 1.30 0.10 GeV

0m = 0.90 0.05 GeV

F = 92.4 5.0 MeVU(3) SU(3)

[ Kaiser & Leutwyler’00 ]


InputsInputs

2

r 2 r 2m m NLO

Fc = d + 1+

8

VM = 776 16 MeV Parameters Parameters needed needed at LO in 1/Nat LO in 1/NCC

(appearing only NLO in S-P)

AM = 1.23 0.04 GeVr

S SM M = 0.98 -1.3 GeVr

P PM M = 1.30 0.10 GeV

0m = 0.90 0.05 GeV

F = 92.4 5.0 MeV

(appearing at LO+NLO in S-P)

rSM = 0.98 -1.3 GeV

rP M = 1.30 0.10 GeV

1

C

r 2 2m m N

C

1d = d

N

U(3) SU(3)

SD matching up to NLO

Short-distance matching at LO


Parameters Parameters needed needed up to NLO in up to NLO in

1/N1/NCC


770 = MeVResultsResults(for comparisson;

exactly scale independent expression)

• Contributions:

3 r8 = -0.08 + 0.54 - 0.72 + 0.68 + 0.38 + 0 + 10 0.1 2 L ( )

U(3)SU(3) tree V S A P[ Kaiser & Leutwyler’00 ]




• Contributions:

3 r8 = -0.08 + 0.54 - 0.72 + 0.68 + 0.38 + 0 + 10 0.1 2 L ( )

U(3)SU(3) tree V S A P

• Uncertainties: 3 r

8+0.25 +0.0010- 0.40 - 0.0040

+0.3- 0.5= 0.11 0.08 10 L ( 0.03 0.03 0.12 ) Δ

MSr dm

r F mo MAMV truncationMP

r





• Contributions:

3 r8 = -0.08 + 0.54 - 0.72 + 0.68 + 0.38 + 0 + 10 0.1 2 L ( )

U(3)SU(3) tree V S A P

• Uncertainties: 3 r

8+0.25 +0.0010- 0.40 - 0.0040

+0.3- 0.5= 0.11 0.08 10 L ( 0.03 0.03 0.12 ) Δ

MSr dm

r F mo MAMV truncationMP

r

r8

- 3+0.4- 0.6= ( 0.9 ) 1 L 0( ) to be compared to the PT

result, r8 PT

- 3L ( ) = ( 0.9 0.3 ) 10



ConclusionsConclusions


• Large NC is meaningful: it is possible to control

NLO



NLO

• Systematic expansion of QCD amplitudes in 1/NC



NLO


• General analysis of the 2-body FF



NLO



• General structure of (t) (dispersive analysis)



NLO




• Short-distance matching order by order in 1/NC



NLO





• Full recovering of PT at low q2:

-Example of L8



NLO






-Example of L8

• Manifestation of the uncertainty origin and full control of the “saturation” scale



NLO






-Example of L8

• Manifestation of the uncertainty origin and full control of the “saturation” scale

• Straight-forward extension to (p6) LECs


• How is it possible to compute hadronic loops?

(Why and how it works? How loops do not blow up at high/low energies? …)

• How is the transition from high to low energy QCD?

(How can the d.o.f. change from Goldstones Resonances pQCD Continuum?

How do we have this progressive change in the amplitudes? …)

• How can we relate hadronic and quark-gluon parameters?

Energy regimes? Weinberg sum-rules? Narrow-width approximations,

do they have some systematic physics behind or they just fix “experimental” numbers?

How well do we understand hadronic How well do we understand hadronic interactions?interactions?


QCD expansion in 1/NQCD expansion in 1/NCC

?? QCD at any qQCD at any q22

(MESONS(MESONS))


Resonance FF, does it make any sense?

(1(1stst))

VMDVMD WSRWSR

(2(2ndnd))

[Weinberg’67]…

[Cata & Peris’ 02]

[Pich, Rosell & SC’04]

(3(3rdrd))[Pich, Rosell & SC;

forthcoming]

a1

a1

R

R’

R

R’

R

R’

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st...

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Transcript of Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st...