Structure of next-to-leading order corrections in 1/NC

1
Structure of next-to-leading order
corrections in 1/NC

• J.J. Sanz Cillero, IPN-Orsay

Hadrons Strings, Trento, July 21st 2006
2
Up
Bottom
Just general QCD properties 4D-QFT description
of hadrons
Very bottom
3

• 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 ?8, NC as fixed
• Pole structure of amplitudes at large NC
  (tree-level)
• Analiticity matching QCD short-distance
  behaviour
• (parton logs as logs OPE)

Callam et al.69 Colleman et al.69 Bando
et al.85 Ecker et al.89
t Hooft 74
p
V,V,
p
, Peris et al.98 Catà et al.05 SC05
4
Why going up to NLO in 1/NC?

• 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 dont
  know it)

5
Large-NC QCD, NLO in 1/NC and NC3 QCD
6
QFT description of amplitudes at large NC

• Infinite number of hadronic states
• Goldstones from the ScSB (special)
• Infinite set of hadronic operators in LS li Oi
  
• (but dont 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)

7
LO in 1/NC (tree-level)
t Hooft 74 Witten 79
p
a1(1260)
r(770)

Imq2
Req2
8
LO NLO in 1/NC (tree-level one-loop)
p
a1(1260)
r(770)

Imq2
Req2
9
LO SLO in 1/NC Dyson-Schwinger summation
(tree-level one-loop ? widths)
p
a1(1260)
r(770)
Imq2
Req2
Unphysical Riemann-sheets
10
Truncation of the large-NC
spectrum
11
Minimal Hadronical Approximation
Knecht de Rafael98

• Lack of precise knowledge on the high-lying
  spectrum
• Relative good knowledge of low-lying states

Large-NC (infinite of d.o.f.)
12
Minimal Hadronical Approximation
Knecht de Rafael98

• Lack of precise knowledge on the high-lying
  spectrum
• Relative good knowledge of low-lying states

Large-NC (infinite of d.o.f.)
Approximate large-NC (finite of d.o.f.
lightest ones-)
13
Ingredients of a
Resonance Chiral Theory (RcT)
Ecker et al.89

• Large NC ? U(nf) multiplets
• Goldstones from ScSB
• MHA First resonance multiplets (RV,A,S,P)
• Chiral symmetry invariance

14
Weinberg79
Gasser Leutwyler84
Gasser Leutwyler85
Ecker et al.89

couplings liRR, liRRR
Moussallam95, Knecht Nyffeler01
Cirigliano et al.06 Pich,Rosell SC,
forthcoming
15
We must build the RcT that best mimics QCD at
large-NC
Weinberg79 Gasser Leutwyler84,85

• Chiral symmetry invariance
• Ensures the right low-energy QCD structure
  (cPT),
• even at the loop level!
• At short-distances
• Demand to the theory the high-energy power
  behaviour prescribed by QCD (OPE)

Catà Peris02 Harada Yamawaki03
Rosell, Pich SC04, forthcoming06
Shifman et al 79
16
s ? -8

• Constraints among the couplings li and masses
  MR at NC?8
• e.g., Weinberg sum-rules

Weinberg67
17
It is possible to develop the RcT up to NLO in
1/NC
RcT at LO
Ecker et al.89,
Catà Peris02
However, LoopsUV Divergences!!
New NLO pieces (NLO couplings)?
Rosell, Pich SC04
Rosell et al.05
Removable through EoM
if proper short-distance
Rosell, Pich SC, forthcoming06
18
Again, one must build the RcT that best mimics
QCD, but now up to NLO in 1/NC -
Natural recovering of one-loop cPT at low
energies - Demanding QCD short-distance power
behaviour
s ? -8

• Constraints among li and masses MR LO NLO
  contribution
• e.g., WSR,

Rosell, Pich SC, forthcoming06
19
However, plenty of problems
Rosell et al.05

• The of different operators is 102 (NOW
  YOU CAN PANIC!!!)
• Even with just the lightest resonances one needs
  30 form-factors Fk(s) to
  describe all the possible intermediate two-meson
  states in PLR(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, Pich SC, forthcoming06
SC05
Bijnens et al.03
20
General properties at NLO in 1/NC
21
Interesting set of QCD matrix elements
SC, forthcoming06

• QCD amplitudes depending on a single kinematic
  variable q2
• Paradigm two-point Green-functions,
• e.g., left-right correllator PLR(q2), scalar
  correllator PSS(q2), also
  two-meson form factors ltM1 M2O0gt F(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.

22
Essentially, we consider amplitude
with an absorptive part of the form
This information determines the
QCD content of the
two-point Green-functions
23

• Exhaustive analysis of the different cases

• Unsubtracted dispersive relations
• Infinite resonance large-NC spectrum
• m-subtracted dispersive relations
• Straight-forward generalization

24
Unsubtracted dispersion relations

• This is the case when P(s)?0 for s?8
• In this case one may use the analyticity of P(s)
  and consider the complex integral
• Providing at LO in 1/NC the correlator expresion

R1, R2,
25

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

• The finite (renormalized) amplitudes contain up
  to doble poles

so the dispersive relation must be performed a
bit more carefully
26
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27
ZOOM
28
e
s
Mk,r2
ZOOM
29
e
s
Mk,r2
ZOOM
30
e
s
Mk,r2
ZOOM
with the finite contribution
31

• where, in addition to the spectral function
  (finite), one needs to specify
  the value of
• Each residue
• Each double-pole coefficient
• Each renormalized mass

32
Whats the meaning of all this is in QFT language?

• Consider separately the one-loop contributions
  P(s)1-loop
• Absorptive behaviour of P(s)1-loop
  P(s)OPE at s?8
• Possible non-absorptive in P(s)1-loop ?
  P(s)OPE at s?8
• (but no physical effect at
  the end of the day)
• Counterterms in P(s)tree behaviour as
  P(s)OPE at s?8

33
If one drops appart the any nasty
non-absorptive contribution in P(s)1-loop P(s)1-
loop fulfills the same dispersion relations as
P(s)LONLO
Same finite function
UV divergences

34

• 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

Finite renormalized couplings
Counter-terms
NNLO in 1/NC
35
leading to the renormalization conditions,
with Dck(1) and Dck(2) setting the
renormalization scheme (for instance,
Dck(1)Dck(2)0 for on-shell scheme )

• Hence, the amplitude becomes finally finite

36
leading to the renormalization conditions,
with Dck(1) and Dck(2) setting the
renormalization scheme (for instance,
Dck(1)Dck(2)0 for on-shell scheme )

• Hence, the amplitude becomes finally finite

On-shell scheme
37
And what about those nasty non-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 P(s)?0 for s?8)

UV divergences
NLO local couplings
38
m-subtracted dispersion relations

• Other Green-functions shows a non-vanishing
  behaviour
• P(s)?sm-1 when s?8
• In that situations, one need to consider not
  P(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

39
To recover the whole P(s) one needs to specify m
subtraction constants
at some reference energy ssO

• These subtractions are not fixed by QCD
• (e.g., in the SM, PVV(sO) is fixed
  by the photon wave-function renormalization)

40

• Providing at LO in 1/NC the pole structures

R1, R2,
41
but at the end of the day, at NLO
one reaches the same kind of renormalization
conditions
and an analogous structure for the renormalized
moment
Finite (from the
spectral function)
Renormalized tree-level
42
but at the end of the day, up to NLO
one reaches exactly the same renormalization
conditions
and an analogous structure for the renormalized
moment
On-shell scheme
Finite (from the
spectral function)
Renormalized tree-level
43
Renormalizability?
44

• RcT descriptions of P(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 P(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
  NCas in our case)
• (see, for instance, the example of QED5 Álvarez
  Faedo06)

45

• General renormalizable structures in other
  matrix elements?
• Appealing!!
• Larger complexity P(s1,s2,)
• Multi-variable dispersion relations, crossing
  symmetry,
• Next step three-point GF and scattering
  amplitudes

46
Conclusions
47

• General QCD properties 1/NC expansion
• Already valuable information
• Decreasing systematic errors
• Increasing accuracy
• Proving that QCDNC3 has to do with QCDNC?8
• MHA

Relevance of NLO in 1/NC
-Introduces systematic uncertainties -Makes
calculation feasible
Nevertheless, at some point the 4D-QFT
becomes unbearably complex
48

• 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 ScSB
• - Short-distance QCD (parton logs as logs
  OPE)
• - Renormalizable structure for P(s)
  amplitudes at NLO in 1/NC
• in terms of a few AdS parameters

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50

• Two-point Green functions
• We focus the attention on the SS-PP with I1

Interest of this correlator

• Chiral order parameter No pQCD contribution
• Isolates the effective cPT coupling L8 (quark
  mass lt-gt pGoldstone mass )
• Less trivial case than the J1 correlators

51
PROGRAM

• Resonance Chiral Theory framework (RcT)
• Construction of the lagrangian

52
PROGRAM

• Resonance Chiral Theory framework (RcT)
• Construction of the lagrangian
• 2-body form-factors at LO in 1/NC
• QCD short-distance constraints on the FF at
  LO in 1/NC

Tree-level
53
PROGRAM

• Resonance Chiral Theory framework (RcT)
• Construction of the lagrangian
• 2-body form-factors at LO in 1/NC
• QCD short-distance constraints on the FF at
  LO in 1/NC
• Derivation of PS-P (dispersive relations)
• QCD short-distance constraints on PS-P up to
  NLO in 1/NC

Tree-level
1-loop
54
PROGRAM

• Resonance Chiral Theory framework (RcT)
• Construction of the lagrangian
• 2-body form-factors at LO in 1/NC
• QCD short-distance constraints on the FF at
  LO in 1/NC
• Derivation of PS-P (dispersive relations)
• QCD short-distance constraints on PS-P up to
  NLO in 1/NC
• Recovering cPT at low energies
• Low energy constants up to NLO in 1/NC L8

Tree-level
1-loop
1-loop
55
RcT lagrangian
56
Ingredients of RcT

• Large NC ? U(nf) multiplets
• Goldstones from ScSB (p,K,h8,h0)
• MHA First resonance multiplets (V,A,S,P)
• Chiral symmetry invariance
• Just O(p2) operators
• Chiral limit

57
Weinberg79
Gasser Leutwyler84
Gasser Leutwyler85
Ecker et al.89

couplings liRR, liRRR
Moussallam95, Knecht Nyffeler01
Cirigliano et al.06 Pich,Rosell SC,
forthcoming
58
2-body form-factors
59
Optical theorem and the 1/NC expansion

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

2

• 1-P cuts asymptotic behaviour

60

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

2

• 2-P cuts asymptotic behaviour??

61
ARGUMENTS
Brodsky Lepage79
62
General FF analysis
p, V, A,
V, A, S,
V
A
p, V, p,
p, V, p,
p, V, A,
V, A, S,
S
P
p, V, p,
p, V, p,
63
SS-PP correlator at one loop
64
The example of L8 SS-PP correlator

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

65
The example of L8 SS-PP correlator

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

which at low energies becomes,
66

• Matching OPE for PS-P

Golterman peris00
67

• Matching OPE for PS-P

Golterman peris00
one gets at low energies,
68

• Matching OPE for PS-P

Golterman peris00
one gets at low energies,
m???
69

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

with the 2-P contributions from dispersion
relations
depending on the correponding couplings li,
fixed before at LO in 1/NC in the FF analysis
70

• Exact definition of the integral

MR2
t
71

• Example pp contribution

72

• Example pp contribution

Tree-level SFF
73

• Example pp contribution

Tree-level SFF
Short-distance SFF (correlator)
74

• Example pp contribution

Tree-level SFF
Short-distance SFF (correlator)
Optical theorem
75

• Example pp contribution

Tree-level SFF
Short-distance SFF (correlator)
Optical theorem
Dispersion relations
76

• 2-particle channels
• Goldstone-Goldstone (pp)
• Resonance-Goldstone (Rp)
• Resonance-Resonance

Suppressed ?Neglected
77
Full recovering of cPT at one loop
78
Low energy expansion at one loop

• Result in cPT within U(nf)

• TO NOTICE
• Exact cancellation of m dependence
• Presence of the massless ln(-q2) from pp loop
• Analytical part (L8 coupling constant)

79

• Tree level

? Analytical LO NLO
80

• Tree level

? Analytical LO NLO

• Intermediate state? pp

• Chiral log NLO

81

• Tree level

? Analytical LO NLO

• Intermediate state? pp

• Chiral log NLO

• Intermediate state? Rp

? Analytical NLO
constant
82

• Tree level

? Analytical LO NLO

• Intermediate state? pp

• Chiral log NLO

• Intermediate state? Rp

? Analytical NLO
constant

• Intermediate state? RR ? NEGLECTED

83
Matching OPE for PS-P(q2) 1/q4 up to
NLO in 1/NC
( ltO4gt not considered, competition ltO4gt vs. NLO)
with
84
Matching OPE for PS-P(q2) 1/q4 up to
NLO in 1/NC
( ltO4gt not considered, competition ltO4gt vs. NLO)
with
85
Matching OPE for PS-P(q2) 1/q4 up to
NLO in 1/NC
( ltO4gt not considered, competition ltO4gt vs. NLO)
with
86
Inputs
Parameters needed at LO in 1/NC
(appearing only NLO in PS-P)
U(3) ? SU(3)
Kaiser Leutwyler00
87
Inputs
Parameters needed at LO in 1/NC
(appearing only NLO in PS-P)
U(3) ? SU(3)
Kaiser Leutwyler00
Parameters needed up to NLO in
1/NC
Short-distance matching at LO
(appearing at LONLO in PS-P)
SD matching up to NLO
88
Results
(for comparisson
exactly scale independent expression)

• Contributions

tree
pp
Vp
Sp
Pp
Ap
U(3)?SU(3)
Kaiser Leutwyler00
89
Results
(for comparisson
exactly scale independent expression)

• Contributions

tree
pp
Vp
Sp
Pp
Ap
U(3)?SU(3)
Kaiser Leutwyler00

• Uncertainties

MV
MSr
dmr
F
mho
MA
truncation
MPr
90
Results
(for comparisson
exactly scale independent expression)

• Contributions

tree
pp
Vp
Sp
Pp
Ap
U(3)?SU(3)
Kaiser Leutwyler00

• Uncertainties

MV
MSr
dmr
F
mho
MA
truncation
MPr
to be compared to the cPT result,
91
Results
(for comparisson
exactly scale independent expression)

• Contributions

tree
pp
Vp
Sp
Pp
Ap
U(3)?SU(3)
Kaiser Leutwyler00

• Uncertainties

MV
MSr
dmr
F
mho
MA
truncation
MPr
to be compared to the cPT result,
92
Results
(for comparisson
exactly scale independent expression)

• Contributions

tree
pp
Vp
Sp
Pp
Ap
U(3)?SU(3)
Kaiser Leutwyler00

• Uncertainties

MV
MSr
dmr
F
mho
MA
truncation
MPr
to be compared to the cPT result,
93
Conclusions
94

• Large NC is meaningful it is possible to control
  NLO

95

• Large NC is meaningful it is possible to control
  NLO
• Systematic expansion of QCD amplitudes in 1/NC

96

• Large NC is meaningful it is possible to control
  NLO
• Systematic expansion of QCD amplitudes in 1/NC
• General analysis of the 2-body FF

97

• Large NC is meaningful it is possible to control
  NLO
• Systematic expansion of QCD amplitudes in 1/NC
• General analysis of the 2-body FF
• General structure of P(t) (dispersive
  analysis)

98

• Large NC is meaningful it is possible to control
  NLO
• Systematic expansion of QCD amplitudes in 1/NC
• General analysis of the 2-body FF
• General structure of P(t) (dispersive
  analysis)
• Short-distance matching order by order in 1/NC

99

• Large NC is meaningful it is possible to control
  NLO
• Systematic expansion of QCD amplitudes in 1/NC
• General analysis of the 2-body FF
• General structure of P(t) (dispersive
  analysis)
• Short-distance matching order by order in 1/NC
• Full recovering of cPT at low q2
• -Example of L8

100

• Large NC is meaningful it is possible to control
  NLO
• Systematic expansion of QCD amplitudes in 1/NC
• General analysis of the 2-body FF
• General structure of P(t) (dispersive
  analysis)
• Short-distance matching order by order in 1/NC
• Full recovering of cPT at low q2
• -Example of L8
• Manifestation of the uncertainty origin
  and full control of the saturation
  scale

101

• Large NC is meaningful it is possible to control
  NLO
• Systematic expansion of QCD amplitudes in 1/NC
• General analysis of the 2-body FF
• General structure of P(t) (dispersive
  analysis)
• Short-distance matching order by order in 1/NC
• Full recovering of cPT at low q2
• -Example of L8
• Manifestation of the uncertainty origin
  and full control of the saturation
  scale
• Straight-forward extension to O(p6) LECs

102
How well do we understand hadronic interactions?

• 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?

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104
QCD expansion in 1/NC ? QCD
at any q2
(MESONS)
105
Resonance FF, does it make any sense?
p
r
p,r,a1
(1st)
Weinberg67

p
VMD
WSR
R
R
p
(2nd)
Cata Peris 02
Pich, Rosell SC04
p
R
R
a1
R
r
(3rd)
p
Pich, Rosell SC forthcoming
R