Amp & Logs Amplitudes and Logarithms in Chiral ...home.thep.lu.se/~bijnens/talks/stockholm16.pdfAmp...
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Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
Recursionrelations
Conclusions
Backup: largeN (O(N))
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Amplitudes and Logarithms in ChiralPerturbation Theory
Johan Bijnens
Lund University
http://thep.lu.se/∼bijnens
http://thep.lu.se/∼bijnens/chpt.html
Stockholm, 30 June 2016
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
Recursionrelations
Conclusions
Backup: largeN (O(N))
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Overview
Somewhat different audience than I usually have
Usually I am “the high loop guy”
Recent work: loops and lattice effects: finite volume,partial quenching, twisted boundary conditions, staggeredChPT,. . .
This talk:
Introduction to Chiral Perturbation Theory/Effective fieldtheoryMain part: leading logarithms to high ordersSome musings about recursion relations and ChPT
Amp & Logsin ChPT
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ChPT/EFT
Logarithms
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Conclusions
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Chiral Perturbation Theory
Exploring the consequences of
the chiral symmetry of QCDand its spontaneous breaking
using effective field theory techniques
Derivation from QCD:H. Leutwyler,On The Foundations Of Chiral Perturbation Theory,Ann. Phys. 235 (1994) 165 [hep-ph/9311274]
For references to lectures see:http://www.thep.lu.se/∼bijnens/chpt.html
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
Recursionrelations
Conclusions
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Effective field theory
Choose your degrees of freedom
Construct the most general Lagrangian
Too many (∞) parameters
Need an ordering principle: power counting
Can be combined with renormalizable interactions as wellbut then need to be very careful about what exactly youexpand in(Discussion about EFT for Higgs)
But still: new terms at every order in your expansion,polynomial in masses and momenta typically notdetermined
Makes unitarity and amplitude methods a bit more trickyto use
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
Recursionrelations
Conclusions
Backup: largeN (O(N))
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Chiral Perturbation Theory
A general Effective Field Theory:
Relevant degrees of freedom
A powercounting principle (predictivity)
Has a certain range of validity
Chiral Perturbation Theory:
Degrees of freedom: Goldstone Bosons from spontaneousbreaking of chiral symmetry
Powercounting: Dimensional counting in momenta/masses
Breakdown scale: Resonances, so about Mρ.
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
Recursionrelations
Conclusions
Backup: largeN (O(N))
5/72
Chiral Perturbation Theory
A general Effective Field Theory:
Relevant degrees of freedom
A powercounting principle (predictivity)
Has a certain range of validity
Chiral Perturbation Theory:
Degrees of freedom: Goldstone Bosons from spontaneousbreaking of chiral symmetry
Powercounting: Dimensional counting in momenta/masses
Breakdown scale: Resonances, so about Mρ.
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
Recursionrelations
Conclusions
Backup: largeN (O(N))
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Chiral Symmetry
Chiral Symmetry
QCD: Nf light quarks: equal mass: interchange: SU(Nf )V
But LQCD =∑
q=u,d,s
[i qLD/ qL + i qRD/ qR −mq (qRqL + qLqR)]
So if mq = 0 then SU(3)L × SU(3)R .
Spontaneous breakdown
〈qq〉 = 〈qLqR + qRqL〉 6= 0
SU(3)L × SU(3)R broken spontaneously to SU(3)V
8 generators broken =⇒ 8 massless degrees of freedomand interaction vanishes at zero momentum
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
Recursionrelations
Conclusions
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Goldstone Bosons
Power counting in momenta: Meson loops, Weinbergpowercounting
rules one loop example
p2
1/p2
∫
d4p p4
(p2)2 (1/p2)2 p4 = p4
(p2) (1/p2) p4 = p4
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
Recursionrelations
Conclusions
Backup: largeN (O(N))
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Chiral Perturbation Theories
Which chiral symmetry: SU(Nf )L × SU(Nf )R , forNf = 2, 3, . . . and extensions to (partially) quenched
Or beyond QCD
Space-time symmetry: Continuum or broken on thelattice: Wilson, staggered, mixed action
Volume: Infinite, finite in space, finite T
Which interactions to include beyond the strong one
Which particles included as non Goldstone Bosons
My general belief: if it involves soft pions (or soft K , η)some version of ChPT exists
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
Recursionrelations
Conclusions
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Lagrangians: Lowest order
U(φ) = exp(i√2Φ/F0) parametrizes Goldstone Bosons
Φ(x) =
π0
√2
+η8√6
π+ K+
π− − π0
√2
+η8√6
K 0
K− K 0 −2 η8√6
.
LO Lagrangian: L2 =F 204 {〈DµU
†DµU〉+ 〈χ†U + χU†〉} ,
DµU = ∂µU − irµU + iUlµ ,left and right external currents: r(l)µ = vµ + (−)aµ
Scalar and pseudoscalar external densities: χ = 2B0(s + ip) quark masses viascalar density: s = M+ · · ·
〈A〉 = TrF (A)
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
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Conclusions
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Lagrangians: Lagrangian structure
2 flavour 3 flavour PQChPT/Nf flavour
p2 F ,B 2 F0,B0 2 F0,B0 2
p4 l ri , hri 7+3 Lri ,H
ri 10+2 Lri , H
ri 11+2
p6 c ri 52+4 C ri 90+4 K r
i 112+3
p2: Weinberg 1966
p4: Gasser, Leutwyler 84,85
p6: JB, Colangelo, Ecker 99,00
➠ Li LEC = Low Energy Constants = ChPT parameters➠ Hi : contact terms: value depends on definition of cur-rents/densities➠ Finite volume: no new LECs➠ Other effects: (many) new LECs
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
Recursionrelations
Conclusions
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Chiral Logarithms
The main predictions of ChPT:
Relates processes with different numbers of pseudoscalars
Chiral logarithms
includes Isospin and the eightfold way (SU(3)V )
Unitarity included perturbatively
m2π = 2Bm+
(
2Bm
F
)2 [ 1
32π2log
(2Bm)
µ2+ 2l r3(µ)
]
+ · · ·
M2 = 2Bm
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
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Leading logarithms
Main question: can we get information on high orders?
This part:
Leading logarithmsWeinberg’s argumentO(N + 1)/O(N)AnomalySU(N)× SU(N)/SU(N)Nucleon
Collaborators: Lisa Carloni, Karol Kampf, Stefan Lanz,Alexey A. Vladimirov
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
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References
JB, L. Carloni, Nucl.Phys. B827 (2010) 237-255 [arXiv:0909.5086]
Leading Logarithms in the Massive O(N) Nonlinear SigmaModel
JB, L. Carloni, Nucl.Phys. B843 (2011) 55-83 [arXiv:1008.3499]
The Massive O(N) Non-linear Sigma Model at High Orders
JB, K. Kampf, S. Lanz, Nucl.Phys. B860 (2012) 245-266 [arXiv:1201.2608]
Leading logarithms in the anomalous sector of two-flavourQCD
JB, K. Kampf, S. Lanz, Nucl.Phys. B873 (2013) 137-164 [arXiv:1303.3125]
Leading logarithms in N-flavour mesonic ChiralPerturbation Theory
JB, A.A. Vladimirov, Nucl.Phys. B91 (2015) 700 [arXiv:1409.6127]
Leading logarithms for the nucleon mass
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
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Leading logarithms (LL)
BEWARE: leading logarithms can mean very different things
Take a quantity with a single scale: F (M)
Subtraction scale in QFT (in dim. reg.) is logarithmic
L = log (µ/M)
F = F0 +(
F 11 L+ F 1
0
)
+(
F 22 L
2 + F 21 L+ F 2
0
)
+(
F 33 L
3 + · · ·)
+ · · ·Leading Logarithms: The terms Fm
m Lm
The Fmm can be more easily calculated than the full result
µ (dF/dµ) ≡ 0
Ultraviolet divergences in Quantum Field Theory arealways local
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
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Renormalizable theories
Loop expansion ≡ α expansion
F = α+(
f 11 α2L+ f 10 α
2)
+(
f 22 α3L2 + f 21 α
3L+ f 20 α3)
+(
f 33 α4L3 + · · ·
)
+ · · ·fji are pure numbers
µdF
dµ≡ F ′, µ
dα
dµ≡ α′, µ
dL
dµ= 1
F ′ = α′ +(
f 11 α2 + f 11 2α
′αL+ f 10 2αα′)
+(
f 22 α32L+ f 22 3α
′α2L2 + f 21 α3 + 3f 21 α
′α2L+ 3f 20 α′α2)
+(
f 33 α33L2 + f 33 4α
′α3L3 + · · ·)
+ · · ·α′ = β1α
2 + β2α3 + · · ·
0 = F ′ =(
β1 + f 11)
α2 +(
2β1f11 + 2f 22
)
α3L
+(
β2 + 2β1f10 + f 21
)
α3 +(
3β1f22 + 3f 33
)
α4L2 + · · ·f 11 = −β1, f 22 = β2
1 , f 33 = −β31 , . . .
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
15/72
Renormalizable theories
Loop expansion ≡ α expansion
F = α+(
f 11 α2L+ f 10 α
2)
+(
f 22 α3L2 + f 21 α
3L+ f 20 α3)
+(
f 33 α4L3 + · · ·
)
+ · · ·fji are pure numbers
µdF
dµ≡ F ′, µ
dα
dµ≡ α′, µ
dL
dµ= 1
F ′ = α′ +(
f 11 α2 + f 11 2α
′αL+ f 10 2αα′)
+(
f 22 α32L+ f 22 3α
′α2L2 + f 21 α3 + 3f 21 α
′α2L+ 3f 20 α′α2)
+(
f 33 α33L2 + f 33 4α
′α3L3 + · · ·)
+ · · ·α′ = β1α
2 + β2α3 + · · ·
0 = F ′ =(
β1 + f 11)
α2 +(
2β1f11 + 2f 22
)
α3L
+(
β2 + 2β1f10 + f 21
)
α3 +(
3β1f22 + 3f 33
)
α4L2 + · · ·f 11 = −β1, f 22 = β2
1 , f 33 = −β31 , . . .
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
15/72
Renormalizable theories
Loop expansion ≡ α expansion
F = α+(
f 11 α2L+ f 10 α
2)
+(
f 22 α3L2 + f 21 α
3L+ f 20 α3)
+(
f 33 α4L3 + · · ·
)
+ · · ·fji are pure numbers
µdF
dµ≡ F ′, µ
dα
dµ≡ α′, µ
dL
dµ= 1
F ′ = α′ +(
f 11 α2 + f 11 2α
′αL+ f 10 2αα′)
+(
f 22 α32L+ f 22 3α
′α2L2 + f 21 α3 + 3f 21 α
′α2L+ 3f 20 α′α2)
+(
f 33 α33L2 + f 33 4α
′α3L3 + · · ·)
+ · · ·α′ = β1α
2 + β2α3 + · · ·
0 = F ′ =(
β1 + f 11)
α2 +(
2β1f11 + 2f 22
)
α3L
+(
β2 + 2β1f10 + f 21
)
α3 +(
3β1f22 + 3f 33
)
α4L2 + · · ·f 11 = −β1, f 22 = β2
1 , f 33 = −β31 , . . .
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
16/72
Renormalizable theories
Leading logs known as soon as β1 is known
F (M) = α(
1− αβ1L+ (αβ1L)2 − (αβ1L)
3 + · · ·)
+ · · ·=
α(µ)
1 + α(µ)β1 log(µ/M)+ · · · = α(M) + · · ·
running coupling constant
Generalizes to lower leading logarithms as well
Multiscale problems: many other terms possible
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
17/72
Renormalization Group
Can be extended to other operators as well
Underlying argument always µdF
dµ= 0.
Gell-Mann–Low, Callan–Symanzik, Weinberg–’t Hooft
In great detail: J.C. Collins, Renormalization
Relies on the α the same in all orders
LL one-loop β1
NLL two-loop β2 and one-loop f 10
In effective field theories: different Lagrangian at eachorder
The recursive argument does not work
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
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Weinberg’s argument for leading logarithms
Weinberg, Physica A96 (1979) 327
Two-loop leading logarithms can be calculated using onlyone-loop: Weinberg consistency conditions
ππ at 2-loop: Colangelo, hep-ph/9502285
General at 2 loop: JB, Colangelo, Ecker, hep-ph/9808421
Proof at all orders
using β-functions: Buchler, Colangelo, hep-ph/0309049
with diagrams: JB, Carloni, arXiv:0909.5086
Extension to nucleons: JB, Vladimirov, arXiv:1409.6127
First mesonic case whereloop-order = power-counting (or ~) order
Later nucleon case whereloop-order 6= power-counting (or ~) order
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
19/72
Weinberg’s argument
µ: dimensional regularization scale
d = 4− w
loop-expansion ≡ ~-expansion
Lbare =∑
n≥0
~nµ−nwL(n)
L(n) =∑
i
c(n)i Oi
c(n)i =
∑
k=0,n
c(n)ki
wk
c(n)0i have a direct µ-dependence
c(n)ki k ≥ 1 only depend on µ through c
(m<n)0i
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
20/72
Weinberg’s argument
Lnℓ ℓ-loop contribution at order ~n
Expand in divergences from the loops (not from thecounterterms) Lnℓ =
∑
k=0,l1wk L
nkℓ
Neglected positive powers: not relevant here, but shouldbe kept in general
{c}nℓ all combinations c(m1)k1j1
c(m2)k2j2
. . . c(mr )kr jr
with mi ≥ 1,such that
∑
i=1,r mi = n and∑
i=1,r ki = ℓ.
{cnn} ≡ {c(n)ni }, {c}22 = {c(2)2i , c(1)1i c
(1)1k }
L(n) = n
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
21/72
Weinberg’s argument
Mass = 0 + 0 1
+ 0 0 0 1 0
1
2 + 2 1
1
· · ·
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
22/72
Weinberg’s argument
~0: L00
~1:
1
w
(
µ−wL100({c}11) + L111)
+ µ−wL100({c}10) + L110
Expand µ−w = 1− w log µ+1
2w2 log2 µ+ · · ·
1/w must cancel: L100({c}11) + L111 = 0this determines the c11i
Explicit log µ: − log µ L100({c}11) ≡ log µ L111
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
22/72
Weinberg’s argument
~0: L00
~1:
1
w
(
µ−wL100({c}11) + L111)
+ µ−wL100({c}10) + L110
Expand µ−w = 1− w log µ+1
2w2 log2 µ+ · · ·
1/w must cancel: L100({c}11) + L111 = 0this determines the c11i
Explicit log µ: − log µ L100({c}11) ≡ log µ L111
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
23/72
Weinberg’s argument
~2:1
w2
(
µ−2wL200({c}22) + µ−wL211({c}11) + L222)
+1
w
(
µ−2wL200({c}21) + µ−wL211({c}10) + µ−wL210({c}11) + L221)
+(
µ−2wL200({c}20) + µ−wL210({c}10) + L220)
1/w2 and log µ/w must cancelL200({c}22) + L211({c}11) + L222 = 02L200({c}22) + L211({c}11) = 0
Solution: L200({c}22) = −1
2L211({c}11)
L211({c}11) = −2L222
Explicit log µ dependence (one-loop is enough)1
2log2 µ
(
4L200({c}22) + L211({c}11))
= −1
2L211({c}11) log2 µ .
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
23/72
Weinberg’s argument
~2:1
w2
(
µ−2wL200({c}22) + µ−wL211({c}11) + L222)
+1
w
(
µ−2wL200({c}21) + µ−wL211({c}10) + µ−wL210({c}11) + L221)
+(
µ−2wL200({c}20) + µ−wL210({c}10) + L220)
1/w2 and log µ/w must cancelL200({c}22) + L211({c}11) + L222 = 02L200({c}22) + L211({c}11) = 0
Solution: L200({c}22) = −1
2L211({c}11)
L211({c}11) = −2L222
Explicit log µ dependence (one-loop is enough)1
2log2 µ
(
4L200({c}22) + L211({c}11))
= −1
2L211({c}11) log2 µ .
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
23/72
Weinberg’s argument
~2:1
w2
(
µ−2wL200({c}22) + µ−wL211({c}11) + L222)
+1
w
(
µ−2wL200({c}21) + µ−wL211({c}10) + µ−wL210({c}11) + L221)
+(
µ−2wL200({c}20) + µ−wL210({c}10) + L220)
1/w2 and log µ/w must cancelL200({c}22) + L211({c}11) + L222 = 02L200({c}22) + L211({c}11) = 0
Solution: L200({c}22) = −1
2L211({c}11)
L211({c}11) = −2L222
Explicit log µ dependence (one-loop is enough)1
2log2 µ
(
4L200({c}22) + L211({c}11))
= −1
2L211({c}11) log2 µ .
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
24/72
All orders
~n:1
wn
(
µ−nwLn00({c}nn) + µ−(n−1)wLn11({c}n−1n−1) + · · ·
+µ−wLnn−1 n−1({c}11) + Lnnn
)
+1
wn−1· · ·
1/wn, log µ/wn−1, . . . , logn−1 µ/w cancel:n∑
i=0
i jLnn−i n−i ({c}ii ) = 0 j = 0, .., n − 1.
Solution: Lnn−i n−i ({c}ii ) = (−1)i(
n
i
)
Lnnn
explicit leading log µ dependence and divergence
logn µ(−1)n−1
nLn11({c}n−1
n−1)
Ln00({c}nn) = −1
nLn11({c}n−1
n−1)
Amp & Logsin ChPT
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ChPT/EFT
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O(N + 1)/O(N)
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Backup: largeN (O(N))
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All orders
~n:1
wn
(
µ−nwLn00({c}nn) + µ−(n−1)wLn11({c}n−1n−1) + · · ·
+µ−wLnn−1 n−1({c}11) + Lnnn
)
+1
wn−1· · ·
1/wn, log µ/wn−1, . . . , logn−1 µ/w cancel:n∑
i=0
i jLnn−i n−i ({c}ii ) = 0 j = 0, .., n − 1.
Solution: Lnn−i n−i ({c}ii ) = (−1)i(
n
i
)
Lnnn
explicit leading log µ dependence and divergence
logn µ(−1)n−1
nLn11({c}n−1
n−1)
Ln00({c}nn) = −1
nLn11({c}n−1
n−1)
Amp & Logsin ChPT
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ChPT/EFT
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Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
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SU(N)×SU(N)/SU(N)
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Backup: largeN (O(N))
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All orders
~n:1
wn
(
µ−nwLn00({c}nn) + µ−(n−1)wLn11({c}n−1n−1) + · · ·
+µ−wLnn−1 n−1({c}11) + Lnnn
)
+1
wn−1· · ·
1/wn, log µ/wn−1, . . . , logn−1 µ/w cancel:n∑
i=0
i jLnn−i n−i ({c}ii ) = 0 j = 0, .., n − 1.
Solution: Lnn−i n−i ({c}ii ) = (−1)i(
n
i
)
Lnnn
explicit leading log µ dependence and divergence
logn µ(−1)n−1
nLn11({c}n−1
n−1)
Ln00({c}nn) = −1
nLn11({c}n−1
n−1)
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
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Conclusions
Backup: largeN (O(N))
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Mass to ~2
~1: 0 =⇒ 1
~2: 1 0
1
=⇒ 2
but also needs ~1: 0 0
0
=⇒ 1
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SU(N)×SU(N)/SU(N)
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Mass to order ~3
0 0 0
0
0 0
0
0
0 0
1 0
1
1 0
1
1
0
0
0
1
2 0
2
1
1
0
11
Amp & Logsin ChPT
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SU(N)×SU(N)/SU(N)
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Mass to order ~6
0
(1)
0
(2)
0
(3)
0
(4)
0
(5)
0
(6)
0
0
(7)
0
0
(8)
0
0
(9)
0
0
(10)
0
0
(11)
0
0
(12)
0
0
(13)
0
0
(14)
0
0
(15)
0
0 0
(16)
0
0 0
(17)
0
0 0
(18)
0
0 0
(19)
0
0 0
(20)
0
0 0
(21)
0
0 0
(22)
0
0
0
0
(23)
0
0
0
0
(24)
0
0
0
0
(25)
0
0
0
0
(26)
0
0
0
0
(27)
0
0
0 0
0
(28)
0
0
0 0
0
(29)
0
0
0
0
0
0
(30)
1
(31)
1
(32)
1
(33)
1
(34)
1
(35)
0
1
(36)
0
1
(37)
0
1
(38)
0
1
(39)
0
1
(40)
0
1
(41)
0
1
(42)
0
1
(43)
0
1
(44)
0
1
(45)
0
1
(46)
0
1
(47)
0
1
(48)
0
1
(49)
0
1
(50)
0
0 1
(51)
0
0 1
(52)
0
0 1
(53)
0
0 1
(54)
0
0 1
(55)
0
0 1
(56)
0
0 1
(57)
0
0 1
(58)
0
0 1
(59)
0
0 1
(60)
0
0 1
(61)
0
0 1
(62)
0
0 1
(63)
0
0
0
1
(64)
0
0
0
1
(65)
0
0
0
1
(66)
0
0
0
1
(67)
0
0
0
1
(68)
0
0
0
1
(69)
0
0
0
1
(70)
0
0
0
1
(71)
0
0
0
1
(72)
0
0
0
1
(73)
0
0
0
1
(74)
0
0
0 0
1
(75)
0
0
0 0
1
(76)
0
0
0 0
1
(77)
0
0
0 0
1
(78)
0
0
0
0
0
1
(79)
2
(80)
2
(81)
2
(82)
2
(83)
0
2
(84)
0
2
(85)
0
2
(86)
0
2
(87)
0
2
(88)
0
2
(89)
0
2
(90)
0
2
(91)
0
2
(92)
0
2
(93)
1
1
(94)
1
1
(95)
1
1
(96)
1
1
(97)
1
1
(98)
1
1
(99)
1
1
(100)
1
1
(101)
0
0 2
(102)
0
0 2
(103)
0
0 2
(104)
0
0 2
(105)
0
0 2
(106)
0
0 2
(107)
0
1 1
(108)
0
1 1
(109)
0
1 1
(110)
0
1 1
(111)
0
1 1
(112)
0
1 1
(113)
0
0 2
(114)
0
1 1
(115)
0
1 1
(116)
0
1 1
(117)
0
1 1
(118)
0
1 1
(119)
0
1 1
(120)
0
1 1
(121)
0
0
0
2
(122)
0
0
0
2
(123)
0
0
0
2
(124)
0
0
1
1
(125)
0
0
1
1
(126)
0
0
1
1
(127)
0
0
1
1
(128)
0
0
0
2
(129)
0
0
1
1
(130)
0
0
1
1
(131)
0
0
1
1
(132)
0
0
1
1
(133)
0
1
0
1
(134)
0
1
0
1
(135)
0
1
0
1
(136)
0
1
0
1
(137)
0
1
0
1
(138)
0
1
0
1
(139)
0
1
0
1
(140)
0
0
1
1
(141)
0
1
0
1
(142)
0
0
0 0
2
(143)
0
0
0 1
1
(144)
0
0
0 1
1
(145)
0
0
0 1
1
(146)
0
0
1 0
1
(147)
0
0
1 0
1
(148)
0
0
1 0
1
(149)
0
0
0 1
1
(150)
0
0
1 0
1
(151)
0
0
0
0
1
1
(152)
0
0
0
1
0
1
(153)
0
0
1
0
0
1
(154)
3
(155)
3
(156)
3
(157)
0
3
(158)
0
3
(159)
0
3
(160)
0
3
(161)
0
3
(162)
0
3
(163)
1
2
(164)
1
2
(165)
1
2
(166)
1
2
(167)
1
2
(168)
1
2
(169)
1
2
(170)
1
2
(171)
1
2
(172)
0
0 3
(173)
0
0 3
(174)
0
0 3
(175)
0
1 2
(176)
0
1 2
(177)
0
1 2
(178)
0
1 2
(179)
0
1 2
(180)
0
1 2
(181)
0
1 2
(182)
0
1 2
(183)
0
1 2
(184)
0
1 2
(185)
1
1 1
(186)
1
1 1
(187)
1
1 1
(188)
1
1 1
(189)
1
1 1
(190)
1
1 1
(191)
0
0
0
3
(192)
0
0
1
2
(193)
0
0
1
2
(194)
0
0
1
2
(195)
0
0
1
2
(196)
0
1
0
2
(197)
0
1
0
2
(198)
0
1
0
2
(199)
0
1
1
1
(200)
0
1
1
1
(201)
0
1
1
1
(202)
0
1
1
1
(203)
0
1
1
1
(204)
0
1
1
1
(205)
0
1
0
2
(206)
0
1
1
1
(207)
0
0
1
2
(208)
0
1
1
1
(209)
0
1
1
1
(210)
0
1
1
1
(211)
0
1
1
1
(212)
0
0
0 1
2
(213)
0
0
1 0
2
(214)
0
0
1 1
1
(215)
0
0
1 1
1
(216)
0
0
1 1
1
(217)
0
0
1 1
1
(218)
0
1
0 1
1
(219)
0
1
0 1
1
(220)
0
1
0 1
1
(221)
0
1
0 1
1
(222)
0
0
0
1
1
1
(223)
0
0
1
0
1
1
(224)
0
1
0
1
0
1
(225)
4
(226)
4
(227)
0
4
(228)
0
4
(229)
0
4
(230)
1
3
(231)
1
3
(232)
1
3
(233)
1
3
(234)
1
3
(235)
2
2
(236)
2
2
(237)
2
2
(238)
0
0 4
(239)
0
1 3
(240)
0
1 3
(241)
0
1 3
(242)
0
2 2
(243)
0
2 2
(244)
0
1 3
(245)
0
2 2
(246)
1
1 2
(247)
1
1 2
(248)
1
1 2
(249)
1
1 2
(250)
1
1 2
(251)
1
1 2
(252)
0
0
1
3
(253)
0
0
2
2
(254)
0
1
0
3
(255)
0
1
1
2
(256)
0
1
1
2
(257)
0
1
1
2
(258)
0
1
2
1
(259)
0
1
2
1
(260)
0
1
2
1
(261)
0
1
1
2
(262)
0
2
0
2
(263)
0
1
1
2
(264)
0
1
2
1
(265)
1
1
1
1
(266)
1
1
1
1
(267)
1
1
1
1
(268)
1
1
1
1
(269)
0
0
1 1
2
(270)
0
0
1 2
1
(271)
0
1
0 1
2
(272)
0
1
1 0
2
(273)
0
1
1 1
1
(274)
0
1
1 1
1
(275)
0
1
1 1
1
(276)
0
1
1 1
1
(277)
0
0
1
1
1
1
(278)
0
1
0
1
1
1
(279)
0
1
1
0
1
1
(280)
5
(281)
0
5
(282)
1
4
(283)
1
4
(284)
2
3
(285)
2
3
(286)
0
1 4
(287)
0
2 3
(288)
1
1 3
(289)
1
1 3
(290)
1
2 2
(291)
1
2 2
(292)
0
1
1
3
(293)
0
1
2
2
(294)
0
1
3
1
(295)
0
2
1
2
(296)
1
1
1
2
(297)
1
1
1
2
(298)
1
1
1
2
(299)
0
1
1 1
2
(300)
0
1
1 2
1
(301)
1
1
1 1
1
(302)
0
1
1
1
1
1
(303)
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
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General
Calculate the divergence
rewrite it in terms of a local Lagrangian
Luckily: symmetry kept: we know result will besymmetrical, hence do not need to explicitly rewrite theLagrangians in a nice formLuckily: we do not need to go to a minimal LagrangianSo everything can be computerizedThank Jos Vermaseren for FORM
We keep all terms to have all 1PI (one particle irreducible)diagrams finite
Amp & Logsin ChPT
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ChPT/EFT
Logarithms
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Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
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Conclusions
Backup: largeN (O(N))
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Massive O(N) sigma model
O(N + 1)/O(N) nonlinear sigma model
Lnσ =F 2
2∂µΦ
T∂µΦ+ F 2χTΦ .
Φ is a real N + 1 vector; Φ → OΦ; ΦTΦ = 1.Vacuum expectation value 〈ΦT 〉 = (1 0 . . . 0)Explicit symmetry breaking: χT =
(
M2 0 . . . 0)
Both spontaneous and explicit symmetry breakingN-vector φ
N (pseudo-)Nambu-Goldstone Bosons
N = 3 is two-flavour Chiral Perturbation Theory
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
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Backup: largeN (O(N))
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Massive O(N) sigma model: Φ vs φ
Φ1 =
√
1− φTφF 2
φ1
F...φN
F
=
( √
1− φTφF 2
φF
)
Gasser, Leutwyler
Φ5 =1
1 + φTφ4F 2
(
1− φTφ2F 2
φF
)
Φ3 =
1− 12φTφF 2
√
1− 14φTφF 2
φF
Weinberg only mass term
Φ4 =
cos√
φTφF 2
sin√
φTφF 2
φ√φTφ
CCWZ
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
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Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
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Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
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Backup: largeN (O(N))
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Massive O(N) sigma model: Checks
Need (many) checks:
use many different parametrizations
compare with known results:
M2phys = M2
(
1− 1
2LM +
17
8L2M + · · ·
)
,
LM =M2
16π2F 2log
µ2
M2
Usual choice M = M.
large N but massive results more hiddenColeman, Jackiw, Politzer 1974
JB, Carloni, Kampf, Lanz: mass to 6 loops
Amp & Logsin ChPT
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Logarithms
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Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
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SU(N)×SU(N)/SU(N)
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Backup: largeN (O(N))
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Results
M2phys = M2(1 + a1LM + a2L
2M + a3L
3M + ...)
LM = M2
16π2F 2 logµ2
M2
i ai , N = 3 ai for general N
1 − 12
1− N2
2 178
74− 7N
4+ 5 N2
8
3 − 10324
3712
− 113N24
+ 15 N2
4− N3
4 243671152
839144
− 1601 N144
+ 695 N2
48− 135 N3
16+ 231 N4
128
5 − 8821144
336612400
− 1151407 N43200
+ 197587 N2
4320− 12709 N3
300+ 6271 N4
320− 7 N5
2
6 19229646676220800
158393809/3888000 − 182792131/2592000 N
+1046805817/7776000 N2 − 17241967/103680 N3
+70046633/576000 N4 − 23775/512 N5 + 7293/1024 N6
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
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Results
Fphys = F (1 + a1LM + a2L2M + a3L
3M + ...)
i ai for N = 3 ai for general N1 1 −1/2 + 1/2N
2 −5/4 −1/2 + 7/8N − 3/8N2
3 83/24 −7/24 + 21/16N − 73/48N2 + 1/2N3
4 −3013/288 47/576 + 1345/864N − 14077/3456N2 + 625/192N3
−105/128N4
5 206014751840
−23087/64800 + 459413/172800 N − 189875/20736N2
+546941/43200N3 − 1169/160N4 + 3/2N5
6 − 69228787466560
−277079063/93312000 + 1680071029/186624000 N
−686641633/31104000 N2 + 813791909/20736000 N3
−128643359/3456000 N4 + 260399/15360N5 − 3003/1024N6
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
Backup: largeN (O(N))
34/72
Results
〈qiqi〉 = −BF 2(1 + c1LM + c2L2M + c3L
3M + ...)
M2 = 2Bm χT = 2B(s 0 . . . 0)s corresponds to uu + dd current
i ci for N = 3 ci for general N
1 32
N2
2 − 98
3N4
− 3N2
8
3 92
3N2
− 3N2
2+ N3
2
4 − 1285128
145N48
− 55N2
12+ 105N3
32− 105N4
128
5 46 3007N480
− 1471N2
120+ 557 N3
40− 1191 N4
160+ 3 N5
2
Anyone recognize any funny functions?
Amp & Logsin ChPT
Johan Bijnens
ChPT/EFT
Logarithms
LL
Weinberg’sargument
O(N + 1)/O(N)
Masses, decay
Other expan-sions/Numerics
Other work
Anomaly
SU(N)×SU(N)/SU(N)
Nucleon
Conclusions
Recursionrelations
Conclusions
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Alternative expansions
LM =M2
16π2F 2log
µ2
M2
LM =M2
phys
16π2F 2log
µ2
M2phys
Lphys =M2
phys
16π2F 2phys
logµ2
M2phys
For masses expansion in LM best, but no general obviouschoice
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Numerical results
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
1.01
0 0.05 0.1 0.15 0.2 0.25
M2 π
/M2
M2 [GeV
2]
0
1
2
3
4
5
6
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
1.01
0 0.05 0.1 0.15 0.2 0.25
M2 π
/M2
M2π [GeV
2]
0123456
Left:M2
phys
M2 = 1 + a1LM + a2L2M + a3L
3M + · · ·
F = 90 MeV, µ = 0.77 GeV
Right:M2
phys
M2 = 1 + c1Lphys + c2L2phys + c3L
3phys + · · ·
Fπ = 92 MeV, µ = 0.77 GeV
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Numerical results
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
1.18
0 0.05 0.1 0.15 0.2 0.25
Fπ
/F
M2 [GeV
2]
0
1
2
3
4
5
6
1
1.05
1.1
1.15
1.2
0 0.05 0.1 0.15 0.2 0.25
Fπ
/F
M2π [GeV
2]
0
1
2
3
4
5
6
Left:Fphys
F= 1 + a1LM + a2L
2M + a3L
3M + · · ·
F = 90 MeV, µ = 0.77 GeV
Right:Fphys
F= 1 + c1Lphys + c2L
2phys + c3L
3phys + · · ·
Fπ = 92 MeV, µ = 0.77 GeV
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Numerics: π-π scattering
1
1.2
1.4
1.6
1.8
2
2.2
2.4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
a0 0/a
0 0tr
ee
M2phys [GeV
2]
0
1
2
3
4
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
a2 0/a
2 0tr
ee
M2phys [GeV
2]
0
1
2
3
4
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Other results
We also did: vector and scalar form-facors
Bissegger, Fuhrer, hep-ph/0612096 Dispersive methods, masslessΠS to five loops
Kivel, Polyakov, Vladimirov, 0809.3236, 0904.3008, 1004.2197, 1012.4205,
1105.4990
In the massless case tadpoles vanishhence the number of external legs needed does not growAll 4-meson vertices via Legendre polynomialscan do divergence of all one-loop diagrams analyticallyalgebraic (but quadratic) recursion relationsmassless ππ, FV and FS to arbitrarily high orderlarge N agrees with Coleman, Wess, Zumino
large N is not a good approximation
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Other results
Leading chiral logarithms of KS → γγ and KS → γl+l− attwo loops, K. Ghorbani, 1403.6791
K → ππ at two-loop order and K → nπ,M. Buchler, hep-ph/0504180, hep-ph/0511087
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Anomaly for O(4)/O(3)
JB, Kampf, Lanz, arXiv:1201.2608
LWZW = −Nc
8π2ǫµνρσ
{
ǫabc
(
1
3Φ0∂µΦ
a∂νΦ
b∂ρΦ
c− ∂µΦ
0∂νΦ
a∂ρΦ
bΦc)
v0σ
+(∂µΦ0Φa− Φ
0∂µΦ
a)v
aν∂ρv
0σ +
1
2ǫabc
Φ0Φavbµv
cν∂ρv
0σ
}
.
A(π0 → γ(k1)γ(k2)) =
ǫµναβ ε∗µ1 (k1)ε
∗ν2 (k2) k
α1 k
β2 Fπγγ(k
21 , k
22 )
Fπγγ(k21 , k
22 ) =
e2
4π2FπF Fγ(k
21 )Fγ(k
22 )Fγγ(k
21 , k
22 )
F : on-shell photon; Fγ(k2): formfactor;
Fγγ nonfactorizable part
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Anomaly for O(4)/O(3)
Done to six-loops
F = 1 + 0− 0.000372 + 0.000088 + 0.000036 +0.000009 + 0.0000002 + . . .
Really good convergence
Fγγ only starts at three-loop order (could have been two)
Fγγ in the chiral limit only starts at four-loops.
The leading logarithms thus predict this part to be fairlysmall.
Fγ(k2): plot
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Anomaly for O(4)/O(3)
0.88
0.9
0.92
0.94
0.96
0.98
1
0 0.05 0.1 0.15 0.2 0.25 0.3
Fγ(-Q2)
Q2 [GeV2]
123456
VMD
Leading logs small, converge fast
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γ3π
Experiment 1: F 3πexp = 12.9 ± 0.9± 0.5 GeV−3
Experiment 2: F 3π0,exp = 9.9± 1.1 GeV−3
Theory lowest order: F 3π0 = 9.8 GeV−3
Theory (LL only)F 3πLL0 = (9.8−0.3+0.04+0.02+0.006+0.001+. . .) GeV−3
good convergence
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SU(N)× SU(N)/SU(N)
SU(N)× SU(N)/SU(N) (vector and scalar)
Mass, Decay constants, Form-factors
Meson-Meson, γγ → ππ
No luck with guess for general N-dependence either
Four different parametrizations of a unitary matrix used
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Mass
i ai for N = 2 ai for N = 3 ai for general N
1 −1/2 −1/3 −N−1
2 17/8 27/8 9/2 N−2− 1/2 + 3/8N2
3 −103/24 −3799/648 −89/3 N−3 + 19/3 N−1− 37/24 N − 1/12 N3
4 24367/1152 146657/2592 2015/8 N−4− 773/12 N−2 + 193/18 + 121/288 N2
+41/72 N4
5 −8821/144 −27470059186624
−38684/15 N−5 + 6633/10 N−3− 59303/1080 N−1
−5077/1440 N − 11327/4320 N3− 8743/34560 N5
6∗ 19229646676220800
129027731639331200
7329919/240 N−6− 1652293/240 N−4
−4910303/15552 N−2 + 205365409/972000
−69368761/7776000 N2 + 14222209/2592000 N4
+3778133/3110400 N6
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Meson-meson scattering
M(s, t, u) =[
trT aT bT cT d + trT aT dT cT b]
B(s, t, u)
+[
trT aT cT dT b + trT aT bT dT c]
B(t, u, s)
+[
trT aT dT bT c + trT aT cT bT d]
B(u, s, t)
+ δabδcdC (s, t, u) + δacδbdC (t, u, s) + δadδbcC (u, s, t) .
Two functions needed
Two-loops known exactly JB, J. Lu, arXiv:1102.0172, JHEP 03
(2011) 028
Leading logs done to five loops
7 different channels (ππ has I=0,1,2)
No obvious pattern, not even large N
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Nucleon
In this sector very little done at two-loop and none higher(here N3LO)
Mass:p5 Birse, McGovern, hep-ph/9807384, hep-lat/0608002
p6 Schindler, Djukanovic, Gegelia, Scherer, nucl-th/0611083, 0707.4296
gA to p5 via RGE Bernard, Meissner, hep-lat/0605010
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Nucleon Lagrangian
We use the heavy-baryon approach: explicit powercounting
IR-regularization plus relativistic Lagrangian used as acheck
LO Lagrangian is order p (mesons p2):
L(0)Nπ = N (ivµDµ + gAS
µuµ)N
Propagator is order 1/p (mesons 1/p2)
Loops add p2 just as for mesons
Massless case: vertices with more legs needed (mesonicargument fails)
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Nucleon Lagrangian
Different parametrizations for mesons
Two different p2 Lagrangians:Bernard-Kaiser-Meissner:
L(1)πN = Nv
[
(v ·D)2−D·D−igA{S·D,v ·u}2M + c1tr (χ+)
+(
c2 − g2A
8M
)
(v · u)2 + c3u · u +(
c4 +14M
)
iǫµνρσuµuνvρSσ
]
Nv
Ecker-Mojzis:
L(1)Nπ = 1
MN[
− 12 (DµD
µ + igA{SµDµ, vνuν})+A1tr (uµu
µ)
+A2tr(
(vµuµ)2)
+ A3tr (χ+) + A5iǫµνρσvµSνuρuσ
]
N
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Nucleon loops
set ~n ∼ pn+1 for meson-nucleon
set ~n ∼ pn+2 for mesons
Introduce a RGO renormalization group order ≈max power of 1/w
same p-order can be different RGO, e.g.
both p5, left RGO 1, right RGO 2
Note: same equations, if no tree level contributionnext-to-leading log also calculable
For nucleon can have fractional powers of quark masses
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Nucleon loops
set ~n ∼ pn+1 for meson-nucleon
set ~n ∼ pn+2 for mesons
Introduce a RGO renormalization group order ≈max power of 1/w
same p-order can be different RGO, e.g.
both p5, left RGO 1, right RGO 2
Note: same equations, if no tree level contributionnext-to-leading log also calculable
For nucleon can have fractional powers of quark masses
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Results
M: nucleon mass, m: pion mass, L =m2
(4πF )2log
µ2
m2
Mphys = M + k2m2
M+ k3
πm3
(4πF )2+ k4
m4
(4πF )2Mln
µ2
m2
+k5πm5
(4πF )4ln
µ2
m2+ ...
= M +m2
M
∞∑
n=1
k2nLn−1 + πm
m2
(4πF )2
∞∑
n=1
k2n+1Ln−1,
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Results
k2 −4c1M
k3 −32g
2A
k434
(
g2A + (c2 + 4c3 − 4c1)M
)
− 3c1M
k53g2
A
8
(
3− 16g2A
)
k6 −34
(
g2A + (c2 + 4c3 − 4c1)M
)
+ 32c1M
k7 g2A
(
−18g4A +
35g2A
4 − 44364
)
k8278
(
g2A + (c2 + 4c3 − 4c1)M
)
− 92c1M
k9g2A
3
(
−116g6A +
2537g4A
20 − 3569g2A
24 + 556091280
)
k10 −25732
(
g2A + (c2 + 4c3 − 4c1)M
)
+ 25732 c1M
k11g2A
2
(
−95g8A +
5187407g6A
20160 − 449039g4A
945 +16733923g2
A
60480 − 2987855211935360
)
gA ↔ −gA: only even powers
k2n peculiar structure
Drop g3A then can calculate k12
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Results
r2 −4c1M
r3 −32g
2A
r434
(
g2A + (c2 + 4c3 − 4c1)M
)
− 5c1M
r5 −6g2A
r6 5c1M
r7g2A
4
(
−8 + 5g2A − 72g4
A
)
r8253 c1M
r9g2A
3
(
−116g6A +
647g4A
20 − 457g2A
12 + 1740
)
r1072536 c1M
r11g2A
2
(
95g8A − 1679567g6
A
20160 +451799g4
A
3780 − 320557g2A
15120 + 89646760480
)
r121754 c1M
everything rewritten in terms of physical pion mass
Simpler expression
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Results
Conjecture:
M = Mphys +3
4m4
phys
log µ2
m2phys
(4πF )2
(
g2A
Mphys− 4c1 + c2 + 4c3
)
− 3c1(4πF )2
µ2∫
m2phys
m4phys(µ
′)dµ′2
µ′2.
Take now known result for pion mass, k14 and k16calculable
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Numerics
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0 0.02 0.04 0.06 0.08 0.1
Mph
ys-M
[GeV
]
m2 [GeV2]
012345
M = 938 MeV,
c1 = −0.87 GeV−1
c2 = 3.34 GeV−1
c3 = −5.25 GeV−1
gA = 1.25
µ = 0.77 GeV
F = 92.4 MeV
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Numerics
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
2 4 6 8 10 12 14 16
Mph
ys-M
[GeV
]
n
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Conclusions Leading Logarithms
Leading logarithms can be calculated using only one-loopdiagrams
Results for a large number of quantities for mesons
Look at the (very) many tables, we would be veryinterested in all-order conjectures
Nucleonmass as the first result in the nucleon sector
Work in progress for other nucleon quantities
Exploratory steps done for gravity (Relefors)
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Nonlinear sigma model
The nonlinear U(N) sigma model recursion relations arederived in
Kampf, Novotny, Trnka, 1212.5224
Kampf, Novotny, Trnka, 1304.3048
Cheung, Kampf, Novotny, Shen, Trnka, 1509.03309
Can this be generalized to the massive U(N) nonlinear sigmamodel?
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Underlying assumptions
The amplitude can be written in an ordered form(colour-ordered, flavour-ordered,. . . )
Ma1a1...ann =
∑
σ∈Sn/Zn
〈taσ(1) . . . taσ(n)〉Mσ(p1, . . . , pn)
Mσ(p1, . . . , pn) = M(pσ(1), . . . , pσ(n))
The ordered amplitude consists of planar graphs
Scalar particles (no spin)
Interactions involve only an even number of particles
All particles have the same mass
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Consequences
M(p1, . . . , pn) = f (m2, {pij · pkl})pij =
∑
k=i ,j
pk , pii = pi
The tree level amplitude is thus independent of thedimension
Every internal propagator has momentum pij as a sum ofan odd number of consecutive external momenta
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Massive to massless
pi with p2i = m2, propagators are 1/(p2ij −m2) and∑
i pi = 0
go to 5 dimensions
set q = (0, 0, 0, 0,m) and pi = (pi , 0) so p2i = m2,q2 = −m2 and q · pi = 0
q2i+1 = p2i+1 + q and q2i = p2i − q
qi with q2i = 0, propagators are 1/q2ij and∑
i qi = 0
Caveat 1: do not use simplifications in the amplitude in 5dimensions since in the numerator when going backqij · qkl → pij · pkl since it came from ∂µφ∂
µφ and in thedenominator q2ij → p2ij −m2
Caveat 2: It’s not clear whether that is compatible withthe recursion relations in general
Amp & Logsin ChPT
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Massive to massless
pi with p2i = m2, propagators are 1/(p2ij −m2) and∑
i pi = 0
go to 5 dimensions
set q = (0, 0, 0, 0,m) and pi = (pi , 0) so p2i = m2,q2 = −m2 and q · pi = 0
q2i+1 = p2i+1 + q and q2i = p2i − q
qi with q2i = 0, propagators are 1/q2ij and∑
i qi = 0
Caveat 1: do not use simplifications in the amplitude in 5dimensions since in the numerator when going backqij · qkl → pij · pkl since it came from ∂µφ∂
µφ and in thedenominator q2ij → p2ij −m2
Caveat 2: It’s not clear whether that is compatible withthe recursion relations in general
Amp & Logsin ChPT
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Massive U(N) nonlinear sigma model
The previous method allows to remove masses frompropagators and external momenta
What about mass-dependent interaction terms?
Massive U(N) nonlinear sigma model:
L =F 2
4
⟨
∂µU∂µU†⟩
+F 2M2
4
⟨
U + U†⟩
Usual parametrization U = exp (iΦ)
Choose: U = 1− Φ2/2 + iΦ√
1− Φ2/4 then the massterm has no interaction vertices
(Φ =√2φ/F )
Amp & Logsin ChPT
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Conclusions
Very short intro to EFT and ChPT
Leading logarithms for a number of EFT known to a highnumber of loops
First steps towards recursion relations for the massiveU(N) nonlinear sigma model
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Large N
Power counting: pick L extensive in N ⇒ F 2 ∼ N, M2 ∼ 1
︸ ︷︷ ︸
2n legs⇔ F 2−2n ∼ 1
Nn−1
⇔ N
1PI diagrams:NL = NI −
∑
n N2n + 12NI + NE =
∑
n 2nN2n
}
⇒ NL =∑
n(n − 1)N2n − 12NE + 1
diagram suppression factor:NNL
NNE /2−1
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Large N
diagrams with shared lines are suppressed
each new loop needs also a new flavour loop
in the large N limit only “cactus” diagrams survive:
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large N: propagator
Generate recursively via a Gap equation
( )−1 = ( )−1 + + + + + · · ·
⇒ resum the series and look for the pole
M2 = M2phys
√
1 + NF 2A(M
2phys)
A(m2) = m2
16π2 logµ2
m2 .
Solve recursively, agrees with other result
Note: can be done for all parametrizations
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large N: Decay constant
= + + + + + · · ·
⇒ and include wave-function renormalization
Fphys = F√
1 + NF 2A(M
2phys)
Solve recursively, agrees with other result
Note: can be done for all parametrizations
Amp & Logsin ChPT
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large N: Vacuum Expectation Value
= + + + + + · · ·
〈qq〉phys = 〈qq〉0√
1 + NF 2A(M
2phys)
Comments:
These are the full∗ leading N results, not just leading log
But depends on the choice of N-dependence of higherorder coefficients
Assumes higher LECs zero ( < Nn+1 for ~n)
Large N as in O(N) not large Nc
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Large N : ππ-scattering
Semiclassical methods Coleman, Jackiw, Politzer 1974
Diagram resummation Dobado, Pelaez 1992
A(φiφj → φkφl) =A(s, t, u)δijδkl + A(t, u, s)δikδjl + A(u, s, t)δil δjk
A(s, t, u) = A(s, u, t)
Proof same as Weinberg’s for O(4)/O(3), group theoryand crossing
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Large N : ππ-scattering
Cactus diagrams for A(s, t, u)
Branch with no momentum: resummed by
Branch starting at vertex: resum by
= + + + + + · · ·
The full result is then
+ + + · · ·
Can be summarized by a recursive equation
= +
Amp & Logsin ChPT
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Large N : ππ scattering
y = NF 2A(M
2phys)
A(s, t, u) =
sF 2(1+y)
− M2
F 2(1+y)3/2
1− 12
(
sF 2(1+y)
− M2
F 2(1+y)3/2
)
B(M2phys,M
2phys, s)
or
A(s, t, u) =
s−M2phys
Fphys
1− 12
s−M2phys
F 2phys
B(M2phys,M
2phys, s)
M2 → 0 agrees with the known results
Agrees with our 5-loop results