Amp & Logs Amplitudes and Logarithms in Chiral ...home.thep.lu.se/~bijnens/talks/stockholm16.pdfAmp...

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions

Backup: largeN (O(N))

1/72

Amplitudes and Logarithms in ChiralPerturbation Theory

Johan Bijnens

Lund University

[email protected]

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


2/72

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

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


3/72

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


4/72

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


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


6/72

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


7/72

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


8/72

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


9/72

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

Logarithms

Recursionrelations

Conclusions


10/72

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


11/72

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


12/72

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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


13/72

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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


14/72

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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


18/72

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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


20/72


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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


21/72


Mass = 0 + 0 1

+ 0 0 0 1 0

1

2 + 2 1

1

· · ·

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


22/72


~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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


23/72


~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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


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

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


25/72

Mass to ~2

~1: 0 =⇒ 1

~2: 1 0

1

=⇒ 2

but also needs ~1: 0 0

0

=⇒ 1

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


26/72

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

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


27/72

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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


28/72

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

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


29/72

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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


30/72

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

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


31/72

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

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


32/72

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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


33/72

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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


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


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


35/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


36/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


37/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


38/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


39/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


40/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


41/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


42/72


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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


43/72


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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


44/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


45/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


46/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


47/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


48/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


49/72

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)

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


50/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


51/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


52/72

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,

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


53/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


54/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


55/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


56/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


57/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

LL


O(N + 1)/O(N)

Masses, decay


Other work

Anomaly

SU(N)×SU(N)/SU(N)

Nucleon

Conclusions

Recursionrelations

Conclusions


58/72

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)

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


59/72

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?

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


60/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


61/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


62/72

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

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


63/72

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

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


64/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


65/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


66/72

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:

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


67/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


68/72

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

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


69/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


70/72

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

Amp & Logsin ChPT

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


71/72

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

Johan Bijnens

ChPT/EFT

Logarithms

Recursionrelations

Conclusions


72/72

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

Amp & Logs Amplitudes and Logarithms in Chiral ...home.thep.lu.se/~bijnens/talks/stockholm16.pdfAmp...

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