PHYSICS BEYOND �PT

A. Pich

IFIC, Univ. Valencia { CSIC

� F�(t): F. Guerrero, A.P., PLB 412 (1997) 382

F. Guerrero, PRD 57 (1998) 4136

D. G�omez{Dumm, A.P., J. Portol�es, PRD 62(2000) 054014

� F ��

S(t): E. Pallante, A.P., hep-ph/0007208

� "0=": E. Pallante, A.P., PRL 84 (2000) 2568;

E. Pallante, A.P., hep-ph/0007208

E. Pallante, A.P., I. Scimemi, IFIC/00-31

G. Ecker etal., PLB 477 (2000) 88

� FK�

S(t) ; ms: M. Jamin, J.A. Oller, A.P., hep-ph/0006045

M. Jamin, J.A. Oller, A.P., IFIC/00-32

� KL ! �+��: D. G�omez{Dumm, A.P., PRL 80 (1998) 4633

EVOLUTION FROM MW TO MK

Scale Fields E�. Theory

MW

W;Z; ; g

�; �; e; �i

t; b; c; s; d; u

Standard Model

<�mc

; g ; �; e; �i

s; d; uL(nf=3)

QCD , L�S=1;2e�

MK

; �; e; �i

�;K; �ChPT

?

?

OPE

NC !1

�PT

Chiral Symmetry: [M � diag(mu;md;ms) = 0]

LQCD := i �qL/DqL+ i �qR/DqR ; qT = (u; d; s)

qL;R ! gL;R qL;R ; gL;R 2 SU(3)L;R

SU(3)L SU(3)R ! SU(3)V + 8 0� Goldstones

h�qjLqiRi U = exp

�ip2�=f

�; U ! gR U g

yL

� =

0BBB@q

12�0+

q16� �+ K+

�� �q

12�0+

q16� K0

K� �K0 �q

23�

1CCCA

Low{Energy Expansion: (p2n ; mnq ) L =

PnL2n

L2 = f2

4 hD�UyD�U + 2B0

�UyM+MyU

�i

= D��+D��

� �M2��+��+ � � �

+ 16f2

��+

$D� �

�� �

�+$D ���

�+ � � �

M2�

mu+md=

M2

K0

ms+md=

M2

K+

ms+mu= B0 = �h�qqi

f2

O(p4) �PT

i) L4 at tree level (Gasser{Leutwyler)

L4 = L1 hD�UyD�Ui2 + L2 hD�U

yD�Ui hD�UyD�Ui+ L3 hD�U

yD�UD�UyD�Ui + L4 hD�U

yD�Ui hUy�+ �yUi+ L5 hD�U

yD�U�Uy�+ �yU

�i + L6 hUy�+ �yUi2

+ L7 hUy�� �yUi2 + L8 h�yU�yU + Uy�Uy�i� i L9 hF ��

RD�UD�U

y+ F��

LD�U

yD�Ui + L10 hUyF ��

RUFL��i

F��

J� @�J� � @�J� � i[J�; J�] ; J� = l�; r� ; � � 2B0M

ii) L2 at one loop (Unitarity)

T4 � p4na log(p2=�2) + b(�)

o

� Chiral Logarithms unambiguously predicted

� Li's �xed by QCD dynamics

Reabsorb one{loop divergences Lri(�)

iii) Wess{Zumino{Witten term (chiral anomaly)

�0 ! ; � ! ; � ! �� ; � � �

O(p4) �PT COUPLINGS

i Lri(M�)� 103 Source �i

1 0:4� 0:3 Ke4, �� ! �� 3=32

2 1:4� 0:3 Ke4, �� ! �� 3=16

3 �3:5� 1:1 Ke4, �� ! �� 0

4 �0:3� 0:5 Zweig rule 1=8

5 1:4� 0:5 FK : F� 3=8

6 �0:2� 0:3 Zweig rule 11=144

7 �0:4� 0:2 Gell-Mann{Okubo, L5;8 0

8 0:9� 0:3 MK0 �MK+, L5, 5=48

(ms � m̂) : (md �mu)

9 6:9� 0:7 hr2i�V 1=4

10 �5:5� 0:7 � ! e� �1=4

Li = Lr

i(�) + �i

�D�4

32�2

n2

D�4 + E � log (4�) � 1o

Lr

i(�2) = Lr

i(�1) +

�i

(4�)2log

��1

�2

�

�� � 4�f� � 1:2 GeV Li � f2�=4

�2�� 2� 10�3

Li'S

FRO

M

RESO

NANCE

EXCHANGE

i

Lr i(M�)

V

A

S

S1

�1

Total

Totalb)

1

0:4�0:3

0:6

0

�0:2

0:2

0

0:6

0:9

2

1:4�0:3

1:2

0

0

0

0

1:2

1:8

3

�3:5�1:1

�3:6

0

0:6

0

0

�3:0

�4:9

4

�0:3�0:5

0

0

�0:5

0:5

0

0:0

0:0

5

1:4�0:5

0

0

1:4a)

0

0

1:4

1:4

6

�0:2�0:3

0

0

�0:3

0:3

0

0:0

0:0

7

�0:4�0:2

0

0

0

0

�0:3

�0:3

�0:3

8

0:9�0:3

0

0

0:9a)

0

0

0:9

0:9

9

6:9�0:7

6:9a)

0

0

0

0

6:9

7:3

10

�5:5�0:7

�10:0

4:0

0

0

0

�6:0

�5:5

a)

Input

b)

LV 1

=

LV 2=2=

�LV 3=6=

LV 9=8=

�L

V

+A

10

=6=

f2 �=(16M2 V)

PION FORM FACTOR

h�0��j�d �uj0i �p2 F�(s)

�p�� � p�0

��

� O(p4) �PT:

F�(s) = 1+2Lr

9(�)

f2�s� s

96�2f2�A

�M2

�s;M2

�

�2

�

A(x; y) � log y+8x� 53+ �3

xlog

��x+1�x�1

�; �x �

p1� 4x

� NC !1: (1 Resonance only)

�

�

�

�

�

+

�

�

�

F�(s) = 1+FVGV

f2�

sM2

��s=

M2�

M2��s

(lims!1 F�(s) = 0)

� O(p4) �PT + NC !1:

F�(s) =M2

�

M2��s

� s96�2f2�

A

�M2

�s;M2

�

�2

�

� Omn�es Summation:

F�(s) =M2

�

M2��s

exp

�� s96�2f2�

A

�M2

�s ;

M2�

M2�

��

Omn�es Problem

ImAIJ(s+ i�) =

1

2

Xn

h(��)IJ jT yjni hnjOji(q)i

ImAIJ =

�ImAI

J

�2�

= e�i�IJ sin �IJ A

IJ = ei�

IJ sin �IJ A

I�J

= sin �IJ jAIJ j = tan �IJ ReA

IJ

AI

J(s) =

n�1Xk=0

(s� s0)k

k!

dkAI

J

dsk

����s=s0

+(s� s0)n

�

Z 1

4M2

�

dz

(z � s0)ntan �I

J(z)ReAI

J(z)

z � s� i�

AIJ(s) = exp

8<:n�1Xk=0

(s� s0)k

k!

dk

dsklog

nAIJ(s)

o���s=s0

+(s� s0)

n

�

Z 14M2

�

dz

(z � s0)n

�IJ(z)

z � s� i�

)

� I;J(s; s0) AIJ(s0)

� Rho Width: Dyson Summation

(a) (b)

(c) (d) (e)

F�(s) � M2�

M2��s�iM���(s)

��(s) = �(s� 4M2�) �

3�

M� s

96�f2�

��(s) = 144 MeV [exp: (150:7� 1:1) MeV]

�11(s) = arctan

�M� ��(s)Ms

��s

�� s �3�

96�f2�+ � � �

F�(s) =M2

�

M2��s�iM���(s)

exp

�� s96�2f2�

Re

�A

�M2

�s;M2

�

M2�

���

0.5 1 1.5

√s (GeV)

−1

0

1

log

|FV|2

CLEO−II dataALEPH dataFit (Mρ = 0.776 GeV)

Guerrero and Pich (Mρ = 0.776 GeV)

τ decay data vs theory

−0.5 0 0.5 1 1.5

s/ √(|s|) (GeV)

−0.5

0

0.5

1

1.5

log

|FV

|2

Fit τ data (Mρ = 0.776 GeV)

Guerrero and Pich (Mρ = 0.776 GeV)

e+ e

− data vs theory (fit to τ decay data)

SCALAR FORM FACTOR

h�i(p0)j�uu+ �ddj�k(p)i � �ik F�S (t)

F�S (t) = F�

S (0)n1+ g(t) +O(p4)

o

g(t) =t

f2

��1�M2

�

2t

��J��(t) +

1

4�JKK(t) +

M2�

18t�J��(t)

+ 4(Lr

5+ 2Lr

4)(�) +5

4(4�)2

�ln

�2

M2�

� 1

�� 1

4(4�)2lnM2

K

M2�

�

�JPP(t) = 1(4�)2

n2� �P ln

��P+1�P�1

�o; �P �

q1� 4M2

P

t

F�S (t) =0(t; t0)F

�S (t0)�0(t; t0)F

�S (0) f1+ g(t0)g

t0 g(t0) <0(M2K; t0) jF �

S(M2

K)=F �

S(0)j

0 0 1.45 1.45

M2�

0.042 1.40 1.46

2M2�

0.091 1.34 1.46

3M2�

0.15 1.26 1.45

4M2�

0.26 1.11 1.40

M2K

0:54� 0:46 i � 1 1.61

jF�S (M

2K)=F

�S (0)j = 1:55� 0:10

�+� �T(KL ! �+��)T(KS ! �+��)

� "K + "0K

�00 �T(KL ! �0�0)

T(KS ! �0�0)� "K � 2"0K

"K = (2:280� 0:013)� 10�3 ei�"K

�"K = (43:47� 0:51)�

Re

"0K"K

!� 1

6

8<:1�

����� �00�+�

�����29=; = (19:3� 2:4)� 10�4

DIRECT CP VIOLATION

104Re("0K="K) = 33� 11 NA31 (1988)

32� 30 E731 (1988)

20� 7 NA31 (1993)

7:4� 5:9 E731 (1993)

28:0� 4:1 KTeV (1999)

14:0� 4:3 NA48 (1999{2000)

K ! 2� ISOSPIN AMPLITUDES

j��i : J = 0 , CP = + , I = 0 ; 2 (Bose)

A[K0! �+��] � A0 ei �0+

1p2A2 e

i �2

A[K0! �0�0] � A0 ei �0 �

p2A2 e

i �2

A[K+! �+�0] � 3

2A2 e

i �2

�I = 1=2 Rule :

! � Re (A2)Re (A0)

� 122

Strong Final State Interactions :

�0 � �2 = 45� � 6�

"0K = �ip2ei (�

2��0) !�Im (A0)Re (A0)

� Im (A2)Re (A2)

�

SHORT DISTANCES

L�S=1eff =� GFp

2Vud V

�us

Pi Ci(�) Qi(�)

Q1 =�s�u�

�V�A

�u�d�

�V�A

Q2 = (su)V�A (ud)V�A

Q3;5 = (sd)V�AP

q(qq)V�A

Q4;6 =�s�d�

�V�A

Pq(q�q�)V�A

Q7;9 = 32(sd)V�A

Pqeq (qq)V�A

Q8;10 = 32

�s�d�

�V�A

Pqeq(q�q�)V�A

q > � : Ci(�) = zi(�)� yi(�)�Vtd V

�ts=Vud V

�us

�O(�n

stn) , O(�n+1

stn) , [t � log (M=m)] (Munich / Rome)

q < � : h��jQi(�)jKi ?

Physics does not depend on �

Buras et al

"0K"K

� Im�V �tsVtd

� hP (1=2) � P (3=2)

i

P (1=2) = rXi

yi(�) hQi(�)i0 (1�IB)

P (3=2) =r

!

Xi

yi(�) hQi(�)i2

Experiment: r =GF !

2j"jRe (A0)

! =Re (A2)Re (A0)

� 122 ; Re(A0) = 3:37� 10�7 GeV

Theory: hQi(�)i � hQiivs Bi(�)

"0K"K

��110MeVms(2GeV)

�2 �B(1=2)6 (1�IB)� 0:4B

(3=2)8

�

B(1=2)6 = 1:0� 0:3 ; B

(3=2)8 = 0:8� 0:2

IB � �+�0 = 0:25� 0:08 0:16

�0{� MIXING (Ecker et al)

�0 � �3+ "�0� �8 A0 �! AIB2 / "�0� A0

A(K0 ! �0�0) � A(K0 ! �3�3) + 2 "�0�A(K0 ! �8�3)

IB =Im (AIB

2)

! Im (A0)=

2p2 "

�0�

3p3!

O(p2) CHPT: m̂ � (mu +md)=2

"(2)�0�

=

p3 (md�mu)4 (ms�m̂) IB � 0:13

O(p4) CHPT: "�0� = "(2)�0�

+ "(4)�0�

IB � �+�0 = 0:16� 0:03

"(4)�0�

="(2)�0�

/ f(3L7+ L8(�)) + � logsg

�0: "(4)�0�(L7)="

(2)�0�

= 1:10 ; a0: "(4)�0�(L8)="

(2)�0�

= �0:83

RECENT

THEO

RETICAL

ESTIM

ATES

Group

B(1=2)

6

B(3=2)

8

"0 K="K

�104

Munich

1:0�0:3

0:8�0:2

9:2+6:8

�4:0

(G)

1:4!32:7

(S)

Rome

1:0�1:0

0:71�0:13

8:1+10:3

�9:5

(G)

�13!37

(S)

Trieste

1:3

0:84

22�8

(G)

9!48

(S)

Dortmund

1:50!1:62

B(1=2)

6

=1:72

6:8!63:9

(S)

Dubna{DESY

1:0

1:0

�3:2!3:3

(S)

Granada{Lund

2:5�0:4

1:35�0:20

34�18

Montpellier

2:99�0:34

1:70�0:39

25:5�8:4

Taipei

1:5

1:5

7!16

Valencia

1.55

0.92

17�6

O(GF p2) �PT

L�S=12 =

nG8 f

4 h�L�L�i+G27 f

4

�L�23L

�

11+2

3L�21L

�

13

�

+ e2f6GEW h�UyQUio

GR �� GFp2VudV �usgR ; � � �(6�i7)=2 ; L� = �iUyD�U

A0 =p2f�

��G8+

1

9G27

�(M2

K �M2� )�

2

3f2�e

2GEW

�

A2 =2

9f�n5G27 (M

2K �M2

� )� 3f2�e2GEW

o

�0 = �2 = 0

h�(K ! 2�) + �I

iExp

: g8 � 5:1 ; jg27=g8j � 1=18

NC !1: g8 =�35C2 � 2

5C1+ C4

��16L5

�h�i(�)

f3�

�2C6(�)

g27 =35(C2+ C1) ; e2 gEW =�3

�h�i(�)

f3�

�2C8(�)

- Equivalent to standard calculations of Bi

- � dependence only captured for Q6;8

- Good estimate of Im (gI) . Large1NC

corrections

to Re (gI) (Pich { de Rafael '91)

1{Loop �PT

(Kambor et al '90)

� Large enhancement of A0 (� 40{50%)

� �I 6= 0 (still �0 < �0exp)

� Loop correction dominated by infrared

log (M�) terms from �{� loops (FSI)

� O(p4) local terms �xed at NC !1� Not included in Lattice { 1=NC estimates

(�I = 0 at NC !1)

Overlooked in Standard predictions of "0="

Higher Orders ?

FINAL STATE INTERACTIONS

Watson Theorem: AI � A[K ! (��)I ] � AI ei�I

SU(3): AI = (M2K �M2

� ) aI

Omn�es: aI(M2K) = I(M

2K; s0) aI(s0)

I(s; s0) � exp

((s� s0)

�

Zdz

(z � s0)

�I(z)

(z � s� i�)

)

� ei�I(s) <I(s; s0)

� �I(s) = �I(s)j�� below inelastic threshold

� Arbitrary number of subtractions

(n)I(s; s0) = exp

�Pn(s; s0) +

(s� s0)n

�

Zdz

(z � s0)n�I(z)

(z � s� i�)

�

� All{order resummation of infrared chiral logs (same 8n)

� Polynomial ambiguity ) Short{distance information

<0 = 1:55� 0:10 ; <2 = 0:92� 0:03

K ! 2�

AI � A[K ! (��)I ] � AI ei�I = (M2

K �M2�) aI e

i�I

aI(s) = aI(0)n1+ gI(s) +O(p4)

o

g(s) =s

f2

��1� M2

�

2s

��J��(s) +

1

4�JKK(s) +

M2�

18s�J��(s)

+ 4(Lr

5+2Lr

4)(�) +5

4(4�)2

�ln

�2

M2�

� 1

�� 1

4(4�)2lnM2

K

M2�

�

g(8)0 (s) =

s

f2

��1� M2

�

2s

��J��(s) �

1

4

�1�

M2K

s

��JKK(s) +

M2�

18s�J��(s)

+ C85(�) +

3

4(4�)2

�ln

�2

M2�

� 1

�+

1

4(4�)2lnM2

K

M2�

�

g(27)0 (s) =

s

f2

��1� M2

�

2s

��J��(s) �

3

2

�1�

M2K

s

��JKK(s)�

M2�

2s�J��(s)

+ C275 (�)� 1

2(4�)2

�ln

�2

M2�

� 1

�+

3

2(4�)2lnM2

K

M2�

�

g2(s) =s

f2

��12

�1� 2M2

�

s

��J��(s)

+ �C275 (�)� 1

2(4�)2

�ln

�2

M2�

� 1

��

INCLUDING FSI

"0K"K

��110MeVms(2GeV)

�2 �B(1=2)6 (1�IB)� 0:4B

(3=2)8

�

� B(1=2)6

����NC!1

� B(3=2)8

����NC!1

� 1

� B(1=2)6 � B

(1=2)6

����NC!1

� <0 � 1:55

� B(3=2)8 � B

(3=2)8

����NC!1

� <2 � 0:92

� IB � 0:16�<2=<0 � 0:09

(Wolfe{Maltman: 0:08� 0:05)

"0

"= (17� 6)� 10�4

SUMMARY

� �PT � Low{Energy Symmetry Constraints

- Known infrared logarithms

- Unknown chiral couplings

Short Distances

� 1=NC: QCD Matching

Resonance Exchanges

� Omn�es: FSI (elastic unitarity)

Exponentiation of chiral logarithms

� Unitarity: Coupled channels

� Many applications:

- Estimates of chiral couplings

- Hadronic Form Factors / Matrix Elements

- "0="- ms

- KL ! �+��