LEPTON PAIR PRODUCTION AS A PROBE OF TWO PHOTON EFFECTS IN EXCLUSIVE

PHOTON-HADRON SCATTERING

Pervez HoodbhoyQuaid-e-Azam University

Islamabad

OUTLINE OF TALK

• INTRO: 1. Nucleon Form Factors And GPDs 2. Why Does Rosenbluth Fail? • RADIATIVE CORRECTIONS

• TWO-PHOTON EFFECTS

• AN ASYMMETRY OBSERVABLE

• CALCULATION FOR LARGE-t

• SUMMARY AND OPEN QUESTIONS

Nucleon Electro-Magnetic Form Factors

- Fundamental ingredients in “Classical” nuclear theory

- A testing ground for theories that construct nucleons

- Spatial distribution of charge, magnetization

- Wavelength of probe can be tuned by selecting Q2: < 0.1 GeV2 integral quantities (charge radius,…) 0.1-10 GeV2 internal structure of nucleon > 20 GeV2 pQCD scaling

- Additional insights can be gained from the measurement of the form factors of nucleons embedded in the nuclear medium

- Implications for binding, equation of state, EMC, precursor to QGP

Sachs Charge and Magnetization Form Factors GE and GM

with E (E’) incoming (outgoing) energy, scattering angle, anomalous magnetic momentIn the Breit (centre-of-mass) frame the Sachs FF can be written

as the Fourier transforms of the charge and magnetization radial density distributions

GE and GM are often alternatively expressed in the Dirac (non-spin-flip) F1 and Pauli (spin-flip) F2 Form Factors

2 2

22( , ) 2 tan / 21

EM

MM

G GG

dE

d

2 2

3 4

' cos / 2

4 sin / 2M

E

E

2

2 21

Q =

(1 ) 4M E

E M

G GG GF F

M

Rosenbluth separation method

One-photon exchange elastic electron-

nucleon cross section

Method : at fixed Q2, vary angle (or equivalently )

and plot reduced cross section

versus

One-photon theorist’s view

Polarization transfer method

Method : measure ratio of sideways ( ) to

longitudinal ( ) recoil polarization of proton

(absolute normalization drops out !)in one-photon exchange approximation :

Rosenbluth vs polarization transfer measurements of GE/GM of proton

Jlab/Hall A Polarization

data

Gayou et al. (2002)

SLAC

Rosenbluth data

Two methods, two different results !

Speculation : missing radiative corrections

Speculation : there are radiative corrections to Rosenbluth

experiments that are important and are not included missing correction : linear in not strongly Q

2 dependent

GE term is proportionally smaller at large Q2

if both FF scale in same wayeffect more visible at large Q2

Q2 = 6 GeV2

Basics Of QED Radiative Corrections

(First) Born approximation

Initial-state radiation Final-state radiation

Cross section ~ dω/ω => integral diverges logarithmically: IR catastrophe

Vertex correction => cancels divergent terms; Schwinger (1949)

)}(2

1

36

17)1)(ln

12

13{(ln

2,)1( 2

2

exp f

m

Q

E

E

eBorn

Multiple soft-photon emission: solved by exponentiation,Yennie-Frautschi-Suura (YFS), 1961

e )1(

Radiative correction diagrams

bremsstrahlung

vertex

corrections

2 photon

exchange box

diagrams

Two-Photon Exchange

• 1-2 interference is of the order of

=e2/4=1/137

(in usual calculations of radiative corrections,

one photon is ‘hard’ and one is ‘soft’)

• Due to the sharp decrease of the FFs, if the

momentum is shared between the two photons,

the 2contribution can become very large

Qualitative estimationof Two-Photon exchange ( for ed)

Form factors → quark counting rules: Fd ~ t-5 and FN~t-2

For t = -4 GeV2,

For d, 3He, 4He, 2effect should appear at ~1 GeV2,for protons ~ 10 GeV2

Calculation of soft part at nucleon level

LET : sum of soft contributions from the partonic calculation has to match

the soft contributions at nucleonic level

Pictorially :

To satisfy the LET, one has to include the

soft-photon contributions from the cats’ ears diagrams

soft soft softsoft

P 'P

l

l

q

Proposal: use real photons to investigate 2-photon effects. To get more insight take an extreme case where the proton structure is relatively well-understood.

2

2

( ,0,0, )

( , sin ,0, cos )

( , sin ,0, cos )

(1,sin cos ,sin sin ,cos )

(1, sin cos , sin sin , cos )

M

M

q

P

P M

l

l

2

2

2

cos 12

s t

m

M t

Ms

P 'P

l

l

q

4

For massless leptons, helicity

v (

is conser

)

ved:

( )M

l u l

4

M 5

4v ( ) ( )

Ml u l

4

M

*

5

(0,cos cos sin ,co

Condition

0

2 0

on

s cos cos , s

is that:

This is satisfied

n )

0

b :

i

y

i

l

i

l l

l

P 'P

l

l

q

P 'P

l

l

q

1 1 1

1

1 1

2

3

2

1

2

( , , , , )

( 1) ( 1)2 ( )

sin( )

Now use

( , , , , ) ( , , ,

ONE PHOTO

symmetry arguments to ge

N EXCHANG

, )

( , , , ,

E

t:

i i i i

A A l l

e i t e Me esi e F

A A l l A l l

tt M

A A

t

l

1

2

3

1 3/

1 1

2

1

2

) ( , , , , )

( , , , , ) ( , , , , )

Extreme limit: - gives:

8tan ( )

( )

l A l l

s t M

eA i s F

A A l l

tt

A l l

2 2 20

1

2

2

2 6 2 22,1 6

2 2, ,1 1

, ,

4, 0

1(

24 1) ( 1)csc

2

The coefficients are:

Here is dimensionless:

( 1)cot

THE ASYMMETRY: ( , ) ( , )

64 ( )cos

(1 )

n

s s

s

s s

ns n

a

a

a

t

M

a

s e F ta

A

A nM

A

IS ZERO

Typical suppressed diagrams

1 2 1 2

† † † †1 2 1 2 1 2

1 2

1

2

(1 )

1,2,3 is colour. ( ) 6 (1 )

1( , ) ( ) ( ) ( ) ( )3

PION STATE:

a a a aP

dx dx dx x x

a x f x x

dxx x u x d x u x d x

x x

1Px

2PxP

• Assume transverse momentum of quarks is negligible• Assume lowest Fock state dominates at large -t

PP

l q

l

2x

1x

2y

1yPP

l q

l

2x

1x

2y

1y

1 1

A typical denominator, expanded to leading s, looks like:

1

( ) 1 2 cos sin ( 1)cos

1 1Decompose: ( ) ( )

Principal parts cancel between diagrams.

Ward identity satisfied in all

s x y i

P i xx i x

possible ways.

1x

2 2

3 2

5 2 2

2

1 2 1 2 1 1

1 1 1 1

( , , , , )

4 2cot

(1 )

( 1) ( 1)

( , ) ( , ) ( )

1 and

(

sin 24

)

TWO PHOTON EXCHANGE RESULTS

i iu d

i i

i

i

A A l l

i e e e se i e

M

i e e e

x x y y x ydx dy

y y

ey y

y y

221

(1 )sin4

ie

A Quick Aside: Charge Conjugation

• C operation - interchange of particle with its antiparticle.

• C symmetry in classical physics - invariance of Maxwell’s equations under change in sign of the charge, electric and magnetic fields.

• C symmetry in particle physics - the same laws for a set of particles and their antiparticles: collisions between electrons and protons are described in the same way as collisions between positrons and antiprotons. The symmetry also applies for neutral particles.

• C: Even or odd symmetry.

• Example: particle decay into two photons, for example o 2 by the electromagnetic force. Photon is odd under C symmetry; two photon state gives a product (-1)2 and is even. So, if symmetry is exact, then 3 photon decay is forbidden. In fact it has not been observed.

• C symmetry holds in strong and electromagnetic interactions.

A Quick Aside: Charge Conjugation – cont’d

2 2 2

2 2 2

2 2 2

Now use th

( , , , , ) ( , , , , )

( , , , , ) (

ese arguments to g

, , , , )

( , , , , ) ( , , , ,

:

)

et

A A l l A l l

A A l l A l l

A A l l A l l

1

1

1

.

C C

C C

C C

etc

,s

1 2

20

* ,s * ,s ,s1 2 1 2

,s

2,1

,

4 ( )cos sin

( ) ( ) ( )

(A A A A )

( , )

( ) ( ) at large .

The integral becomes constant at large .

iu d

s

s

e e M xe dx cc

t F t xx xxt

F t t

t

A

t

is p

Defin

ropor

e charge a

tional to

( , ) :

(

the di

, )

symmetry

fference in count rates between ele

( , ) ( , )

( , ) ( , )

( , )ctrons and posi on .

tr s

l

l

q

l

l

q

1q 2qx1P

(1-x1)P

1 4M GeV

PP

l q

l

2x 2y

1y

3x 3y

1x

PP

l q

l

2x 2y

1y

3x 3y

1x

For the proton case: 6 12 diagrams:

• Assume transverse momentum of quarks is negligible• Assume lowest Fock state dominates at large -t

1 2 3 1 2

† † † † †1 2 3

3

1 2 3 1 3

3

2

1

1 2 31 2 3

6,

24

1( ) ( ) ( ) ( ) ( ) 0

(1 )

Asymptotically, ( , ,

6

)

( , )

abca b c b c

dx dx dx dx x x

P

x

x x x x x x

u x u x d x d x u x

dxx x x

x x x

1Px

2PxP

3Px

5

Define auxiliary vector for proton:

(0,cos , , sin ) 0

Spi

2n

2matrix:

PP

P

P

P

i

2 2

2 3 2( )1

7 3 2

2 2 2 3 3 3 1 2 3 1 2 3 1 1 12

1 2 3 1 2 3 1 1 1

( , , , , )

64 21 cot ( 1) ( 1)

(1

TWO PHOTON EX

)

( , , )

CHAN

( , , ) ( )

( )

GEN Nl l

i i i i is

A A l l

i e e se e i e e e

M

e x y e x y x x x y y y x x ydx dy

x x x y y y y y y

2 2 2

2

2

2

2

2 2

2 2

( , , , , ) ( , , , , )

( , , , , ) ( , , , , )

( , , , , ) ( , ,

11 and sin 2 (1 )sin

44

Now use symmetry arguments to ge

,

t:

i i

A A l l A l l

A A l l A l l

A A l

e ey y

l A

, )l l

,s * ,s * ,s ,s1 2 1 2

,s

21

2 2

2 2 2 3 3 3 1 2 3 1 2 3 1 1 12

1 2 3 1 2 3 1 1 1

2,1

,

64cos sin

(A

( ) ( )

( , , ) ( , , ) ( )

A A A

)

)

(

( , )

is

N

Ns

s

e Me

t F t t

e x y e x y x x x y y y x x ydx dy

x x x y y y y y y

A

SUMMARY

• Real photons are used to probe nucleon structure.

• Real photons are easily available at many labs.

• At large-t the proton structure is much simpler.

• The expression for the asymmetry is very compact.

• The size of the signal is large at modest –t.

• Only F1 form-factor considered here: F2 involves spin-flip which is zero for massless, collinear quarks.

OPEN QUESTIONS

• How big will Sudakov effects be?

• Will the next order calculation (few thousand diagrams!) change the angular structure?

• Will it dominate the present calculation?