Optical theorem - DESYjung/qcd_collider_physics_2005/lecture14.pdf · Page 1 H. Jung, QCD & Collider Physics, Lecture 14 WS 05/06 Optical theorem in optics: derived from conservation

Page 1H. Jung, QCD & Collider Physics, Lecture 14 WS 05/06

Optical theoremin optics: derived from conservation of energyin Quantum Mechanics: derived from conservation of probability

¾tot=

s!14¼

sImA(s; t = 0)


Regge trajectories

Amplitude for exchange of meson with mass M and spin J:

plot mesons as function of mass and spindefine trajectory

➔ expand trajectory:

Ames(s; t) » Aj(t)Pj(cosµt)

» Pj(cosµt)

t¡M2

s!1» sJ

®(t)

®(t) = ®(0)+ ®0(t)

3H. Jung, QCD & Collider Physics, Lecture 14 WS 05/06

Factorization of Hard Diffraction

Factorization in hard diffraction (J. Collins Phys.Rev.D57:30513056,1998, Erratum

ibid.D61:019902,2000):

diffractive pdfs behave similar to usual pdfsno assumptions on Regge factorization

collinear factorizationDGLAP evolutionfor Q2 sufficiently large, while are fixeduse full machinery of NLO DGLAP evolution...

d¾ =X

i

Zd»f

(D)i (»; xIP ; t;¹)d¾̂i+non¡ leading power of Q

xBj ; xIP ; t


Diffractive PDFs

FD(4)2 (¯;Q2; xIP ; t) =

X

i

Z 1

¯

dz

zCi³¯z

´fDi (z; xIP ; t;Q

2);

fDi (z; xIP ; t;Q2) =

fIP=p(xIP ; t) ¢ fIPi (z;Q2)


Diffractive dijet production

use diffractive pdfs, obtained from F2Dpredict cross section in diffractive DIS

➔ x section is described


Diffractive Factorization is broken

use diffractive pdfs also for photo production dijetspredicted cross section ~ factor 2 too largesimilar effect seen in protonproton collisions

➔ factorization is broken


Understanding diffraction

simplest model for Pomeron: 2 gluon system


2 gluon exchange: qq diagram

Calculate Born diagram: scattering on a quark...

M = ¹v(k +Q)°¹=k

k2(igs°¯T )

=k ¡ =u¡ =l

(k ¡ u¡ l)2(igs°®T ) ¢ v(u¡ k)

¢ ¡i

(l + u)2g¯¯0

¡i

l2g®®0¹v(p)(igs°¯0T )

=p¡ =u¡ =l

(p¡ u¡ l)2(igs°®0T )v(p¡ u)

Levin Wusthoff, PRD 50 4306 91994)


2gluon exchange: qq diagrams

also crossed diagrams need to be included, but not all contribute ...

Levin Wusthoff, PRD 50 4306 91994)


Diffractive quarkantiquark production (transv.)

perform calculation for

using approximations for small

1/k4 dependence, squared of gluon density

d¾T

dM2dtdk2 jt=0=

X

f

e2f®em¼2®2s

12

1

M4

(1¡ 2k 2

M2 )q1¡ 4k 2

M2

£IT (Q

2;M2; k2)¤2

IT = ¡Z

dl2

l2FG(xIP ; l2)

24M2 ¡Q2

M2 +Q2+

l2 + k 2

M2 (Q2 ¡M2)q

(l2 + k 2M2 (Q2 ¡M2))2 + 4k4 Q

2

M2

35

IT =

·4Q2M4

k2(M2 +Q2)3+ bt

@

@k2

¸xIPG(xIP ; k2

Q2 +M2

M2)

°¤p! q¹qp

l

Bartels, Lotter Wusthoff, hepph/9602363


Diffractive quarkantiquark production (long.)

perform calculation for

using approximations for small

1/k4 dependence, squared of gluon density, when usung IL2

°¤p! q¹qp

l


d¾L

dM2dtdk2 jt=0=

X

f

e2f®em¼

2®2s3

4

Q2M2

k2

M2

1q1¡ 4k 2

M2

£IL(Q

2;M2; k2)¤2

IL(Q2;M2; k2) = ¡

Zdl2

l2FG(xIP ; l2)

24 Q2

(M2 +Q2)¡ k2 Q2

M2

q(l2 + k 2

M2 (Q2 ¡M2))2 + 4k4 Q2

M2

35

IL =

·Q2M2(Q2 ¡M2)

k2(M2 +Q2)3+ bl

@

@k2

¸xIPG(xIP ; k2

Q2 +M2

M2):

higher twist ...


Energy dependence of diffractive qq

strong energy dependence due to squared gluon densitytest for different gluon densities x section comparison:

¾qq » 46pb

¾resIP » 1100pb

jung hepph/9809373

0:1 < y < 0:7

5 < Q2 < 80

xIP < 0:05

pjetst > 2GeV



Azimuthal angle in diffractive qq

azimuthal dependence is different compared to bosongluon fusion process:

2gluon: jets are perpendicular to scattering planeBGF: jets are in the scattering plane



2 gluon exchange: qqg final states

many more diagrams contributecalculations for different final state configurations exist:

➔ small, gluon has large pt➔ large, gluon has small pt

calculations are very trickyBUT not a real NLO calculation exists, including the virtual corrections to

Bartels, Jung, Wusthoff, EPJC 11 (1999) 111

°p! q¹qg + p

mq¹q

mq¹q

q¹q


Contributions to F2D

different contributions to different regions of phase spaceNOTE: long contribution can be larger than transverse...

..... higher twist larger than leading twist

GolecBiernat, Wusthoff PRD 60 114023 (1999)


Diffractive Dijets and 2gluon approx.

dijets:

0:1 < y < 0:7

4 < Q2 < 80

xIP < 0:05

pjetst > 4GeV


Diffraction and hard scattering factorization

soft

hard IP

➔ hard perturbative 2gluon exchange

➔ hard jet close to rapidity gap

➔ NOT included in DGLAP

IP

➔ diffraction is in initial condition➔ start Q2 evolution from Q0➔ DGLAP of F2

D3 etc...


Towards understanding of diffraction

Cutting rules (AGK) extended to QCDRelate diffraction, saturation and multiple scatterings All from the same amplitude, but different factors:

+1 Diffraction 4 Saturation+2 Multiple Interactions

Extended now also to pp !!!!further work needed ...

➔ HERA is the place to understand MI !!!!

➔ Towards the description of “everything” !!!!!

Bartels, Kowalski, SabioVera


Toy Model for AGKAbramovsky Gribov Kanchely cutting rules (Sov.J.Nucl.Phys. 18, 308 (1974))

AGK formulated before QCD... no treatment of color ...where is relation of diffraction – multiple scatterings – saturation coming from ?single parton exchange:

2parton exchange:


Saturation nonlinear evolution

f(x; k2) = f0(x; k2)



f(x; k2) = f0(x; k2) +K1 f



f(x; k2) = f0(x; k2) +K1 f ¡ 1

R2K2 f2

evolution equation including recombination effects:

GribovLevinRyskin equation (Phys.Rep. 100 1(1983))

BalitskyKovchegov equation (NPB 463, 99 (1996), PRD 60 (1999) 034008, D62 (2000) 074018)


Parton saturation

number of gluons in long. phase space :

occupation area: nr of gluons x (trans size)2

saturation starts when:

define saturation scale:

dx=x

xg(x; ¹2)dx=x= xg(x; ¹2)d logx

xg(x; ¹2)1

¹2

®s(¹2)

¹2xg(x; ¹2) ¸ ¼R2

Q2s(x) »

®s(Q2s)xg(x;Q

2s)

¼R2


Saturation scales

saturation scale depends on size of gluon density:

➔ large gluon ... large Qs

➔ small gluon ... small Qs

Q2s(x) »

®s(Q2s)xg(x;Q

2s)

¼R2J. Collins, ...M. Lublinsky in HERALHC proceedings 2006


Geometric Scaling

remember F2(x,Q2)strong scaling violations visible

Stasto et al PRL 86 (2001) 596


Geometric Scaling

Define new variable:

scaling observed form small x<0.01dependence on saturation scale

¿(x) =Q2

Q2s(x)

Stasto et al PRL 86 (2001) 596


Diffraction Saturation Multiple Interactions

Proton AntiProton

Multiple Parton Interactions

PT(hard)

Outgoing Parton

Outgoing Parton

Underlying Event Underlying Event

Outgoing Parton

Outgoing Parton

from R. Field

What is the underlying event (UE), multiple parton interactions (MI)?➔ Everything, except the LO process we're currently interested in

parton showersadditional remnant – remnant interactions

✗ NOT pileup events (luminosity dependent)


Underlying event – Multiple Interaction

p?min ! 01=p4?min

Basic partonic perturbative cross section

➔ diverges faster than as and exceeds eventually total inelastic (nondiffractive) cross section

¾hard(p2?min) =

Z

p2?min

d¾hard(p2?)dp2?

dp2?


Underlying event – Multiple Interaction

¾hard(p2?min) =

Z

p2?min

d¾hard(p2?)dp2?

dp2?

hni = ¾hard(p?min)¾nd

p?min ! 01=p4?min

Basic partonic perturbative cross section

➔ diverges faster than as and exceeds eventually total inelastic (nondiffractive) cross section, resulting in more than 1 interaction per event (multiple interactions, MI).

➔ Average number of interactions per event is given by:

It depends how soft interactions are treated, BUT also on the

parton densities and factorization scheme !!!!!!!!


Multiple Interactions at HERAJ. Turnau, L Lönnblad, HERALHC workshop 2006

multiple interactions also in DIS forward jets at large Q2 ?

forward jet production


Understanding diffraction

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Optical theorem - DESYjung/qcd_collider_physics_2005/lecture14.pdf · Page 1 H. Jung, QCD & Collider Physics, Lecture 14 WS 05/06 Optical theorem in optics: derived from conservation