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Dynamics of color in the soft and hard phenomena in hadron ...

**Dynamics** **of** **color** **in** **the** s**of**t **and** **hard** diffractive **phenomena** **in****hadron** -nucleus collisions .L.Frankfurt , TAU10th International Workshop IWHSS10,Venice March . 14-17, 2010

Outl**in**e●Fundamental properties **of** high energy processes **in** QCD.●●☛☛☛Fluctuations **of** strengths **of** **in**teraction **in** a projectile w.f. around averagevalue **in**elastic s**of**t diffractive processes . Experimental facts .QCD factorization **the**orems for **the** **in**teraction **of** spatially small **color**lessdipole **and** **hard** diffractive **phenomena** . Experimental facts.Specific properties **of** **hadron** states produced **in** diffractive processes.Measurement **of** novel GPDs **of** various **hadron**s**in** **hadron** **in**duced processes **of** nucleons(nuclei).Conclusions

Fundamental properties **of** high energy processes **in** QCD .●Asymptotic freedom: coupl**in**g constant is small for **the** small space-time **in**tervals.α s (Q 2 ) = (4π/b)/ ln(Q 2 /λ 2 QCD)b = 11N c3− 2n f3●At moderately large energies **in**clud**in**g **the** ones probed at HERA **the** **in**teraction **of** **color**neutral, spatially small quark dipole with a **hadron**(nuclear) target T is unambiguously calculable**in** QCD =QCD factorization **the**orems.QCD factorization **the**orem for **the** **in**teraction **of** small size **color** s**in**glet wavepackage **of** quarks **and** gluons.[σ(d, x) = π23 α s(Q 2 eff )d 2 xG N (x, Q 2 eff )+ 2 ]3 xS N (x, Q 2 eff )Q 2 eff = λ/d 2 , λ =4÷ 10 Baym, Blattel, LF, MS, 93, LF,Miller, MS 93

Coherence phenomenon **in** **the** s**of**t **in**elastic diffraction.t m**in****in** **the** **in**elastic diffraction:h + T → a + rapidity gap + Trapidly decreases with energy.h≠aThe amplitudes characteriz**in**g contributions **of** states “a” are coherent.Equivalent statement is that coherence length is rapidly **in**creas**in**g with energy:L c =2E h /(M 2 a − M 2 h)- uncerta**in**ty pr**in**ciple⇓Quark-gluon configurations **in** **the** wavefunctions **of** energetic **hadron**s are frozen.

S**of**t **in**elastic diffraction at =0 **in** quantum mechanics|h〉 = a 1 |1〉 + a 2 |2〉habsorber withsame absorptionfor “1” **and** “2”|f**in**al〉 = λ(a 1 |1〉 + a 2 |2〉) =λ |h〉honly elastic scatter**in**g|h〉 = a 1 |1〉 + a 2 |2〉habsorber withdifferent absorptionfor “1” **and** “2”|f**in**al〉 = λ 1 a 1 |1〉 + 2 a 2 |2〉)h+h’= c 1 |h〉 + c 2 |h ′ 〉elastic scatter**in**g+**in**elastic diffraction

Due to a slow space-time evolution **of** **the** projectile wave function one can treat **the** **in**teraction asa superposition **of** **in**teraction **of** configurations **of** different strengths . This was understood byPomeranchuk & Fe**in**berg(QED); Good **and** Walker; Pumpl**in** &Miett**in**en (preQCD). In QCDdifferent strengths are mostly due to different sizes **of** different configurations.Size/orientation **of** **the** **color** distribution with**in** a wave function **of** sufficiently energetic **hadron**varies lead**in**g to **the** fluctuations **of** **the** strengths **of** its **in**teraction with a target . Illustrations:pNN = 3q + 3qg + 3q+ ! + ...●●●●rtrvs● ●rtrDescription **of** processes **in** terms **of** fluctuations **of** **color** is reasonable for total cross sections**and** for **the** **in**elastic diffraction at very small t.6

The coherence **of** contributions **of** different configurations **in** **the** projectile wave function Ph(σ) -probability that projectile **in**teracts with target with cross section σ:∫= P (σ)σ n dσ∫∫P (σ)dσ =1Total cross section :σ tot =< σ >=P (σ)σdσSecond moment is fixed by Miettenen- Pumpl**in** relation (1978):w σ = − < σ > 2< σ > 2➠S**of**t **in**elastic diffraction arises due to fluctuations close to average cross section:No fluctuations = no **in**elastic diffraction!!Fur**the**r constra**in**ts (Baym, Blattel, Frankfurt, Strikman (BBFS) 93)The form **of** Ph(σ) at small σ follows from QCD :P h (σ) σ→0 ∝ σ n q−1Here nq is **the** number **of** valence quarks with**in** **the** projectile **hadron**.Sum rule for for p 2 H diffraction

Cross-section probability for pions Pπ(σ) **and** nucleons PN(σ) as extracted from experimentaldata. Pπ(σ=0) is compared with **the** perturbative QCD prediction (BBFS93).

If **the**re were no fluctuations **of** strength - **the**re will be no **in**elastic diffraction at t=0:dσ(pp→X+p)dtdσ(pp→p+p)dt| t = 0=! tot [mb]!#" !Tevatron! tot!#" !Tevatron! tot! tot! totLHC∫(σ − σtot ) 2 150 P (σ)dσ 150LHC! tot [mb]σ 2 tot10050100≡ ω σTevatronvarianceLHC! tot0.40.30.20.1" !0010 2 10 3 10 4 10 5 0, i.e., **the** average area repre-10 2 LHC# s [GeV] 10 3 10 4 10 5 0Size **of** both small **and**Thelargearea **of**configurations **the** **in**ner **and** outer disk at given energy is proportional to(a)sents **the** average cross section (b) Tevatrongrows with energy - still **the**re is a correlation betweentot, **the** difference (r**in**g) **the** range **of** **the** fluctuations . (b) Theσ **and** parton distributions -smaller σ, **hard**er quark distribution.–dependence **of** **the** total cross section tot (left –axis) **and** **the** dispersion (right –axis), as predicted by a(a)!#" !TevatronFig. 3: (a) Graphical representation **of** **the** cross section distributions **in** diffraction# at s **the** [GeV] Tevatron **and** LHC energy.slow variation **of** **the** diffractive cross section with energy.The area **of** **the** **in**ner **and** outer disk at given energy is proportional to! tot! totLHC! tot [mb](b)500010 2 10 3 10 4 10 5Tevatron" !Fig. 3: (a) Graphical representation **of** **the** cross section distributions **in** diffraction at **the** Tevatron **and** LH(b)15010050(a), i.e., **the** average area represents**the** average cross section tot, **the** difference 9 (r**in**g) **the** range **of** **the** fluctuations . (b) TheThe area **of** **the** **in**ner **and** outer disk at given energy is proportional to! tot [mb](b)15010050Tevatron# s [GeV]Fig. 3: (a) Graphical representation **of** **the** cross section distributions **in** diffraction at **the** Tevatron **and** LHC energy.0010 2 10 3 10 4 10 5Regge–based parametrization **of** tot [10] **and** a parametrization **of** **the** **in**elastic diffractive cross section **in**el ,measured up to **the** Tevatron energy [9]. The weak energy dependence **of** **the** width **of** **the** r**in**g **in** figure (a) reflects **the**# s [GeV]! tot" !order–**of**–magnitude **of** **the** effect, as well as its energy dependence. Our basic assumption is thatLHC! tot" !0.40.30.40.30.20.20.10.1" !" !LHC! tot" !0.40.30.20.1, i.e., **the** average a

→→√ s = 14 T eVP(σ)[ mb −1 ]0.020.0150.01√0.020.02s = 30 GeV0.0150.015→0.01 0.01√ s = 2 TeV0.020.0150.010.020.0150.010.0050.005 0.0050.0050.00500 00 00 10 20 30 0 40 01050 102060 203070 304080 0 40 5010 90 50 60 100 020 60 70 10 110 30 70 80 12040 80 130 90 50 100 90 60 40 100 110 70 50 120 80 110 60 130 90 120 701σ[ mb ]The 30 GeV curve is result **of** **the** analysis (Baym et al 93) **of** **the** FNAL diffractive pp **and** pd datawhich expla**in**s FNAL diffractive pA data (LF, Miller, Strikman 93-97). The 14 **and** 2TeV curves areguesses based on match**in**g with fixed target data **and** collider diffractive data.10

Critical test - coherent diffraction **of**f nucleiσ diff (A) =∫d 2 B[2 ] −[∫] ] 2dσP (σ) [ ∫dσP (σ) ∑ n∫ ∞F (σ, B) = 1 − e − 1 2σT (B)T (B) =−∞ρ A (B, Z)dZHere **the** direction **of** **the** beam is Ẑ **and** **the** distance between**the** projectile **and** **the** nuclear center is ⃗R = B ⃗ + ZẐM =∫d 2 Be i⃗q t· ⃗B 〈h | F (σ, B) | X〉dσ diffdt∫(0 ◦ )=π dσP (σ)f 2 (σ) − π∫ (1 f(σ) ≡ BdB − e − σ 2 T (B))[∫dσP (σ)f(σ)] 2

Fluctuations near average strength dom**in**ate for all nuclei for total cross section **of** diffraction ( asignificant enhancement **of** smaller size contribution for A ~ 200) - need special triggers to look forconfigurations with σ Full expression.σ apprdiff = ω σ〈σ〉 24∫d 2 BT 2 (B)e−〈σ〉T (B)L.F,J.Miller,M.S PRL 93

The cross section **of** coherent diffraction dissociation**of** protons **and** neutrons on nuclei as a function **of** A. The solidl**in**es are **the** **the**oretical prediction based on **the** above eqn.Total cross section data are from **the** FNAL emulsion **and** 4 He jettarget experiments. The FNAL data on **the** reactionn+A → p π - +A (a small fraction **of** **the** total diffractive crosssection) is presented as stars have similar A-dependence for allmasses **and** provide a good **in**dication **of** **the** overall trend **of** **the** A-dependence. The **the**oretical prediction for coherent diffraction on4He is given by **the** dashed l**in**es.A- dependence for exclusivechannel is reproducedσ(A) ∝ A n ; n(A = 16) ≈ 0.8, n(A = 200) ≈ 0.4 for s~ 400 GeV 2LHC n=0.27 **and** e.m. contribution dom**in**ates Guzey & Strikman 08

A- dependence for exclusivechannel is reproducedσ(A) ∝ A n ; n(A = 16) ≈ 1.05, n(A = 200) ≈ .65

Regime **of** complete absorptionIf projectile is absorbed with 100/% probability amplitudes **of** diffractive processes are proportional to **the**overlapp**in**g **in**tegral between wave functions **of** **in**itial **and** f**in**al states. For **the** forward scatter**in**g **the**overlapp**in**g **in**tegral differs from 0 if **in**itial **and** f**in**al states co**in**cide. Thus elastic scatter**in**g **of**f target T**of** radius RT with **the** amplitude A= s(2 π RA 2 ) is allowed only. This physics is analogue **of** **the**Fraunh**of**er diffraction **of**f black screen **in** optics.If **color** filter**in**g is neglected **hadron** scatter**in**g **of**f heavy nuclei would be close to black disc regime forcentral impact parameters ➙ **in**elastic diffraction from **the** projectile scatter**in**g**of**f nuclear edge ➙ σdiff ∝ A 1/3 reached at **the** LHC

$.* 7'#$0"/ !.%$%&3B& $.* /*"5'&+ /%+"#'$.6'( "!!#%A'6"$'%& '&, $.* *22*($) %2 IJ@ #"5'"$'%& ("&"+"'& 1* "1)%#1*5 '& $.* "6!/'$05* 2%# $.* )("$$*#'&+ %2 $.* )6"//=)'

H1 **and** ZEUS observed processes **of** diffractive electroproduction **of** vector mesons.γ ∗ + p → V + pV=ω,ρ,φ,J/ψγ ∗ + p → J/ψ + rap gap + XPractically all regularities predicted by QCD factorization **the**orems **and** DGLAPapproximation were observed at HERA

HERA data confirm **in**crease **of** **the** cross sections **of** small dipoles predicted by pQCD2(d , x, Q ) (mb)qqN! −454035Hard 30Regime252015105""Match**in**g Region= 4 x = 0.0001= 10x = 0.001x = 0.01S**of**tRegime000.10.20.30.40.5d (fm)0.60.70.80.91The **in**teraction cross-section, ˆσ for CTEQ4L, x = 0.01, 0.001, 0.0001,λ =4, 10. Based on pQCD expression for ˆσ at small d t , s**of**t dynamics atlarge b, **and** smooth **in**terpolation. Provides a good description **of** F 2p atHERA **and** J/ψ photoproduction.Frankfurt, Guzey, McDermott, MS 2000-200118Provided a reasonable prediction for σL

Motivations for **the** **hard** exclusive **hadron** **in**duced processes with nucleons **and** nuclei✴Small size **color**less wave package **of** quarks **and** gluons weakly **in**teracts with **hadron**(nucleus) target**in** a wide **in**terval **of** energies. Follows from gauge **in**variance **and** asymptotic freedom. As a resultcross section for **the** **in**teraction **of** this wave package is unambigously calculable **in** QCD **in** a widek**in**ematical range. This QCD factorization **the**orem allows to evaluate cross sections **of** variety **of****hard** diffractive processes observed at FNAL, HERA, TJNAF.✴Ano**the**r important feature is that gluon dipole - N cross section is 9/4 times larger than quarkdipole - N cross section-important for evaluation **of** BDR at LHC.✴Analysis **of** **hard** diffractive processes allows to extract variety **of** GPDs which conta**in****in**formation on **hadron**(nucleus) quark-gluon structure.✴Go**in**g beyond one dimensional image **of** nucleon - GPDs & correlations **in** **the** wave functions **of**baryons **and** mesons✴To **in**vestigate **the** multiparton structure **of** **hadron**s **and** how it is different for mesons **and** baryons

Discovery **of** high energy CT=⇒ Need to trigger on small size configurations at high energies.Two ideas:⋄ Select special f**in**al states:diffraction **of** pion **in**to two high transversemomentum jets - an analog **of** **the** positronium **in**elastic diffraction. Qualitatively- from **the** uncerta**in**ty relation d ∼ 1/p t (jet)⋄ ⋄ Select a small **in**itial state - diffraction **of** longitud**in**ally polarized virtualphoton **in**to mesons. Employs **the** decrease **of** **the** transverse separation betweenq **and** ¯q **in** **the** wave function **of** γL ∗, d ∝ 1/Q.QCD factorization for **the**se states **in**cludes CT .20

π + N(A) → “2 high p t jets ′′ + N(A)Mechanism:Pion approaches **the** target **in** a frozen small size q¯q configuration**and** scatters elastically via **in**teraction with G target (x, Q 2 ).**the** first analysis for πp scatter**in**g R**and**a(80), nuclear effects - Bertsch, Brodsky,Goldhaber, Gunion (81), pQCD treatment: Frankfurt, Miller, MS (93)❖❖First attempt **of** **the** **the**oretical analysis **of** πN process - R**and**a 80 - power law!qq(1-z)PzP!!, kdependence **of** pt **of** **the** jet (wrong power)First attempt **of** **the** **the**oretical analysis **of** πA process - Brodsky et al 81 -exponential suppression **of** pt spectra, weak A dependence (A 1/3 )❖ pQCD factorization A(N) **the**orem - Frankfurt, A(N) Miller, MS 93; elaborated argumentsrelated to factorization 2003A(π + N → 2 jets + N)(z, p t , t = 0) ∝∫d 2 dψ q¯qπ (z, d)σ ”q¯q”−N(A) (d, s) exp(ik t · d),t-ktd = r q t − r ¯q t ,ψ q¯qπ (z, d) ∝ z(1 − z) d→0 is **the** light-cone q¯q pion wave function.M.Strikman21

pion wave function is equal to **the** asymptotic change one. **in**identities **the** wave **and**where **the** function condition isx **of** 1 ,x **the** **the** 2 light f**in**alcone state. fraction Here M**of** **in**t **the** pion momentumvertex for **the** three gluon **in**teraction, **the** **color** neutrality **of**2Thus **the** ratio is determ**in**ed by **the** **color** content (m **of** **color** carried by an exchangedπ + N(A) → “2 high p t jets ′′ gluon: k+ N(A) 1 /p k 2 /p rec l 2 t . A factor **of** t l t ispresent t )/(1z) **in** **the** numerator, is **the** with mass 2 t orig**in**at**in**g **of** **in**termediate from **the** vertices**in** **the** WW representation **and** lstate, .**the** pion wave function, **and** **the** dijet f**in**al state. In **the** derivationit is helpful to use **the** observation that effectively2flow **in** **the** pion wave function **and** **the** quark**and** **color** m rec **and**isby**the** **in**variant Thus we mass arrive **of** **the** 2 t from **the** **in**tegrationover quark momenta **in** **the** pion wave atfunction. **the** recoil equation Allsystem **in** all this **in** **the****the** dependence **of** energy denom**in**ators on **the** fraction **of**x 2 2 Feynman amplitude mechanism. suppressed In **the** by **the** region factor l 2 t **of** /( 2 t ) **in**tegration 3 . Ano**the**r casetribute is shown.1zt /. Evidently similar reason**in**g is applicable **in** com-l 2 22t /Moccurs2 jet onewhenmayl 2 neglect by M 2 t 2 t . Then this diagram(2 jet)will**in** be suppressed**the** denom**in**atoraspion momentum carried by quarks **and** gluons. So Pion approaches **the** target **in** a frozen m˜ 2kput**in**g amplitudes to lead**in**g order **in** 1t small size q¯q configurations **and** all ordersMechanism:**in**compared to T 1 atas compared to l x**and** scatters 2 least by one power **of** s without **the** largeelastically via **in**teraction 1 1 with .k 1 zp 2 m 2G target (x, Q 26q zx 1 •••, 33asymptotic freedom. This is nontrivial s**in**ce **the**se factor ln 2 2t /(1z). properties So one obta**in**s are T **of**ten violated 22 **in** **the** literature.t / QCD . But here we restrict ourselves to **the**while that **the** near-mass-shell ). l**in**e b is **in**dependent **of****the** first analysis for πpdiagramsscatter**in**g Fig. 6 is suppressedR**and**a(80),by a factornuclearl 2 t / 2 t as com-**the** to Feynman thatall **in** Fig.effects - Bertsch, Brodsky,found thatpared produced mechanism 1. This **the**particlespower-type is a higher **in**suppression**the** twist **in**termediateifcorrec-states that 0Goldhaber, Gunion (81), **the** pionpQCD wave function treatment: is non-perturbative, Frankfurt, **and** may beMiller, a MS (93) s ln 2 t /l 2 t .T 2 F2 38T 1 F 2 1 1z1z z (1/ 2Repeat**in**g **the** same reason**in**g as **in** **the** estimate **of** **the**1z 2 ln z t ) analysis (z,l **of** 2 LO t )d 2 corrections. l t dz. In this case, ano**the**r factor **of**terms **of** TIt follows from **the** requirement **of** positivity **of** energies **of**1a ,T 1b , **and** remember**in**g that x 2 2 1/ 4 Similar reason**in**g helps to prove that **the** contribution **of**t / we t arises from **the** **in**tegration over z. Hence we haveachieve **the** estimate T 3 2 s x 1 G A 1z (x 1 ,x 2 ,zln1z 2t )/ 4 t . It is **in**structiveto **in**vestigate whe**the**r **the** Feynman mechanism,tion to **the** PQCD contribution. The Feynman mechanism 2 . 24 1. We can now calculate m˜ 2 isNLO s correction if **the** perturbative high momentum taildirectly is **in** terms **of** **the** lightfur**the**r suppressed **in**cluded **in** **the** bypion **the**wave requirement function. **of** a lack **of** coll**in**earwhere **the** lead**in**g quark anti-quark carries a fraction **of** **the**cone momenta **of** (1-z)P **the** qq¯ , kto pion momentum Ano**the**r contributionq radiation—see to T 4 arises ! pair **in** **the**t**in**termediate state:Here F 2 (i) for i8,3 is **the** Casimir operator for octet ! **and**Dom**in**ant triplet representations diagram2**the**from discussion **the** sum **of** Feynm**and**iagrams **in** which **the** gluon exchange between **the** qbelow.pion momentum z close to 1 but high momentum jets are**of** **color** group SU(3)zPc . The ratio!qm˜ l t 2T 2 /T 1 is 0.5 for z1/2, rema**in**s nearly constant for zz k 1tl t 2formed by **the** action **of** a f**in**al state **in**teraction, may compete**and** q¯ **in** **the** beam occurs dur**in**g **the** **in**teraction with **the**D. Gluontarget, admixture see Figs. to8, **the** 9, **and** wave 10. The functions naive expectation **of** **in**itialis**and** that f**in**alwith **the** PQCD description. In this case transverse momenta1k 21z 1t . 27such terms, which amount to hav**in**g a gluon exchanged dur**in**g**the** very short **in**teraction time 4 characteristic **of** **the** twostates—T**of** constituents l t **in** **the** pion wave function are expected to.50.3 **and** **in**creases to 9/8 at z0,1. This term is addi-regime. suppressed For certa**in**tybylet**the** us model Sudakov-type **the** Feyn-form**the** factor time**and** order**in**g bygluon exchange process occurr**in**g at high energies, must be -kbe equal to **the** mean transverse momenta **of** partons **in** **the** The Feynman diagram Comb**in****in**g correspond**in**g Eqs. 25,27 to Fig. we7obta**in**conta**in**stvery small **in**deed.non-perturbativetionallyThe **in**tent correspond**in**g **of** this section is to use **the** **the**qq¯ analytic g configuration propertiestor factor for l**in**e a is given by**in**man mechanism**the** byform assum**in**g factorthat w 2 —see recoil**the** system discussion quark below.**of** **the** scatter**in**g amplitude to show that2A(N)lT**the** pion wave function **in**teract**in**g with t A(N)z **the** k 1tl t 24 is negligible.quarks **in**anti-quark with momentum 1z close to 0. With**in** this1z x **the**Instead **of** calculat**in**g **the** sum **of** **the** imag**in**ary parts **of** all **of**f**in**al state. 1,28**the** Inamplitudes, tak**in**g **the** we will imag**in**ary prove that part this sum **of**vanishes **the** amplitude, by analyz**in**g**the**model we will obta**in** FeynmanC. F**in**aldiagramsstate **in**teractionfor **the** term**of**T**the** 2 , butqq¯ pair—T **in**termediate 3state analytic must properties conta**in** **of** **the** a **hard** important on-shell diagrams. quark Each **and** while a that **of** l**in**e c is given bywith **the** region **of** **in**tegration def**in**ed by **the** Feynmanconsidered diagram conta**in**s a product **of** **in**termediatestatequark**hard** on-shell gluon.**and** which, Butanti-quarksuch when propagator.a state us**in**g cannotAt Eq. high energies, 5, be produced leads **the**se to bymechanism. A simple The dimensional **in**teractionevaluation with **the** **of** target termgluons T may occur before**the** **in**teraction between quarks∫ 3 due a s**of**t almost propagators on-shell are controlled quark **in**by **the** **in**itial terms **of**state, highest so power **the**re **of** is anto **the** Feynman mechanism with**in** **the** Gribov representation **in** d**the** 2 f**in**aldψπ q¯q state, **and** **the** related(z, d)σ additional x”q¯q”−N(A)suppression (d, s) factor, exp(ik caused t · d),Examplesamplitudes are denoted **the**as T 3 , seesuppressedFigs. 5 **and** 6.diagramsx 1 1 by l **the**t 21 2p•p x 1 , **and** as to be shown haverapid d = r q t − r ¯q t , ψ q¯q z k poles1tl decrease **in** **the**t 2**of** m 2complex x 2 jetshows that it is suppressed by **the** powers **of** t . The contribution**of** **the** region l 2 tour **of** **in**tegration. The sign **of** **the** term conta**in****in**g () 2**the** non-perturbative1 plane which are located on one side **of** **the** con-pion wave function with **in**creas**in**g x1z 2 . 29t /(1z)M 2 jet has been consideredeach propagator unambiguously follows from **the** directionsabove—it is additionally suppressed for **the** Feynman mechanismby **the** restriction **of** **the** region **of** **in**tegration over z.**in**tegral is **of**Therefore**the** formThus our next discussion is restricted by **the** consideration **of****the** contribution **of** **the** region, l 2 t /(1z)M 2 jet :T 3 1 t2 z,l t 2 1l t 22A(π + N → 2 jets + N)(z, p t , t = 0) ∝π (z, **of**d) pion∝ **and**z(1 target− momenta. z) d→0 If we iscan **the** showlight-cone that **the** typical q¯q pion wave function.x 2 1 l t 2 z k 1tl t 21z tz1z 2.M.Strikman30In order for **the** term T 3a to compete with T 1a we need tohave l t t , k 1t t —o**the**rwise T 3a will be additionallysuppressed by **the** power **of** 2 t , s . These k**in**ematics causeEq. 30 to yield **the** result x 2 2 t shown. /.This argument can be carried out for all comb**in**ations **of**094015-13diagrams represented by Fig. 5. For example, ano**the**r attachment**of** gluons, **in** which **the** gluon k 1 is absorbed by **the**quark, corresponds to **in**terchang**in**g z with 1z, **and** **the**re-M 2 **the** pro**of** would be complete.2L. FRANKFURT, **in**t G. M A. MILLER, We now consider **the** Feynman graphs, start**in**g with Fig.2 jet AND M. STRIKMAN PHYSICAL REVIEW D 65 0940159. Once aga**in** we compute **the** imag**in**ary part **of** **the** graph FIG. 8. A contribution to T 4b . The target gluon absorbs a gluon**of** pion wave function. Only one diagram **of** **the** eight that occur is1z d2 l t dz.311dx 1x 1 aix 1 bi ,,0 32FIG. 7. A time order**in**g that contributes to T 4 . The qq¯ g state**in**teracts with **the** target. Only a s**in**gle diagram **of** **the** eight where agluon **in**teracts with quarks **in** a pion fragmentation region that con-Calculation accounts for energy-momentum conservation, gauge **in**variance, QCD evolution **and**FIG. 4. Contribution to T 2b from **the** qq¯ g **in**termediate FIG. 6. state. **and** Contribution consider **the**to **in**termediate T 3b . A gluon state asfrom be**in**g**the** on **the** two-gluon energy fieldIn **the** aboveforeformulasleadsThe **in**teractiontowe**the** sameuse **of** **the**resultone Brodsky-Lepage targetfor x 2 .gluonEvidentlyfieldconven-tion for **the** 2 is valid **in** **the** lead**in**g thiswithresultan exchangedfor**of** **the** target gluonshell.**in**teracts The propagatorwith **the**forhigh**the** l**in**emomentuma has **the** factorcomponent **of** **the**xdef**in**ition **the** **in**termediate **of** wavestates. functions s ln There 2 2t /**and** QCD approximation also.is also reta**in**a diagram terms **in** which f**in**al qq¯ **the**pair gluons wave function. Only a s**in**gle diagram **of** **the** eight thatThus we consider **the** second situation: lmaximally s**in**gular fromwhen **the** target z→1. arePower crossed, count**in**g **and** 2 t kano**the**r is 2 1t simple: group 2 t . In this**in**contribute which **the**is exchangedshown.case, **the** **in**itial pion wave function conta**in**s a **hard** quark,**and** we discussgluon**hard** radiativeis emittedcorrectionby **the****in**anti-quark.**the** next orderOnly**of**one **of** 16 diagramsthat contribute is shown.s . This is **the** typical situation **in** which **the**re are extra 094015-12 **hard**l**in**es, as compared with **the** dom**in**ant terms, **and** one obta**in**sa suppression factor 1/ 2 t which could be compensated by 094015-11**the** d 2 k t **in**tegral. However, this **in**tegral does not produce2ln 2 t / QCD because **the** region **of** **in**tegration is too narrow.So this contribution is at most **the** non-lead**in**g-order NLOcorrection over s . But we restrict ourselves by **the** lead**in**gorderLO contribution only.FIG. 5. Contribution to T 3a . The high momentum component **of****the** f**in**al qq¯ pair **in**teracts with **the** two-gluon field **of** **the** target.Only a s**in**gle diagram **of** **the** eight that contribute is shown.**the** factor l t 2 /(1z) t 2 is from **the** gluon exchange **in** **the**222x 1 , because **the** quark momenta **in** **the** f**in**al state **and** **in** **the**pion wave function are not connected with **the** target momentum.The propagator **of** l**in**e c has **the** factork 2 q 1 2 m q 2 x 2 z•••x 1 z•••.34Here q 1 is **the** momentum **of** **the** jet (z, t ) **and** ••• denotes**the** terms which are **in**dependent **of** x 1 . The last equation isobta**in**ed from us**in**g Eqs. 5,7. The results 33,34 showthat **the** diagram **of** Fig. 9 takes on **the** ma**the**matical form **of****the** **in**tegral 32. Thus this term vanishes.We also consider **the** diagram **of** Fig. 10. In this case **the**reare three propagators a,b,c that have a term proportionalto x 1 , but **the** coefficients are not all positive. The propaga-x 1 p k 1 2 x 1 z•••,35k 2 q 2 2 m 2 q x 2 1z•••1zx 1 •••.36At **the** same time, **the** coefficient multiply**in**g x 1 **in** **the** propagatorb gluon production has no def**in**ite sign. Thus for

The f**in**al answer isA(π + N → 2 jets + N)(z, p t ,t= 0) ∝∫d 2 dψ q¯qπ σ q¯q−N(A) (d, s) exp(ip t d)d = r q t − r ¯q t ,ψ q¯qπ (z, d) ∝ z(1 − z) d→0is **the** quark-antiquark Fock component **of** **the**meson light cone wave functionPlane wave **in** **the** f**in**al state - faster onset **of** scal**in**g than for VM productionProportionality **of** **the** amplitude to **the** gluon nuclear density (actually GPD) which for small x isnot l**in**ear **in** A - change **of** mean**in**g **of** CT for high energies. Dependence on t is **the** same as fordiffractive vector meson production at HERA. Gluon distribution **in** impact parameter space isnoticeably more narrow than valence quark distribution.23

Predictions A-dependence: A 4/3 [GA (x, k 2 t )AG N (x, k 2 t )] 2where x = M 2 dijet/sdσ(u)du ∝ φ2 π(z) ≈ u 2 (1 − u) 2 where u = E jet1 /E πkt dependencedσd 2 ∝ 1k t ktn,n≈ 8 for x ~ 0.02Absolute cross section is also predicted

E-791 (FNAL) experiment (PRL 2001) at E**in**c π =500 GeVFirst experimental observation **of** high energy CT for pion **in**teraction (Ashery 2001):π +A →”jet”+”jet” +A. Confirmed predictions **of** pQCD (LF,Miller, Strikman93) forA-dependence, distribution over energy fraction, u carried by one jet, dependence on pt(jet), etc♥Coherent peak is well resolved♥♥ Observed A-dependence A 1.61±0.08 [C → P t]FMS prediction A 1.54 [C → P t] for large k t & extra smallenhancement for **in**termediate k t .For s**of**t diffraction **the** Pt/C ratio is ∼ 7 times smaller!!(An early prediction Bertsch, Brodsky, Goldhaber, Gunion 81σ(A) ∝ A 1/3 )In s**of**t diffraction **color** fluctuations are also important lead**in**g toσ s**of**t diffr (π + A → X + A) ∝ A .7discussed **in** **the** 1stpart **of** **the** talkMiller Frankfurt &S, 9325

dσdu ∝ φ2 π (u, Q2 ) = 36u 2 (1 − u) 2 1.0 + a 2 C 3/22 (2u − 1) + a 4 C 3/224 (2u − 1)prediction(π wave funct) 2Q 2 (π f.f.) ∼ 4k 2 t (jet)Squeez**in**g occurs already before **the**k t :lead**in**g term (1-u)u dom**in**ates!!!a 2 = a 4 = 0 → Asymptoticstrong squeez**in**g **in** π form factor forQ 2 =6 GeV 2t : a 2 = 0.30 ± 0.05, a 4 = (0.5 ± 0.1) · 10 −2 → TransitionNote that enhancement **of** u ➛(0,1) tail for moderate kt may expla**in** **the** observed by BaBarurs already before **the** lead**in**g term (1-z)z dom**in**atecollaboration slow decrease with Q **of** **the** form factor for **the** process: γ*(Q^2)+γ➙π 0.2616

♥♥♥♥ k −nt dependence **of** dσ/dk 2 t ∝ 1/k 7.5tfor k t ≥ 1.7GeV/c close to **the**QCD prediction - n ∼ 8.0 for **the** k**in**ematics **of** E971or higher terms **in**Gegenbauer expansion???CT is easier to probe for mesons than for baryons as only two quarks have to come closeMeson is not as much **of** a rope (camel) as a baryon **and** can be easier put through a needleDijet production with quantum numbers different from that for projectile can be run by COMPASSus**in**g hydrogen at a couple **of** energies to measure decrease **of** \alpha'R for GPDs with **in**crease **of**kt as expected **in** QCD.M.StrikmanCT has been observed also **in** **the** diffractive photoproduction **of** J/ψ at FNAL **and** smallereffects **in** **the** exclusive electroproduction **of** pions **of**f nuclei at TJNAF.

Implications for fixed target physics.Wide distribution over **the** sizes **in** **the** wave function **of** energetic **hadron**s **and** **in** **the**strength **of** **in**teractions with a target. Nonzero probability **of** small size quark-gluonconfigurations has been established experimentally **in** **color** transparency **phenomena** ➙existence **of** “superstrong” strong **in**teraction follows from **the** probability conservation.Post-selection. To vary strengths **of** **hadron**-**hadron** (nucleus) _ **in**teractions by trigger**in**g forcerta**in** **hadron**ic f**in**al states. Example: selection **of** q q g **hadron**ic state with quantumnumbers 1 -+ enhances **the** contribution **of** superstrong strong **in**teraction. Experimentalsignature would be slower dependence on atomic number.Prob**in**g QCD **in** **the** **hard** diffractive **phenomena** where strong **in**teraction becomesweak-**color** coherence **and** **color** screen**in**g **phenomena**. Effective methods **of**measur**in**g **of** generalized parton distributions **in** nucleons **and** nuclei.

Inelastic diffraction is promis**in**g laboratory for **the** hunt for new **hadron**ic statesIt is impossible to kill **in**f**in**ite number **of** **color** multipoles without **in**f**in**ite number**of** degrees **of** freedom. So constituent quark models **of** a **hadron** are a oversimplification.How to hunt for **the** **hadron**s which can not be described **in** terms **of** quark models.● Inelastic forward diffraction is due to **color** fluctuations i.e. arises from non averagequark-gluon configurations **in** **the** projectile w.f. Larger **and** smaller configurations areenhanced ➜ production **of** states with nodes is enhanced, quark-gluon structure **of** **hadron**icstates produced **in** **the** **in**elastic diffraction should differ from **hadron**ic states typical for quarkmodels **of** a **hadron**.●Wave function **of** diffractively produced state has two but not 3 dimensional rotationalsymmetry s**in**ce configurations **of** different transverse size **in**teract differently. Production **of**states with different angular momenta are allowed for **the** forward scatter**in**g. All diffractivestates should have helicity 0, so **the**y are aligned along projectile momentum direction.● A-dependence **of** resonance production measures average strength **of** **the**ir **in**teraction.σ apprdiff = ω σ〈σ〉 24∫d 2 BT 2 −〈σ〉T (B)(B)e

Vary**in**g transverse momenta **of** produced **hadron**s ➜postselects size **of** projectileChange **of** dependence on atomic number would allow to establish transition from superstrongstrong **in**teraction (larger than average size **of** projectile -slower dependence on atomic number )to CT regime (small size **of** projectile faster dependence on atomic number).Uncerta**in**ty relation based guess - very small kt - trigger preferentially on larger thanaverage size configurations; large kt on small size configurations. Study us**in**g rescal**in**g **of** ktfor fixed longitud**in**al fractions. Examples: p →BM (ΛK + ,...), π→3πp B pBMM./"$0!σ(k t ) ∝ A n(k t),)-+12%. 345!/36,++$&'($0+#$!!$%$&'()*" !

Factorization:First quantitative studies: Kumano, MS, **and**Sudoh PRD 09; Kumano &MS arXiv:0909.1299,bt’dbt’dPhys.Lett. 2010GPDc (meson)GPDc (baryon)Ne (baryon)Ne (meson)ttIf **the** upper block is a **hard** (2 →2 ) process, “b”,“d”,“c” are **in** small size configurations as well asexchange system (qq, qqq). Can use CT argument **of** **the** pro**of** **of** QCD factorization **of** meson exclusiveproduction **in** DIS (Coll**in**s, LF, Strikman 97)⇓M NN→NπB = GP D(N → B) ⊗ ψ i b ⊗ H ⊗ ψ d ⊗ ψ c31

Best bet for test**in**g **the**se ideas **in** **the** near future:π − A → π − π + (π − )A ∗●easier to squeeze● Significant rates - observed at FNAL for A=pdlcoh ~60 fm●freez**in**g is 100% effective for p**in**c > 100 GeV/c -bAc●Hence a high sensitivity to **the** size (σcluster- N)15 mb10 mbT (A)0.115 mb20 mbσ eff= 25 mbIf two scales **in** pion w.f. (one**of** explanations **of** γγ*→π 0rates) steps **in** T(kt π )●0.0310 20 50 100 200 300COMPASS has data on tapeAObservation **of** CT implies that GPDs can be measured **in** **hadron** **in**duced processes forpions, kaons, baryons **in** **the** k**in**ematics complementary to DIS studies **of** nucleon GPDs

Many o**the**r **in**terest**in**g channels to explore multiparton structure **of** nucleon **and** mesonsqqP-t’P,Δ, N*-t’/s’~1/2πPπP, Δ, N* Λ,ΣGPDN!qqqP,Δ, N*Pqq!, ρ,η, ϕK,K*(N→M)PMPN, Δ, N*-t=constpp → pN + M(π, η, ππ)pp → p∆ + M(π, η, ππ)pp → pΛ + K +π − p → pπ + MGPD (N→B)π − p → π − π − ∆ ++ ,π − p → π − π + ∆ 0 ,π − p → π − π 0 p,π − p → π − p +(π 0 π 0 − forward low p t )33

ConclusionsData on **in**elastic s**of**t diffraction observed at fixed target at FNAL prefer **color**fluctuations around average value. Open way for search for **the** superstrong strong**in**teraction , **of** resonances beyond constituent quark models.Evidence for onset **of** CT **in** **the** diffractive pion production **of** double jets at FNAL, **in** J/\psi diffractivemeson production at FNAL , **in** exclusive meson electroproduction demonstrates fruitfulness **of** **the**concept **of** spatially small dipole. Observation **of** CT **in** reactions with pions (COMPASS),antiprotons/protons would allow studies **of** Generalized Parton Distributions **of** various **hadron**s **in****hadron**ic **in**teractionsHav**in**g few h**and**les **in** **the** diffractive production **of** few **hadron**s by protons **and** pions will allow to**in**vestigate transitions between regimes **of** superstrong strong **in**teraction **and** squiz**in**g **of** **hadron**s**and** CT regime, dynamics **of** QCD **in**teractions at **the** **in**terface between **hard** **and** s**of**t QCD,explore **the** quark-gluon structure **of** various mesons **and** baryons.