Probing photon structure in DVCS on a photon target

0.12 MB·English

by  M. El Beiyad

9 Probing photon structure in DVCS on a photon target
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M. El Beiyada,b, S. Friotc, B.Pirea, L. Szymanowskid, S. Wallonb
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a CPhT, E´cole Polytechnique, CNRS, Palaiseau, France
n bLPT, Universit´e Paris XI, CNRS, Orsay, France
a cIPN, CNRS, Orsay, France
J
dSoltan Institute for Nuclear Studies, Warsaw, Poland
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Abstract
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The factorization of the amplitude for the deeply virtual Compton scattering (DVCS) process
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p γ∗(Q)γ → γγ at high Q2 is demonstrated in two distinct kinematical domains, allowing to
e define the photon generalized parton distributions and the diphoton generalized distribution
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amplitudes. Both these quantities exhibit an anomalous scaling behaviour and obey new inho-
[
mogeneous QCD evolution equations.

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Key words: QCD,Exclusiveprocesses,Factorization, GeneralizedPartonDistributions
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PACS: 12.38.Bx,12.20.Ds,14.70.Bh
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. 1. Motivation
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The parton content of the photon has been the subject of many studies since the
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seminal paper by Witten [1] which allowed to define the anomalous quark and gluon
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v distributionfunctions.Recentprogressesinexclusivehardreactionsfocusongeneralized
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X partondistributions(GPDs),whicharedefinedasFouriertransformsofmatrixelements
between different states, such as hN′(p′,s′)|ψ¯(−λn)γ.nψ(λn)|N(p,s)i and their crossed
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a versions, the generalized distribution amplitudes (GDAs) which describe the exclusive
hadronization of a q¯q or gg pair in a pair of hadrons, see Fig. 1. In the photon case,
these quantities are perturbatively calculable [2,3] at leading order in α and leading
em
logarithmic order in Q2. They constitute an interesting theoretical laboratory for the
non-perturbative hadronic objects that hadronic GPDs and GDAs are.
2. The diphoton generalized distribution amplitudes
Defining the momenta as q =p−Qs2n, q′ = Qs2n, p1 =ζp, p2 =(1−ζ)p, where p and
n are two light-cone Sudakov vectors and 2p·n=s, the amplitude of the process
PreprintsubmittedtoElsevier 13January2009
gg **((qq)) gg ((pp ))
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W 2 = (q+q’) 2 = 0
− q2 = Q 2 large CCFF GDA
gg ((qq’’)) gg ((pp22 ))
g *(q) g (q’)
2 CF
t = (p − p ) = 0
2 1
2 2
− q = Q large
g (p ) g (p ) GPD
1 2
Fig.1.Factorizations oftheDVCSprocessonthephoton.
∗ ′ ′
γ (Q,ǫ)γ(q ,ǫ)→γ(p1,ǫ1)γ(p2,ǫ2) (1)
may be written as A=ǫµǫ′νǫ1∗αǫ∗2βTµναβ. In forwardkinematics where (q+q′)2 =0, the
tensorial decomposition of Tµναβ reads (see [3])
1 1 1
gµνgαβWq+ gµαgνβ +gναgµβ −gµνgαβ Wq+ gµαgνβ −gµβgαν Wq.
4 T T 1 8 T T T T T T 2 4 T T T T 3
(cid:16) (cid:17) (cid:16) (cid:17)
Atleadingorder,thethreescalarfunctionsWq canbewritteninafactorizedformwhich
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is particularly simple when the factorization scale M equals the photon virtuality Q.
F
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Wq is then the convolutionWq = dzCq(z)Φq(z,ζ,0) of the coefficient function Cq =
1 1 V 1 V
R0
e2 1 − 1 with the anomalous vector GDA (z¯=1−z,ζ¯=1−ζ) :
q z 1−z
(cid:16) (cid:17)
Φq(z,ζ,0)= NCe2q log Q2 z¯(2z−ζ)θ(z−ζ)+ z¯(2z−ζ¯)θ(z−ζ¯)
1 2π2 m2 (cid:20) ζ¯ ζ
z(2z−1−ζ) z(2z−1−ζ¯)
+ θ(ζ−z)+ θ(ζ¯−z) .
ζ ζ¯ (cid:21)
Conversely,Wq is the convolutionofthe function Cq =e2 1 + 1 with the axialGDA:
3 A q z z¯
(cid:0) (cid:1)
Φq(z,ζ,0)= NCe2q log Q2 z¯ζθ(z−ζ)− z¯ζ¯θ(z−ζ¯)− zζ¯θ(ζ −z)+ zζθ(ζ¯−z)
3 2π2 m2 (cid:20) ζ¯ ζ ζ ζ¯ (cid:21)
and Wq = 0. Note that these GDAs are not continuous at the points z = ±ζ. The
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anomalousnatureofΦq andΦq comesfromtheirproportionalitytolog Q2,whichreminds
1 3 m2
us of the anomalous photon structure functions. A consequence is that d Φq 6= 0;
dlnQ2 i
consequentlytheQCDevolutionequationsofthediphotonGDAsobtainedwiththehelp
of the ERBL kernel are non-homogeneous ones.
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3. The photon generalized parton distributions
We now look at the same process in different kinematics, namely q =−2ξp+n, q′ =
(1+ξ)p, p1 = n, p2 = p1+∆ = (1−ξ)p , where W2 = 12−ξξQ2 and t = 0. The tensor
Tµναβ is now decomposed on different tensors with the help of three functions Wq as
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(see [2]):
1 1 1
gµαgνβWq+ gµνgαβ +gανgµβ −gµαgνβ Wq+ gµνgαβ −gµβgνα Wq
4 T T 1 8 T T T T T T 2 4 T T T T 3
(cid:16) (cid:17) (cid:16) (cid:17)
These functions can also be written in factorized forms which have direct parton model
interpretations when the factorization scale M is equal to Q . The coefficient func-
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tions are Cq = −2e2 1 ± 1 and the unpolarized Hq and polarized Hq
V/A q x−ξ+iη x+ξ−iη 1 3
(cid:16) (cid:17)
anomalous GPDs of quarks inside a real photon read :
Hq(x,ξ,0)= NCe2q θ(x−ξ)x2+(1−x)2−ξ2
1 4π2 (cid:20) 1−ξ2
x(1−ξ) x2+(1+x)2−ξ2 Q2
+θ(ξ−x)θ(ξ+x) −θ(−x−ξ) ln ,
ξ(1+ξ) 1−ξ2 (cid:21) m2
Hq(x,ξ,0)= NCe2q θ(x−ξ)x2−(1−x)2−ξ2
3 4π2 (cid:20) 1−ξ2
1−ξ x2−(1+x)2−ξ2 Q2
−θ(ξ−x)θ(ξ+x) +θ(−x−ξ) ln .
1+ξ 1−ξ2 (cid:21) m2
Similarly as in the GDA case, the anomalous generalized parton distributions Hq are
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proportional to ln Q2. Consequently, the anomalous terms Hq supply to the usual ho-
m2 i
mogeneousDGLAP-ERBLevolutionequationsofGPDsanon-homogeneoustermwhich
changes them into non-homogeneous evolution equations.
WedonotanticipatearichphenomenologyofthesephotonGPDs,butinthecaseofa
high luminosity electron - photon collider which is not realistic in the near future. How-
ever, the fact that one gets explicit expressions for these GPDs may help to understand
the meaning of general theorems such as the polynomiality and positivity [4] constrains
or the analyticity structure [5]. For instance, one sees that a D-term in needed when
expressing the photon GPDs in terms of a double distribution. One also finds that, in
theDGLAPregion,H1(x,ξ)issmallerthanitspositivityboundbyasizeableandslowly
varying factor, which is of the order of 0.7−0.8 for ξ ≈0.3.
This work is supported by the Polish Grant N202 249235, the French-Polish scientific
agreement Polonium and the grant ANR-06-JCJC-0084.
References
[1] E.Witten,Nucl.Phys.B120(1977) 189.
[2] S.Friotet.al.,Phys.Lett.B645(2007) 153andNucl.Phys.Proc.Suppl.184,35(2008).
[3] M.ElBeiyadet.al.,Phys.Rev.D78(2008) 034009andAIPConf.Proc.1038,305(2008).
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[4] B.Pire,J.SofferandO.Teryaev,Eur.Phys.J.C8(1999)103.
[5] I.V.Anikinet.al.,Phys.Rev.D76,056007(2007).M.Diehlet.al.,Eur.Phys.J.C52(2007)919.
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