The evolution of $\sigma^{\gamma P}$ with coherence length

0.32 MB·

by  Allen Caldwell

Date: January19,2016
The evolution of σγP with coherence length
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Allen Caldwell
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1 Max Planck Institute for Physics (Werner-Heisenberg-institut)

] Munich, Germany
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Abstract
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4 Assumingthe formσγP ∝ lλeff atfixed Q2 forthe behaviorof thevirtual-photonprotonscat-
0 tering cross section, where l is the coherence length of the photon fluctuations, it is seen that the
1. extrapolatedvaluesofσγP fordifferentQ2 crossforl ≈ 108 fm. Itisarguedthatthisbehavioris
0 notphysical,andthatthebehaviorofthecrosssectionsmustchangebeforethiscoherencelengthl
6 isreached.Thiscouldsetthescalefortheonsetofsaturationofpartondensities.
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1 Introduction
According to quantum field theory, the microscopic world is a dynamic environment where short-lived
statesareconstantlybeingcreatedandannihilated. Increasingtheresolutionofaprobeusedtosensethis
environment,weseeevermorestructureuntil,weanticipate,atsomesmallenoughdistancescalewesee
universal distributions. It has already been observed that the energy dependence of the scattering cross
sectionofhighenergyhadrons becomesuniversal[1]andthatthesizeofthecrosssectiononlydepends
on the number of valence quarks in the scattering particles and the center-of-mass energy. We study in
this paper the energy dependence of scattering cross sections for virtual photons on protons, where the
interactioncrosssectionisdominatedbythestronginteraction. Weanticipatethatinahighenergylimit,
thescatteringcrosssectionsofvirtualphotonswillalsoachieveauniversalbehaviorsincetheinteraction
willhaveassourcethephotonfluctuating intoaquark-antiquark pair.
Indeepinelastic scattering ofelectrons onprotonsatHERA,thestrongincrease intheprotonstruc-
turefunction F withdecreasing x forfixed, large, Q2 isinterpreted asanincreasing density ofpartons
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intheproton, providing morescattering targets fortheelectron. Thisinterpretation relies onchoosing a
particularreference frametoviewthescattering -theBjorkenframe. Intheframewheretheprotonisat
rest, it is the state of the photon or weak boson that differs with varying kinematic parameters. For the
bulk of the electron-proton interactions, the scattering process involves a photon, and we can speak of
different states of the photon scattering on a fixed proton target. What is seen is that the photon-proton
crosssection risesquickly withphotonenergy ν forfixedvirtuality Q2 [2]. Weinterpret thisasfollows:
astheenergy ofthephoton increases, timedilation allowsshorter livedfluctuations ofthephoton tobe-
comeactiveinthescatteringprocess,therebyincreasingthescatteringcrosssection. Weusetheconcept
of coherence length of the short-lived photon states to analyze the behavior of the photon-proton cross
section.
Thisanalysisisanupdateoftheworkreportedin [3]andhasbeeninspiredbythespace-timepicture
ofGribov[4].
2 Coherence Length
TheHandconvention [5]isusedtodefinethephotonfluxyielding therelation:
Q4(1−x)
FP(x,Q2) = σγP (1)
2 4π2α(Q2+(2xM )2)
P
where FP is the proton structure function, α is the fine structure constant and M is the proton mass.
2 P
Q2 andxarethestandard kinematicvariablesusedtodescribe deepinelastic scattering.
The behavior of σγP is studied in the proton rest frame in terms of the coherence length [6] of the
photon fluctuations, l, and the virtuality, Q2. The physical picture isgiven inFig. 1, where the electron
acts as a source of photons, which in turn acts as asource of quarks, antiquarks and gluons. Weexpect
thescattering crosssection toincrease withcoherence lengthsincethephoton hasmoretimetodevelop
structure. In the proton rest frame, the proton is a common scattering target for the incoming partons
independent fromthevaluesofQ2 andl.
Werecallthedefinitionofthecoherence length, l[7]:
~c
l ≡
∆E
1
g
e proton
e
Figure1: Photonfluctuations scattering ontheprotonintheprotonrestframe.
where ∆E, the change in energy of the photon as it fluctuates into a system of quarks and gluons, is
givenby
m2+Q2
∆E ≈ (2)
2ν
whereν isthephotonenergy andmisthemassofthepartonic state. ForQ2 ≫ m2,wehave
Q2
∆E ≈ (3)
2ν
and
2ν~c ~c
l ≈ ≈ . (4)
Q2 xM
P
Note that M only appears because of the definition of x and the value of the coherence length is not
P
dependent on the proton mass. The coherence length increases linearly with the photon energy and
inversely proportional to the virtuality. In the limit Q2 → 0, the dependence of the coherence length
willbe modified since the m2 term cannot be ignored. Wewilltherefore limit our study of the limiting
behavior ofthecrosssection totheregimeQ2 ≥ 1GeV2 whereweexpectthatEq.3shouldbevalid.
3 Data Sets
The analysis relies primarily on the recently released combination of HERA electron-proton neutral-
current scattering data from the ZEUS and H1 experiments [8]. This is complemented by the small-x
data from the E665 [9]and NMC[10] experiments. Weuse data in the kinematic range: 0.01 < Q2 <
500 GeV2, x < 0.01 and 0.01 < y < 0.8 for a global analysis and a more restricted range when
studying the extrapolated behavior of the cross sections. The HERA data are presented in the form of
reducedcrosssections
xQ4 dσ2 y2
σ = ≈ F − F
red 2πα2Y dxdQ2 (cid:20) 2 Y L(cid:21)
+ +
where Y = 1+(1−y)2. The structure function F can be ignored in the (x,Q2) range selected for
+ 3
the analysis. We calculate F from the reduced cross section assuming R = F /(F − F ) = 0.25,
2 L 2 L
andlimitthevalues ofy tobesmaller than 0.8tocontrol thepossible error duetothis assumption. The
values ofxarerestricted tobeless than 0.01 toensure thatwearedealing withlarge coherence lengths
compared to the proton size. The total experimental uncertainties are used (statistical and systematic
added in quadrature on a point-by-point basis). The use of correlated systematics was investigated but
foundnottogivesignificantdifferencesforthisanalysis. Inanycase,wearelookingmoreforqualitative
behavior.
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In total, 45 E665 data points, 13 NMC data points, 23 HERA e P data points, and (115, 160, 75,
276)HERAe+P datapointswithE = (460,575,820,920) GeVsatisfiedtheselection criteria.
p
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3.1 Bin-centering andfurther analysis
Thedatafrom thedifferent experiments arereported insomecasesatfixedvalues ofxand varying Q2,
inothercasesatfixedQ2 andvarying x,oratfixedQ2 andvarying y. Thisisnoproblem forthemodel
fittingprocedure,butitdoesmakedatapresentationcumbersome. Thedataweretherefore‘bin-centered’
bymovingdatatofixedvaluesofQ2 usingaparametrization (seebelow)asfollows:
Fpred(Q2,x)F (Q2,x)
F (Q2,x) = 2 c 2 (5)
2 c Fpred(Q2,x) Sexpt
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whereS isanormalizationfactorfortheexperimentinquestionthatresultedfromtheglobalfit. The
expt
followingQ2 valueshavebeenused: 0.25,0.4,0.65,1.2,2,3.5,6.5,10,15,22,35,45,90,120 GeV2.
c
3.2 Functional form forglobalfit
A global fit of all the data satisfying the selection criteria was performed to have a functional form for
thebin-centering. Thefunctional formforthevirtual-photon protoncrosssectionwasconstructed using
thefollowingreasoning.
3.2.1 Q2 dependence
Forcompactphotonfluctuations,themaximumvalueofσγP isgivenbythesizeoftheprotonmultiplied
by α, giving roughly 200 µbarn. However, pQCD calculations have pointed to the property of ‘color
transparency’forsmalldipoles[11],indicatingthatatlargeQ2thecrosssectionshouldbehaveasσγP ∝
1/Q2. I.e., the proton appears almost transparent for small dipoles with the cross section proportional
to the size of the photon. The photon state will have a maximum size which is expected to be set by
the mass of the lightest vector meson. An effective mass was used as a free parameter in the fits. We
thereforehavethefollowingfactorinourparametrization:
M2
σ
0Q2+M2
where σ is expected to be a typical hadronic cross section (multiplied by α). This term has two free
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parameters(M2,σ ).
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3.2.2 ldependence
As discussed in the introduction, we expect (and already know from looking at previous data) that the
cross sections will grow with increasing coherence length. We also know that at very small Q2, the
energy dependence of the photon-proton cross section is close to that for hadron-hadron scattering [2],
whilethecrosssectionrisesmoresteeplywithenergyatlargerQ2. Thetransitioninthebehaviorsetsin
aroundQ2 = 1GeV2 [2,3]. Wetherefore usethefollowingfactortomodeltheldependence:
l ǫeff
(cid:18)l (cid:19)
0
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with
ǫ = ǫ Q2 ≤ Q2
eff 0 0
ǫ = ǫ +ǫ′ln(Q2/Q2) Q2 ≥ Q2
eff 1 1 1
ln(Q2/Q2)
ǫ = ǫ +(ǫ −ǫ ) 0 Q2 < Q2 < Q2
eff 0 1 0 ln(Q2/Q2) 0 1
1 0
Thistermhasfivefreeparameters(ǫ ,ǫ ,ǫ′,Q2,Q2). Thecoherence lengthismeasuredinfm.
0 1 0 1
3.2.3 PredictionforF
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Thephoton-proton crosssectionistherefore fullyparametrized withtheeightparameters as
M2 l ǫeff(ǫ0,ǫ1,ǫ′,Q20,Q21)
σγp = σ , (6)
0Q2+M2 (cid:18)l (cid:19)
0
Theprediction for F (and subsequently the reduced cross section) is then achieved using Eq. (1), with
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the addition of one parameter for each data set to account for relative normalization uncertainties. The
bin-centereddataareshowninFig.2. Asisseeninthefigure,thedatacoverseveralordersofmagnitude
incoherence length atthesmallervaluesofQ2,reducing tooneorderofmagnitude atQ2 = 120GeV2.
Itisalsoclearthattheslopeofthecrosssectionincreases withincreasing Q2.
Photon-Proton Cross Section
-1
10
-2
10
-3
10
2 3 4 5
10 10 10 10
Figure 2: Bin-centered photon-proton cross section data, shown in alternating colors for clarity. The
valuesofQ2 inGeV2 foragivensetofpointsaregivenattheleftedgeoftheplot.
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4 Global Fit Results
TheBayesian Analysis Toolkit (BAT)[12]was used to extract the 15-dimensional posterior probability
distribution of the parameters (the 8 parameters for the photon-proton cross section and additionally 7
normalizations) assuming Gaussian probability distributions for all data points. The prior probability
distributions chosen are given in Table 1. For the parameters of the photon-proton cross section, the
priors reflect the knowledge that exists on the parameters. For the normalization parameters, the priors
reflecttheexperimentally quoteduncertainties.
Table 1: Prior probability definitions for the parameters in the fit function and results of the global fit
to the data. The notation x ∼ G(◦|µ,σ) means that x is assumed to follow a Gaussian probability
distribution centered on µ with variance σ2. The parameter boundaries are also given. The parameter
valuesattheglobalmodeaswellasthe68%smallestmarginalizedintervalsfromtheglobalfitaregiven
inthelasttwocolumns.
Parameter priorfunction globalmode 68%interval
σ σ ∼ G(◦|0.07,0.02) 0.01 < σ < 0.2 0.062 0.059−0.067
0 0 0
M2 M2 ∼ G(◦|0.75,0.5) 0.1 < M2 < 2.0 0.63 0.59−0.67
0
l l ∼ G(◦|1.0,0.5) 0.1 < l < 2.0 1.6 1.54−1.77
0 0 0
ǫ ǫ ∼ G(◦|0.09,0.01) 0.05 < ǫ < 0.2 0.106 0.102−0.110
0 0 0
ǫ ǫ ∼ G(◦|0.2,0.2) 0.1 < ǫ < 0.3 0.156 0.152−0.160
1 1 1
′ ′ ′
ǫ ǫ ∼ G(◦|0.05,0.02) 0.0 < ǫ < 0.1 0.052 0.051−0.054
Q2 Q2 ∼ G(◦|1.0,1.0) 0.01 < Q2 < 2.0 0.37 0.33−0.41
0 0 0
Q2 Q2 ∼ G(◦|3.0,3.0) 2.0 < Q2 < 10.0 3.13 2.96−3.36
1 1 1
S S ∼ G(◦|1.0,0.018) 0.9 < S < 1.1 0.97 0.958−0.982
E665 E665
S S ∼ G(◦|1.0,0.025) 0.9 < S < 1.1 0.94 0.093−0.096
NMC NMC
Se−p S ∼ G(◦|1.0,0.015) 0.9 < Se−p < 1.1 0.997 0.988−1.006
S S ∼ G(◦|1.0,0.015) 0.9 < S < 1.1 1.020 1.014−1.030
e+p460 e+p460
S S ∼ G(◦|1.0,0.015) 0.9 < S < 1.1 1.014 1.008−1.024
e+p575 e+p575
S S ∼ G(◦|1.0,0.015) 0.9 < S < 1.1 1.009 1.002−1.020
e+p820 e+p820
S S ∼ G(◦|1.0,0.015) 0.9 < S < 1.1 0.998 0.992−1.008
e+p920 e+p920
ThevaluesoftheparametersattheglobalmodearegiveninTable1togetherwiththe68%smallest
intervalsforthe1Dmarginalizeddistributions. Thefitwassatisfactoryandwasfurtherusedtobin-center
thedata points. Theposterior probability at theglobal modecorresponded toa χ2 value of 761 for 707
fitteddatapoints.
Thebin-centered datawerethenfittothesimpleform
l λeff(Q2)
σ(l,Q2) = σ (Q2) (7)
1
(cid:18)1 fm(cid:19)
forfixedvalues ofQ2. Somesample fitsareshown inFig3. These two-parameter fitswere universally
good, indicating that at fixed Q2 values, the data are completely consistent with a simple power law
behavior.
Theparameters resulting from these fits are shown inFig. 4, together with the expectation from the
formula used in the global fit of the data. It is seen that the global fit does a decent job or representing
the data, in particular the Q2 dependence. The dependence on the coherence length has the expected
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Photon-Proton Cross Section Photon-Proton Cross Section
0.1837 0.0629
0.1506 0.0508
0.1175 0.0386
0.0845 0.0264
0.0514 0.0143
2 3 4 2 3
10 10 10 10 10
-2
x 10
Photon-Proton Cross Section Photon-Proton Cross Section
0.0139 0.38
0.0112 0.312
0.0086 0.243
0.006 0.174
0.0033 0.106
2 3 2
10 10 10
Figure 3: Fits of the data using the function given in (7) for selected values of Q2. The values of Q2
are given in the panels, as well as the number of data points fit and the value of χ2 using the best-fit
parameters. The band indicates the 68 % credible interval from the fit function. The error bars on the
pointsarefromadding thestatistical andsystematicuncertainties inquadrature.
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features: atsmall Q2, the exponent λ ≈ 0.1, avalue compatible with the results from hadron-hadron
eff
scattering data. For values above a few GeV2, the exponent λ ∝ ln(Q2), with a transition region in
eff
between
0.35
0.275
0.2
0.125
0.05
2
1 10 10
-2
10
-3
10
2
1 10 10
Figure 4: Values of the parameters λ and σγp(l = 1 fm) = σ (see Eq. 7) as a function of Q2. The
eff 1
values expected from theglobal fittothedata (see Eq.6)isshownasthegreen lines. Theuncertainties
onthefitparameters aretypicallysmallerthanthensizeofthesymbols.
From Fig. 4, we note that while the cross section at l = 1 fm decreases approximately as 1/Q2 for
the larger values of Q2, the dependence on the coherence length becomes steeper with increasing Q2.
An extrapolation of the cross sections will result in the cross section for high Q2 photons eventually
becoming larger than the cross section for lower Q2 photons. We study this for Q2 > 3 GeV2, where
the data in Fig. 4 indicate that we have reached the simple scaling behavior of the cross section. An
extrapolation of thecross sections isshowninFig. 5andreveals that forl ≈ 108 fm, theextrapolations
merge.
5 Discussion
At small values of the coherence length, the photon-proton scattering cross section scales roughly as
1/Q2, as expected from color transparency. However, as the coherence length increases, the scattering
cross sections increase at different rates, and the scattering cross section for an initially small photon
configuration willgrowtobecomparable tothatforasmallerQ2,orlarger, photonconfiguration unless
thegrowthofthecross section ismodified. Ifthephoton statesbecome comparable insize, theyshould
evolveinthesamewaysothatthecrosssectionsevolveuniformlywithcoherencelength,independently
of their initial size, and with the same energy dependence as typical for hadronic cross sections. This
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Photon-Proton Cross Section
-1
10
-2
10
-3
10
3 5 7 8
10 10 10 10
Figure5: ExtrapolationofthefitfunctionstolargerlforvaluesofQ2intherange3.5 ≤ Q2 ≤ 90GeV2.
Thewidthofthebandsrepresentthe68%credibleintervalsforthefunctionsfromthedatafitsdescribed
in the text. The values of Q2 range from 3.5 GeV2 (top curve) to 90 GeV2 (bottom curve) with the
intermediate curvescorresponding tointermediate Q2 values.
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hadron-like energy dependence is observed for the smallest Q2 values investigated (the energy depen-
denceisthesameasforhadron-hadronscattering). Wethereforeexpectthattheslopeofthecrosssection
withcoherencelengthshouldflattenasafunctionofcoherencelength;thiscouldbeanindicationforsat-
urationofthepartondensities. Itisalsopossiblethatthepowerlawbehaviorofthecrosssectionchanges
withcoherence length withoutsaturation [16]butthiswouldagainimplyaninteresting change ingluon
dynamics. Theneweffectsshouldsetinbelowl = 108 fmtoavoidthecrosssectioncrossing. Giventhe
relation giveninEq.4,thismeansthatachange shouldsetinforx > 10−9. Firstsignsofthechangein
slopewouldpresumablysetinconsiderablyearlier. Theapproachtosaturationandafundamentallynew
state of matter is therefore perhaps within reach of next generation lepton-hadron colliders [13–15], an
excitingprospect.
6 Acknowledgments
I would like to thank Matthew Wing and Henri Kowalski for many interesting discussions concerning
thisanalysis.
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