- N () -- -- N or m - N () -- -- N or m

A

LU

() =

- +

- N () -- -- N or m - N () -- -- N or m

,

-- -- -- -- has to b e normalized for different helicity states to the factors N or m and N or m. The normalization can b e done either by the corresp onding numb er of DIS events or by the measured luminosity. Since the normalization could effect a constant-term p 0 in the asymmetry, it is useful to check the normalization on consistency. In the following, b oth metho ds are compared. The normalization p er numb er of DIS events is the standard metho d and was applied in the previous sections (e.g. figure 4.13). The kinematic b oundaries and acceptance cuts for the DIS event selection were given in 3.1.2. In order to apply the luminosity metho d,

57


LU

A

A

LU

0.6

11.55 / 7 2 / ndf 0.09515 ± 0.03468 p0 -0.2029 ± 0.05067 p1 0.06173 ± 0.04876 p2

0.4

0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -3 -2 -1 0 1 2 3 (rad) -3 -2 -1 0

2 / ndf 6.418 / 7 -0.03452 ± 0.04167 p0 -0.2445 ± 0.06021 p1 0.05209 ± 0.05945 p2

1

2

3 (rad)

Figure 5.5: The BSA for neon, p erformed by using the luminosity metho d, for the coherent (left) and incoherent (right) part. the pro duct of the rate in the luminosity monitor, the live time of trigger 21 and of the burst length has to b e calculated. The BSA extracted by using the luminosity metho d instead is shown in figure 5.5 for the coherent and incoherent part. A comparison b etween b oth normalization metho ds is given in the following table: Norm p er DIS 0.065±0.035 -0.203±0.051 0.062±0.049 -0.062±0.042 -0.244±0.060 0.048±0.059 Norm p er Lumi 0.095±0.035 -0.203±0.051 0.062±0.049 0.035±0.042 -0.244±0.060 0.052±0.059

Coherent

Incoherent

const p Asin LU Asin 2 LU const p Asin LU Asin 2 LU

0

0

Summarizing, it is shown that the metho d of normalization do es not effect the BSAs Asin and Asin 2 , as exp ected. The constant term p0 is shifted up by ab out one sigma LU LU in the luminosity metho d.

5.5 Study of different Fit Metho ds
5.5.1 The Influence of Binning
In order to estimate the effect of binning for the azimuthal angle for the BSA, a sin calculation with different numb er of -bins is p erformed. In figure 5.6 A LU and Asin 2 LU are plotted in dep endence on the numb er of -bins. For the coherent part the average

58


sin(2) ) LU

sin( ALU ) (A

0.05 0

A

sin( ) LU

(A

0.1

sin(2 ) ) LU

0.15


sin(2) >=0.05± LU

0.02

0.1

sin(2) > LU

= 0.053 ± 0.024

-0

-0.05 -0.1 -0.15 -0.2 -0.25


sin() LU >=-0.206±0.021

-0.1


sin() > LU

= -0.241 ± 0.027

-0.2

-0.3
-0.3 8 9 10 11 12 13 Number of -bins

8

9

10

11

12 13 Number of -bins

Figure 5.6: Asin and Asin 2 in dep endence on the numb er of -bins for the coherent LU LU (left) and incoherent (right) part. Asin (Asin 2 ) is -0.206 (0.05) with an uncertainty of ±0.021 (±0.02). The numb er of LU LU -bins varied from 8 to 13. For the incoherent part the average A sin (Asin 2 ) is -0.241 LU LU (0.053) with an uncertainty of ±0.027 (±0.024). In comparison to the usual fit metho d with 10 -bins the difference is less then 0.01 for A sin and less than 0.02 for Asin 2 , i.e. LU LU can b e neglected at the present level of statistics.

5.5.2 The Fit Function
The standard fit-function is given as f () = p0 + p1 sin + p2 sin 2 (5.2)

In order to study the influence of additional harmonics, the following fit functions are applied: f () = p1 sin f () = p0 + p1 sin + p2 sin 2 + p3 sin 3 + p4 sin 4 f () = p0 + p1 cos + p2 cos 2 (5.3) (5.4) (5.5)

The corresp onding BSA results are shown in figure 5.7 for the coherent part. Note that the amplitudes can not change by using additional parameters in the fit-function, e.g. shown in the top part in figure 5.7 for A sin : LU p1 sin A
sin LU sin LU

p0 + p1 sin + p2 sin 2 A

= -0.203 ± 0.051

= -0.208 ± 0.05

The results of this study can b e summarized in the following:

59


2 / ndf 16.75 / 8 0 ± 1.414 p0 -0.2084 ± 0.05048 p1
0.4

0.4

2 / ndf 11.53 / 7 0.06476 ± 0.03473 p0 -0.2031 ± 0.05072 p1 0.06222 ± 0.04883 p2

LU

A

0.2

A 0.2 0 -0.2

0

-0.2

LU

-0.4 -3

Fit: p1sin()
-2 -1 0 1 2 3 (rad)

p0 + p1sin() + p2sin(2)
-0.4 -3 -2 -1 0 1 2 3 (rad)

0.4

2 / ndf p0 p1 p2 p3 p4

0.06601 -0.1989 0.05529 -0.0754 0.0513

± ± ± ± ±

7.921 / 5 0.03473 0.05086 0.04897 0.04996 0.04793

0.4

2 / ndf 26.33 / 7 0.06596 ± 0.035 p0 0.07158 ± 0.04813 p1 0.07052 ± 0.04982 p2

LU

A

0.2

A 0.2 0 -0.2

0

-0.2 p0 + p1sin() + p2sin(2) + p3sin(3) + p4sin(4 ) -3 -2 -1 0 1 2 3 (rad)

LU

p0 + p1cos() + p2cos(2)
-0.4 -3 -2 -1 0 1 2 3 (rad)

-0.4

Figure 5.7: The BSA results with different fit functions for the coherent part. ·A
sin 3 LU

and A

sin 4 LU

are given as A A
sin 3 LU sin 4 LU

= -0.051 ± 0.048

= -0.075 ± 0.05

A theoretical interpretation of these higher harmonics is only known for proton and not for nuclei. In addition they have to o small statistics and therefore they will not b e further discussed here. · The cos -dep endence is negligible within the statistical errors. Note that a nonzero cos amplitude is forbidden from theory. The same conclusions hold for the incoherent sample.

60


5.5.3 The Anti-symmetrization Fit Metho d
The anti-symmetrization fit metho d is characterized by mapping the region [- , 0] to the region [0, ], which pro jects out the odd part in in the BSA. Hence the even harmonics disapp ear, esp ecially acceptance effects. This means, anti-symmetrized BSAs, by ansatz, are not sensitive (anymore) to a p ossible constant term p 0 , as this is also an even contribution. For the b eam-spin asymmetry on an unp olarized target, two different formulae exist for the anti-symmetrization: 1st way: A
LU

can b e calculated for each -bin as A = 1 < |P | >
- - N ()- N (-) -- -- N or m - - N (-) N () -- + -- -- -- N or m N or m

LU

- +

- - N ()- N (-) -- -- N or m - - N () N (-) -- + -- -- -- N or m N or m

(5.6)

In terms of the cross-section d = d dU d
ev en in

+

dL P d

L

(5.7)

odd in

the ab ove defined asymmetry turns out to b e theoretically correct: A
LU

1P = PL

dL L d - dU d

(-PL ) +
dU d

dL d

=

dL d dU d

(5.8)

2nd way: Alternatively, ALU can b e calculated -bin: - - - 1 N () ALU = - - < |P | > N () - - - 1 N () ALU = - - < |P | > N () In terms of cross sections - P - - ALU = - P
dL d dU d dL d dU d

separately for each helicity state and - - N (-) - + N (-) - - N (-) - + N (-) - - = -ALU (5.9)

(5.10)

=

(5.11)

also the 2nd way to calculate the anti-symmetrized BSA is correct. In comparison with the standard (non-symmetrized) fit metho d b oth ways share the fact that lo cal acceptance effects cancel out. In addition, the second way provides the

61


0.1

6.866 / 3 2 / ndf -0.2176 ± 0.05076 p0 0.06373 ± 0.04893 p1

3.183 / 3 2 / ndf -0.2742 ± 0.05976 p0 0.03898 ± 0.05887 p1
-0

LU

A

-0 -0.1 -0.1 -0.2 -0.2

A -0.3 -0.4

-0.3

0.5

1

1.5

2

2.5

3 (rad)

LU

0.5

1

1.5

2

2.5

3 (rad)

Figure 5.8: The BSA obtained via the anti-symmetrization metho d for the coherent (left) and incoherent (right) part. advantage that any helicity-dep endent acceptance effects (as e.g. b eam envelop es and/or vertex p ositions dep ending on helicity) drop out, as long as they are even in . The final BSA, in the second way of antisymmetrization, can b e obtained as the weighted average of b oth measurements: -- -i -i - - w ALU - wk Ak i,k LU ALU = , (5.12) - -i k w +w i,k where i(k ) runs over helicity p erio ds (). The weighting can b e done either by 1 normalization w = N or m or by the statistical error w = (ALU )2 , which in fact should b e equivalent. Note that although the statistical uncertainty p er bin decreases by ab out 2, the resulting uncertainties of the fit stays the same, as the numb er of bins was halved. The result of the anti-symmetrization metho d via the 2nd way is plotted in figure 5.8. The asymmetries ALU are fitted with the function p0 sin + p1 sin 2.
1 Note that the weighting is done by the statistical error w = (ALU )2 for each helicity state. The result of this anti-symmetrization metho d in comparison with the standard (non-symmetrized) fit metho d is given in the following table:

non-symmetrized anti-symmetrized

Coherent Asin = -0.20 ± 0.05 LU Asin = -0.22 ± 0.05 LU

Incoherent Asin = -0.24 ± 0.06 LU Asin = -0.27 ± 0.06 LU

Summarizing, the BSA results obtained via the anti-symmetrization fit metho d are in go o d agreement with the results of the standard fit metho d.

62


N/(1000*NDIS)

0.22 0.2

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.8 0.85 0.9 0.95 1 1.05

Parallel Antiparallel

1.1

1.15

1.2 E/P

Figure 5.9: Distribution of the ratio b etween the lepton energy dep osited in the calorimeter E and the momentum of the lepton P , derived from the x-slop e difference b etween the front and the back track, for the parallel and antiparallel helicity state.

5.6 The Influence of the Calorimeter on the Results
5.6.1 Study of the Photon Energy Reconstruction
Since the calorimeter is the only detector able to measure the energy and p osition of the photon, it is very imp ortant for this analysis. In order to check the reconstruction of the photon energy from the calorimeter measurement, the ratio b etween the lepton energy dep osited in the calorimeter E and the momentum of the lepton P is used. The latter is derived from the x-slop e difference b etween the front and the back track. The distribution for the value E /P for the lepton sample is shown in figure 5.9 for the parallel and antiparallel helicity state. The mean value is extracted by a gaussian fit function and the result is given as E = 0.991 ± 0.001, P M ean b eing the same for parallel as well as antiparallel helicity. Hence there is no difference in the energy reconstruction from the calorimeter when comparing b oth helicity states.

5.6.2 Study of the Photon Position Reconstruction
Another imp ortant quantity for this analysis is the resolution of the p osition reconstruction of the calorimeter. In order to cross check the p osition measurement of the

63


LU

LU

A

A

0.4

0.4

No Calo shift 5mm Calo shift

No Calo shift 5mm Calo shift

0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -3 -2 -1 0 1 2 3 (rad) -3 -2 -1 0 1 2 3 (rad)

Figure 5.10: Comparison b etween the results achieved with and without -5 mm correction of the photon p osition in the top part of the calorimeter for the coherent (left) and incoherent sample (right). calorimeter, the reconstructed cluster p osition for charged tracks can b e used. Note that the p osition reconstruction in the calorimeter b ehaves similarly for leptons and photons, b ecause the photons used in this analysis convert into lepton pairs in the preshower detector, which is lo cated in front of the calorimeter. Investigating the difference y track - ycalo separately for the top and the b ottom part of the calorimeter reveals that the mean of this distribution in the top part is ab out zero while it is ab out -5 mm in the b ottom part for the year 2000 [Ell]. The influence on the asymmetry ALU of this p osition shift is calculated by using a -5 mm correction of the photon p osition in the top part of the calorimeter. A comparison b etween the results achieved with and without the correction is shown in figure 5.10 for the coherent and incoherent sample. The results for the BSAs are given in the following table. No Correction 0.065±0.035 -0.203±0.051 0.062±0.049 -0.062±0.042 -0.244±0.06 0.048±0.059 5mm Correction 0.055±0.035 -0.199±0.051 0.072±0.049 0.045±0.042 -0.242±0.06 0.046±0.06

Coherent

Incoherent

const p Asin LU Asin 2 LU const p Asin LU Asin 2 LU
sin 2 LU

0

0

The influence on A

sin LU

(A

) is less than 0.005 (0.011).

64


rec. - gen. (rad)

1.4 1.2 1 0.8 0.6 0.4 0 0.002 0.004 0.006 0.008 0.01 0.012 * (rad)

Figure 5.11: Average difference b etween generated and reconstructed in dep endence on .

5.7 Smearing Effects
Smearing effects are mainly of interest as they change the relative direction of the real photon with resp ect to that of the virtual photon. Therefore they change the value of the azimuthal angle as well as the p olar angle and the momentum transfer tc . The smearing of the photon can b e studied by using MC and comparing the generated photon direction with the reconstructed photon direction. In figure 5.11 the average difference b etween the generated and the reconstructed is shown in dep endence of . The lower cut for was set to 3 mrad, in order to reduce smearing effects in . The remaining smearing for an asymmetry has b een obtained in a MC simulation in 4 bins in t. The generated asymmetry amplitude is 0.5. It is shown in figure 5.12 that smearing effects reduce the generated asymmetry by ab out 0.02 (0.04) in the coherent (incoherent) part. Hence an relative error of 4% (8%) of the asymmetry amplitude in the coherent (incoherent) sample is estimated.

5.8 Determination of the Background Contribution
The background for DVCS originates from the pro duction of photons pro duced in a semi-inclusive or exclusive background (= exclusive reaction other than DVCS). It was derived from MC studies that the main part stems from 0 meson pro duction ( 80%) and the rest is dominated by photons from the decay of the meson. The general

65


A 0.5 0.4 0.3 0.2 0.1 0 0

sin LU

0.6

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4 0.45 2 tc (GeV )

Figure 5.12: Reconstructed Asin from a MC simulation with a generated asymmetry LU amplitude of 0.5, for different tc bins.

problem in the data analysis is not only the exclusive 0 -or pro duction, which are almost indistinguishable from BH-DVCS under certain kinematic constraints, but also semi-inclusive pro cesses can leak into the exclusive sample. Since there are statistical fluctuations in the measured energy dep osition in the calorimeter, the energy of the single photon cluster can b e overestimated such that the missing mass gets closer to the exclusive range. The fraction of the background in dep endence of t is studied in [Kra]. The upp er limit for the coherent sample is 7% and it increases to 14% for the incoherent sample. In order to determine the uncertainty of the b eam-spin asymmetry results caused by the background contribution, the asymmetry of the background has to b e estimated. Since the largest contribution stems from the 0 meson, it is imp ortant to determine the b eam-spin asymmetry in semi-inclusive 0 pro duction, which is found to b e less than 3% [HERMES04a]. In summary the uncertainty caused by background contributions is approximately given as: Abkg = 0.03 (t) where denotes the fraction of the dataset that originates from background: =
Nbkg Ntot

5.9 Combined Systematic Uncertainties
In addition to the statistical uncertainty, as given in equation 4.9, the estimation of the systematic uncertainties is also imp ortant. Therefore the results describ ed in the

66


previous sections are used. The total systematic uncertainty arises from the following uncertainties: · Photons from the semi-inclusive background and exclusive reactions other than DVCS: It was derived from MC studies that the main part stems from 0 meson ( 80%) and the rest is dominated by photons from the decay of the meson. For the coherent (incoherent) part the upp er limit of the uncertainty is 0.003 (0.005), as discussed in section 5.8. · Detector smearing: The relative error is estimated to b e 4% (8%) of the asymmetry amplitude in the coherent (incoherent) sample, as explained in section 5.7. These values include binning effects as discussed in section 5.5.1. · Uncertainty of the p olarization measurement: The measurement of the b eamp olarization is done by two p olarimeters (see section 3.1.) in parallel. Usually, the p olarization value is taken from the longitudinal p olarimeter (LPOL) since it has a smaller systematic uncertainty (1.6%). Only if there are some problems in the LPOL during op eration, the values of the transverse p olarimeter (TPOL) with a systematic uncertainty of 3.4% are taken. The resulting systematic uncertainty is 1.9% of the b eam p olarization. Hence the uncertainty of the asymmetry is 0.004 for the coherent part and 0.005 for the incoherent part. · The systematic uncertainty of a vertical calorimeter shift (alignment) was found to b e 0.005 (0.003) for the coherent (incoherent) part, as describ ed in section 5.6.2. The combined systematic uncertainty is obtained by adding in quadrature the uncertainties listed ab ove. The systematic uncertainty for the coherent part is dA and for the incoherent part dA
sin LU

(sy st) = 0.011

(5.13) (5.14)

sin LU

(sy st) = 0.021.

5.10 Study of the Constant Term
In the fits to extract the BSAs, as shown in figure 4.13, a non-negligible constant term p0 is found for the coherent part (p0 = 0.065 ± 0.035), as well as for the incoherent part (p0 = -0.062 ± 0.041). Note that the sign of p 0 is different for each part and that consequently the sum of b oth parts in figure 4.6 shows no constant term p 0 . For the coherent part, p0 is two times the statistical error (2 ), which corresp onds to a probability of 95% not to b e caused by statistical effects. Since there is no known theoretical reason for a constant term, which could explain a -indep endent different

67


Ratio

Ratio

2 / ndf 49.72 / 45 8.365 ± 0.1642 p0 + + -

16 14 12 10 8 6

12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 Runnumber

Figure 5.13: Ratio of coherent DVCS to DIS events p er fill. The vertical lines indicate different helicity running p erio ds.

cross-section b etween parallel and antiparallel helicity states, the only remaining reason can b e due to systematic differences. Because of the -indep endence, p 0 is defined by the difference b etween the ratio of DVIC S for b oth helicity states, given as DS 1 p0 =

-- - - - ND V C S -- - ND I S -- - - - ND V C S -- - ND I S

- +

- - - -- ND V C S - -- ND I S - - - -- ND V C S - -- ND I S

(5.15)

In order to find the systematic reason for p 0 , several systematic studies are p erformed in the following.

5.10.1 The Time Dep endence of p

0

Since there is no physical reasons for a different ratio of DVIC S events, a strong evidence DS for a time-dep endent change of some detector prop erties exists. For example, a different ratio could b e caused by a change in the detector efficiency b etween two run p erio ds, which corresp ond to different b eam helicity states. In order to estimate the time dep endence, the ratio DVIC S is calculated p er fill for the coherent sample, as shown in figure DS 5.13. In particular, the region from run 14000 to run 16000 shows a significantly low ratio.

68


5.10.2 Study of Systematic Influences
In order to identify a systematic origin for the non-zero value of p 0 , a numb er of studies has b een done: · The standard normalization is done p er DIS events. In addition, the normalization p er luminosity is p erformed as describ ed in section 5.4. The result by using the luminosity is shown in figure 5.5 with p0 = 0.095 ± 0.035. Since p 0 do es not disapp ear, p0 can not b e caused by wrong normalization. · A different photon energy reconstruction of the calorimeter b etween b oth helicity states was also considered. Since the energy reconstruction E /P is the same for b oth helicity states, as describ ed in section 5.6.1, it could not cause a constant term. In addition there is also no difference in E /P , dep ending on the p osition in the calorimeter. · The photon p osition reconstruction was reviewed and describ ed in 5.6.2. The result, calculated with a correction of -5 mm in the top part of the calorimeter, gives a constant term p0 reduced by 0.01, as describ ed in section 5.6.2. · Another p ossibility could arise from a variation of the efficiency in the ho doscop e H0. In order to calculate the H0 inefficiency, the numb er of DIS events was counted by using trigger-18 in comparison to trigger-21. Since the only difference b etween trigger-18 and trigger-21 is the required signal in H0, the efficiency is given by the ratio of the numb er of trigger-18 events with and without trigger-21. The H0 efficiency is calculated for a grid of 2â2cm 2 across the H0 scintillator plane using the lepton tracks that passed trigger-18. The correction is done by weighting the resp ective events with the inverse efficiency for each run p erio d. The results show a negligible influence (less than 2%) for p 0 as well as for Asin and Asin 2 . LU LU · Usually the threshold of the calorimeter is set at 3.5 GeV for high density runs and 1.4 GeV for standard-density runs. In fact, the threshold was also set for several high-density runs at 1.4, which could have an influence on the acceptance. In order to estimate this influence, the BSA is calculated without these runs and compared to the standard result. The differences for A sin and Asin 2 are negligible. For p0 LU LU the difference is ab out 0.006 and can b e included in its systematic uncertainty. · Since high density runs with neon are usually following standard ABS density hydrogen data taking, a p ossible systematic deviation should b e also b e seen in the hydrogen data. Therefore, the hydrogen data of the according time p erio ds is analyzed by using the same cuts as in the neon analysis. In the hydrogen results no constant term app ears. · In addition to the studies discussed ab ove, the following kinematic variables are reviewed for differences in the distributions b etween b oth helicity states: Q 2 , W 2 ,

69


2 , xB j , yB j , tc , , lepton , P , , E , Mx , zvtx , tvtx and the b eam energy. Summarizing, a significant variation can b e only seen in t c and as well as in the Mx distribution, illustrated in figure 5.14

A combined systematic uncertainty for p 0 is obtained by adding in quadrature the uncertainties of the photon p osition and the H0 inefficiency. From that the systematic uncertainty for p0 is given as dAp0 (sy st) = 0.012. (5.16) LU This is clearly not covering the observed 2 deviations of p 0 . As a non-zero value for p0 can only b e due to exp erimental systematic problems, it must at present b e assigned to unknown sources of systematic uncertainties.

70


p0

0.4 0.3 0.2 0.1 -0 -0.1

-0.2 -0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 2 -t c (GeV )

p0 0.2 0.1 0 -0.1 -0.2 -0.3 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 (rad) p0 0.4 0.2 0 -0.2 -0.4 -0.6 -2 -1 0 1 2 M2 (GeV ) x
2

Figure 5.14: The constant term p0 in dep endence of the kinematic variables t c , 2 Mx for full sample. 71



and


6 Summary of the Results
In this chapter the b eam-spin asymmetries extracted in Deeply Virtual Compton Scattering using a neon target and their kinematic dep endences are presented . The interpretation and a comparison with hydrogen data will b e discussed in the next chapter. All data for this study was accumulated during the 2000 running p erio d of HERA using a p olarized p ositron b eam with an average p olarization of 54%. The measurements amount to an integrated luminosity of ab out 82 pb -1 . For the mostly coherent (incoherent) part the exclusive single photon sample contains 1394 (909) events in the parallel helicity state and 1470 (1102) in the antiparallel helicity state. The asymmetries have b een extracted in dep endence on the variables x their kinematic constraints, given as: 0.03 < xB j -tc Q2 < < > 0.35 0.7 GeV 1 GeV2
Bj

, -tc , Q2 within

2

A separation b etween coherent and incoherent exclusive sample is done at -t c = 0.045, whereby the coherent (incoherent) sample is constrained as -t c < 0.045 (-tc >= 0.045). The average values of the kinematic variables for the coherent and incoherent exclusive sample are: Coherent 0.068 0.019 GeV 1.75 GeV2 Incoherent 0.107 0.163 GeV2 2.76 GeV2

xB j -t c Q2

2

6.1 Coherent Sample
The result for A
LU

is plotted in figure 6.1 in dep endence on using the fit function f () = p0 + p1 sin + p2 sin 2. (6.1)

72


A

LU

0.6 0.4 0.2 0 -0.2 -0.4 -3

f()=p0+p1sin()+p2sin(2)

11.535 / 7 2 / ndf 0.065 ± 0.035 p0 -0.203 ± 0.051 p1 0.062 ± 0.049 p2
-2 -1 0 1 2 3 (rad)

Figure 6.1: Beam spin asymmetry on unp olarized neon (2000) in 10 bins of for the coherent sample. The three-parameter fit function f () and its co efficients are also shown.

Note that the parameters p0 , p1 and p2 corresp ond to the constant term, to the amplitude Asin and to the amplitude Asin 2 . A clear left/right asymmetry is visible. This is related LU LU to the negative amplitude of Asin with a statistical significance of 4 . The amplitude LU Asin 2 is p ositive with a very small statistical significance of 1.3 and therefore no LU kinematic dep endences are shown for it. In addition there is a p ositive constant term p 0 with a statistical significance of 1.9 . The asymmetry A sin is clearly non-zero: LU A
sin LU

= -0.203 ± 0.051(stat) ± 0.011(sy st)

with a significance of 3.9 when quadratically combining statistical and systematic uncertainties. The extracted amplitudes Asin for the coherent sample are shown in figure 6.2 in deLU p endence on xB j , -tc , Q2 , whereby the resp ective other two variables were integrated over in each case. The numerical results and the average kinematics of the resp ective other two variables are presented in the app endix (table 9.1). The dep endences on x B j and Q2 , shown in the top and b ottom panel, app ear to b e not very strong within the statistical uncertainty. Lo oking at the dep endence on -t c in the middle panel, the very low tc region (-tc 0.03) seems to have a significant non-zero asymmetry.

73


sin LU

A

0.1

-0

-0.1

-0.2

-0.3

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

x

Bj

A

sin LU

-0

-0.1

-0.2

-0.3

-0.4 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

-tc (GeV )
sin LU

2

A

-0

-0.1

-0.2

-0.3

-0.4

1

2

3

4

5
2

6

Q (GeV )

2

Figure 6.2: A xB j , -tc , Q2 .

sin LU

for the coherent sample in dep endence on the kinematical variables

74


A

LU

0.4 0.2 0 -0.2 -0.4 -0.6 -3

f()=p0+p1sin()+p2sin(2)

2 / ndf p0 p1 p2
-2

6.589 / 7 -0.062 ± 0.041 -0.244 ± 0.060 0.048 ± 0.059
-1 0 1 2 3 (rad)

Figure 6.3: Beam spin asymmetry on unp olarized neon (2000) in ten bins of for the incoherent sample. The three-parameter fit function f () and its co efficients are also shown.

6.2 Incoherent Sample
For the incoherent sample the extracted asymmetry A LU is shown in figure 6.3, fitted with the same function as in the case of the coherent sample (equation 6.1). A significant negative amplitude of Asin is visible: LU A
sin LU

= -0.244 ± 0.06(stat) ± 0.021(sy st)

with a significance of 3.8 when adding quadratically statistical and systematic uncertainties. A2 sin is found to b e compatible with zero and therefore no kinematic LU dep endences are shown. The constant term p 0 is small and negative with a significance of 1.5 . The extracted amplitudes Asin for the incoherent sample are shown in figure 6.4 in LU dep endence on xB j , -tc , Q2 . The numerical results are presented in the app endix (table 9.2) in conjunction with the corresp onding average kinematic values. Because of the small statistics no significant dep endence on x B j , -tc and Q2 can b e seen.

75


A
0 -0.1 -0.2 -0.3 -0.4 -0.5 0

sin LU

0.1

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

Bj

A

sin LU

-0

-0.1

-0.2

-0.3

-0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-tc (GeV )
sin LU

2

A

0 -0.1 -0.2 -0.3 -0.4 -0.5 2 4 6 8
2

10

Q (GeV )

2

Figure 6.4: A

sin LU

for the incoherent sample in dep endence on x

Bj

, -tc , Q2 .

76


7 Interpretation in the Light of Theoretical Models
In the following, the results obtained on neon in the last chapter are compared with hydrogen results and nuclear effects predicted from theoretical mo dels are discussed. Since hydrogen consists only of one proton in comparison to a neon nucleus (Ne-20) with 10 protons and additional 10 neutrons, some nuclear effects can b e exp ected, as explained in section 2.5.

7.1 Comparison with Hydrogen Data
The hydrogen data was accumulated in the running p erio d 2000 and analyzed using the same kinematic cuts as for neon, describ ed in section 4.1. Note that the hydrogen results are compatible with released results [Now05, Ell] as shown in section 4.3. According to the fact that the neon data is separated in a coherent and incoherent sample, also the hydrogen data is separated into the same t c regions: -tc < 0.045 GeV2 for the coherent sample and -tc > 0.045 GeV2 for the incoherent sample. In the following table a comparison b etween the results for neon and hydrogen is presented. Neon 0.065±0.035 -0.203±0.051 0.062±0.049 -0.062±0.042 -0.244±0.060 0.048±0.059 Hydrogen -0.011±0.036 -0.152±0.051 -0.021±0.050 -0.048±0.026 -0.213±0.037 0.010±0.037
sin LU

Coherent

Incoherent

const p Asin LU Asin 2 LU const p Asin LU Asin 2 LU

0

0

For b oth coherent and incoherent part the amplitudes A within one sigma of the statistical uncertainty.

on neon and hydrogen agree

sin In figure 7.1 the comparison of ALU and Asin 2 in dep endence of tc is shown. The LU binning in tc has b een chosen, since it is the b est variable to discriminate b etween different pro cesses. In the b ottom part the corresp onding ratios of neon over proton

77


sin LU

sin 2 LU

0.1

A

0.3 0.25 0.2

A

Neon
Hydrogen

-0

-0.1

Neon
Hydrogen

0.15 0.1 0.05

-0.2

-0.3

0 -0.05

-0.4 0 0.1 0.2 0.3 0.4 0.5 -0.1 0
2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4
2

-tc (GeV )
sin sin /A LU,hyd LU,neon sin 2 sin 2 /A LU,hyd LU,neon

tc (GeV )
25 20 15 10 5

4 3.5 3 2.5 2 1.5

A

A
1 0.5 0 -0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0 -5 -10 -15 0

tc (GeV )

2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4
2

tc (GeV )

Figure 7.1: Comparison of the asymmetry amplitude A sin (left panel) and Asin 2 (right LU LU panel) vs. tc for neon (filled squares) and hydrogen (empty squares). In the b ottom part the ratios of neon over proton asymmetry are shown as A sin ,neon /Asin ,hyd and LU LU
2 2 Asin ,neon /Asin ,hyd . Note that error bars with a larger value than the y-axis are cut LU LU off and hence not shown in their full length.

asymmetry are shown. Note that error bars with a larger value than the y-axis are cut off and therefore not shown in their full length. Lo oking at the left panel, the asymmetry Asin is systematically slightly larger for neon than for hydrogen in all t c bins although LU fully compatible within error bars. Also the asymmetries A sin 2 seems to b e larger LU for neon than for hydrogen in a similar way. Because the latter are compatible with zero, the ratio
A
sin 2 LU,neon sin 2 ALU,hy d

will not b e further discussed here. In the app endix (table 9.3)
sin LU,neon

the numerical results of A

and A

sin LU,hy d

with the average values of the kinematic

variables are given for each tc bin. A comparison of Asin b etween neon and hydrogen in LU dep endence on the variables xB j and Q2 is presented in figure 7.2. No significant effect Aneon is found within statistical uncertainties. A study of the ratio LU d in dep endence on hy
A

the azimuthal angle , presented in figure 7.3, shows no visible harmonics for all three

LU

78


sin LU

A

A
0 -0.1 -0.2 -0.3 -0.4 -0.5 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0.05 0

-0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35

x

sin LU

0.1

0.1

Bj

x

Bj

sin LU

0.3 0.2 0.1

A

A
0 -0.1 -0.2 -0.3 -0.4 -0.5 1 2 3 4 5
2

0 -0.1 -0.2 -0.3 -0.4 -0.5 6

sin LU

2
2

4

6

8
2

10

Q (GeV )

Q (GeV^2)

Figure 7.2: Comparison of Asin for the kinematic variables xB j and Q2 for neon (filled LU squares) and hydrogen (op en squares). The asymmetries are shown for the coherent sample (-tc < 0.045 GeV2 ) in the left panels and for the incoherent sample (-t c > 0.045 GeV2 ) in the right panels. tc bins. Hence no dep endent nuclear effects are found.

79


0.00<-tc<0.045
hyd neon /A LU LU

10

A
5 0 -5 -10

-3

-2

-1

0

1

2

3

(rad)

0.045<-tc<0.18
hyd neon LU /A LU

10

A

5

0

-5

-10

-3

-2

-1

0

1

2

3

(rad)

0.18<-tc<0.7
hyd neon LU /A LU

10

A

5

0

-5

-10

-3

-2

-1

0

1

2

3

(rad)

Figure 7.3: The ratio

A

neon LU Ahy d LU

in dep endence on for all three tc bins.

80


7.2 Mo del Calculations and Interpretation
Exploratory studies using GPDs were p erformed on the generalized EMC effect, i.e. the mo difications of the nuclear GPDs with resp ect to the free-nucleon (more precisely, deuteron) GPDs, normalized to their resp ective form factors [GS03, LT05a]. In particular it was shown in [LT05a] that the role of partonic transverse degrees of freedom, accounted by a consideration of nucleon off-shellness, is enhanced in the generalized Anucleus EMC effect. In the mo del of [LT05a] an enhancement of the ratio LUoton for higher t pr
A

is predicted. An influence on A is also predicted by Guzey and Strikman [GS03], and describ ed in section 2.5. It is shown, apart from the combinatorial enhancement of the asymmetries b ecause of the neutron contribution, that there are two additional effects: while Fermi motion of the nucleons enhances the asymmetries, the presence of the incoherent scattering for larger t reduces the asymmetry. Following the study of Anucleus Guzey and Strikman, the ratio of neon to proton asymmetry ( ALU oton ) is significantly Pr larger than unity for coherent nuclear DVCS (exp ected to b e close to the factor of two for t 0). In the case of higher t the inclusion of the incoherent contribution should decrease the ratio of the asymmetries, and part.
Anucleus LU Apr oton LU
LU

sin LU

LU

1 is exp ected for the incoherent

From the results of Asin given ab ove a difference of 0.051 (0.031) b etween neon and LU proton is obtained for the coherent (incoherent) part. The extracted ratio is 1.34±0.77 (1.15±0.48) for the coherent (incoherent) part. In figure 7.4 a comparison of the extracted ratio with the theoretical mo dels of Guzey/Strikman [GS03] and Liuti/Taneja [LT05a], as explained in section 2.5, is shown in dep endence on t c . Note that in the theoretical mo dels b oth coherent and incoherent contribution are used for the calculations. In fact, the statistical error is yet to o large to distinguish b etween the theoretical mo dels. Summarizing, the asymmetry is in all t c bins slightly higher for neon than for proton. This seems to b e in go o d agreement to all known theoretical results [GS03, LT05a]. Because of limited statistics no significant conclusions on the b ehavior of asymmetry ratios vs. tc can b e extracted.

81


sin LU,hyd

Ratio ALU,neon/A

4

sin

Model GS total Model LT total

3

2

1

0

-1 0

0.1

0.2

0.3

0.4

0.5 2 tc (GeV )

Figure 7.4:

Comparison of the ratio

A

sin LU,neon Asin ,hy d LU

vs.

t

c

to theoretical mo dels of

Guzey/Strikman (GS) and Liuti/Taneja (LT).

82


8 Summary and Outlook
The aim of this thesis was to study and to extract the b eam-spin asymmetry in hard electropro duction of real photons off neon. The data presented has b een accumulated by the HERMES exp eriment at DESY, scattering the HERA 27.6 GeV p olarized p ositron b eam on an unp olarized neon gas target. Attributed to the interference b etween the Bethe-Heitler pro cess and the deeply-virtual Compton scattering (DVCS) pro cess, the asymmetry gives access to the latter at the amplitude level. Its description is expressed in the theoretical framework of generalized parton distributions (GPDs), which offers a p ossibility to determine the total angular momentum carried by the quarks in the nucleon. The measurement of DVCS on the proton has shown the p ossibility to provide a sensitive test of current mo dels of GPDs. Similarly, studying DVCS on a nucleus op ens access to the prop erties of quark and gluon matter inside nuclei by measuring the mo dification of particle correlations enco ded in GPDs due to the nuclear environment. The DVCS reaction on neon pro ceeds through two different pro cesses, the coherent pro cess that involves the nucleus as a whole and the incoherent pro cess as the reaction on a single nucleon. Since coherent and incoherent pro cesses contribute to the photon pro duction cross section, b oth parts have to b e separated. In the present analysis the leading amplitude Asin of the b eam-spin asymmetry is obtained for the coherent and LU the incoherent sample. The extraction of these asymmetries has b een describ ed in detail and the results are presented as functions of one of the variables x B j , tc and Q2 . In addition, several systematical studies are p erformed in order to determine consistency and uncertainties of the obtained results. An explicit extraction of non-leading order effects is not p ossible, since the amplitudes of the higher harmonics are not significant within the present exp erimental uncertainty. Finally, although the amplitudes of the leading harmonic A sin have significant values, LU their interpretation is still difficult. With the present Hermes detector it is not p ossible to make a separation b etween the following pro cesses: · coherent BH-DVCS on neon, · incoherent BH-DVCS on neutron, · incoherent BH-DVCS on proton, · incoherent BH-DVCS on the neutron with excitation of a nuclear resonance.

83


· incoherent BH-DVCS on the neutron with excitation of a nuclear resonance. Hence there is presently no way to clearly disentangle the different contributions. In order to solve this problem and to improve the exclusivity of the DVCS pro cess, the installation of a recoil detector is currently under preparation [HERMES04b, Kra]. In summary, the amplitudes of the leading harmonics for the coherent and the incoherent sample are extracted and a clear evidence for a b eam-spin asymmetry in each part has b een found. In addition, a comparison b etween proton and neon asymmetries has shown a slightly higher asymmetry for neon than for proton, which is in agreement with theoretical predictions for nuclear DVCS. In the near future, the analysis of BSAs will b e extended to other nuclei. Hence, p ossible nuclear effects could b ecome more visible, including dep endences of the atomic numb er. In fact, with more statistics a distinction b etween different theoretical mo dels for nuclear mo dels seems p ossible. In addition, also the theoretical development in the field of nuclear DVCS probably needs more time. A b etter theoretical interpretation may eventually allow, through comparison to data of higher precision and also more complete kinematics information, to extract new information ab out the dynamical interplay of complex hadronic systems.

84


9 Appendix
Tables of Results for the Coherent Sample
xB j 0.03 0.06 0.09 bin - 0.06 - 0.09 - 0.20 xB j 0.049 0.073 0.110 xB j 0.058 0.071 0.081 xB j 0.055 0.075 0.106 -t (GeV 2 ) 0.016 0.019 0.027 Q
2

c

(GeV 2 ) 1.29 1.83 2.79 (GeV 2 ) 1.47 1.84 2.07 (GeV 2 ) 1.26 1.95 3.20

Asin LU -0.196 -0.236 -0.166 Asin LU -0.196 -0.305 -0.109

± ± ± ± ± ± ± ±

stat. 0.078 0.083 0.119 stat. 0.078 0.086 0.104

± sy st. ± 0.011 ± 0.012 ± 0.010 ± sy st. ± 0.011 ± 0.014 ± 0.008 ± ± ± ± sy st. 0.011 0.010 0.014

-tc bin 0 - 0.014 0.014 - 0.026 0.026 - 0.045 Q2 1.0 1.6 2.5 bin - 1.6 - 2.5 - 6.0

-t

c

(GeV 2 ) 0.009 0.019 0.034 (GeV 2 ) 0.017 0.020 0.026

Q

2

-t

c

Q

2

Asin LU -0.214 -0.158 -0.298

± stat. ±0.069 ±0.090 ±0.131

Table 9.1: The asymmetry Asin of the coherent sample p er kinematic bin at the reLU sp ective average kinematic value.

85


Tables of Results for the Incoherent Sample
xB j 0.03 0.08 0.13 bin - 0.08 - 0.13 - 0.35 xB j 0.059 0.102 0.182 xB j 0.099 0.111 0.121 xB j 0.067 0.108 0.185 -t (GeV 2 ) 0.157 0.148 0.191 (GeV 2 ) 0.075 0.174 0.374 (GeV 2 ) 0.135 0.162 0.222 Q
2

c

(GeV 2 ) 1.62 2.60 4.58 (GeV 2 ) 2.40 2.85 3.50 (GeV 2 ) 1.48 2.80 5.23

Asin LU -0.255 -0.203 -0.355 Asin LU -0.305 -0.155 -0.273 Asin LU -0.138 -0.379 -0.215

± ± ± ± ± ± ± ± ± ± ± ±

stat. 0.098 0.104 0.119 stat. 0.089 0.103 0.130 stat. 0.096 0.095 0.131

± sy st. ± 0.022 ± 0.018 ± 0.030 ± sy st. ± 0.026 ± 0.015 ± 0.023 ± sy st. ± 0.014 ± 0.031 ± 0.019

-tc bin 0.045 - 0.12 0.12 - 0.25 0.25 - 0.7 Q2 1.0 2.0 3.8 bin - 2.0 - 3.8 - 10

-t

c

Q

2

-t

c

Q

2

Table 9.2: The asymmetry Asin of the incoherent sample p er kinematic bin at the LU resp ective average kinematic value.

Table of Results for the Comparison of Neon and Hydrogen
-tc bin 0 - 0.045 0.045 - 0.18 0.18 - 0.7 xB j 0.07 0.10 0.12 -t (GeV ) 0.02 0.10 0.31
sin LU 2

c

Q

2

(GeV ) 1.7 2.5 3.3

2

Neon A ± stat. ± sy st. -0.203±0.051±0.011 -0.295±0.075±0.025 -0.157±0.099±0.015
sin LU

Hydrogen A ± stat. ± sy st. -0.152±0.051±0.010 -0.257±0.045±0.022 -0.124±0.065±0.012
sin LU

Table 9.3: Comparison of A

b etween neon and hydrogen p er tc bin.

86


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Acknowledgements
This thesis could not have b een written without the help of many p eople and I would like to thank all of them for their supp ort. First and foremost, I would like to thank my advisor Dr. Wolf-Dieter Nowak who encouraged me to b egin this work at HERMES and always found the time and patience to discuss all questions and to listen to my presentations. I am also grateful for his careful reading of this manuscript and his detailed comments and suggestions on it. My sp ecial thanks go to Prof. Dr. Butz who agreed on sup ervising this dissertation as my official advisor at the University of Leipzig. I want to express my gratitude to all Hermes memb ers, esp ecially to Elke Aschenauer and Delia Hasch, for their supp ort throughout the work on the diploma thesis. In particular I would like to thank Frank Ellinghaus, who intro duced me in the DVCS analysis, Hayg Guler and Frank Zhenyu Ye for their helps on my data analysis and for all the inspiring discussions on physics as well as on unrelated topics. Outside the DVCS group I b enefited from discussions with Achim Hillenbrand and Markus Diefenthaler, who help ed me with their patient advices and with software problems of all kind. I would like to thank Jim Stewart and Eduard Avetisyan for their knowledge and explanation of the Hermes detectors. My thanks also go to all Hermes memb ers who are not mentioned ab ove and I very much appreciated the friendly athmosphere in the entire group, who made work and life in Hamburg so enjoyable. Finally, I would like to thank my friends and my family for their supp ort and understanding.

91


Declaration
This do cument is the result of my own work. Material from the published or unpublished work of otheres, which is referred to in the do cument, is credited to the author in the text.

Selbst¨ andigkeitserkl¨ arung
Hiermit erkl¨ ich, die vorliegende Diplomarb eit selbst¨ are andig und nur unter Verwendung der angegeb enen Hilfsmittel und Quellen angefertigt zu hab en.

Leipzig, den 16.08.2005

92