Phenomenological Relationships Between the EMC Effect and Short-Range Correlations

Amy LovellOffice of Science, Summer Undergraduate Laboratory Internship Program

Rensselaer Polytechnic Institute

Thomas Jefferson National Accelerator FacilityNewport News, Virginia

July 29, 2011

Prepared in partial fulfillment of the requirements of the Office of Science, Department of Energy’s Summer Undergraduate Laboratory Internship Program under the direction of Douglas W. Higinbotham in the Hall A Division of the Thomas Jefferson National Accelerator Facility.

Participant: ________________________________________Signature

Research Advisor: ________________________________________Signature

Abstract

Phenomenological Relationships Between the EMC Effect and Short-Range Correlations. AMY LOVELL (Rensselaer Polytechnic Institute, Troy, New York, 12180) DOUGLAS W. HIGINBOTHAM (Thomas Jefferson National Accelerator Laboratory, Newport News, VA).

High-energy scattering ratios of deep inelastic per-nucleon cross sections of various nuclei were expected to be unity. In 1983, however, the European Muon Collaboration discovered this was not true, and a myriad of models were derived to explain this deviation from unity, which became known as the EMC effect. In 2011, a connection was made between the EMC effect and the plateaus observed in the cross section ratios from elastic scattering of the same nuclei, which were believed to be caused by short-range correlations (SRCs). The slope of the EMC region was shown to be directly proportional to the SRC scaling factor (the magnitude of the plateau where SRCs dominate): the scaling factors increased as the slopes became more negative. The focus of this project was to search for phenomenological relations between the EMC effect and SRCs, where knowledge of either the slope of the EMC effect or the SRC scaling factor would allow the discernment of the other. To accomplish this, previously measured per-nucleon cross section ratios for 3He, 4He, and 12C were plotted for the inelastic region, fitted, and compared with the SRC scaling factors in the elastic region. The missing area (the area between the EMC data and unity) within the deep inelastic region and the extra area (that area bounded by the x-axis and the SRC scaling factor) were calculated for these ratios. It was found that this missing area was directly proportional to the extra area, through a multiplicative constant. Thus, the two regions can be related through a sum which states that the area within the EMC region plus a constant times the area in the SRC region equals one. This relation further emphasizes the notion that the EMC effect and SRCs are not disjoint phenomena but, rather, are closely linked results of the same underlying physics. Also, this sum rule adds a more fundamental reasoning to the observation that the slopes of the EMC region and the SRC scaling factors are directly proportional. In the future, this relationship will allow a concise explanation to be given as to the nature of these effects and will give insight into what is occurring within the nucleus that causes these per-nucleon cross section ratios to differ from unity.

Introduction

The European Muon Collaboration found in 1983 that the ratio of the per-nucleon cross

sections of iron to deuterium differed from unity, falling below one. This became known as the

EMC effect and was later confirmed by the Stanford Linear Accelerator Center (SLAC) [1]. The

unexpected data spanned the range of Björken-x from approximately 0.3 to 0.7 (where Björken-

x, xB, is defined as Q2/(2mω); where Q2 is the four-momentum transfer, m is the mass of a

proton, and ω is the difference between the energy of the electron beam and the measured

scattered electron energy). The per-nucleon cross section ratios began at unity and then dipped,

reaching a minimum close to xB = 0.7, after which they once more rose towards unity and above,

as the quasi-elastic (QE) region was reached (see Fig. 1).

Although it was initially believed that the EMC effect was dependant on either the

number of nucleons within the nucleus (A) or the density of the nucleus (as the experimental data

from SLAC displayed the EMC effect increasing with both nuclear density and number of

nucleons [1]), more precise data from the Thomas Jefferson National Accelerator Facility

(Jefferson Lab) using light nuclei, A ≤ 12, concretely showed otherwise. While the effect was

found to be the weakest in 3He (data was observed to be closest to unity), the other three targets,

4He, 9Be, and 12C, displayed the same magnitude of the effect, despite having different densities

and number of nuclei. 9Be and 12C were found to have nearly the same slope, even though the

average density of 9Be is much lower than that of 12C. Thus, the A-scaling and density-scaling of

the EMC effect broke down in light nuclei – leading to the hypothesis that, perhaps, the effect

had to do more with local density than average density [2].

In separate experiments at high values of xB (typically greater than 1.3), it was found that

the per-nucleon cross section ratios were independent of Björken-x. This effect was used to

study short-range correlations (SRCs) between the nucleons within the nucleus, as their wave

functions have a finite probability of overlapping at the small distances found within the nucleus.

These correlations were then discovered to be the cause of the high momentum tail of nucleon

momentum distributions within the atomic nuclei [3]. Experimentation showed that non-zero

plateaus formed in this region (Fig. 1), which was outside of any theory. Even though

protons were not thought to vary from element to element, it was not hypothesized that a

2H nucleus would be correlated to an 56Fe nucleus. The magnitude of this plateau is the SRC

scaling factor.

Although data of the EMC effect and SRCs had traditionally both been plotted against

Björken-x, the two were never expected to be correlated – the EMC effect having to do with

deep inelastic scattering (DIS) while SRCs only concerned quasi-elastic scattering. However, in

a conversation between Doug Higinbotham and Hugh Montgomery, the two regions of Björken-

x (xB < 1 and xB > 1) were discussed and shown taped together, giving rise to the idea that neither

should be considered a separate entity [4]. Plots of the EMC slopes against the SRC scaling

factors further correlated the two regions. The magnitude of the scaling factors increased as the

slopes became more negative. This seemed to imply that the two effects stem from the same

underlying physics, and that, in knowing the value of either the slope or the scaling factor, one

would be able to predict the unknown quantity [5]. Further relationships between the two effects

could lead to new, deeper understandings of the physics behind what happens to a nucleon in the

nucleus.

Materials and Methods

In knowing that the slopes of the EMC effect and SRC scaling factors were directly

proportional, the conclusion drawn was that the unexpected areas around these slopes and scaling

factors were similarly correlated. Because it had been expected that the per-nucleon ratios for x B

< 1 would have a constant value of one, there was missing area in the region between the

horizontal line at unity and the line formed by the EMC effect. Likewise, the expectation of the

SRC region to fall to zero gave rise to extra area for xB > 1, bounded by the x-axis and the line

formed by the SRC scaling factor plateau. Using a linearly simplified picture of the per-nucleon

cross section ratio of an unspecified element to 2H, as shown in Fig. 2, an attempt was made to

equate the two areas, through the use of a function, f(x), where the correlation would take the

form

∫.3

.7

σEMC ( x )∗f (x ) dx=∫1.3

1.9

σSRC ( x )∗f ( x ) dx .(1)

Here, σEMC(x) is the per-nucleon cross section ratio for xB < 1 and σSRC(x) is the per-nucleon cross

section ratio for xB > 1. Various forms of f(x) were tested, including xa, eax (-5 ≤ a ≤ 5), ln(x),

and 1/ln(x).

Using data from Hall C at Jefferson Lab [6], the per-nucleon cross section ratios for 3He,

4He, 9Be, and 12C to 2H were plotted from approximately xB = 0.3 to xB = 0.8 and fitted. These

fitted lines, σEMC(x), were then used in the left hand side of Eq. 1 along with the SRC scaling

factors from [4] for σSRC(x) in the right hand side. The aforementioned functions, f(x), were

again utilized to equate the data.

With the EMC lines from [6], the area between unity and σEMC(x) was calculated between

0.3 ≤ xB ≤ 0.7 for 3He, 4He, and 12C. The area bounded by the x-axis and the SRC scaling factor

(from [4]), for 1.3 ≤ xB ≤ 2, was calculated for the same three targets. Fig. 3 shows the

simplified ratios used to calculate the missing and extra areas for 3He, 4He, and 12C. The missing

area from the EMC region was plotted as a function of the extra area in the SRC region (Fig. 4),

and a line was fitted through the three points (along with a fourth point, corresponding to the

missing EMC and extra SRC areas for 2H, both of which were zero).

Data points from [6] were normalized so that each of the EMC lines had a y-intercept of

1, as overall normalization does not change the value of the slope – it merely raises or lowers all

of the points simultaneously. [5] Thus, for the DIS region, a line of the form

σ EMC (x )=mEMC∗x+1(2)

was used. The form of the equation for the elastic scattering region was

σ SRC ( x )=C A(3)

where CA is the SRC scaling factor.

Because the slope of the EMC effect (mEMC) and the SRC scaling factor (CA) have been

shown to be linearly correlated, experimental values of mEMC and CA were used to solve for a

constant, k, that would relate the two magnitudes, in the form

−mEMC=k∗CA .(4)

The average k was then computed. Values for mEMC were derived from [1, 2, 5] and the values of

CA came from [4, 5, 7, 8]. The known experimental values of CA (for 3He, 4He, 9Be, 12C, 27Al,

56Fe, and 197Au) were then used to solve for the corresponding EMC slopes, and the known

experimental EMC slope values (for the same nuclei) were used to solve for the corresponding

SRC scaling factors. These calculated values were then compared with the most recent

experimental value for each target in the form of a ratio of theoretical results to experimental

results. Fig. 5 shows the ratios of the slopes, and Fig. 6 shows the ratio of the plateaus.

Experimental slopes calculated from [1] and scaling factors from [7] (where the corresponding

slope or scaling factor had not previously been measured) were used in Eq. 4 to solve for the

unknown scaling factor or slope, respectively. Slopes were calculated for 63Cu, and scaling

factors were calculated for 40Ca and 108Ag.

Results

It was found that x-2 correlated the missing EMC area and the extra SRC area when the

linearly simplified scattering ratio was analyzed (although the SRC area was greater by a factor

of two). However, this same relationship did not work when the EMC slopes from [6] and the

SRC scaling factors from [4] were analyzed by the same method. The multiplicative factor was

not constant when comparing the two regions using ratio data for 3He, 4He, and 12C, no matter

which of the functions was applied to Eq. 1. However, when just the missing areas were plotted

as a function of the extra areas for these three nuclei (along with 2H) as shown in Fig. 6, there

was a strict linear correlation. Because of this linear correlation, the following sum rule was

devised:

1=∫0

1 σ A(x )σ D(x )

dx+m∫1

2 σ A(x )σ D(x )

dx (5)

where the per-nucleon cross section ratio taken from 0 to 1 is equivalent to Eq. 2 and the ratio

from 1 to 2 is equivalent to Eq. 3. In expanding this integral – using Eq. 2 and Eq. 3 – one has

−mEMC=2∗m∗CA .(6)

Combining Eq. 4 and Eq. 6, it is seen that,

m= k2

.(7)

Here, k was found to be 0.0585 ± 0.0081, leading to a value of 0.02825 ± 0.00405 for m, the

correlation factor. Using this value and the known SRC scaling factors, Eq. 6 was solved for the

EMC slopes, values of which can be found in Table 1, and were then compared with the

experimentally observed EMC slopes. Fig. 7 shows the known EMC slopes and the calculated

EMC slopes, along with error lines – greater than and less than ten percent of the highest and

lowest experimental data. This same analysis was done using the known EMC slopes and Eq. 6

to solve for the SRC scaling factors (Table 2). Fig. 8 shows the experimentally observed SRC

scaling factors along with the calculated scaling factors and two lines showing a plus and minus

ten percent error. All theoretical and calculated values for 3He are shown in Fig. 9, along with

the errors.

Using this model, only 58% of the slopes from the EMC region were calculated within

ten percent of an experimentally observed slope, and only 53% of the SRC scaling factors were

calculated within ten percent of an experimentally observed plateau values. Throughout the data,

calculations for 3He consistently produced higher slopes (considering the absolute value of the

slope) and lower scaling factors than the experimental data had shown. When these calculations

for 3He are not taken into account, 73% of the slopes are accurate within ten percent, and 60% of

the scaling factors are accurate within ten percent. In using only the current data for 3He, a value

of 0.01935 ± 0.00085 for the correlation factor gives the correct values for the slopes and scaling

factors within ten percent of the experimentally known data.

Because the theoretical data for 3He differed consistently from the experimental data, k

was again calculated, this time without taking into account the 3He data. In this case, k was

found to be 0.0625 ± 0.0015, leading to a value of 0.03125 ± 0.00075 for m. When 3He is not

considered, 73% of the theoretically calculated data for the EMC slope falls within ten percent of

an experimental value for each given nuclei, and 80% of the data for the SRC scaling factor falls

within ten percent of an experimental value. (When 3He is considered – using the same value of

m – 58% of the EMC slopes are accurate within ten percent, and 71% of the SRC scaling factors

are accurate within the same range.) Reasons for this subsequent calculation are discussed in the

next section.

Using the experimental slopes for 40Ca and 108Ag calculated from [1], Eq. 6 was solved

for the scaling factors. The theoretical scaling factor for 40Ca is predicted to be 5.86, and the

scaling factor for 108Ag is likewise predicted to be 7.38. Similarly, using the experimental

scaling factor for 63Cu from [7], the theoretical EMC slope is predicted to be -0.3423.

Discussion/Conclusion

As was exemplified in the previous section, the theoretical calculations using 3He do not

fit the experimental data – when using this sum model and assuming an error of ten percent

greater than and less than the range of accepted values. However, the uncertainty in the

measurements of the per-nucleon cross sections of 3He to 2H is much greater than ten percent.

For a slope of -0.0700, the uncertainty is ± 0.0290, which is over four times the uncertainty

percentage for the remaining slopes. When this is taken into account, one of the theoretical

slopes does fall within the experimental error (the slope calculated using the scaling factor from

[8]), as shown in Fig. 9.

This large error displays a need for more accurate data for the cross sections of 2H and

3He. For heavier (solid) targets, the exact densities can be known very precisely, leading to

extremely accurately applied corrections. On the other hand, it is difficult to know the exact

density of the cryogenic targets as the beam traverses the length of the target. If the target’s

temperature increases locally, its density changes, which amplifies the intricacy in applying

corrections. More accurate cross sections would increase the accuracy of the per-nucleon ratios,

leading to an increase in accuracy of the EMC slopes and SRC scaling factors.

As previously mentioned, if 3He is taken independently, a value of 0.01934 ± 0.00085 for

the correlation factor gives the experimentally observed EMC slopes and SRC scaling factors

within ten percent of the most current experiment values. There are numerous reasons for this

discrepancy between the correlation factor of 3He and the one used for the rest of the nuclei, the

two most notable being either a flaw within the sum model or a flaw within the isospin

corrections for asymmetric nuclei. Fig. 3 in [2] shows the raw data for 3He along with the same

data after isospin corrections were applied. The author readily admits that there is still

uncertainty in this correction because of the uncertainty in the understanding of the nuclear

structure function of the neutron [2]. The heavier nuclei need less corrections of this type (with

symmetric nuclei needing none of these corrections), and thus, the slopes (and scaling factors) of

the heavier nuclei used in the above calculations have less uncertainty than the slope of 3He. It is

because of this inherent uncertainty that 3He was not included in the final calculation of the

correlation factor.

If there is a flaw within the isospin correction for 3He (and assuming that this model is

correct), we should be able to predict the proper corrected slope for 3He. Using the SRC scaling

factor from [7] in Eq. 6, the predicted value of the EMC slope for 3He is -0.1375 ± 0.0188. This

is larger than the corrected slope extracted from [2], as seen in Fig. 10, but it does follow the

trend of the EMC effect for 4He and heavier nuclei (as would be expected from the model).

In the future, not only should more data be taken using the cryogenic targets for more

accurate results, experiments should also span the range of Björken-x from 0 to 2, to collect data

in the EMC and SRC regions during the same run. This would give a more constant

normalization to the data from each target, which, in turn, could give a more accurate

phenomenological correlation between the two regions. If this sum model is correct, more

accurate data would lead to a more precise calculation of the correlation factor – leading to

greater ability to accurately predict unknown values of either the EMC slope or the SRC scaling

factor.

Because Eq. 5 uses the area over the entire DIS region whereas the EMC slopes are only

falling through the range of 0.3 ≤ xB ≤ 0.7, after the original data analysis, the integral was

modified to reflect this smaller region. However, this would only change the constants within

Eq. 5. With this new range, the correlation factor would be multiplied by a constant that depends

on the limits of integration. Also, the sum would no longer equate to one; instead, it would

equate to the positive difference between the limits of integration. Likewise, the region of

integration over the SRC scaling factor can be modified to reflect only the region where the cross

section ratios are independent of Björken-x (generally beginning at approximately xB = 1.3).

Again, this modification only multiplies the correlation factor by a constant.

Because these changes only scale the correlation factor, the previous argument for the

normalization of all EMC lines is increased. If each target had a unique y-intercept, more

variables would be introduced into the calculation of the EMC slopes, increasing the difficulty of

solving for the unknown values. By fixing the y-intercept at 1, the direct correlation between the

slopes and scaling factors is maintained. Since it is the EMC slope or the SRC scaling factor that

is being predicted, the overall normalization does not matter, as long as it is kept constant for all

nuclei.

However, changing the cuts in the range of data does change how effective this specific

relationship is. Using the cuts that were referenced in [2] to refit the slopes, Eq. 5 only held for

50-60% of the data. Instead of 3He being the only outlier, 4He also did not fit within the model,

along with several other values for the various nuclei. To make the same sum correlation hold,

the lines of the EMC effect had to be normalized with a y-intercept of 1.1, and the correlation

factor rose to approximately 0.0397. While this new relationship accurately correlates the data,

it perhaps leads one to wonder why certain cuts were originally chosen. Yet, it is still interesting

to note that the two effects can be linearly correlated, even with different cuts.

While it could be sufficient to simply state that the measureable results of the two effects

are linearly proportional, this would not lead to a deeper understanding of the underlying physics

of these two regions. There is hope that, in providing this sum as a phenomenological

relationship between the areas of the EMC and SRC regions, a more in depth study will be

undertaken in order to explain the physics that correlates these effects. Before 2011, the EMC

effect and short-range correlations had been thought of as entirely separate entities, but now,

connections are being made that could definitively shed light on the similarities of the two. On

the surface, this correlation allows the prediction of the SRC scaling plateau if the slope of the

EMC effect is known, along with the converse. On a deeper level, however, the relationship

between these two phenomena can lead to a more profound understanding of the close bond

between nuclear physics and particle physics, as well as the small region where their boundaries

are blurred.

Acknowledgements

I would like to thank my mentor, Doug Higinbotham, for all of the help that he has given

me this summer. Without his knowledge, guidance, and expertise, this project would have never

arrived at this point; he gave me information when I needed it, a direction to follow when I was

lost, and the confidence to make a couple of paths of my own. I am also extremely grateful to

the Office of Science, Department of Energy, and Jefferson Lab for giving me this wonderful

opportunity this summer, along with all of the SULI education staff for making this program

possible. A special thanks goes to Lisa Surles-Law for putting together all of our activities this

summer, making it possible for me to do my best work and to Brita Hampton for reminding me

to enjoy myself while getting all of this work accomplished.

References

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Phys. Rev. D 49 (1994) 4348.

[2] J. Seely, et al., New Measurements for the European Muon Collaboration Effect in Very

Light Nuclei, Phys. Rev. Lett. 103 (2009) 202301.

[3] K.Sh. Egiyan, et al., Observation of Nuclear Scaling in the A(e,e’) Reaction at xB > 1 (2008)

arXiv:nucl-ex/0301008v1.

[4] D.W. Higinbotham, J. Gomez, and E. Piasetzky, Nuclear Scaling and the EMC Effect,

(2010).

[5] L.B. Weinstein, E. Piasetzky, D.W. Higinbotham, J. Gomez, O. Hen, and R. Shneor, Short

Range Correlations and the EMC Effect, Phy. Rev. Lett. 106 (2011) 052301.

[6] J. Seely, A. Daniel, et al., General Information Experiment E03103 (nucl-ex/0904.4448),

June 25, 2009, https://hallcweb.jlab.org/experiments/E03103/.

[7] N. Fomin, et al., New measurements of high-momentum nucleons and short-range structures

in nuclei, (2011).

[8] L.L. Frankfurt, M.I. Strikeman, D.B. Day, and M. Sargsyan, Evidence for short-range

correlations from high Q2 (e,e’) reactions, Phys. Rev. C 48 (1993) 2451.

Figure 1: Plot of the per-nucleon cross section ratio for an unspecified element to 2H. The EMC region – deep inelastic

scattering – is shown in green (0.3 ≤ xB ≤ 0.7), the quasi-elastic region in black (0.7 < xB ≤ 1.3), and the region of short-

range correlations is shown in blue (xB > 1.3). This data most closely resembles the ratio of 3He to 2H.

(DataFunction.root)

Figure 2: This plot shows a simplified diagram of the per-nucleon cross section ratio of an unspecified element to 2H (with

the curves approximated as straight lines). Here, the slope of the EMC region from 0.3 ≤ xB ≤ 0.7 is -0.5 and the SRC

scaling factor, CA, is 2. (Areas.root)

Figure 3: A plot showing the simplified versions of the per-nucleon ratios for 3He, 4He, and 12C, along with the ratio that

was expected for all of the data. Slopes were derived from [2] and plateau scaling factors were from [4]. The slopes were

-0.0809, -0.2843, and -0.2889 for 3He, 4He, and 12C respectively, and the scaling factors were 2, 4, and 4.8, for the same

elements, respectively. (SimpleRatios.root)

Figure 4: This plot shows the correlation between the missing area below unity for the EMC region and the extra area

between the SRC scaling factor plateau and the x-axis. The slope of the best fit line of the four data points is 0.0075 and

the y-intercept is -0.0006. Among these four points, 3He is clearly the outlier. (CorrelatedAreas.root)

Figure 5: This plot shows the ratio of theoretically calculated EMC slopes to experimentally observed EMC slopes as a

function of the number of nucleons. The experimental values are for 3He, 4He, 9Be, and 12C are from [2], and the values

for 27Al, 56Fe, and 197Au are from [1]. The references in the legend denote from which source the SRC scaling factors came

(SRC scaling factors were then used to calculate the theoretical slope values). Red lines indicate the deviation from unity

of each of the accepted slopes when error is taken into account. (SlopeRatio.root)

Figure 6: This plot shows the ratios of theoretically calculated SRC scaling factors to experimentally observed SRC

scaling factors as a function of the number of nucleons. The experimental values for 3He, 4He, 9Be, 12C, and 197Au are from

[7], the value for 27Al is from [8], and the value for 56Fe is from [4]. The references in the legend denote from which source

the EMC slopes came (these slopes were then used to calculate the theoretical value of the scaling factors). Red lines

indicate the deviation from unity of each of the accepted scaling factors when error is taken into account.

(PlateauRatio.root)

Target Source Experimental Scaling Factor

Theoretical Slope

He-3 [5] 1.97 -0.1231He-3 [8] 1.70 -0.1063He-3 [4] 2.00 -0.1250He-3 [7] 2.20 -0.1375He-4 [5] 3.80 -0.2375He-4 [8] 3.30 -0.2063He-4 [4] 4.00 -0.2500He-4 [7] 3.79 -0.2369Be-9 [7] 4.12 -0.2575C-12 [5] 4.75 -0.2969C-12 [8] 5.00 -0.3125C-12 [4] 4.80 -0.3000C-12 [7] 5.06 -0.3163Al-27 [8] 5.30 -0.3313Fe-56 [5] 5.58 -0.3488Fe-56 [8] 5.20 -0.3250Fe-56 [4] 5.70 -0.3563Au-197

[8] 4.80 -0.3000

Au-197

[7] 5.52 -0.3450

Table 1: This table shows the known experimentally determined SRC scaling factors (sources list from which piece of

literature the scaling factor came) and the corresponding calculated EMC slopes. Slopes in green fall within ten percent

of an experimentally observed slope, and those in red do not fall within this ten percent range. Experimental slopes can

be found in Table 2.

Figure 7: These six plots show the experimentally observed EMC slopes (in black) and the theoretically calculated slopes (in green). The thick red lines denote plus and minus

ten percent on either side of the highest and lowest experimental slopes, respectively. Dashed lines denote data from [1], dotted from [2], dashed-dotted from [4], long dashes from

[5], dashed with three dots from [7], and medium dashes from [8]. Exact slopes (and the SRC scaling factors from which they were calculated) are given in Table 1.

(Slopes1.root)

Target Source

Experimental Slope Theoretical Scaling Factor

He-3 [2] -0.0809 1.29He-3 [5] -0.0700 1.12He-4 [1] -0.1904 3.05He-4 [2] -0.2177 3.48He-4 [5] -0.1970 3.15Be-9 [1] -0.2382 3.81Be-9 [2] -0.2843 4.55Be-9 [5] -0.2430 3.89C-12 [1] -0.3297 5.28C-12 [2] -0.2889 4.62C-12 [5] -0.2920 4.67Al-27 [1] -0.3084 4.93Al-27 [5] -0.3250 5.20Fe-56 [1] -0.3681 5.89Fe-56 [5] -0.3880 6.21Au-197 [1] -0.4184 6.69Au-197 [5] -0.4090 6.54

Table 2: This table shows the known experimentally determined slopes of the EMC effect (sources list from which piece

of literature the slope was calculated) and the corresponding calculated SRC scaling factor. Scaling factors in green fall

within ten percent of an experimentally observed scaling factor, and those in red do not fall within this ten percent range.

Experimental scaling factors can be found in Table 2.

Figure 8: These six plots show the experimentally observed SRC plateaus (in black) and the theoretically calculated SRC scaling factors (in blue). The thick red lines

denote plus and minus ten percent on either side of the highest and lowest experimental plateaus, respectively. Dashed lines denote data from [1], dotted from [2], dashed-

dotted from [4], long dashes from [5], dashed with three dots from [7], and medium dashes from [8]. Exact scaling factors (and the EMC slopes from which they were

calculated) are given in Table 2. (Plateaus2.root)

Figure 9: These plots shows the experimentally observed EMC slopes and SRC plateaus (in black) and the theoretically

calculated EMC slopes (in green on the left) and SRC plateaus (in blue on the right) for 3He. Thick red lines denote plus

and minus the experimental error on the highest and lowest experimentally observed slopes and plateaus. Dotted lines

denote data from [2], dashed-dotted from [4], long dashes from [5], dashed with three dots from [7], and medium dashes

from [8]. (Helium3.root)

Figure 10: This plot shows the cross section ratio of 3He from [6]. Filled black circles show the ratio of 3He to 2H after the

correction for isospin had been applied. The solid green line depicts the slope of the EMC effect as predicted by the

model discussed in this paper. It has been normalized to the approximate range of the isospin-corrected data. The red

squares indicate the raw data for this cross section ratio. (Helium3Correction.root)