Visual Representation of Correlation Use the Online Library to find five visual representations of correlations. Discuss what i

Quark correlations in the ground and excited nucleon states via the photo-absorption sum rules

Sergo Gerasimov1,∗

1Joint Institute for Nuclear Research, Dubna, Russia

Abstract. The account of the dynamical correlations of quarks composing the nucleons is necessary for relevant description of nucleon and nuclear interac- tions. The present work is devoted to the use of the technique of known integral sum rules for the cross-sections and amplitude of the photo-hadron processes. In particular, a kind of the visual representation and quantification of the va- lence quark correlation characteristics referring to the photo-excitation of the nucleon resonances can be presented.

1 Introduction: the past experience

Instead of well-known non-relativistic dipole sum rules [1] for atomic and nuclear photo- effect,derived in the center-of-mass of the non-relativistic target systems:

σn(E1) = ∫ ∞

thr dω ωn σE1(ω),

where: n = −2 refers to Kramers-Heisenberg sum rule (SR) for static electric-dipole polar- izability of a given quantum system; n = −1 is the bremsstrahlung-weighted SR, depending of charged “parton” correlation in a given system; n = 0 is the famous Thomas-Reiche-Kuhn SR, known as a precursor of the Quantum Mechanics itself, we should accept the picture of the transverse quark current densities of the fast moving nucleons. Just this framework gives fruitful and widely used approach for the description of the electromagnetic structure parameters of the nucleon.

In this paper, we return to the application of the current algebra technique in the infinite momentum frame used previously in the derivation of the Cabibbo–Radicati [2] sum rule and in the derivation of the earlier known from the dispersion theory [3–5] and current algebra [6] sum rules for the anomalous magnetic moments of the nucleon. In this way, a certain number of relations will be obtained directly depending on the flavor quark correlation in the ground nucleon state and in the states including an overlap of the ground and excited states of the nucleon.

2 Sum rules with relativistic spin degrees of freedom

In the relativistic theories the operators of electric dipole moments depend on the spin and following formally to the pz → ∞ techniques derivation of the Cabibbo-Radicati [2] or the ∗e-mail: [email protected]

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© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).

GDH s.r. derivation in the infinite momentum frame [6] we can derive the relations

4π2α( 1 3 < ~D2 > −(

κN 2mN

)2) = ∫

dν ν σrestot (ν), (1)

4π2α[ 1 3 < ~D2 >] =

∫ ∞

thr

dν ν

(σresp (ν), (2)

where σp(a) refers to the interaction cross section of the polarized nucleon and polarized photon with parallel (or anti-parallel) spins, κn-anomalous magnetic moment of the nucleon in the natural units,

D̂ = ∫

~xρ̂(~x)d3 x = 3∑

j=1

Qq( j)~d j. (3)

The replacement of the constant κ2 by its integral form illustrates and fulfills the essential element of the relationship and interdependence of the description of the static characteristics of a given system and known properties of its excited states.

The defined operators Qq( j) and ~d j are the electric charges and configuration variables of point-like interacting quarks in the infinite-momentum frame of the bound system. Finally, we relate the electric dipole moment operator correlations, successively for the proton, the neutron and the pure "isovector-nucleon" part equal for both nucleons and the isovector part of the mean-squared radii operators entering the Cabibbo-Radicati sum rule, which all are sandwiched by the nucleon state vectors in the "infinite – momentum frame", with experi- mentally measurable data on the resonance parts of the photoabsorption cross sections on the proton and neutron presently known below ∼ 2 GeV. The listed operator mean values are parameterized as follows [7]

RV = 1 6

(< r21 >P − < r 2 1 >N ) =

1 3

(α− 1 2 β), (4)

JP = 1 3 < D̂2 >P=

8 27 α +

1 27 β +

8 27 γ−

8 27 δ, (5)

JN = 1 3 < D̂2 >N =

2 27 α +

4 27 β +

2 27 γ−

8 27 δ, (6)

JV = 1 3 < D̂2 >V =

2 12 α +

1 12 β +

2 12 γ−

4 12 δ, (7)

where < ~d1 2 >=< ~d2

2 >=α,< ~d3

2 >=β, < ~d1 · ~d2 >=γ, < ~d1 · ~d3 >=< ~d2 · ~d3 >=δ and indices

"1" and "2" refer to the like quarks, and "3" to the odd quark. Evaluation of the relativistic electric dipole moment fluctuation was carried out with the

available compilation [8] of the resonance pion-photoproduction data on the proton and neu- tron AP(N)1/2 and A

P(N) 3/2 the numerical values we calculated and all integrals over photoexcited

nucleon resonances were taken in the narrow resonance approximation, when

Jresp(a) ' 4πmn|Ares3/2(1/2)|

2

m2res − m2n , (8)

where mn(res) is the nucleon (or resonance) mass. Solving the system of the linear equations and evaluating the RV, JP,N,V with the help of experimentally known partial amplitudes of main photo-excited resonances, we find our final results for the numerical values α, β and the opening angle θ12 and θ13 between vectors ~d1 and ~d2 and vectors ~d1 and ~d3, lying in the plane transverse to the "infinite momentum"’ pz→∞ vector [9]:

α1/2 = 0.75 ± 0.06 fm, β1/2 = 0.77 ± 0.12 fm, θ12 ' 120 0, θ13 ' θ12 ∼ 120

0.

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We note that we could use the isoscalar input

JS = 1 3 < D̂2 >S =

1 108

(2α + β + 2γ + 4δ) (9)

instead of the isovector one in Eq. (7). The numerical values of the parameters α,β,γ,δ would turn out to be the same due to the use of the standard isotopic symmetry relations relating JV and JS with the JP and JN .

Summarizing the results of the global approach via the use of the sum rules, we obtain a kind of averaged picture across the spectrum pattern with the presumably dominating influ- ence of the first ∆(1236)-resonance with its magnetic-dipole photo-excitation of the nucleon spin degrees of freedom, which is also hiding explication of the qualitative isospin-depending features of the (mainly, electric-dipole) 2-nd and (the electric-quadrupole) 3-rd resonance photo-excitation regions. The value of effective dipole moments, determining the excitation cross section of the "electric-dipole-type" nucleon resonances, includes apparently the pionic degrees of freedom of the constituent quarks. This is supported by the fundamental principle of the charge symmetry of strong interaction dynamics. The participation of the pion degrees of freedom plays the role in the dynamical formation of the quark-diquark spatially looking nucleon structure. We suggest to illustrate it numerically choosing the experimental data on the resonance photo-absorption data on the J P = 3/2−, P13(1500) and J P = 3/2−, D33(1600)- resonances,composing the region of the "second" resonance area. Performing the evaluations analogous the earlier presented for all the resonances with the data from the PDG table [8], we obtain [9] the results differing from earlier cited

α1/2 = 0.82 fm, β1/2 = 0.53 fm, θ12 ' 130 0, θ13 ' 70

0, θ23 ∼ 160 0.

A pronounced asymmetry of the sides of the triangles Oud and Ouu

Cminud Cuu

' 0.8 fm 1.5 fm

(10)

can testify to the validity of the quark-diquark model of the considered resonances though diquarks are not apparently strongly bound. On the other hand, the obvious need to fulfill the charged symmetry requirements implies a significant role of the isospin-dependent quark- quark potentials, that is the role of the interactions caused by the pion exchanges, for instance. Ultimately, however, both factors can be the sides of the same phenomenon.

References

[1] J.S. Levinger, H.A. Bethe, Phys. Rev. 78, 115 (1950) [2] N. Cabibbo, L.A. Radicati, Phys. Lett. 19, 678 (1966) [3] S.B. Gerasimov, Sov. J. Nucl. Phys 2, 430 (1966) [4] S.B. Gerasimov, Sov. J. Nucl. Phys 5, 1263 (1967) [5] S.D. Drell, A.C. Hearn, Phys. Rev. Lett. 16, 908 (1966) [6] M. Hosoda, K. Yamamoto, Prog. Theor. Phys. 36, 425 (1966) [7] S.B. Gerasimov, Czech. J. Phys. 55, A209 (2005) [8] K.A. Olive, et al. Chin. Phys. G 38, 090001 (2014) [9] S. B. Gerasimov, J. Phys.: Conf. Ser. 678 012009 (2016)

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Use of a real‑life practical context changes the relationship between implicit body representations and real body measurements Lize De Coster1*, Pablo Sánchez‑Herrero2, Jorge López‑Moreno2,3 & Ana Tajadura‑Jiménez1

A mismatch exists between people’s mental representations of their own body and their real body measurements, which may impact general well‑being and health. We investigated whether this mismatch is reduced when contextualizing body size estimation in a real‑life scenario. Using a reverse correlation paradigm, we constructed unbiased, data‑driven visual depictions of participants’ implicit body representations. Across three conditions—own abstract, ideal, and own concrete body— participants selected the body that looked most like their own, like the body they would like to have, or like the body they would use for online shopping. In the own concrete condition only, we found a significant correlation between perceived and real hip width, suggesting that the perceived/real body match only exists when body size estimation takes place in a practical context, although the negative correlation indicated inaccurate estimation. Further, participants who underestimated their body size or who had more negative attitudes towards their body weight showed a positive correlation between perceived and real body size in the own abstract condition. Finally, our results indicated that different body areas were implicated in the different conditions. These findings suggest that implicit body representations depend on situational and individual differences, which has clinical and practical implications.

The way we perceive our body has been shown to be different from what our actual body looks like, creating a mismatch between our body size/shape and the mental representations of our body (body representations)1–4. These distortions have been observed for both explicit (conscious body representations that are thought to be primarily constructed through visual perception of the body and cognitive-affective factors) and implicit (unconscious body representations that arise through somatosensation such as the sense of position5 and tactile perception6, proprioception, and interoception) body representations7–9. Furthermore, both explicit and implicit distorted self-body representations have been linked to general well-being, as well as clinical disorders such as body dysmorphia disorder and anorexia nervosa10–14.

A separate distinction that has been used to refer to bodily self-representations is the difference between online and offline body representations. Thus far, research has largely focused on online body representations15,16, defined as representations that are updated in real-time through continuous multisensory integration17,18, a process associated with activation in posterior parietal and premotor regions19–23. However, far less focus has been directed at the study of offline self-body representations, defined as abstract models of our body stored in memory that guarantee the spatial coherence of the body18, even though they have been shown to be important for our long-term body representations24,25 and are thought to code for more permanent aspects of the body such as its appearance and its motor repertoire18. Contrary to online body representations, representing what the body is ‘currently like’, these offline representations represent what the body is ‘usually like’ by storing long- term information about size, shape, etc.17.

OPEN

1DEI Interactive Systems Group, Department of Computer Science and Engineering, Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganés 28911, Spain. 2Seddi Labs, Madrid, Spain. 3Multimodal Simulation Lab, Department of Computer Science and Architecture, Computer Systems and Languages, Statistics and Operative Investigation, Universidad Rey Juan Carlos, Madrid, Spain. *email: [email protected]

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In a recent study, offline full body representations of one’s self were investigated in a data-driven, visual, and unbiased way using a novel implementation of the reverse correlation technique26. This psychophysiological method has its roots in signal detection theory and auditory perception27–29 and has been widely used to create visual depictions of mental representations of faces30, sex and body shapes31, personal identity32, personality traits33, and racial groups34. Importantly, this reverse correlation technique allowed the authors to investigate implicit offline body representations (offline body representations that are accessed without explicitly asking about them), contrary to the vast amount of research that has examined these representations explicitly1–4. In their experiments, the authors presented female participants with a two-alternative forced choice classification task, where participants were asked to choose which of two images was most similar to their own body. These images were taken from a large set of variations of a base image of a female body by superimposing random noise patterns on this base image. Importantly, the bodies observed in the reverse correlation task are not based on participants’ real bodies. This is done intentionally, to avoid biasing the results and placing assumptions on what the participants’ mental representations of their own body should look like, and thus preserving the data-driven nature of the task as much as possible26. The process was repeated with a different question, where participants were asked to choose which of two images was most similar to the typical body of someone of their age and gen- der. The corresponding noise patterns of the selected images were then used to create two resulting classification images (CIs) that were regarded as the visualizations of the implicit mental representations of participants’ own and typical body. No correlations were observed between perceived hip width of these CIs and the real hip widths of participants (although results indicated that visual body representations of participants with negative attitudes towards their bodies showed wider hip widths than those of participants with more positive attitudes), suggesting that no relationship exists between people’s real body size and an offline, pictorial representation of their body26.

These findings, using the reverse correlation technique for the first time to depict implicit and offline self- representations in an unbiased and pictorial manner, are in line with previous research indicating that body size estimates are more reflective of attitudes towards one’s own body rather than real body size35 (but see36), and various studies showing that a mismatch exists between own and perceived body size1,2,4,7,8,37–43 (but see44). Thus far, research has largely focused on comparing perceived and real body measurements, although the importance of someone’s ideal body representation (internalized ideals about one’s own physical appearance45) and its dis- crepancy with actual body size has also received attention46–49. One study, for example, suggested that symptoms of anorexia nervosa might be related to distorted attitudes towards their ideal or desired body rather than visual distortions that lead to an overestimation of own body weight50. Importantly, body size estimation in these studies has been performed by framing the task of matching own and perceived body in a general, abstract way (abstract body representations; e.g. when asking whether a perceived body matches your own without framing this question within a specific context). The question arises whether the match between real and perceived body size changes when people’s body size estimation has real-life practical implications and influences real-world decision making (concrete body representations; e.g. when asking whether a perceived body matches your own while providing a practical context in which to consider this question). While research has shown that higher- level self-beliefs and attitudes26,51,52, as well as socio-cultural factors such as social comparison39,53–56 and achieve- ment expectations and the pressures to be slim57, influence self-body representations in healthy populations and patients with anorexia nervosa, we suggest that higher-level beliefs about the environment might also influence self-body representations (possibly due to exchanges between these body representations and the environment through active inference58–60). One practical, real-life scenario, where participants are asked to judge perceived body size in relationship to a specific, well-known practical use, is provided by the online clothes shopping expe- rience. A recent study suggests that the discrepancy between real and perceived body size persists when making an explicit choice about which model participants would use in an online clothes shopping environment61, even though it has been shown that these environments would greatly benefit (e.g. more purchase intentions) from an increased resemblance between the consumer and the online model/avatar62. The question remains whether the same is true when offline body representations are accessed implicitly.

In the current study, we employed the reverse correlation paradigm to investigate the relationship between real and perceived body size when participants had to select their own abstract (i.e. the body that looked most like their own)26, ideal (i.e. the body that looked most like the body they would like to have), and own concrete (i.e. the body that looked most like the body they would use for online clothes shopping) body, as well as the influence of several individual differences (bodily self-esteem, personality traits, body dissatisfaction) on these choices. Importantly, the reverse correlation technique, previously employed to construct visual depictions of participants’ own body in an abstract context26, allowed us to do the same with participants’ ideal body representation, as well as with the body they would use in a real-life context (i.e. own concrete body representation), using a data-driven method measuring implicit offline body representations for the first time. We hypothesized that framing own body estimation in a practical context would increase the relationship between perceived body measurements based on implicit offline body representations and real own body measurements, contrary to the absence of such a relationship when no such context is provided26 or when this relationship is measured explicitly61.

Additionally, we examined the relationship between all three implicit body representations (own abstract, ideal, and own concrete). While there is ample evidence (collected using explicit body estimation measures) for a discrepancy between own abstract and ideal perceived body representations and the role of this mismatch in body dissatisfaction45, the link between our own concrete (e.g. online clothes shopping) and own abstract/ ideal body representations remains poorly understood. Understanding potential differences between mental representations of body size according to different contexts is important given that different neural processes might be engaged, which may have important clinical and practical implications. For example, the discrepancy between ideal and own body size estimation seems to be important in the understanding of anorexia nervosa44, but the relationship with body size estimation in a concrete, practical context has not yet been investigated. Furthermore, the link between own abstract, ideal, and own concrete body perception plays an important role

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in the design and development of online avatars for a wide range of scenarios. Evidence from 3D online game environments, for example, suggests that people prefer idealized avatars in such environments63. Based on this research, we hypothesized that own concrete and ideal body representations would show a stronger relationship than own concrete and own abstract body representations. Furthermore, in line with research that has shown that body size/shape estimation is affected by bodily self-esteem26, personality traits61, and body (dis)satisfaction with certain body parts42,64, we hypothesized that body estimation in the current study would be influenced by these psychological differences.

Finally, we aimed to explore which body areas were predictive of the different body representations (own abstract, ideal, own concrete), by using a statistical test based on random field theory specifically adapted to reverse correlation CIs65. This technique, which identifies clusters of pixels that predict participants’ choices, has been successfully employed to identify intensity and unpleasantness of pain expressions66, and facial trustworthiness30, for example. When it comes to body perception, researchers have used eye tracking as a method to explore the areas of the body that people look at (i.e. pay attention to) when estimating their body size. These studies have shown that people who are able to accurately determine their body size are more likely to look at informative body areas (such as the waist and upper thigh gap), while people who overestimate their body size (including women suffering from anorexia nervosa) tend to focus on less informative areas such as the face and upper parts of the torso67. Additionally, a recent study indicated that key areas for accurate self-assessment are located on the edges of the torso and waist on either side of the body68. Importantly, it seems that an equal division of attention towards both sides of the body is necessary, given that a preference for the right side of the body is associated with overestimation of own body size. While we did not use eye tracking in the current study, we wanted to explore whether similar patterns could be observed using the statistical test mentioned earlier in the paragraph, applied to all three body representations. As shown in previous research using explicit body size measures67,68, we hypothesized that diagnostic areas of our CIs would depend on participants’ under- and overestimation of own body measurements. Consequently, we also expected that the relationship between real and perceived hip width would be different for under- compared to over-estimators as a result of using different diagnostic areas to perform the estimation.

Results Correlations between real and perceived hip width. Correlations were calculated between perceived hip width in all conditions (own abstract, ideal, own concrete) and participants’ real hip width measurements. Perceived own abstract (r = 0.19, p = 0.155, BF10 = 0.451) and ideal (r = 0.11, p = 0.419, BF10 = 0.231) hip width did not correlate with real hip width. Perceived own concrete hip width, however, showed a significant negative cor- relation with real hip width (r = -0.27, p = 0.050, BF10 = 1.09; see Fig. 1), suggesting that the hip width of CIs in the own concrete body condition was larger for participants with smaller hip widths (and vice versa). Perceived hip widths off all three conditions did not correlate with each other (own abstract—ideal: r = 0.08, p = 0.551, BF10 = 0.200; own abstract—own concrete: r = 0.02, p = 0.895, BF10 = 0.170; ideal—own concrete: r = -0.05, p = 0.690, BF10 = 0.182). Note that the BF for the correlation between real and perceived hip width in the own abstract condition does not meet the threshold to support the absence of any effects (although it is very close to reaching this threshold; BF10 < 1/3), while the BF for the own concrete condition does not meet the threshold to reject the null hypothesis (BF10 > 3). These findings should therefore be interpreted with more caution.

In addition to the analyses across all participants, the correlation analyses were performed separately for under- versus over-estimators. Under-/over-estimators were defined as participants whose perceived hip width was smaller/larger than their real hip width. For each condition, we investigated whether the relationship between real and perceived hip width interacted with group (under- versus overestimation). For own abstract body, under-estimators showed a significantly positive correlation between real and perceived width (n = 20, r = 0.89, p < 0.001, BF10 > 100), while no relation was observed for over-estimators (n = 35, r = 0.21, p = 0.223, BF10 = 0.43; interaction with group, F(1,51) = 27.30, p < 0.001, BF10 > 100; see Fig. 2). No effects of under- versus overestima- tion were found for ideal and own concrete body.

Psychological traits. To investigate the influence of psychological traits on body representations, mixed linear effects models were run (see Supplementary Table  S1 for model details). An H1 model with perceived hip width as dependent variable that included participants’ real hip width, condition (own abstract, ideal, own concrete), their interaction, and the interaction between condition and Big 5 Conscientiousness significantly improved model fit (AIC = 333.95, χ2 = 10.27, p = 0.016; AICH0 = 357.08). Big 5 Conscientiousness significantly predicted perceived hip width for own concrete (β = 0.31, SE = 0.13, t(53) = 2.44, p = 0.018) but not own abstract (β = − 0.03, SE = 0.10, t(53) = − 0.34, p = 0.738) and ideal (β = − 0.15, SE = 0.10, t(53) = − 1.49, p = 0.143) body con- ditions (see Fig. 3a), indicating that participants who reported to be more conscientious showed wider perceived hip width for the own concrete body.

We additionally ran linear models with perceived hip width as dependent variable, and real hip width, psy- chological traits, and their interaction as predictors for each condition separately (see Supplementary Table S2 for model details). A model that included BESAA Weight scores, as well as their interaction with real hip width, significantly improved model fit for the model in the own abstract condition (AIC = 62.78, χ2 = 8.03, p = 0.034; AICH0 = 58.30). A trending interaction was observed between BESAA Weight scores and real hip width when predicting perceived hip width for the own abstract body judgment (β = − 0.07, SE = 0.03, t(51) = − 1.95, p = 0.057; see Fig. 3b), suggesting that a negative relationship was found between real and perceived hip width for partici- pants with more positive attitudes towards their body weight when producing own abstract body CIs, while the reverse relationship was observed for participants with more negative body weight attitudes. For the ideal and own concrete condition, no psychological trait scores significantly improved model fit.

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No influence of the psychological traits was found on the differences between the body representations (own abstract—ideal, own abstract—own concrete ideal—own concrete).

Diagnostic areas. All participants showed clear clusters for all three conditions. Figure 4a–c represent the CIs of the own abstract, ideal, and own concrete body respectively, across all participants. The areas in red indi- cate the areas that were significantly correlated with each judgment (zcrit ≥|2.3|, p < 0.05). For the own abstract body, a large cluster in the upper body, skewed to the right, was observed (including the right hip and waist, torso, neck, and right arm), as well as small clusters around the left leg and both of the feet. For the ideal body, small clusters were observed around the legs and arms. Finally, the own concrete body showed clusters located in the left leg, both of the arms, and parts of the right neck and face. A similar cluster analysis was performed by calculating the CIs across all participants who under-/over-estimated perceived compared to real hip width. The resulting images for each group were subtracted from each other for visualization purposes (e.g. underes- timation–overestimation displays the areas that were uniquely sign

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