#  The 2175$\,{\rm\AA}$ Interstellar Extinction Bump: Is the Wavelength
Variable?

Qian Wang1,2, X.J. Yang1,2, and Aigen Li2
1Hunan Key Laboratory for Stellar and Interstellar Physics and School of
Physics and Optoelectronics,
Xiangtan University, Hunan 411105, China
2Department of Physics and Astronomy, University of Missouri, Columbia, MO
65211, USA
xjyang<EMAIL_ADDRESS>
###### Abstract

The most striking characteristics of the mysterious 2175$\,{\rm\AA}$
extinction bump, the strongest spectroscopic absorption feature seen on the
interstellar extinction curve, are the invariant central wavelength and
variable bandwidth: its peak position at 2175$\,{\rm\AA}$ is remarkably
constant while its bandwidth varies from one line of sight to another.
However, recent studies of the lines of sight toward a number of Herbig Ae/Be
stars have revealed that the extinction bump exhibits substantial shifts from
the canonical wavelength of 2175$\,{\rm\AA}$. In this work we revisit these
lines of sight and take a physical approach to determine the ultraviolet (UV)
extinction curve for each line of sight. It is found that the wavelengths of
the derived UV extinction bumps are around 2200$\,{\rm\AA}$ and the scatters
are considerably smaller than that of the previous study based on the same set
of Herbig Ae/Be stars, consistent with the conventional wisdom. Nevertheless,
the scatters are still appreciably larger than that associated with the
nominal bump position of 2175$\,{\rm\AA}$. This is discussed in the context
that Herbig Ae/Be stars are not well-suited for interstellar extinction
studies.

###### keywords:

ISM: dust, extinction — ISM: lines and bands — ISM: molecules

††pagerange: The 2175$\,{\rm\AA}$ Interstellar Extinction Bump: Is the
Wavelength Variable?–The 2175$\,{\rm\AA}$ Interstellar Extinction Bump: Is the
Wavelength Variable?††pubyear: 2023

##  1 Introduction

The 2175$\,{\rm\AA}$ extinction bump was first detected nearly six decades ago
by Stecher (1965). It shows up as the most prominent spectral feature on the
interstellar extinction curve. It is widely seen in the Milky Way and nearby
galaxies, including the Large Magellanic Cloud, several regions in the Small
Magellanic Cloud, and M31 (see Whittet 2022). Very recently, it was detected
by the James Webb Space Telescope (JWST) in a distant galaxy at redshift
$z\approx 6.71$ (Witstok et al. 2023).

Despite nearly 60 years’ extensive observational, theoretical and experimental
studies, the exact carrier of the 2175$\,{\rm\AA}$ extinction bump remains
unidentified, although various candidate materials have been proposed,
including small graphitic grains (Stecher & Donn 1965, Draine 1988),
polycyclic aromatic hydrocarbon (PAH; Joblin et al. 1992, Li & Draine 2001,
Steglich et al. 2013), and carbon buckyonions composed of spherical concentric
fullerene shells (Chhowalla et al. 2003, Iglesias-Groth et al. 2003, Ruiz et
al. 2005, Li et al. 2008). More recently, Ma et al. (2020) found that a new
carbon allotrope known as T-carbon, formed by substituting each atom in
diamond with a carbon tetrahedron (Sheng et al. 2011), exhibits a prominent
peak around 2175$\,{\rm\AA}$ in its electronic absorption spectrum.

The difficulty in identifying the carrier of the 2175$\,{\rm\AA}$ extinction
bump is mainly related to its striking characteristics: while its strength and
width vary with environment, its peak position is quite invariant: the central
wavelength of this feature varies by only $\pm$0.46% (2$\sigma$) around
2175$\,{\rm\AA}$ (4.6$\,{\rm\mu m}^{-1}$), while its full width half maximum
(FWHM) varies by $\pm$12% (2$\sigma$) around 469$\,{\rm\AA}$
($\approx$1$\,{\rm\mu m}^{-1}$). The bump width appears to show a strong
correlation with the environment. In general, broad bumps are often seen in
dense, quiescent environments, while narrower bumps are mainly seen in diffuse
regions and regions of recent early-type star formation (Fitzpatrick & Massa
1986; hereafter FM86). There is no substance that can explain the
observational fact that the central wavelength of the 2175$\,{\rm\AA}$
extinction bump basically does not change with the environment. For example,
although increasing the size of graphite grains can broaden the bump, its peak
shifts to longer wavelength which is inconsistent with the observational
characteristic of a stable peak wavelength (Draine & Malhotra 1993, Mathis
1994).

Recently, Blasberger et al. (2017) analyzed the UV spectra of 26 Herbig Ae/Be
(HAeBe) stars obtained with the International Ultraviolet Explorer (IUE) and
determined the UV extinction curves for the lines of sight toward these stars.
Contrary to the conventional wisdom, the extinction bumps derived by
Blasberger et al. (2017) do not always peak at 2175$\,{\rm\AA}$. Instead, they
exhibit significant shifts of the central wavelength of the extinction bump.
If confirmed, this would be an important breakthrough in the interstellar
extinction studies, particularly, this will provide valuable insight into the
carriers of the 2175$\,{\rm\AA}$ bump. This motivates us to revisit all these
26 sight lines. In §2 we briefly describe the sample and method. The results
are presented and discussed in §3. Our major conclusion is summarized in §4.

##  2 The Sample and Method

We consider the sample of Blasberger et al. (2017), which consists of 26 pre-
main-sequence stars of spectral type A2 and earlier. All have significant UV
absorption (equivalent width, $\mathrm{EW}\lower
2.15277pt\hbox{$\buildrel>\over{\sim}$}100\,{\rm\AA}$) around
2175$\,{\rm\AA}$. The sample and the stellar parameters are listed in Table 1.

Following Blasberger et al. (2017), we derive the UV extinction curve for each
line of sight by comparing $F_{\lambda}^{\rm obs}$, the stellar spectrum
observed by the IUE which suffers dust extinction, with $F_{\lambda}^{\rm
int}$, the intrinsic stellar spectrum which is free of dust extinction. The
IUE spectra were extracted from the Mikulski Archive for Space Telescopes,111
https://archive.stsci.edu/iue/ by combining data obtained with three cameras:
the short-wavelength camera (SWP) which spans the wavelength range of
1200–1970$\,{\rm\AA}$, and the two long-wavelength cameras (LWP and LWR) which
span the wavelength range of 1970–3200$\,{\rm\AA}$.

We approximate the intrinsic stellar spectrum by the stellar model atmospheric
spectrum of Castelli & Kurucz
(2004).222http://kurucz.harvard.edu/grids/gridxxxodfnew Let $A_{\lambda}$ be
the extinction at wavelength $\lambda$, and $A_{V}$ be the visual extinction.
The observed IUE spectrum and the intrinsic stellar spectrum are related
through dust extinction

$F_{\lambda}^{\rm obs}=F_{\lambda}^{\rm
int}\exp\left(-\frac{A_{\lambda}}{A_{V}}\frac{A_{V}}{1.086}\right)~{}.$ (1)

Let $F_{\nu}^{\rm RK}$ be the Kurucz model atmospheric flux at stellar surface
(in unit of $\,{\rm erg}\,{\rm s}^{-1}\,{\rm cm}^{-2}\,{\rm Hz}^{-1}$). At an
Earth-stellar distance of $d$, the intrinsic, extinction-free flux (in unit of
$\,{\rm erg}\,{\rm s}^{-1}\,{\rm cm}^{-3}$) would be

$F_{\lambda}^{\rm int}=F_{\nu}^{\rm
RK}\times\frac{c}{\lambda^{2}}\times\left(\frac{R_{\star}}{d}\right)^{2}~{},$
(2)

where $c$ is the speed of light.

Following Fitzpatrick & Massa (1988; hereafter FM88) and Cardelli et al.
(1989), we represent the wavelength-dependence of extinction by an analytical
formula consisting of three parts,

$\frac{A_{\lambda}}{A_{V}}=c_{1}+c_{2}x+c_{3}D\left(x;\gamma,x_{0}\right)+c_{4}F(x)~{},$
(3)

where $x\equiv\lambda^{-1}$ is the inverse wavelength; $c_{1}+c_{2}x$ is the
linear background; $D\left(x;\gamma,x_{0}\right)$, a Drude function of width
$\gamma$ (in unit of $\,{\rm\mu m}^{-1}$) peaking at $x_{0}$ (also in unit of
$\,{\rm\mu m}^{-1}$), characterizes the 2175$\,{\rm\AA}$ extinction bump and
is defined as

$D\left(\mathrm{x};\gamma,x_{0}\right)=\frac{x^{2}}{\left(x^{2}-x_{0}^{2}\right)^{2}+x^{2}\gamma^{2}}~{},$
(4)

and $F(x)$ is the far-UV (FUV) nolinear rise at $x>5.9\,{\rm\mu m}^{-1}$ as
described by

$F(x)=\begin{cases}0.5392(x-5.9)^{2}+0.05644(x-5.9)^{3}&x\geq
5.9\,\mu\mathrm{m}^{-1}~{},\\\
0&\mathrm{x}<5.9\,\mu\mathrm{m}^{-1}~{}.\end{cases}$ (5)

The Castelli & Kurucz (2004) stellar atmospheric model is characterized by
four parameters: effective temperature $T_{\rm eff}$, gravity log $g$, metal
abundance ${\rm\left[M/H\right]}$, and microturbulence $\xi$. The Castelli &
Kurucz (2004) stellar model spectral library spans a range in $T_{\rm eff}$
from 35,00 K to 50,000 K, ${\rm\left[M/H\right]}=-1.5,-1.0,-0.5,0.0$, and
$0.5$ (relative to solar abundance), log $g$ from 0.0 dex to 5.0 dex, and
$\xi$ = 2.0$\,{\rm km}\,{\rm s}^{-1}$. For a source of given spectral type
(therefore $T_{\rm eff}$), to select the most appropriate stellar model
atmospheric spectrum, we try all the possible metallicities and gravities and
select those that best fit the measured spectrum.

For each object, we make use of the Levenberg–Marquardt algorithm to fit the
observed spectrum (see eq. 1) by minimizing ${\chi}^{2}$. The best-fit
parameters $A_{V}$, $\gamma$, $x_{0}$, $c_{1}$, $c_{2}$, $c_{3}$, and $c_{4}$
are listed in Table 2. During the fitting process, we find that the bump width
$\gamma$ and peak position $x_{0}$ are relatively independent of the other
parameters and can be accurately determined. Therefore, we first fix $\gamma$
and $x_{0}$ and fit the observed IUE spectrum to search for best-fitting
$c_{1}$, $c_{2}$ and $c_{4}$. We then fix $\gamma$, $x_{0}$, $c_{1}$, $c_{2}$
and $c_{4}$ and re-fit the observed spectrum to determine $c_{3}$ and $A_{V}$.
Finally, we take these values as initial “guesses” and fit the observed
spectrum again so that we derive a full set of model parameters (i.e.,
$A_{V}$, $c_{1}$, $c_{2}$, $\gamma$, $x_{0}$, $c_{3}$, and $c_{4}$). We repeat
this last step three times so that we achieve the minimum $\chi^{2}$.

##  3 Results and Discussion

For each object, we show in Figures 1–6 the observed IUE spectrum
($F_{\lambda}^{\rm obs}$) in comparison with the Kurucz model-based,
extinction-free “intrinsic” spectrum ($F_{\lambda}^{\rm int}$), and with the
best-fit, dust-obscured Kurucz model spectrum ($F_{\lambda}^{\rm
int}\exp\left\\{-A_{\lambda}/1.086\right\\}$). Also shown are, for each line
of sight, the derived extinction curve and its three components, i.e., the
linear background, the 2175$\,{\rm\AA}$ bump, and the FUV rise.

Figures 1–6 demonstrate that our approach is successful in closely reproducing
the obseved IUE spectra in the wavelength range of
$\sim\,$1200–3200$\,{\rm\AA}$ for 25 (of 26) sources. The only exception is HD
95881. As shown in Figure 6, our approach could not fit the observed IUE
spectrum in the wavelength range of $\sim\,$1200–3200$\,{\rm\AA}$. Our model
spectrum at $\sim\,$1600–2000$\,{\rm\AA}$ appreciably exceeds the observed IUE
spectrum. Nevertheless, if we confine ourselves to 1600–3200$\,{\rm\AA}$, our
model has no problem in closely explaining the observed spectrum. We have also
tried to fit the IUE spectra of all other 25 sources exclusively in the
1600–3200$\,{\rm\AA}$ wavelength range. It is found that the derived peak
wavelength and width of the extinction bump are essentially the same as that
derived from fitting the IUE spectra over the entire 1200–3200$\,{\rm\AA}$
wavelength range (see Table 2).

We show in Figure 7 the histogram of the central wavelengths ($\lambda_{0}$)
of the 2175$\,{\rm\AA}$ extinction bump derived for all 26 sources. Except for
HD 95881 for which $\lambda_{0}$ is derived from fitting the IUE spectrum over
$\sim\,$1600–2000$\,{\rm\AA}$, for all other 25 sources, we determine
$\lambda_{0}$ from the IUE spectra over 1200–3200$\,{\rm\AA}$. Figure 7
reveals that the peak wavelengths of the extinction bump of our sample of 26
sources cluster around 2200$\,{\rm\AA}$, with a standard deviation of only
$\sim\,$33$\,{\rm\AA}$. The line of sight toward HD 100546, with
$\lambda_{0}\approx 2299\,{\rm\AA}$, has the longest peak wavelength. If we
exclude HD 100546, the median peak wavelength of our sample becomes
$\lambda_{0}\approx 2201\pm 27\,{\rm\AA}$. Figure 7 demonstrates that although
the bumps of our sample peak at relatively longer wavelengths, their peak
wavelengths are rather stable and the scatters are small. Compared with that
derived by Blasberger et al. (2017), the peak wavelengths derived here are
appreciably less scattered.

To determine the peak wavelengths of the extinction bump of these sources,
Blasberger et al. (2017) took an approach differing from ours. They fitted the
IUE spectra around the 2175$\,{\rm\AA}$ extinction bump in the wavelength
range of 1600–3200$\,{\rm\AA}$. They took the Castelli & Kurucz (2004) model
atmospheric spectrum ($F_{\nu}^{\rm RK}$), modulated by a power law to
represent the broadband extinction, and absorbed by a feature with a Drude
profile. They assumed the measured flux spectrum $F_{\lambda}^{\rm obs}$ as a
function of wavelength $\lambda$ to take the following form:

$F_{\lambda}^{\rm obs}=F_{\nu}^{\rm RK}\times
N\times\left(\frac{\lambda}{2000{\rm\AA}}\right)^{-\alpha}\times\exp\left(\frac{-A}{\pi}\frac{\lambda^{2}}{(\lambda^{2}-\lambda_{0}^{2})+\lambda^{2}w^{2}}\right)~{},$
(6)

where $N$ is the normalization factor, $\alpha$ is the power law slope, and in
the exponent, $A$ represents the amplitude of the absorption feature, $w$ its
width, and $\lambda_{0}$ its central wavelength. Comparing the approach of
Blasberger et al. (2017) with ours (see eq. 1), it is apparent that the major
difference is that they assumed the attenuation caused by the broadband
extinction (i.e., the linear background extinction component
$c_{1}+c_{2}\lambda^{-1}$) take a power-law form $\lambda^{-\alpha}$, while we
take a physical approach in which the attenuation caused by the linear
background extinction is $\exp\left\\{-c_{1}+c_{2}\lambda^{-1}\right\\}$.
Apparently, it has to be justified if the power-law $\lambda^{-\alpha}$ is a
valid approximation of the attenuation
$\exp\left\\{-c_{1}+c_{2}\lambda^{-1}\right\\}$. In Figure 8 we compare the
peak wavelengths of the extinction bumps derived here with that of Blasberger
et al. (2017). Clearly, the peak wavelengths derived by Blasberger et al.
(2017) were systematically longer than that derived here from a more physical
approach.

We argue that, based on a physical fitting of the observed IUE spectra of 26
interstellar lines of sight, the peak wavelengths of the extinction bump are
more or less stable around $\sim\,$2200$\,{\rm\AA}$, although they are longer
than the nominal 2175$\,{\rm\AA}$. We note that, very recently, an extinction
bump peaking at $\sim\,$2263${}^{+20}_{-24}\,{\rm\AA}$ was discovered by JWST
in JADES-GS+53.15138-27.81917, a galaxy at $z\approx 6.71$ (Witstok et al.
2023). This is notably longer than the nominal 2175$\,{\rm\AA}$. Draine &
Malhotra (1993) have investigated the effects on the 2175$\,{\rm\AA}$
extinction bump of changes in graphite size and shape. They found that, while
small graphite grains do produce a pronouced extinction bump, the central
wavelength of the bump is correlated with the bump width. In Figure 9 we
compare the central wavelengths with the widths of the extinction bumps
derived here. Contrary to Draine & Malhotra (1993), no correlation is found.
This is in agreement with Fitzpatrick & Massa (1986) who found large
variations in width of the extinction bump with minimal, and uncorrelated,
variations in the central wavelength.

Finally, we note that, although the extinction bump positions derived here are
considerably less scattered than that of Blasberger et al. (2017), they are
still appreciably more scattered than that of the classical studies.
Fitzpatrick & Massa (1986) studied the IUE data of 45 stars and derived a
stable bump peak of $\sim\,$2174.4$\,{\rm\AA}$ and a $2\sigma$ scatter of
$\sim\,$10$\,{\rm\AA}$. Based on the IUE data of 417 stars, Valencic et al.
(2004) placed the peak of the extinction bump at $\sim\,$2179$\,{\rm\AA}$ with
a $1\sigma$ scatter of only $\sim\,$5$\,{\rm\AA}$. The discrepency between the
present work and that by Fitzpatrick & Massa (1986) and Valencic et al. (2004)
may arise from the validity of HAeBe stars as an interstellar extinction
indicator. It is well recognized that HAeBe stars are surrounded by
circumstellar disks. Free-free emission from these disks extends well into the
V-band, thus adding to the measured $E(B-V)$ value normally attributed to
interstellar reddening (e.g., see Garrison 1978, Schild 1978, Witt & Cottrell
1980).

Furthermore, HAeBe stars are frequently embedded in nebulosity. When they are
used for extinction studies, scattered nebular continuum could enter the
relatively large apertures ($\sim\,$10′′$\times$20′′) of the IUE
spectrometers, adding to the apparent stellar flux and changing the stellar
spectral energy distribution (SED); also, a substantial fraction of the dust-
related extinction may arise in the nebulosity itself, where the dust
properties may not be representative of typical interstellar dust. It is not
entirely clear why the nebulosity and the local extinction in the vicinity of
HAeBe stars cause the bump positions to systematically shift to longer
wavelengths. We argue that this is probably related to the non-standard
scattering and extinction properties of the local (i.e., nebular and
circumstellar) dust. Due to grain growth, the nebular and circumstellar dust
grains around HAeBe stars are generally much larger than interstellar grains
(e.g., see Li & Lunine 2003) and therefore they cause less obscuration in the
UV. This would effectively raise the flux level observed at shorter
wavelengths and therefore cause the bump to shift to longer wavelengths.

Moreover, depending on the geometries and orientations of the circumstellar
disks surrounding those HAeBe stars, the observed SEDs of the stars may suffer
additional changes due to absorption in the disk. In particular, unresolved
(at IUE resolution) absorptions in Fe II/Fe III multiplets arising in the disk
can alter the apparent SEDs of HAeBe stars in the
$\sim\,$2000–3500$\,{\rm\AA}$ wavelength region, potentially affecting the
appearance of the 2175$\,{\rm\AA}$ extinction bump.

Finally, many HAeBe stars are actively accreting material from the
circumstellar disk. Accretion shocks from the infalling material impacting on
the stellar photosphere can give rise to UV excess radiation that is more
readily observed in cooler accreting stars but which can also alter the
intrinsic photospheric SEDs of hotter stars.

In view of these caveats of HAeBe stars as an effective indicator for
interstellar extinction, the discovery of variable peak positions by
Blasberger et al. (2017) for the 2175$\,{\rm\AA}$ extinction bump is
questionable. We should stress that, even if HAeBe stars were well-suited for
interstellar extinction studies, as demonstrated earlier in this paper, our
physical approach leads to a much less scattered extinction bump, also
indicating that the discovery of variable peak positions by Blasberger et al.
(2017) is questionable.

##  4 Summary

We have determined the UV extinction curves for a sample of interstellar lines
of sight toward 26 Herbig Ae/Be stars, by comparing the observed IUE spectrum
with the Kurucz stellar model atmospheric spectrum for each object. The
extinction is represented by an analytical formula consisting of a linear
background, a Drude function for the extinction bump, and a FUV nonlinear
rise. It is found that, peaking at $\sim\,$2200$\,{\rm\AA}$, the peak
positions of the extinction bumps are rather stable and the scatters are
small. This is consistent with the conventional wisdom that while the strength
and width of the interstellar extinction bump vary with environment, its
central wavelength is quite invariant.

## Acknowledgements

We thank B.T. Draine, Q. Li, Q. Lin and the anonymous referee for valuable
suggestions. QW and XJY are supported in part by NSFC 12122302 and 11873041.
AL is supported in part by NASA grants 80NSSC19K0572 and 80NSSC19K0701.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding authors.

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Table 1: Stellar Parameters

Object Spectral Type† E(B-V)† $T_{\rm eff}$ $L_{\star}/L_{\odot}$ Distance
Stellar Radius $R_{\star}/R_{\odot}$ References‡ (mag) (K) (pc) (cm) BD+30549
B8:p 0.59 12022 12.59 350 5.67E+10 (6) BD+404124 B2Ve 0.98 22000 5900 980
3.67E+11 (1) CD-4211721 B0IVe 1.58 30000 8710 400 2.39E+11 (5) HD 31293 A0Ve
0.13 9800 57.5 144 1.83E+11 2.62 (3) HD 36917 B9III/IV 0.17 10000 245.5 375
3.62E+11 (1) HD 36982 B1.5Vp 0.37 20000 1659.59 375 2.38E+11 3.42 (3) HD 37903
B3IV 0.31 23677 3311.31 820 2.37E+11 (7) HD 58647 B9IV 0.14 10500 912 546
6.33E+11 (1) HD 95881 A1/A2III/IV 0.13 9000 7.6 118 7.91E+10 (1) HD 97048
A0Vep 0.31 10000 44 150 1.53E+11 (1) HD 97300 B9V 0.24 10700 37 188 1.23E+11
(1) HD 100546 B9Vne 0.48 10500 32 103 1.19E+11 (1) HD 132947 B9V 0.13 10500
93.3 565 2.02E+11 3.1 (4) HD 141569 B9.5V 0.09 9800 30.9 116 1.34E+11 (1) HD
141926 B2nne 0.74 28000 20118.7 1254 6.75E+11 9.7 (2) HD 145554 B9V 0.20 10471
42.66 137 1.38E+11 (7) HD 145631 B9V 0.20 10471 44.67 141 1.41E+11 (7) HD
147009 A0V 0.28 10471 42.66 130 1.38E+11 (7) HD 147701 B5III 0.71 15488 204.17
140 1.38E+11 (7) HD 147889 B2III/IV 1.05 21877 1995.26 140 2.16E+11 (6) HD
149404 O8.5Iab(f)p 0.69 32000 810 2.36E+12 33.9 (8) HD 149914 B9.5IV 0.26
10471 128.82 159 2.39E+11 (7) HD 151804 O8.5Iab(f)p 0.36 34000 580 2.36E+12 34
(8) HD 179218 A0Ve 0.11 9640 182 254 3.35E+11 (1) HD 200775 B2Ve 0.61 18600
8912.5 429 6.30E+11 (1) V699 Mon B7IIne 0.70 13800 616.6 800 2.99E+11 4.3 (2)

†

Taken from Blasberger et al. (2017).

‡

Seok & Li (2017), (2) Verhoeff et al. (2012), (3) Alecian et al. (2013), (4)
Fairlamb et al. (2015), (5) Boersma et al. (2009), (6) Hamaguchi et al.
(2005), (7) Hernández et al. (2005), (8) Lamers et al. (1995).

Table 2:  Extinction Model Parameters

Object $A_{V}$ $x_{0}$ $\gamma$ $c_{1}$ $c_{2}$ $c_{3}$ $c_{4}$
$\mathcal{X}^{2}$ (mag) ($\,{\rm\AA}$) ($\mu$m-1) BD+30549 1.32 ± 0.00 2161.7
± 0.01 1.06 ± 0.01 0.14 ± 0.02 0.23 ± 0.00 1.15 ± 0.01 0.39 ± 0.03 6.39
BD+404124 2.68 ± 0.00 2213.2 ± 0.01 1.20 ± 0.02 1.24 ± 0.03 0.20 ± 0.01 1.95 ±
0.01 0.15 ± 0.11 157.25 CD-4211721 2.34 ± 0.00 2223.8 ± 0.01 1.16 ± 0.04 2.00
± 0.14 0.50 ± 0.04 1.99 ± 0.02 0.01 ± 1.46 128.07 HD 31293 0.92 ± 0.00 2130.6
± 0.03 1.06 ± 0.03 -0.15 ± 0.03 0.34 ± 0.01 0.50 ± 0.02 0.10 ± 0.04 13.87 HD
36917 0.50 ± 0.00 2221.3 ± 0.02 1.06 ± 0.02 2.98 ± 0.07 0.02 ± 0.01 1.40 ±
0.05 0.01 ± 0.04 6.82 HD 36982 1.31 ± 0.00 2222.2 ± 0.01 1.28 ± 0.01 1.79 ±
0.01 0.01 ± 0.00 1.20 ± 0.01 0.09 ± 0.01 6.48 HD 37903 1.21 ± 0.00 2177.8 ±
0.00 0.94 ± 0.01 0.36 ± 0.01 0.12 ± 0.00 0.66 ± 0.00 0.10 ± 0.01 1.72 HD 58647
1.00 ± 0.00 2166.8 ± 0.01 1.03 ± 0.01 0.48 ± 0.02 0.05 ± 0.00 0.57 ± 0.01 0.01
± 0.01 6.13 HD 95881 0.65 ± 0.00 2244.2 ± 0.01 1.22 ± 0.01 -0.18 ± 0.05 0.31 ±
0.01 2.00 ± 0.03 0.01 ± 1.81 10.67 HD 97048 1.43 ± 0.00 2204.6 ± 0.01 1.20 ±
0.01 1.25 ± 0.02 0.16 ± 0.00 0.93 ± 0.01 0.04 ± 0.02 6.58 HD 97300 1.53 ± 0.00
2204.6 ± 0.01 1.22 ± 0.01 1.34 ± 0.01 0.07 ± 0.00 0.93 ± 0.01 0.16 ± 0.01 7.07
HD 100546 0.99 ± 0.01 2298.8 ± 0.02 1.23 ± 0.02 0.14 ± 0.02 0.01 ± 0.00 0.68 ±
0.01 0.01 ± 0.01 4.82 HD 132947 0.84 ± 0.01 2231.5 ± 0.01 0.88 ± 0.01 0.24 ±
0.02 0.01 ± 0.00 0.50 ± 0.01 0.02 ± 0.01 5.26 HD 141569 0.93 ± 0.00 2164.6 ±
0.01 0.92 ± 0.01 0.71 ± 0.02 0.01 ± 0.00 0.50 ± 0.01 0.09 ± 0.01 5.1 HD 141926
2.54 ± 0.00 2201.9 ± 0.00 1.17 ± 0.01 0.68 ± 0.03 0.28 ± 0.01 1.46 ± 0.01 0.01
± 0.08 175.57 HD 145554 0.60 ± 0.00 2175.6 ± 0.01 1.11 ± 0.01 1.86 ± 0.03 0.01
± 0.00 1.81 ± 0.02 0.01 ± 0.01 4.62 HD 145631 0.73 ± 0.00 2197.5 ± 0.02 0.80 ±
0.02 2.00 ± 0.04 0.01 ± 0.01 0.45 ± 0.02 0.01 ± 0.02 28.17 HD 147009 1.49 ±
0.00 2219.1 ± 0.01 1.13 ± 0.01 1.23 ± 0.02 0.10 ± 0.00 1.07 ± 0.01 0.33 ± 0.02
5.47 HD 147701 1.66 ± 0.00 2157.9 ± 0.01 1.04 ± 0.01 1.83 ± 0.01 0.16 ± 0.00
1.25 ± 0.01 0.34 ± 0.03 7.41 HD 147889 2.65 ± 0.00 2191.8 ± 0.01 1.26 ± 0.04
1.60 ± 0.02 0.13 ± 0.00 2.39 ± 0.01 0.22 ± 0.04 38.89 HD 149404 1.65 ± 0.00
2204.6 ± 0.00 0.89 ± 0.01 1.81 ± 0.02 0.34 ± 0.01 1.21 ± 0.01 0.03 ± 0.04
10.07 HD 149914 1.34 ± 0.00 2216.9 ± 0.01 1.02 ± 0.01 0.85 ± 0.01 0.13 ± 0.00
0.84 ± 0.01 0.19 ± 0.02 6.44 HD 151804 0.80 ± 0.00 2201.1 ± 0.01 0.66 ± 0.01
4.46 ± 0.04 0.46 ± 0.01 0.64 ± 0.01 0.01 ± 0.03 11.89 HD 179218 1.40 ± 0.00
2189.5 ± 0.01 1.05 ± 0.01 1.01 ± 0.01 0.01 ± 0.00 0.50 ± 0.01 0.22 ± 0.01 3.47
HD 200775 1.61 ± 0.00 2190.9 ± 0.01 1.20 ± 0.01 1.57 ± 0.01 0.22 ± 0.00 1.05 ±
0.01 0.12 ± 0.02 5.48 V699 Mon 1.84 ± 0.00 2219.2 ± 0.01 1.30 ± 0.01 1.88 ±
0.02 0.10 ± 0.00 1.92 ± 0.01 0.26 ± 0.03 13.17

Figure 1:  Observed, extinction-suffering IUE spectra ($F_{\lambda}^{\rm
obs}$), “intrinsic”, extinction-free spectra ($F_{\lambda}^{\rm int}$), and UV
extinction curves of the interstellar lines of sight toward BD+30549,
BD+404124, CD-4211721, HD 31293, and HD 36917. The left columns compare
$F_{\lambda}^{\rm obs}$ (green solid lines) with $F_{\lambda}^{\rm int}$
(black solid lines). The middle columns highlight the best fits (red solid
lines) to the observed spectra (green solid lines). The right columns show the
interstellar extinction curves expressed as $A_{\lambda}$/$A_{V}$. The red
line is the FM parametrization at $\lambda^{-1}>3.3\,{\rm\mu m}^{-1}$, which
is the sum of a linear “background” (green line), a Drude bump of width
$\gamma$ peaking at $x_{0}\equiv\lambda_{0}^{-1}$ (blue line), and a nonlinear
FUV rise (black dash-dotted line) at $\lambda^{-1}>5.9\,{\rm\mu m}^{-1}$.

Figure 2:  Same as Figure 1 but for HD 36982, HD 37903, HD 58647, HD 97048,
and HD 97300

Figure 3:  Same as Figure 1 but for HD 100546, HD 132947, HD 141569, HD
141926, and HD 145554.

Figure 4:  Same as Figure 1 but for HD 145631, HD 147009, HD 147701, HD
147889, and HD 149404.

Figure 5:  Same as Figure 1 but for HD 149914, HD 151804, HD 179218, HD
200775, and V699 Mon.

Figure 6:  Same as Figure 1 but for the observed stellar and “intrinsic”
spectra as well as the UV extinction curve of HD 95881 in the
1200–3200$\,{\rm\AA}$ wavelength range (upper panel) and in the
1600–3200$\,{\rm\AA}$ wavelength range (bottom panel).  Figure 7:  Histograms
of the central wavelengths of the 2175$\,{\rm\AA}$ extinction bump for the 26
lines of sight determined in this work (red bars) as well as that derived by
Blasberger et al. (2017; cyan bars).  Figure 8:  Comparison of the central
wavelengths of the 2175$\,{\rm\AA}$ bump derived in this work (orange bars)
with that determined by Blasberger et al. (2017; green bars). The horizontal
bar shows the nominal wavelength of the 2175$\,{\rm\AA}$ extinction bump and
its nominal variation range.  Figure 9:  The central wavelength of the
2175$\,{\rm\AA}$ extinction bump against its width.


$\mathrm{tail}(i_{j,h})=\mathrm{head}(i_{j,h+1\mod
q_{j}}),h=0,\ldots,q_{j}-1$.

  2. (b)

$C_{1},\ldots,C_{N}$ are simple cycles of $G$.

  3. (c)

The cycles $C_{1},\ldots,C_{N}$ form a basis of the cycle space of $G$ (which
is isomorphic to $\mbox{\rm H}_{1}(|G|)$).

14:for $1\leq h\leq N$ do

15: $\Psi_{h}\leftarrow\Theta_{i_{h,0}}\vee\cdots\vee\Theta_{i_{h,q_{h}-1}}$.

16:end for

17:Call Algorithm 6 (Simplicial replacement) with input:

  1. 1.

$\mathcal{Q}$,

  2. 2.

the tuple of $\mathcal{Q}$-closed formulas
$\bm{\Phi}=(\phi_{0},\ldots,\phi_{N})=(\Psi_{1},\ldots,\Psi_{N},\Phi)$,

  3. 3.

$\ell=1$

to obtain a simplicial complex $\Delta_{1}(\bm{\Phi})$ such that
$(J\mapsto|\Delta_{1}(\bm{\Phi}|_{J})|)_{J\subset[N]}$ is homologically
$1$-equivalent (cf. Definition 9) to
$\mathbf{Simp}^{[N]}({\mathcal{R}}(\bm{\Phi}))$ (cf. Notation 11).

18: Using Gauss-Jordan elimination identify a minimal subset
$J\subset\\{1,\ldots,N\\}$ such that the
$\mathrm{span}(\\{\mathrm{Im}(\mbox{\rm
H}_{1}(\Delta_{1}(\bm{\Phi}|_{\\{h\\}}))\rightarrow\mbox{\rm
H}_{1}(\Delta_{1}(\bm{\Phi})))\;|\;h\in J\\})=\mbox{\rm
H}_{1}(\Delta_{1}(\bm{\Phi}))$.

19:Output $(\Psi_{h})_{h\in J}$.

20: The complexity of each $\Psi_{h},h\in J$ is bounded by $(sd)^{O(k^{2})}$,
and the complexity of the algorithm is bounded by $(sd)^{k^{O(1)}}$, where
$s=\mathrm{card}(\mathcal{P})$ and $d=\max_{P\in\mathcal{P}}\deg(P)$.

###### Proof of correctness of Algorithm 7.

The correctness of Algorithm 7 follows from the correctness of Algorithms 2,
3, 5, and 6. ∎

###### Complexity analysis of Algorithm 7.

The complexity upper bound is a consequence of the complexity analysis of the
Algorithms 2, 3, 5, and 6. ∎

###### Proof of Theorem 2.

Theorem 2 follows from the correctness and complexity analysis of Algorithm 7.
∎

###### Proof of Theorem 3.

Theorem 3 follows from the correctness and complexity analysis of Algorithms 4
(Computing one-dimensional subset) and 5 (Conversion of curve segment
representations to closed formulas). ∎

##  5\. Conclusion and open problems

In this paper we have proved the existence of an algorithm with singly
exponential complexity for obtaining a semi-algebraic basis of the first
homology group of of a given closed semi-algebraic set (generalizing existing
algorithm in the case of the zero-th dimensional homology). One obvious open
problem is to generalize this result to the higher homology groups. As an
intermediate problem we can ask for the solution to Conjecture 1. The
techniques developed in the current paper may possibly generalize to prove
Conjecture 1, but there are formidable technical problems to overcome and we
leave this to future work.

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# Safe but Incalculable: Energy-weighting is not all you need

Samuel Bright-Thonney<EMAIL_ADDRESS>Physics Department, Cornell
University, 109 Clark Hall, Ithaca, New York 14853, USA  Benjamin Nachman
<EMAIL_ADDRESS>Physics Division, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  Berkeley Institute for Data Science, University of
California, Berkeley, CA 94720, USA  Jesse Thaler<EMAIL_ADDRESS>Center for
Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA
02139, USA  The NSF AI Institute for Artificial Intelligence and Fundamental
Interactions, U.S.A.

###### Abstract

Infrared and collinear (IRC) safety has long been used a proxy for robustness
when developing new jet substructure observables. This guiding philosophy has
been carried into the deep learning era, where IRC-safe neural networks have
been used for many jet studies. For graph-based neural networks, the most
straightforward way to achieve IRC safety is to weight particle inputs by
their energies. However, energy-weighting by itself does not guarantee that
perturbative calculations of machine-learned observables will enjoy small non-
perturbative corrections. In this paper, we demonstrate the sensitivity of
IRC-safe networks to non-perturbative effects, by training an energy flow
network (EFN) to maximize its sensitivity to hadronization. We then show how
to construct Lipschitz Energy Flow Networks ($L$-EFNs), which are both IRC
safe and relatively insensitive to non-perturbative corrections. We
demonstrate the performance of $L$-EFNs on generated samples of quark and
gluon jets, and showcase fascinating differences between the learned latent
representations of EFNs and $L$-EFNs.

††preprint: MIT-CTP/5641

###### Contents

  1. I Introduction
  2. II Methodologies
    1. II.1 Lipschitz Energy Flow Networks
    2. II.2 Bounds on the Energy Mover’s Distance
  3. III Case Study with Quark/Gluon Jets
    1. III.1 Generated Dataset
    2. III.2 Classification Results
    3. III.3 Confirming the EMD Bound
  4. IV Visualizing Non-Perturbative Effects
  5. V Conclusions and Outlook

##  I Introduction

Infrared and collinear (IRC) safety has played a central role in perturbative
quantum chromodynamics (QCD) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], in particularly
for studying the properties of high-energy jets arising from the fragmentation
of quarks and gluons. By being insensitive to soft and collinear splittings
within a parton shower, an IRC-safe observable has a cross section that can be
well-described by a fixed-order perturbation series in the strong coupling
constant, $\alpha_{s}$. For this reason, IRC safety has been used as a guiding
principle in developing new jet substructure observables [11, 12, 13, 14, 15,
16, 17, 18]. While other classes of calculable observables exist—such as
Sudakov-safe observables [19, 20] and track-based observables [21, 22]111More
generally, there is a broader class of observables that can be calculated
after absorbing perturbative singularities into universal non-perturbative
objects like fragmentation functions [23, 24].—IRC safety has driven the
selection of collider observables for both measurements and searches involving
jets.

Machine learning (ML) methods have significantly extended the sensitivity of
jet tagging, in part by leveraging low-level information [15, 25, 26]. Such
taggers are generally IRC unsafe, which means that they are reliant on the
non-perturbative models in the generators used to train and test them. Since
these non-perturbative models are imperfectly constrained by experimental
data, this has motivated the development of ML observables that are IRC (or
Sudakov) safe [27, 28, 29]. One class of IRC-safe architectures are Energy
Flow Networks (EFNs) [28], which are built on the deep sets framework [30] and
are able to model any permutation-invariant IRC-safe observable. EFNs, along
with extensions that incorporate graph-like structures [31, 32, 33], achieve
IRC safety by linearly weighting particle inputs by their energies.

In this paper, we emphasize that energy-weighting, by itself, does not ensure
trustworthy perturbative calculations for ML observables. While IRC safety
does ensure the existence of a fixed-order perturbation series in
$\alpha_{s}$,222Strictly speaking, this statement depends on the precise
definition one uses for “IRC safety”. In the language of Ref. [10], we really
mean Hölder continuous. it does not guarantee that non-perturbative
corrections will be small. Using EFNs as a representative example, we show how
to train an IRC-safe neural network to be maximally sensitive to non-
perturbative hadronization, thereby constructing an observable that is “safe
but incalculable”.333The opposite case of “unsafe but calculable” observables
can arise in resummed perturbation theory, where there is no order-by-order
$\alpha_{s}$ expansion but nevertheless non-perturbative corrections are
suppressed [19, 20]. As a step towards restoring calculablity, we introduce
Lipschitz Energy Flow Networks ($L$-EFNs), whose bounded gradients ensure
bounded sensitivity to non-perturbative corrections.

The fact that IRC-safe observables can have cross sections with large non-
perturbative corrections is not new, even if it may not be widely appreciated.
The standard (but misleading) lore is that IRC-safe observables should have
non-perturbative corrections that are power-suppressed as $(\Lambda_{\rm
QCD}/E)^{n}$, where $\Lambda_{\rm QCD}$ is the QCD confinement scale, $E$ is
the energy scale of the process in consideration, and $n$ is some integer
power (typically 1 or 2). Already, though, it is known that jet angularities
[34, 35, 36, 37] with angular exponent $\beta\lesssim 1$ have non-perturbative
corrections with $n=\beta$ scaling [38, 8, 19, 19], which turns into
$\mathcal{O}(1)$ effects as $\beta\to 0$. Because there is no general first-
principles understanding of non-perturbative QCD effects, then the cross
section is essentially incalculable (or at least untrustable) if these
corrections grow large.

In the context of IRC-safe ML models, we are not aware of any previous studies
of the general impact of non-perturbative effects. Here, to identify ML
observables with maximal non-perturbative sensitivity, we train an IRC-safe
classifier to distinguish parton-level from hadron-level events. Classifiers
whose cross sections have controlled non-perturbative corrections should be
unable to distinguish between these samples. Instead, we find that EFNs are
highly effective at parton-level versus hadron-level classification, implying
large non-perturbative sensitivity. Our new $L$-EFN architecture reduces this
sensitivity by imposing spectral normalization [39, 40], which is equivalent
to bounding the Lipschitz norm of the network (see related work in Refs. [41,
42]). This approach is motivated by the Kantorovich-Rubinstein duality theorem
[43] and the Energy Mover’s Distance (EMD) [44], which provides a robust way
to estimate the size of non-perturbative effects.

The remainder of this paper is organized as follows. In Sec. II, we introduce
$L$-EFNs and explain how the Lipschitz constraint enforces an EMD bound on
non-perturbative corrections. We then perform a case study in Sec. III to
compare the hadronization sensitivity of EFNs and $L$-EFNs. We investigate the
learned latent representations of ($L$-)EFNs in Sec. IV and conclude in Sec.
V.

##  II Methodologies

###  II.1 Lipschitz Energy Flow Networks

The $L$-EFN architecture we propose in this work is built on top of a standard
EFN, which provides a generic framework for learning IRC-safe observables.
Given a jet with constituent momenta $p_{1},p_{2},\ldots,p_{M}$, an EFN
computes a function of the form:

$\mathrm{EFN}(\\{p_{1},\ldots,p_{M}\\})=F\left(\sum_{i=1}^{M}z_{i}\Phi(\hat{p}_{i})\right),$
(1)

where $z_{i}=p_{T,i}/p_{T,\mathrm{jet}}$ is the constituent momentum or energy
fraction and $\hat{p}_{i}$ is the particle’s angular position relative to the
jet axis. The function $\Phi:\mathbb{R}^{2}\to\mathbb{R}^{\ell}$ maps
individual particles to a latent space of dimension $\ell$. The function
$F:\mathbb{R}^{\ell}\to\mathbb{R}^{d_{\mathrm{out}}}$ maps the latent
representation to the final output. In a standard EFN, the functions $\Phi$
and $F$ are unconstrained and typically implemented as neural networks. The
additive and energy-weighted structure of an EFN guarantees a naturally
permutation-invariant and IRC safe output; see Ref. [28] for further
discussion.

An $L$-EFN extends the EFN setup by constraining $\Phi$ and $F$ to be
$L$-Lipschitz, meaning that

$\displaystyle\|\Phi(\hat{p}_{1})-\Phi(\hat{p}_{2})\|$ $\displaystyle\leq$
$\displaystyle L\|\hat{p}_{1}-\hat{p}_{2}\|,$
$\displaystyle\|F(x_{1})-F(x_{2})\|$ $\displaystyle\leq$ $\displaystyle
L\|x_{1}-x_{2}\|.$

This is effectively a bound on the gradients of these functions, though the
Lipschitz constraint does not require $\Phi$ and $F$ to be everywhere
differentiable. In principle, one could choose different $L$ values for $\Phi$
and $F$, but we keep them the same for simplicity of discussion.

If $\Phi$ and $F$ are neural networks with $L$-Lipschitz activations,444Many
standard activation functions are $1$-Lipschitz, such as ReLU, LeakyReLU, and
Sigmoid. this amounts to a constraint on the spectral norm of their weight
matrices $W^{i}$ [39]:

$\sigma(W^{i}):=\max_{\mathbf{h}\neq\mathbf{0}}\frac{\|W^{i}\mathbf{h}\|_{2}}{\|\mathbf{h}\|_{2}}\leq
L.$ (2)

This can be enforced during training by scaling the weight matrices as
$W^{i}\to LW^{i}/\sigma(W^{i})$ using a computationally efficient estimation
of $\sigma(W^{i})$ [39]. We focus on the $L=1$ case throughout this paper, and
$L$-EFN should be henceforth understood as $L=1$.

For the studies in Sec. III.2 we use the base architectures from the
EnergyFlow package [28, 27], implemented and trained using TensorFlow [45],
Keras [46], and Adam [47]. To enforce the $1$-Lipschitz constraint when
training $L$-EFNs, we replace all linear Dense layers in the networks with
SpectralDense layers from the deel-lip package [48]. Unless otherwise
specified, the $\Phi$ ($F$) networks of all ($L$-)EFNs are implemented as
three-layer fully-connected neural networks with a width of 60 (80) and 0%
(10%) dropout [49]. We use ReLU activations for internal nodes and Sigmoid for
the 1D output node, both of which have a Lipschitz constant of 1. The networks
are trained to learn 0/1 class labels based on the binary cross-entropy loss
function.

As a reference, we also show results for a particle flow network (PFN), which
is IRC unsafe by construction:

$\mathrm{PFN}(\\{p_{1},\ldots,p_{M}\\})=F\left(\sum_{i=1}^{M}\Phi(p_{i})\right).$
(3)

This differs from the EFN in Eq. (1) by the lack of $z_{i}$ weighting and the
use of the unnormalized particle momentum $p_{i}$ instead of just the particle
direction $\hat{p}_{i}$. We also consider a Lipschitz PFN ($L$-PFN) which
imposes a spectral norm constraint on $F$ and $\Phi$.

###  II.2 Bounds on the Energy Mover’s Distance

With the simple addition of a Lipschitz constraint, we can now make precise
statements about the sensitivity of an $L$-EFN to non-perturbative effects.
Here, we leverage a recent geometric language for analyzing collision events
based on the EMD [44], which measures the “work” required to transform one jet
into another. The EMD is a variant of the Earth Mover’s Distance from computer
vision [50, 51, 52, 53, 54] and is equivalent to the 1-Wasserstein metric [55,
56] in certain limits. The EMD provides a pairwise metric distance for IRC-
safe energy flows, which can be used to triangulate the space of jets and
define various geometric quantities.

Consider an additive IRC-safe observable $f$ acting on a jet $\mathcal{J}$:

$f(\mathcal{J})=\sum_{i\in\mathcal{J}}E_{i}\Phi(\hat{p}_{i}),$ (4)

where $\Phi$ has Lipschitz constant $L$. Given a pair of jets
$\mathcal{J}_{1}$, $\mathcal{J}_{2}$, a key result from Ref. [44] places a
bound on the difference between $f(\mathcal{J}_{1})$ and $f(\mathcal{J}_{2})$
based on the EMD between $\mathcal{J}_{1}$ and $\mathcal{J}_{2}$:

$\frac{1}{RL}\left|f(\mathcal{J}_{1})-f(\mathcal{J}_{2})\right|\leq\mathrm{EMD}(\mathcal{J}_{1},\mathcal{J}_{2}),$
(5)

where $R$ is the jet radius. The composition of two Lipschitz functions with
constants $L_{A}$ and $L_{B}$ is also Lipschitz, with constant $L_{A}L_{B}$.
Thus, for an $L$-EFN, we have:

$\frac{1}{RL^{2}}\left|L\text{-EFN}(\mathcal{J}_{1})-L\text{-EFN}(\mathcal{J}_{2})\right|\leq\mathrm{EMD}(\mathcal{J}_{1},\mathcal{J}_{2}).$
(6)

This bound holds for any $\mathcal{J}_{1}$ and $\mathcal{J}_{2}$, and we again
emphasize that we focus on $L=1$ in this paper.

Now, consider $\mathcal{J}_{1}$ and $\mathcal{J}_{2}$ to be the same jet at
parton- and hadron-level, respectively. Of course, parton-level information
cannot be accessed at a real collider, but it can be done in the context of a
parton shower event generator. Then, Eq. (6) gives an upper bound on
modifications of the $L$-EFN observable due to hadronization. This prevents an
$L$-EFN from becoming overly sensitive to non-perturbative physics, and makes
it naturally robust to any unphysical mismodeling artifacts present in Monte
Carlo simulations. By contrast, a plain EFN does not have this constraint,
which explain why it can be arbitrarily sensitive to non-perturbative effects,
as we will see in the following case study.

##  III Case Study with Quark/Gluon Jets

###  III.1 Generated Dataset

To demonstrate the impact of non-perturbative physics on ML algorithms, we
perform a case study involving quark and gluon jets. Following Ref. [57], we
generate samples of $e^{+}e^{-}\to H\to q\bar{q}$ (quark jets) and $gg$ (gluon
jets) using Pythia 8.307. We take a center-of-mass collision energy of
$\sqrt{s}=m_{H}=1$ TeV, and we fix $q=u$ (i.e. up quarks only) and $m_{u}=0$
for simplicity.

In each event, we independently cluster the partons (post-shower) and final-
state hadrons into anti-$k_{T}$ jets with $R=1.0$ [58, 59] and the winner-
take-all (WTA) axis. We match the leading (highest $p_{T}$) hadron-level jet
to the nearest parton-level jet within $\Delta R<0.1$,555If no parton-level
jet is found within this $\Delta R$ cone, the event is skipped. and save the
constituent kinematic information for both jets. We generate a total of
200,000 $e^{+}e\to q\bar{q}$ and 200,000 $e^{+}e^{-}\to gg$ events. Each
dataset is split into training, validation, and test sets with fractions 50%,
25%, and 25%, respectively.

(a)  (b)  (c)

Figure 1:  The spectrum of (a) jet mass, (b) jet constituent multiplicity, and
(c) $p_{T}^{D}$. Shown are the normalized distributions for quarks (blue) and
gluons (orange) at parton-level (filled) versus hadron-level (solid).

(a)  (b)

Figure 2: Score distributions for the (a) EFN and (b) $L$-EFN architectures,
when trained to distinguish parton-level from hadron-level jets. The color
scheme is the same as Fig. 1. Despite being IRC safe, the EFN achieves
excellent parton/hadron separation power. After imposing the Lipschitz
constraint, the $L$-EFN exhibits the desired insensitivity to non-perturbative
effects.

(a)  (b)

Figure 3:  ROC curves for parton- vs hadron-level discrimination for (a) quark
and (b) gluon jets using an EFN/PFN (solid) and an $L$-EFN/$L$-PFN (dashed).
The bands represents the spread from starting with 10 different random
initializations of the neural networks. The results for constituent
multiplicity is shown for reference. Only the $L$-EFN shows the desired lack
of sensitivity to non-perturbative effects.

In Fig. 1, we show the spectrum of three jet substructure observables: the jet
mass, the number of jet constituents, and $p_{T}^{D}$ [60]. The latter
observable is the sum of the squares of the momentum fractions of constituents
within the jet. Of these observables, only the jet mass is IRC safe. Not
surprisingly, the jet mass is the observable with the smallest difference
between the parton-level and hadron-level distributions. For the number of
constituents and $p_{T}^{D}$, non-perturbative corrections are an
$\mathcal{O}(1)$ effect. Nevertheless, these two observables are highly
effective for quark versus gluon jet tagging [61], which is why one has to be
careful about non-perturbative sensitivity in the ML context.

###  III.2 Classification Results

To investigate the impact of Lipschitz constraints on non-perturbative
sensitivity, we train EFNs and $L$-EFNs to distinguish hadron-level from
parton-level jets. For this analysis, we treat the quark and gluon datasets
separately. To verify the stability of our trainings and estimate
uncertainties on performance metrics, we train ten versions of each model with
a different random initializations. In testing, we found that $L$-EFNs train
more stably with a significantly larger batch size than EFNs (10,000 vs. 128)
and a smaller learning rate ($10^{-4}$ vs. $10^{-3}$). This is likely due to
the spectral normalization of the weight matrices at each step, and the
inherent difficulty of distinguishing parton-level from hadron-level jets
given the constraint of Eq. (6). With these modifications, however, all
$L$-EFN trainings converged easily.

Distributions of the learned EFN and $L$-EFN scores are shown in Fig. 2,
considering all four combinations of quark versus gluon and parton-level
versus hadron-level jets. Despite being IRC safe, the EFN is highly effective
at distinguishing parton-level from hadron-level jets. This implies that the
non-perturbative corrections to the EFN observable are large. Assuming that
the EFN architecture and training procedure are sufficiently flexible,666We
did not perform an extensive hyperparameter scan, but we found little gains
from small variations on our setup. then this is an example of an IRC-safe
observable with maximal non-perturbative sensitivity. By contrast, the $L$-EFN
is not very effective at distinguishing parton-level from hadron-level jet, so
according to the hadronization model in Pythia, the non-perturbative
corrections are small.

To better assess the learned separation power, we plot receiver operating
characteristic (ROC) curves in Fig. 3, obtained by placing a sliding cut over
the distributions in Fig. 2. We also indicate their Area Under the Curve (AUC)
scores in the legend. The ROC curves underscore and quantify the trends
observed in Fig. 2. The EFN is an extremely efficient parton-versus-hadron
classifier, whereas the $L$-EFN performs barely better than random chance. We
therefore conclude that the $1$-Lipschitz modification is effective at
suppressing the $L$-EFN’s sensitivity to non-perturbative effects.

As a point of comparison, Fig. 3 also shows ROCs and AUCs for multiplicity,
PFNs and $L$-PFNs – all explicitly IRC-unsafe observables. Unsurprisingly, the
PFN is able to nearly perfectly distinguish parton from hadron jets, and the
$L$-PFN performs only slightly worse. This indicates that the $1$-Lipschitz
constraint does not severely limit the expressiveness of the model, and that
the large gap between the EFN and $L$-EFN is a genuine result of non-
perturbative effects.

(a)  (b)  (c)

(d)  (e)  (f)

Figure 4: Two dimensional distributions of the parton-hadron EMD versus the
difference in observables at parton- and hadron-level. Shown are results for
(left column) the $\beta=1$ angularity $\lambda^{(1)}$, (middle column) an
EFN, (right column) an $L$-EFN, separately for (top row) quark jets and
(bottom row) gluon jets. The EMD is computed using momentum fractions $z_{i}$
in keeping with the $\lambda^{(1)}$/EFN/$L$-EFN calculations, and is thus
dimensionless.

###  III.3 Confirming the EMD Bound

As a cross-check of our analysis, we verify that the learned $L$-EFN satisfies
the EMD bound in Eq. (6). The plots in Fig. 4 compares the difference between
observables at parton level versus hadron level to the EMD between parton- and
hadron-level events.

The top row of Fig. 4 reproduces Fig. 2 of Ref. [44] for the $\beta=1$
angularity:

$\lambda^{(1)}=\sum_{i\in\text{jet}}z_{i}\,\Delta R_{i,\text{jet}},$ (7)

where $z_{i}=p_{T,i}/p_{T,\text{jet}}$ and $\Delta
R_{i,\text{jet}}^{2}=\Delta\eta_{i,\text{jet}}^{2}+\Delta\phi_{i,\text{jet}}^{2}$.
This function is $1$-Lipschitz and the bound is respected and nearly
saturated.

The middle row of Fig. 4 shows the non-perturbative modification to the EFN
score, which clearly violates the theoretical bound. This is expected, since
the EFN is not required to be a Lipschitz function. Indeed, there is no finite
slope (i.e. Lipschitz constant) that would bound the EFN score relative to the
EMD.

By contrast, the bottom row of Fig. 4 demonstrates that the $L$-EFN does
respect the EMD bound. For illustrative purposes, we show the $L$-EFN logits
(i.e. pre-activations on the final layer), since the actual output scores all
lie along the bottom of the plot. Even after performing the logit
transformation, the $L$-EFN does not appear to saturate the bound, unlike what
was seen for the IRC-safe angularities.

(a)  (b)  (c)

(d)  Figure 5:  Angular filters $\Phi(\eta,\phi)$ for parton-level versus
hadron-level discrimination with a latent space of dimension $\ell=1$. The EFN
filters are shown for (a) quark and (c) gluon jets, and similarly for the
$L$-EFN filters for (b) quark and (d) gluon jets. The radial profiles
$\Phi(R=\sqrt{\eta^{2}+\phi^{2}})$ are shown in (e), demonstrating the EFN
sensitivity to highly collinear emissions ($R\lesssim 0.02$). While the jet
radius is $R=1.0$, we have zoomed in to show the collinear structure of the
filters.

(e)

##  IV Visualizing Non-Perturbative Effects

The above case study demonstrates that IRC safety is insufficient to protect
against large non-perturbative modifications, but it gives no immediate
insight into which observables are most susceptible. In this section, we take
a step towards answering this question by training EFNs with a minimal latent
dimension size of $\ell=1$. This corresponds to a single learned filter
$\Phi(\eta,\phi)$ which can be easily visualized, and a simple one-dimensional
classification function $F$.

As in Sec. III.2, we train $\ell=1$ EFNs and $L$-EFNs to discriminate parton-
and hadron-level jets, treating the quark and gluon samples separately.
Perhaps surprisingly, the $\ell=1$ EFNs are still able to achieve AUCs of 0.90
(quarks) and 0.91 (gluons), compared to 0.96 for the full $\ell=60$ models in
Sec. III.2. The $\ell=1$ $L$-EFNs perform similarly well, with AUCs of 0.54
and 0.53, compared to 0.55 and 0.54 previously. We therefore conclude that an
$\ell=1$ analysis is sufficient to capture the leading sources of non-
perturbative sensitivity.

In Fig. 5d, we plot the filter profiles $\Phi(\eta,\phi)$ for the $\ell=1$
($L$-)EFNs (left and middle columns) alongside their angle-averaged radial
profiles (right column). All filters exhibit cylindrical symmetry, as expected
since the generated samples are for unpolarized jets with no preferred angular
orientation. Interestingly, the EFN filters appears to be maximally sensitive
to highly collinear radiation. Despite the jet radius being $R=1.0$, the EFN
filters peak within a radius of $\sim 0.02$, with a plateau going out to
larger scales. The EFN filters also vanish at the origin, reflecting the
consistent presence of a particle at $R=0$ aligned with the WTA jet axis.777If
angles are instead measured with respect to the standard jet momentum axis,
then the $\ell=1$ EFN filters often exhibit anisotropic features.

The $L$-EFN filters, on the other hand, cannot vary so rapidly near the origin
due to the $1$-Lipschitz constraint. Thus, the $L$-EFN cannot be as sensitive
to collinear radiation, while simultaneously ignoring the central particle.

This differing behavior of EFNs and $L$-EFN is consistent with theoretical
expectations about the non-perturbative sensitivity of the angularities:

$\lambda^{(\beta)}=\sum_{i\in\text{jet}}z_{i}\,\Delta
R_{i,\text{jet}}^{\beta},$ (8)

which has Lipschitz constant $L=\beta$. When $\beta\gtrsim 1$, non-
perturbative corrections are suppressed by $\Lambda_{\rm QCD}/E_{\rm jet}$, so
they are relatively small and independent of $\beta$ for high energy jets, in
agreement with the $L$-EFN behavior.

For $\beta\lesssim 1$, though, non-perturbative corrections scale as
$(\Lambda_{\rm QCD}/E_{\rm jet})^{\beta}$. This is unsuppressed as $\beta\to
0$, which yields large differences between parton-level and hadron-level
distributions. Note that in the $\beta\to 0$ limit, the angularities are
dominated by collinear emissions. This explains why the IRC-safe EFN with no
Lipschitz constraint can have large non-perturbative sensitivity coming from
small-angle physics.

(a)  (b)

Figure 6:  Parton versus hadron discriminating power for the angularities
$\lambda^{(\beta)}$ from Eq. (8) as a function $\beta$. Shown are (a) ROC
curves and (b) AUC scores. We also show ROC curves for the $\ell=1$ EFNs and
$L$-EFNs, which agree well with the $\beta\ll 1$ and $\beta\approx 1$ limits.
For simplicity, in (a) we only show quark ROC curves for the $\beta$ scan,
though the gluon results are qualitatively similar.

In Fig. 6, we quantify the parton versus hadron discrimination power of the
angularities as a function of $\beta$. ROC curves are shown for the EFN and
$L$-EFNs for comparison, and we see a rough interpolation where $\beta\to 0$
maps to the EFN, while $\beta\approx 1$ maps to the $L$-EFN. The AUCs
monotonically increase as $\beta$ decreases, peaking around $\beta=0.02$
before losing sensitivity as $\beta\to 0$.

##  V Conclusions and Outlook

In this paper, we demonstrated the susceptibility of IRC-safe neural networks
to large, incalculable non-perturbative corrections. We showed how to make the
network maximally sensitive to these corrections by training it to distinguish
between parton-level and hadron-level jets. In general, there is nothing
constraining this effect in a typical application. Although IRC safety
guarantees that the network’s output is perturbatively calculable in
principle, this is insufficient for robustness to large and potentially mis-
modeled non-perturbative corrections.

To mitigate this issue, we introduced a new neural network architecture, the
$L$-EFN, which has significantly reduced sensitivity to non-perturbative
physics. We achieve this by constraining the networks to be $L$-Lipschitz
functions, which bounds potential non-perturbative modifications according to
the EMD between parton- and hadron-level jets. Using Pythia’s hadronization
model as a proxy for non-perturbative effects, we demonstrated that an $L$-EFN
is barely able to distinguish between the same jet at parton- and hadron-
level. In the future, it would be interesting to explore different
hadronization models and other non-perturbative effects like multiparton
interactions (i.e. underlying event). It would also be interesting to study
the robustness of $L$-EFNs to perturbative uncertainties in parton showers.

The ideas behind $L$-EFNs could be extended to other architectures, such as
graph neural networks and transformers, though it may be more complicated to
derive theoretical bounds similar to Eq. (6). It would also be interesting to
benchmark $L$-EFNs and $L$-EFN-inspired architectures on standard jet tagging
tasks and assess the tradeoff between performance and robustness to
hadronization. In the case of $L$-EFNs, the Lipschitz constant provides an
easily tunable network constraint that balances these considerations. The non-
Lipschitz networks trained to distinguish parton and hadron jets may also be
useful to explore the impact of hadronization where it is predicted to be the
largest. For non-IRC-safe networks, our studies did not reveal dramatic
differences between PFNs and $L$-PFNs, but the flexibility to adjust $L$ may
be beneficial for ablation studies.

With a growing trend to build robust and interpretable neural networks that
are structured with particle physics in mind, it is important to be guided by
both formal and practical considerations. Finding a balance between neat
theoretical constraints and performance demands is inevitably delicate, but
$L$-EFNs present a concrete step towards uniting the power of deep learning
with a reasoned caution against learning spurious or unphysical details of a
simulation. We look forward to continued investigation and development in
future work, and hope to see the ongoing development of new machine learning
tools alongside theoretical tools to better understand them.

###### Acknowledgements.

We thank Rikab Gambhir for feedback on the manuscript and discussions of the
$\beta\to 0$ limit of angularities. SBT is supported by the National Science
Foundation under Award No. PHY-2209443. BN is supported by the U.S. Department
of Energy (DOE), Office of Science under contract DE-AC02-05CH11231. JT is
supported by the DOE Office of High Energy Physics under Grant Contract No.
DE-SC0012567, and by the National Science Foundation under Cooperative
Agreement No. PHY-2019786 (The NSF AI Institute for Artificial Intelligence
and Fundamental Interactions, http://iaifi.org/). This research used resources
of the National Energy Research Scientific Computing Center, a DOE Office of
Science User Facility supported by the Office of Science of the U.S.
Department of Energy under Contract No. DE-AC02-05CH11231 using NERSC award
HEP-ERCAP0021099.

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