DESY 21-043 ISSN 0418-9833

March 2021

Non-planar universal anomalous dimension
of twist-two operators with general Lorentz spin
at four loops in ${\cal N}=4$ SYM theory

B. A<EMAIL_ADDRESS>and V. N<EMAIL_ADDRESS>

aInstitut für Theoretische Physik,
Hamburg Universität,
Luruper Chaussee 149,
Hamburg, Germany
bTheoretical Physics Division
Petersburg Nuclear Physics Institute
Orlova Roscha, Gatchina
188300 St. Petersburg, Russia

Abstract

We compute the non-planar contribution to the universal anomalous dimension of
twist-two operators in ${\cal N}=4$ supersymmetric Yang-Mills theory at four
loops through Lorentz spin eighteen. Exploiting the results of this and our
previous calculations along with recent analytic results for the cusp
anomalous dimension and some expected analytic properties, we reconstruct a
general expression valid for arbitrary Lorentz spin. We study various
properties of this general result, such as its large-spin limit, its small-$x$
limit, and others. In particular, we present a prediction for the non-planar
contribution to the anomalous dimension of the single-magnon operator in the
$\beta$-deformed version of the theory.

###### Contents

  1. 1 Introduction
  2. 2 Computations for fixed values of $j$
  3. 3 Reconstruction
  4. 4 Conclusion
  5. 5 Appendix

##  1 Introduction

In the framework of Quantum Chromodynamics (QCD), the anomalous dimensions of
composite operators provide us with information on the evolution of the parton
distribution functions of the proton through the Dokshitzer-Gribov-Lipatov-
Altarelli-Parisi (DGLAP) [1, 2, 3, 4] equations. The simplest way to obtain
these quantities is the direct computation of the renormalization of the
composite operators which appear in the operator product expansion of two
currents in the framework of electron-proton deep-inelastic scattering. In
QCD, this is usually done in this way. At the present time, such calculations
are performed analytically through the next-to-next-to-next-to-leading order,
i.e., through fourth order in the strong-coupling constant or, equivalently,
through four loops in the corresponding diagrammatic technique [5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

On the other hand, by definition, the anomalous dimension is the quantum
correction to the canonical dimension of the product of two (or more)
elementary fields or composite operators. This approach is usually adopted in
conformal field theories. Both approaches have actively been used after the
discovery of the AdS/CFT correspondence [24, 25, 26] between supergravity in
anti-de Sitter space and four-dimensional conformal field theory,
$\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory. The great interest in
calculations of anomalous dimensions of composite operators comes from
investigations of integrability in the framework of the AdS/CFT
correspondence. In the planar limit, both direct computations [27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] and computations via the
generalized Lüscher corrections [42, 43] were performed [44, 45, 46, 47, 48,
49, 50] at higher orders of perturbation theory to test the Asymptotic Bethe
Ansatz [51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67,
68, 69, 70] as well as the Quantum Spectral Curve relations [71, 72, 73, 74,
75, 76, 77, 78]. The latter allowed one to compute the general result for the
anomalous dimension of the twist-two operators through seven loops [79, 80]
and, for special values of the Lorentz spin $j$, even through eleven loops
[81, 82, 83, 84].

The computation of the non-planar contribution to the universal anomalous
dimension of the twist-two operators in $\mathcal{N}=4$ SYM theory, which is
suppressed by the inverse color factor $N_{c}$ of the gauge group, is of great
interest in the context of gauge/string duality, as it is related to the loop
corrections of string amplitudes. Information on non-planar corrections can be
obtained by direct computation of the appropriate non-planar Feynman diagrams,
via advanced computerized methods of four-loop calculation. Such computations
were performed in Refs. [85, 86, 87]. Alternatively, they can be calculated by
applying the method of asymptotic expansion to the four-point functions of
length-two half-Bogomol’nyi-Prasad-Sommerfield operators [88]. These results
allows us deepen our understanding of the AdS/CFT correspondence in regions
already accessed and to explore new ones. Having the general result for the
non-planar part of the general anomalous dimension of the twist-two operators,
for general value of their Lorentz spin $j$, allows us to study its particular
limits. The most interesting one, of large Lorentz spin $j$, yields the cusp
anomalous dimension, which can also be computed by other methods and which, in
$\mathcal{N}=4$ SYM theory, was obtained to all orders in the planar limit
[68]. Recently, its non-planar part has been established through four loops,
at $\mathcal{O}(g^{8})$, via the Sudakov form factor, numerically in Refs.
[89, 90] and analytically in Ref. [91], and via light-like polygonal Wilson
loops, analytically in Ref. [92]. At four loops in QCD, at
$\mathcal{O}(\alpha_{s}^{4})$ in the strong-coupling constant $\alpha_{s}$,
the quark cusp anomalous dimension in the planar limit was found via the quark
form factor in Ref. [93], its contribution with quartic fundamental color
factor was obtained, again via the quark form factor, in Ref. [94], and the
complete quark and gluon cusp anomalous dimensions were established via their
counterpart in ${\mathcal{N}}=4$ SYM theory in Ref. [92] and via the massless
quark and gluon form factors in Ref. [95]. Moreover, it is of interest to
study modifications of the Balitsky–Fadin–Kuraev–Lipatov (BFKL) [96, 97, 98]
and the double-logarithmic [99, 100] equations due to non-planar
contributions. The latter can easily be obtained from the general result by
means of analytic continuation.

As for the universal anomalous dimension of the twist-two operators in
$\mathcal{N}=4$ SYM theory, the results previously obtained [85, 86, 87] for
the first three nontrivial even values of $j$, for $j=4,6,8$, gave hope for
the feasibility to reconstruct the general result, for generic value of $j$,
by means of special methods based on number theory. In fact, this provides a
strong motivation for us to proceed to higher values of $j$. In a recent
letter [101], we summarized our new results for the next five nontrivial $j$
values, $j=10,\ldots,18$, and performed a completely independent numerical
computation of the cusp anomalous dimensions. In the following, we provide
full details of the calculation and, on top of that, conjecture an analytic
expression for arbitrary value of $j$.

##  2 Computations for fixed values of $j$

We consider the following SU(4)-singlet, twist-two operators within
$\mathcal{N}=4$ SYM theory:

$\displaystyle\mathcal{O}_{\mu_{1},...,\mu_{j}}^{\lambda}$ $\displaystyle=$
$\displaystyle\hat{S}\bar{\lambda}_{i}^{a}\gamma_{\mu_{1}}{\mathcal{D}}_{\mu_{2}}\cdots{\mathcal{D}}_{\mu_{j}}\lambda^{a\;i}\,,$
(1) $\displaystyle\mathcal{O}_{\mu_{1},...,\mu_{j}}^{g}$ $\displaystyle=$
$\displaystyle\hat{S}G_{\rho\mu_{1}}^{a}{\mathcal{D}}_{\mu_{2}}{\mathcal{D}}_{\mu_{3}}\cdots{\mathcal{D}}_{\mu_{j-1}}G_{\phantom{a,\rho}\mu_{j}}^{a,\rho}\,,$
(2) $\displaystyle\mathcal{O}_{\mu_{1},...,\mu_{j}}^{\phi}$ $\displaystyle=$
$\displaystyle\hat{S}\bar{\phi}_{r}^{a}{\mathcal{D}}_{\mu_{1}}{\mathcal{D}}_{\mu_{2}}\cdots{\mathcal{D}}_{\mu_{j}}\phi_{r}^{a}\,,$
(3)

where ${\mathcal{D}}_{\mu_{i}}$ denotes the covariant derivative, the spinors
$\lambda_{i}$ and the field-strength tensor $G_{\rho\mu}$ describe gauginos
and gauge fields, respectively, and $\phi_{r}$ are the complex scalar fields
appearing in ${\mathcal{N}}=4$ SYM theory. The indices $i=1,2,3,4$ and
$r=1,2,3$ refer to the SU(4) and SO(6)${}\simeq{}$SU(4) groups of inner
symmetry, respectively. The symbol $\hat{S}$ implies a symmetrization of each
tensor in the Lorentz indices $\mu_{1},...,\mu_{j}$ and a subtraction of its
traces. The operator in Eq. (1) is familiar from the same type of computations
in QCD. The operators in Eqs. (1)–(3) form the multiplicatively renormalized
operators. Their anomalous dimensions are expressed through the so-called
universal anomalous dimension,

$\gamma_{\mathrm{uni}}(j)\ =\
\sum_{n=1}^{\infty}\gamma_{\mathrm{uni}}^{(n-1)}(j)\,g^{2n}\,,$ (4)

where $g^{2}=\lambda/(16\pi^{2})$ with $\lambda=g^{2}_{\mathrm{YM}}N_{c}$
being the ’t Hooft coupling constant, up to integer argument shifts [102]. In
particular, the universal anomalous dimension $\gamma_{\mathrm{uni}}(j)$ in
Eq. (4) is related to the anomalous dimension of the frequently studied twist-
two operator

$\mathcal{O}^{M}_{\mathcal{Z}}=\mathrm{Tr}\mathcal{Z}{\mathcal{D}}_{\mu_{1}}{\mathcal{D}}_{\mu_{2}}\cdots{\mathcal{D}}_{\mu_{M}}\mathcal{Z}\,,$
(5)

where $\mathcal{Z}$ is one of the scalar fields $\phi_{r}$, which belong to
the $\mathfrak{sl}(2)$ sub-sector of $\mathcal{N}=4$ SYM theory, just by
shifting the argument, as

$\gamma_{\mathrm{uni}}(j)=\gamma_{\mathcal{O}^{M}_{\mathcal{Z}}}(M+2)\,.$ (6)

Non-planar contributions arise from the Feynman diagrams which contain the
following combinations of color structures:

$d^{abcd}=\frac{1}{6}\left[\mathrm{Tr}\left(f^{paq}f^{qbr}f^{rcs}f^{sdp}\right)+\mathrm{five}\
bcd\ \mathrm{permutations}\right]\,.$ (7)

It hence follows that

$d_{44}=d^{abcd}d_{abcd}=\frac{1}{24}N_{c}^{2}(N_{c}^{2}+36)={N_{c}^{4}}\left(\frac{1}{24}+\frac{3}{2}\frac{1}{N_{c}^{2}}\right)=\frac{{N_{c}^{4}}}{32}\left(\frac{4}{3}+\frac{48}{N_{c}^{2}}\right)\,.$
(8)

From Eq. (4) we thus glean that the non-planar (np) four-loop ($n=4$)
contributions of Lorentz spin $j$ to the universal anomalous dimension,
$\gamma_{\mathrm{uni,np}}^{(3)}(j)$, are proportional to $g^{8}/N_{c}^{2}$.
For the reader’s convenience, we recall here the previously computed results
for $j=4,6,8$ [85, 86, 87]:

$\displaystyle\gamma_{\mathrm{uni,np}}^{(3)}(4)$ $\displaystyle=$
$\displaystyle-360\zeta_{5}\,\frac{48}{N_{c}^{2}}\,,$ (9)
$\displaystyle\gamma_{\mathrm{uni,np}}^{(3)}(6)$ $\displaystyle=$
$\displaystyle\frac{25}{9}\left(21+70\zeta_{3}-250\zeta_{5}\right)\frac{48}{N_{c}^{2}}\,,$
(10) $\displaystyle\gamma_{\mathrm{uni,np}}^{(3)}(8)$ $\displaystyle=$
$\displaystyle\frac{49}{600}\left(1357+4340\zeta_{3}-11760\zeta_{5}\right)\frac{48}{N_{c}^{2}}\,,$
(11)

where $\zeta_{n}=\zeta(n)$ is Riemann’s zeta function. We observe that there
are rather simple common factors on the right-hand sides of Eqs. (9)–(11),
which we pulled out. In fact, the prefactors in Eqs. (9)–(11) resemble the
harmonic sums $\sum_{i=1}^{j-2}\frac{1}{i}$ for $j=4,6,8$, with values $3/2$,
$25/12$, $49/20$, and harmonic sums are also expected to appear as building
blocks of $\gamma_{\mathrm{uni,np}}^{(3)}(j)$, as explained below. If this
kind of factorization were preserved for the higher values of $j$, this would
considerably simplify the procedure for finding the general form of the
universal anomalous dimension from the results for fixed values of $j$.

We now proceed to the case of $j=10$, where we encounter two difficulties in
the application of the method established in Refs. [85, 86, 87]. First, it is
necessary to extend the database of all Feynman integrals which can be
contained in the computed Feynman diagrams. Such a database is formed by
reducing the considered Feynman integrals to master integrals. For this
purpose, we use our MATHEMATICA realization of the Laporta algorithm [103].
Second, a new vertex, with two fermion and four gluon lines, arises from the
operator in Eq. (1) starting at $j=5$. It did not appear in our previous
calculations [85, 86, 87], as we exploited properties of the mixing matrix to
simplify our computations. This simplification can be easily explained by
means of well-known results at the leading order. The matrix of anomalous
dimensions for the operators in Eqs. (1)–(3), sandwiched between definite
states (fermions, gauge field, or scalars), has the following form:

$\displaystyle\gamma^{(0)}_{{gg}}$ $\displaystyle=$
$\displaystyle-4S_{1}(j)+\frac{4}{j-1}-\frac{4}{j}+\frac{4}{j+1}-\frac{4}{j+2}\,,\qquad\gamma^{(0)}_{{\lambda
g}}\,=\,\frac{8}{j}-\frac{16}{j+1}+\frac{16}{j+2}\,,$
$\displaystyle\gamma^{(0)}_{{\phi g}}$ $\displaystyle=$
$\displaystyle\frac{12}{j+1}-\frac{12}{j+2}\,,\qquad\gamma^{(0)}_{{g\lambda}}\,=\,\frac{4}{j-1}-\frac{4}{j}+\frac{2}{j+1}\,,\qquad\gamma^{(0)}_{{\lambda\phi}}\,=\,\frac{8}{j}\,,\qquad\gamma^{(0)}_{{\phi\lambda}}\,=\,\frac{6}{j+1}\,,$
$\displaystyle\gamma^{(0)}_{{\lambda\lambda}}$ $\displaystyle=$
$\displaystyle-4S_{1}(j)+\frac{8}{j}-\frac{8}{j+1}\,,\qquad\gamma^{(0)}_{{g\phi}}\,=\,\frac{4}{j-1}-\frac{4}{j}\,,\qquad\gamma^{(0)}_{{\phi\phi}}\,=\,-4S_{1}(j)\,.$
(12)

The matrix which diagonalizes the matrix of anomalous dimensions can be used
for the construction of the following multiplicatively renormalizable
operators:

$\displaystyle\mathcal{O}^{T_{j}}_{\mu_{1},\ldots,\mu_{j}}$ $\displaystyle=$
$\displaystyle\mathcal{O}^{g}_{\mu_{1},\ldots,\mu_{j}}+\mathcal{O}^{\lambda}_{\mu_{1},\ldots,\mu_{j}}+\mathcal{O}^{\phi}_{\mu_{1},\ldots,\mu_{j}}\,,$
(13) $\displaystyle\mathcal{O}^{\Sigma_{j}}_{\mu_{1},\ldots,\mu_{j}}$
$\displaystyle=$
$\displaystyle-2(j-1)\mathcal{O}^{g}_{\mu_{1},\ldots,\mu_{j}}+\mathcal{O}^{\lambda}_{\mu_{1},\ldots,\mu_{j}}+\frac{2}{3}(j+1)\mathcal{O}^{\phi}_{\mu_{1},\ldots,\mu_{j}}\,,$
(14) $\displaystyle\mathcal{O}^{\Xi_{j}}_{\mu_{1},\ldots,\mu_{j}}$
$\displaystyle=$
$\displaystyle-\frac{j-1}{j+2}\mathcal{O}^{g}_{\mu_{1},\ldots,\mu_{j}}+\mathcal{O}^{\lambda}_{\mu_{1},\ldots,\mu_{j}}-\frac{j+1}{j}\mathcal{O}^{\phi}_{\mu_{1},\ldots,\mu_{j}}\,,$
(15)

whose anomalous dimensions are $\gamma_{\mathrm{uni}}^{(0)}(j)$,
$\gamma_{\mathrm{uni}}^{(0)}(j+2)$, and $\gamma_{\mathrm{uni}}^{(0)}(j+4)$,
respectively, where

$\gamma_{\mathrm{uni}}^{(0)}(j)=-4S_{1}(j)\,,$ (16)

with $S_{1}(j)=\sum_{i=1}^{j}\frac{1}{i}$ being the simplest harmonic sum.

Sandwiching the operators in Eqs. (13)–(15) between different states, we
obtain the following set of relations:

$\displaystyle\gamma_{gg}+\gamma_{\lambda g}+\gamma_{\phi g}$ $\displaystyle=$
$\displaystyle\gamma_{\mathrm{uni}}^{(0)}(j)\,,$ (17)
$\displaystyle\gamma_{g\lambda}+\gamma_{\lambda\lambda}+\gamma_{\phi\lambda}$
$\displaystyle=$ $\displaystyle\gamma_{\mathrm{uni}}^{(0)}(j)\,,$ (18)
$\displaystyle\gamma_{g\phi}+\gamma_{\lambda\phi}+\gamma_{\phi\phi}$
$\displaystyle=$ $\displaystyle\gamma_{\mathrm{uni}}^{(0)}(j),$ (19)
$\displaystyle\gamma_{gg}-\frac{1}{2(j-1)}\gamma_{\lambda
g}+\frac{1}{3}\,\frac{j+1}{j-1}\gamma_{\phi g}$ $\displaystyle=$
$\displaystyle\gamma_{\mathrm{uni}}^{(0)}(j+2)\,,$ (20)
$\displaystyle-2(j-1)\gamma_{g\lambda}+\gamma_{\lambda\lambda}+\frac{2}{3}(j+1)\gamma_{\phi\lambda}$
$\displaystyle=$ $\displaystyle\gamma_{\mathrm{uni}}^{(0)}(j+2)\,,$ (21)
$\displaystyle-3\frac{j-1}{j+1}\gamma_{g\phi}+\frac{3}{2(j+1)}\gamma_{\lambda\phi}+\gamma_{\phi\phi}$
$\displaystyle=$ $\displaystyle\gamma_{\mathrm{uni}}^{(0)}(j+2)\,,$ (22)
$\displaystyle\gamma_{gg}-\frac{j+2}{j-1}\gamma_{\lambda
g}+\frac{j+1}{j}\,\frac{j+2}{j-1}\gamma_{\phi g}$ $\displaystyle=$
$\displaystyle\gamma_{\mathrm{uni}}^{(0)}(j+4)\,,$ (23)
$\displaystyle-\frac{j-1}{j+2}\gamma_{g\lambda}+\gamma_{\lambda\lambda}-\frac{j+1}{j}\gamma_{\phi\lambda}$
$\displaystyle=$ $\displaystyle\gamma_{\mathrm{uni}}^{(0)}(j+4)\,,$ (24)
$\displaystyle\frac{j-1}{j+2}\,\frac{j}{j+1}\gamma_{g\phi}-\frac{j}{j+1}\gamma_{\lambda\phi}+\gamma_{\phi\phi}$
$\displaystyle=$ $\displaystyle\gamma_{\mathrm{uni}}^{(0)}(j+4)\,,$ (25)

where, on the left-hand sides, diagonal elements are normalized to unity and
the arguments of the matrix elements are equal to $j$. Thus, if we calculate
the anomalous dimension for the operators in Eq. (1)–(3) at some fixed values
of $j$, we obtain the result for the universal anomalous dimension not only
for $j$, but also for $(j+2)$ and $(j+4)$. In order to obtain
$\gamma_{\mathrm{uni}}(10)$, it is sufficient to compute
$\gamma_{\lambda\lambda}(6)$. In fact, the operator in Eq. (1) for $j=6$ does
contain the vertex operator mentioned above. If we proceed to four loops, we
are able to compute the contribution to the universal anomalous dimension
similarly as in the leading order. The non-planar contribution appears for the
first time at this order because there are no additional contributions from
the renormalization. We already exploited this property in our previous four-
loop calculations.

Having circumvented both difficulties, we obtain the following result for the
anomalous dimension of the operator in Eq. (1) for $j=6$:

$\hat{\gamma}_{\lambda\lambda,\mathrm{np}}^{(3)}(6)=\left(\frac{5148948727}{47628000}-\frac{943}{135}\,{\mathsf{S}}_{2}+\frac{2767689751}{8164800}\,\zeta_{3}-\frac{2071042}{2205}\,\zeta_{5}\right)\frac{48}{N_{c}^{2}}\,,\quad$
(26)

where
${\mathsf{S}}_{2}=\frac{4}{9\sqrt{3}}\mathop{\mathrm{Cl}}\nolimits_{2}(\pi/3)$
[104], with $\mathop{\mathrm{Cl}}\nolimits_{2}$ being Clausen’s function. We
have yet to add the contribution from the counterterm diagrams due to the non-
gauge-invariant massive operator
$m^{2}{\mathcal{A}}^{\mu_{1}}{\mathcal{A}}^{\mu_{2}}{\mathcal{A}}^{\mu_{3}}{\mathcal{A}}^{\mu_{4}}$
discussed in Ref. [87],

$\gamma_{\lambda\lambda,\mathrm{ren},\mathrm{np}}^{(3)}(6)=\left(-\frac{19099}{38880}+\frac{943}{135}\,{\mathsf{S}}_{2}+\frac{146951}{46656}\,\zeta_{3}\right)\frac{48}{N_{c}^{2}}\,.$
(27)

Combining Eqs. (26) and (27), we obtain

$\gamma_{\lambda\lambda,\mathrm{np}}^{(3)}(6)=\left(\frac{47458819}{441000}+\frac{1077703}{3150}\,\zeta_{3}-\frac{2071042}{2205}\,\zeta_{5}\right)\,\frac{48}{N_{c}^{2}}\,,$
(28)

where ${\mathsf{S}}_{2}$ has canceled out as expected.

Inserting Eq. (28) in Eqs. (18), (21), and (24), we obtain the following
expression for the non-planar contribution to the tenth moment of the
universal anomalous dimension of the twist-two operators:†††This result was
obtained in 2018, but not published.

$\gamma_{\mathrm{uni,np}}^{(3)}(10)=\left(\frac{220854227}{1411200}+\frac{27357}{56}\,\zeta_{3}-\frac{579121}{490}\,\zeta_{5}\right)\frac{48}{N_{c}^{2}}\,.$
(29)

Unfortunately, it is not possible to pull out a common multiplier from this
expression, as was possible for Eqs. (9)—(11). So, our expectation regarding
factorization is not confirmed. We note that the results in Eqs. (9)–(11), and
(29) have recently been obtained using an alternative method in Ref. [88], by
performing the asymptotic expansion of the four-point functions for length-two
half-Bogomol’nyi-Prasad-Sommerfield operators.

For the calculation of the higher moments, we employ the FORM [105] package
FORCER [106], which has recently been developed for computations of this type
and was also used to compute anomalous dimensions of twist-two operators in
QCD [21, 22]. By means of this tool, we compute the matrix element
$\gamma_{\phi\lambda}$, i.e. the anomalous dimension of the scalar operator in
Eq. (3) sandwiched between fermionic states, appearing in Eqs. (18), (21), and
(24) and, in a similar way as described after Eqs. (17)–(25), obtain the next
four even moments,

$\displaystyle\gamma_{\mathrm{uni,np}}^{(3)}(12)$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{28337309747461}{144027072000}+\frac{345385183}{571536}\,\zeta_{3}-\frac{54479161}{39690}\,\zeta_{5}\bigg{)}\frac{48}{N_{c}^{2}}\,,$
(30) $\displaystyle\gamma_{\mathrm{uni,np}}^{(3)}(14)$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{9657407179406311}{41493513600000}+\frac{158654990663}{224532000}\,\zeta_{3}-\frac{7399612441}{4802490}\,\zeta_{5}\bigg{)}\frac{48}{N_{c}^{2}}\,,$
(31) $\displaystyle\gamma_{\mathrm{uni,np}}^{(3)}(16)$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{74429504651244877}{280496151936000}+\frac{205108095887}{256864608}\,\zeta_{3}-\frac{1372958223289}{811620810}\,\zeta_{5}\bigg{)}\frac{48}{N_{c}^{2}}\,,\qquad$
(32) $\displaystyle\gamma_{\mathrm{uni,np}}^{(3)}(18)$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{8122582838282649980649377}{27516111512617728000000}+\frac{72169501556777041}{81811377648000}\,\zeta_{3}$
(33)
$\displaystyle{}-\frac{5936819760481}{3246483240}\,\zeta_{5}\bigg{)}\frac{48}{N_{c}^{2}}\,.$

Moreover, we reproduce our previous results in Eqs. (9)–(11) and (29).

##  3 Reconstruction

Equipped with Eq. (29)–(33), we now attempt to reconstruct the general form of
the non-planar contribution to the four-loop universal anomalous dimension,
i.e. to determine the $j$ dependence of the coefficients of $\zeta_{5}$ and
$\zeta_{3}$ and the rational reminder in the ansatz

$\gamma_{{\mathrm{uni}},{\mathrm{np}}}^{(3)}(j)=\left[\gamma_{{\mathrm{uni}},{\mathrm{np}},\zeta_{5}}^{(3)}(j)\,\zeta_{5}+\gamma_{{\mathrm{uni}},{\mathrm{np}},\zeta_{3}}^{(3)}(j)\,\zeta_{3}+\gamma_{{\mathrm{uni}},{\mathrm{np}},{\mathrm{rational}}}^{(3)}(j)\right]\frac{48}{N_{c}^{2}}\,.$
(34)

For this purpose, we use our method for the reconstruction of general results
from results for fixed values of $j$ using number theory, which was proposed
in Ref. [49] and successfully applied to the reconstruction of anomalous
dimensions in Refs. [109, 110, 79, 80]. This method is based on two
observations.

First, we assume that we know all the basis functions which the answer
contains. For anomalous dimensions in $\mathcal{N}=4$ SYM theory, these are
the generalized harmonic sums. They are defined as [111, 112]

$\displaystyle S_{a}(M)$ $\displaystyle=$
$\displaystyle\sum^{M}_{j=1}\frac{[\mathop{\mathrm{sign}}\nolimits(a)]^{j}}{j^{|a|}}\,,$
(35) $\displaystyle S_{a_{1},\ldots,a_{n}}(M)$ $\displaystyle=$
$\displaystyle\sum^{M}_{j=1}\frac{[\mathop{\mathrm{sign}}\nolimits(a_{1})]^{j}}{j^{|a_{1}|}}\,S_{a_{2},\ldots,a_{n}}(j)\,,$
(36)

where the indices $a_{1},\ldots,a_{n}$ may be positive or negative, except for
the value $-1$. The weight or transcendentality $\ell$ of each sum
$S_{a_{1},\ldots,a_{n}}(M)$ is defined as the sum of the absolute values of
its indices,

$\ell=|a_{1}|+\cdots+|a_{n}|\,.$ (37)

The weight of the product of harmonic sums is equal to the sum of their
weights.

For twist-two operators, there is an additional simplification, thanks to the
so-called generalized Gribov-Lipatov reciprocity [1, 2, 107, 108], which
reflects the symmetry of the underlying process under the crossing of
channels. A consequence of this property is that the harmonic sums can enter
the anomalous dimension only in the form of special combinations, which
satisfy the property mentioned above. In practice, this allows one to restrict
the choice of possible basis functions to the smaller number of so-called
binomial harmonic sums, which are defined as [111]

${\mathbb{S}}_{a_{1},\ldots,a_{n}}(M)=(-1)^{M}\sum_{j=1}^{M}(-1)^{j}\binom{M}{j}\binom{M+j}{j}S_{a_{1},...,a_{n}}(j)\,.$
(38)

They have only positive-integer indices, while their transcendentality is the
same as for usual harmonic sums.

The second observation is that, in the expressions for the $j$-dependent
anomalous dimensions already known, the coefficients in front of these sums
are rather simple numbers, usually small integers. In the general case, we
thus obtain a system of Diophantine equations. If the number of equations is
equal to the number of variables, then we can solve the system exactly, but,
in this case, we need to know a lot of fixed values. However, the system of
Diophantine equations can be solved with help of a special method from number
theory even if the number of equations is less than the number of variables.
In this case, we use the Lenstra–Lenstra–Lovasz (LLL) algorithm [113], which
allows one to reduce the matrix obtained from the system of Diophantine
equations to a matrix the rows of which are the solution of the system with
minimal Euclidean norm.

According to the maximal-transcendentality principle [102], anomalous
dimensions of twist-two operators can only contain harmonic sums with the
maximum weight allowed at the respective order of perturbation theory. The
latter is equal to $2\ell-1$ at $\ell$-th order. In our case, of $\ell=4$, the
basis for the four-loop universal anomalous dimension contains all possible
binomial harmonic sums [see Eq. (38)] with weight $7$, and there are
$2^{7-1}=64$ of them. This only applies to
$\gamma_{{\mathrm{uni}},{\mathrm{np}},{\mathrm{rational}}}^{(3)}(j)$ in Eq.
(34). The weight of $\zeta_{i}$ is $i$, so that
$\gamma_{{\mathrm{uni}},{\mathrm{np}},\zeta_{5}}^{(3)}(j)$ and
$\gamma_{{\mathrm{uni}},{\mathrm{np}},\zeta_{3}}^{(3)}(j)$ in Eq. (34) are of
transcendentalities 4 and 2, respectively. The numbers of binomial sums in the
respective bases are $8$ and $2$, respectively.

The general form of the $\zeta_{5}$ part,

$\gamma_{{\mathrm{uni}},{\mathrm{np}},\zeta_{5}}^{(3)}(j)=-\,40\,{\mathbb{S}}_{1}^{2}(j-2)\,,$
(39)

was determined a long time ago [85] from the single input
$\gamma_{\mathrm{uni,np}}^{(3)}(4)$ in Eq. (9), which was sufficient to fix
the two coefficients in the ansatz, and all the subsequent results for
$\gamma_{\mathrm{uni,np}}^{(3)}(j)$, with $j=6,\ldots,18$, confirmed this
assumption. The general form of the $\zeta_{3}$ part was obtained upon the
derivation of $\gamma_{\mathrm{uni,np}}^{(3)}(12)$ in Eq. (30), when five
values were known, enough to fix the eight coefficients in the ansatz, and
reads [101]

$\gamma_{{\mathrm{uni}},{\mathrm{np}},\zeta_{3}}^{(3)}(j)=8\,\Big{(}8\,{\mathbb{S}}_{4}-9\,{\mathbb{S}}_{1,3}-3\,{\mathbb{S}}_{2,2}-4\,{\mathbb{S}}_{3,1}+4\,{\mathbb{S}}_{1,1,2}+5\,{\mathbb{S}}_{1,2,1}-{\mathbb{S}}_{2,1,1}\Big{)},$
(40)

where
$\mathbb{S}_{\boldsymbol{a}}=\mathbb{S}_{\boldsymbol{a}}(j-2)=\mathbb{S}_{\boldsymbol{a}}(M)$.
The results in Eqs. (31)–(33) all satisfy Eq. (40).

To reconstruct the rational part
$\gamma_{{\mathrm{uni}},{\mathrm{np}},{\mathrm{rational}}}^{(3)}(j)$ in Eq.
(34), we exploit three general properties which the non-planar anomalous
dimension is supposed to satisfy. First of all, its large-$j$ limit should not
contain $\ln j$ to a power higher than one [114, 115, 116] in the combinations
of the basis functions. Specifically, this constraint demands the absence of
the six terms $\ln^{7}\\!j$, $\zeta_{2}\ln^{5}\\!j$, $\zeta_{3}\ln^{4}\\!j$,
$\zeta_{4}\ln^{3}\\!j$, $\zeta_{2}\zeta_{3}\ln^{2}\\!j$, and
$\zeta_{5}\ln^{2}\\!j$. Moreover, the result for the four-loop cusp anomalous
dimension [92, 91, 95] fixes the coefficient of the term proportional to $\ln
j$. Specifically, this determines the two coefficients in the front of
$\zeta_{3}^{2}\ln j$ and $\zeta_{6}\ln j$. The second property comes from the
Balitsky–Fadin–Kuraev–Lipatov (BFKL) equation [96, 97, 98], which is exactly
known at the first two orders, in the leading-logarithmic approximation (LLA)
and the next-to-leading-logarithmic approximation (NLLA) [102, 117, 118]. This
implies that, being analytically continued into $j=1+\omega$ (or
$M=-1+\omega$), the universal anomalous dimension of the twist-two operators
in ${\mathcal{N}}=4$ SYM theory should not contain poles in $\omega$ less than
$1/\omega^{\ell-1}$ at $\ell$ loops, which, in our case, leaves the poles
$1/\omega^{k}$ with $k=3,\ldots,7$ and so impose five constraints. The third
constraint is due to the double-logarithmic equation [99, 100] and implies
that, analytically continued to $j=0+\omega$ (or $M=-2+\omega$), the universal
anomalous dimension of twist-two operators in ${\mathcal{N}}=4$ SYM theory
should not contain the highest pole in $\omega$, which excludes the pole
$1/\omega^{7}$ in our case. For the analytic continuation of the harmonic
sums, we rely on our database [119], generated with the help of the FORM [105]
packages harmpol [120] and summer [111], and the collection DATAMINE [121] of
relations between alternating multiple zeta values. So, we have
$8+(6+2)+5+1=22$ equations for $64$ variables.

To gain confidence in our reconstruction procedure, we apply it to recover the
well-known result for the rational part of the planar four-loop anomalous
dimension that respects reciprocity (RR) [37, 45], which, in the basis of the
binomial harmonic sums, reads

$\displaystyle\frac{\gamma_{\mathbb{RR},\mathrm{uni},\mathrm{pl},\mathrm{rat}}^{4-\mathrm{loop}}}{32}$
$\displaystyle=$
$\displaystyle-2\mathbb{S}_{1,2,4}+2\mathbb{S}_{1,5,1}+4\mathbb{S}_{2,2,3}+\mathbb{S}_{2,3,2}-5\mathbb{S}_{2,4,1}+2\mathbb{S}_{3,1,3}-3\mathbb{S}_{3,2,2}-\mathbb{S}_{3,3,1}$
(41)
$\displaystyle{}+4\mathbb{S}_{4,1,2}-4\mathbb{S}_{4,2,1}+2\mathbb{S}_{5,1,1}+4\mathbb{S}_{1,1,1,4}-6\mathbb{S}_{1,1,2,3}-2\mathbb{S}_{1,1,3,2}+4\mathbb{S}_{1,1,4,1}$
$\displaystyle{}-2\mathbb{S}_{1,2,1,3}+9\mathbb{S}_{1,2,2,2}-9\mathbb{S}_{1,2,3,1}+3\mathbb{S}_{1,3,1,2}-5\mathbb{S}_{1,3,2,1}+4\mathbb{S}_{1,4,1,1}-2\mathbb{S}_{2,1,1,3}$
$\displaystyle{}+5\mathbb{S}_{2,1,2,2}-5\mathbb{S}_{2,1,3,1}+8\mathbb{S}_{2,2,2,1}-6\mathbb{S}_{2,3,1,1}+3\mathbb{S}_{3,1,1,2}-\mathbb{S}_{3,1,2,1}-4\mathbb{S}_{3,2,1,1}$
$\displaystyle{}+2\mathbb{S}_{4,1,1,1}-4\mathbb{S}_{1,1,1,2,2}+6\mathbb{S}_{1,1,1,3,1}+2\mathbb{S}_{1,1,2,1,2}-8\mathbb{S}_{1,1,2,2,1}+4\mathbb{S}_{1,1,3,1,1}$
$\displaystyle{}+2\mathbb{S}_{1,2,1,1,2}-4\mathbb{S}_{1,2,2,1,1}+2\mathbb{S}_{1,3,1,1,1}+2\mathbb{S}_{2,1,1,1,2}-2\mathbb{S}_{2,2,1,1,1}$

In fact, we find that the 22 equations described above can correctly fix all
the coefficients of the 64 binomial harmonic sums using the fplll lattice
reduction library [122], in which the LLL algorithm and other similar
algorithms are implemented. It is, therefore, reasonable to expect that our
reconstruction procedure will also work in the case at hand. We find

$\displaystyle\frac{\gamma_{{\mathrm{uni}},{\mathrm{np}},{\mathrm{rat}}}^{(3)}(j)}{4}$
$\displaystyle=$ $\displaystyle
2\mathbb{S}_{5,2}-2\mathbb{S}_{4,3}-4\mathbb{S}_{1,2,4}-2\mathbb{S}_{1,3,3}+2\mathbb{S}_{1,4,2}+4\mathbb{S}_{1,5,1}-6\mathbb{S}_{2,2,3}+2\mathbb{S}_{2,3,2}$
(42)
$\displaystyle{}+4\mathbb{S}_{2,4,1}+12\mathbb{S}_{3,1,3}-6\mathbb{S}_{3,2,2}-2\mathbb{S}_{3,3,1}+2\mathbb{S}_{4,1,2}+6\mathbb{S}_{4,2,1}-12\mathbb{S}_{5,1,1}$
$\displaystyle{}+8\mathbb{S}_{1,1,1,4}+4\mathbb{S}_{1,1,2,3}-8\mathbb{S}_{1,1,3,2}-4\mathbb{S}_{1,1,4,1}-\mathbb{S}_{1,2,1,3}+3\mathbb{S}_{1,2,2,2}$
$\displaystyle{}+6\mathbb{S}_{1,2,3,1}-14\mathbb{S}_{1,3,1,2}-3\mathbb{S}_{1,3,2,1}+9\mathbb{S}_{1,4,1,1}-3\mathbb{S}_{2,1,1,3}-\mathbb{S}_{2,1,2,2}$
$\displaystyle{}+\mathbb{S}_{2,2,2,1}+3\mathbb{S}_{2,3,1,1}-8\mathbb{S}_{3,1,1,2}-8\mathbb{S}_{3,1,2,1}+12\mathbb{S}_{3,2,1,1}+4\mathbb{S}_{4,1,1,1}$
$\displaystyle{}+4\mathbb{S}_{1,1,1,2,2}-12\mathbb{S}_{1,1,1,3,1}+4\mathbb{S}_{1,1,2,1,2}-4\mathbb{S}_{1,1,2,2,1}+8\mathbb{S}_{1,1,3,1,1}$
$\displaystyle{}+8\mathbb{S}_{1,2,1,1,2}+2\mathbb{S}_{1,2,1,2,1}-14\mathbb{S}_{1,2,2,1,1}+4\mathbb{S}_{1,3,1,1,1}+4\mathbb{S}_{2,1,1,1,2}$
$\displaystyle{}+4\mathbb{S}_{2,1,1,2,1}-4\mathbb{S}_{2,1,2,1,1}-4\mathbb{S}_{2,2,1,1,1}\,,$

where
$\mathbb{S}_{\boldsymbol{a}}=\mathbb{S}_{\boldsymbol{a}}(j-2)=\mathbb{S}_{\boldsymbol{a}}(M)$.
Technical details of our evaluation may be found in the Appendix. Although we
believe that our result in Eq. (42) is correct, it needs to be checked, either
directly by calculating the next moment, for $j=20$, or indirectly by
considering some limiting cases. Below, we present our predictions for some
limiting cases of $\gamma_{{\mathrm{uni}},{\mathrm{np}}}^{(3)}(j)$, evaluated
with Eqs. (39), (40), and (42).

First of all, we consider the large-$j$ limit of
$\gamma_{{\mathrm{uni}},{\mathrm{np}}}^{(3)}(j)$, which yields

$\gamma_{\mathrm{uni},\mathrm{np}}^{(3)\
}\overset{j\to\infty}{=}\left(-24\zeta_{3}^{2}-62\zeta_{6}\right)\ln
j+4\zeta_{4}\zeta_{3}-20\zeta_{2}\zeta_{5}-175{\zeta_{7}}\,.$ (43)

We observe that the $j$-independent term in Eq. (43), which takes the
numerical value $-205.37$, nicely agrees with the value
$-B_{\mathrm{np}}^{(4)}/48=(-207.0\pm 3.0)$ we obtained in Ref. [101].

The analytic continuation of $\gamma_{{\mathrm{uni}},{\mathrm{np}}}^{(3)}(j)$
to the BFKL value $M=-1+\omega$ yields‡‡‡We have used $M$ instead of $j=M+2$
to comply with the standard convention in $\mathcal{N}=4$ SYM theory.

$\gamma_{\mathrm{uni},\mathrm{np}}^{(3)}\overset{M=-1+\omega}{=}-\frac{192}{\omega^{2}}{\zeta_{2}}{\zeta_{3}}+\frac{4}{\omega}\left(31{\zeta_{6}}-2{\zeta_{3}}^{2}\right)+2\left(20{\zeta_{2}}{\zeta_{5}}+280{\zeta_{3}}{\zeta_{4}}+69{\zeta_{7}}\right)+\mathcal{O}(\omega)\,.\qquad$
(44)

This implies that the BFKL equation receives a non-planar contribution in the
next-to-next-to-leading logarithmic approximation (NNLLA), which is the third
order of perturbation theory.

In the double-logarithmic limit, $M=-2+\omega$, we have

$\displaystyle\gamma_{\mathrm{uni},\mathrm{np}}^{(3)}$
$\displaystyle\overset{M=-2+\omega}{=}$
$\displaystyle\frac{192{\zeta_{2}}}{\omega^{5}}-\frac{384{\zeta_{2}}}{\omega^{4}}-\frac{48}{\omega^{3}}\left(4{\zeta_{2}}+15{\zeta_{4}}\right)+\frac{288}{\omega^{2}}\left(4{\zeta_{2}}{\zeta_{3}}+5{\zeta_{4}}\right)$
$\displaystyle{}+\frac{4}{\omega}\left(144{\zeta_{2}}-1728{\zeta_{2}}{\zeta_{3}}-24{\zeta_{3}}^{2}-540{\zeta_{4}}+60{\zeta_{5}}-1367{\zeta_{6}}\right)$
$\displaystyle{}+4\left(96{\zeta_{2}}-288{\zeta_{2}}{\zeta_{3}}+722{\zeta_{2}}{\zeta_{5}}+36{\zeta_{3}}^{2}-814{\zeta_{3}}{\zeta_{4}}+799{\zeta_{6}}-56{\zeta_{7}}\right)+\mathcal{O}(\omega)\,.$

As in BFKL case, the double-logarithmic equation receives a non-planar
contribution in the third order of perturbation theory.

Another limit of interest is $M=0+\omega$, which was taken for the planar case
in Ref. [123, 124]. In this limit, we have

$\displaystyle\gamma_{\mathrm{uni},\mathrm{np}}^{(3)}$
$\displaystyle\overset{M=0+\omega}{=}$
$\displaystyle\omega\left(-72\zeta_{2}\zeta_{3}^{2}-\frac{1870}{3}\,\zeta_{8}\right)+\omega^{2}\left(165\zeta_{9}-320\zeta_{4}\zeta_{5}+462\zeta_{3}\zeta_{6}+490\zeta_{2}\zeta_{7}\right)$
(46)
$\displaystyle{}+\omega^{3}\left(\frac{720706}{1583}\zeta_{3}\zeta_{7}-122\zeta_{3}^{2}\zeta_{4}-\frac{87788}{57}\zeta_{2}\zeta_{3}\zeta_{5}-\frac{86900}{1583}\zeta_{5}^{2}+\frac{13564298077}{10827720}\zeta_{10}\right.$
$\displaystyle{}+\left.\frac{25344}{1583}{{{\mathrm{h}}_{73}}}-\frac{709632}{1583}{{{\mathrm{h}}_{91}}}-\frac{75008}{513}{{{\mathrm{h}}_{53}}}\zeta_{2}+\frac{375040}{171}{{{\mathrm{h}}_{71}}}\zeta_{2}\right)+\mathcal{O}(\omega^{4})\,,\quad$

where $\mathrm{h}_{ab}$ are the special numbers related with the multiple
zeta’s (see Ref. [121]).

Finally, also the value of $\gamma_{{\mathrm{uni}},{\mathrm{np}}}^{(3)}(j)$ at
$M=1$ is of special interest because it is related to the single-magnon
operator at $\beta=1/2$ of the SU(2)β spin chain of marginally
$\beta$-deformed $\mathcal{N}=4$ SYM theory [125, 126]. From Eq. (42), we
obtain

$\gamma_{\mathrm{uni},\mathrm{np}}^{(3)}(3)=-160\zeta_{5}\,.$ (47)

It is surprising that, in the non-planar case, only the most transcendental
contribution survives, which can even be found by computing the non-planar
Konishi contribution in Eq. (9) because, for the reconstruction of the general
expression of the most transcendental part, only one fixed value is necessary
to find the coefficient of $\mathbb{S}_{1}^{2}$ in Eq. (39).

##  4 Conclusion

We constructed the general analytic expression for the non-planar contribution
to the four-loop universal anomalous dimensions in $\mathcal{N}=4$ SYM theory,
given in Eq. (42), using the first eight moments in Eqs. (9)–(11) [85, 86, 87]
and (29)–(33) [101] together with information on the large-$j$ [91, 92, 95],
BFKL [102, 117, 118], and double-logarithmic [99, 100] limits, assuming that
the ansatz should include only binomial harmonic sums [see Eq. (38)] in its
coefficients, and solving the resulting system of Diophantine equations with
the help of number theory. We also provided analytic results for various
interesting limits of our general expression, including the large-$j$ limit in
Eq. (43), the BFKL limit in Eq. (44), the double-logarithmic limit in Eq.
(LABEL:DLlimit), the expansion about $M=0$ in Eq. (46), and the result for
$M=1$ in Eq. (47). The latter is particularly interesting, as it represents a
very simple expression for the non-planar contribution to the anomalous
dimension of the single-magnon operator at $\beta=1/2$ of the SU(2)β spin
chain of marginally $\beta$-deformed $\mathcal{N}=4$ SYM theory. It will be
tantalizing to test these conjectures in the future by independent
calculations.

### Acknowledgments

We would like to thank V. S. Fadin for useful discussions. Our computations
were performed in part with resources provided by the PIK Data Centre in PNPI
NRC “Kurchatov Institute.” The research of B.A.K. was supported in part by
BMBF Grant No. 05H18GUCC1 and DFG Grants No. KN 365/13-1 and No. KN 365/14-1.
The research of V.N.V. was supported in part by RFBR Grants No. 16-02-00943-a,
No. 16-02-01143-a, and No. 19-02-00983-a and a Marie Curie International
Incoming Fellowship within the Seventh European Community Framework Programme
under Grant No. PIIF-GA-2012-331484.

##  5 Appendix

Here, we provide more details on the reconstruction procedure and the result,
which we obtain with the help of number theory. As discussed in Section 3, the
ansatz for the general result consists of a linear combination of the binomial
harmonic sums [see Eq. (38)] of weight 7, of which there are $2^{7-1}=64$ in
total,

$\displaystyle\mathrm{Ansatz}_{\mathrm{Rat}}$ $\displaystyle=$
$\displaystyle\Big{\\{}\mathbb{S}_{7},\mathbb{S}_{1,6},\mathbb{S}_{2,5},\mathbb{S}_{3,4},\mathbb{S}_{4,3},\mathbb{S}_{5,2},\mathbb{S}_{6,1},\mathbb{S}_{1,1,5},\mathbb{S}_{1,2,4},\mathbb{S}_{1,3,3},\mathbb{S}_{1,4,2},\mathbb{S}_{1,5,1},\mathbb{S}_{2,1,4},$
(48)
$\displaystyle\mathbb{S}_{2,2,3},\mathbb{S}_{2,3,2},\mathbb{S}_{2,4,1},\mathbb{S}_{3,1,3},\mathbb{S}_{3,2,2},\mathbb{S}_{3,3,1},\mathbb{S}_{4,1,2},\mathbb{S}_{4,2,1},\mathbb{S}_{5,1,1},\mathbb{S}_{1,1,1,4},\mathbb{S}_{1,1,2,3},$
$\displaystyle\mathbb{S}_{1,1,3,2},\mathbb{S}_{1,1,4,1},\mathbb{S}_{1,2,1,3},\mathbb{S}_{1,2,2,2},\mathbb{S}_{1,2,3,1},\mathbb{S}_{1,3,1,2},\mathbb{S}_{1,3,2,1},\mathbb{S}_{1,4,1,1},\mathbb{S}_{2,1,1,3},$
$\displaystyle\mathbb{S}_{2,1,2,2},\mathbb{S}_{2,1,3,1},\mathbb{S}_{2,2,1,2},\mathbb{S}_{2,2,2,1},\mathbb{S}_{2,3,1,1},\mathbb{S}_{3,1,1,2},\mathbb{S}_{3,1,2,1},\mathbb{S}_{3,2,1,1},\mathbb{S}_{4,1,1,1},$
$\displaystyle\mathbb{S}_{1,1,1,2,2},\mathbb{S}_{1,1,1,3,1},\mathbb{S}_{1,1,2,1,2},\mathbb{S}_{1,1,2,2,1},\mathbb{S}_{1,1,3,1,1},\mathbb{S}_{1,2,1,1,2},\mathbb{S}_{1,2,1,2,1},\mathbb{S}_{1,2,2,1,1},$
$\displaystyle\mathbb{S}_{1,3,1,1,1},\mathbb{S}_{2,1,1,1,2},\mathbb{S}_{2,1,1,2,1},\mathbb{S}_{2,1,2,1,1},\mathbb{S}_{2,2,1,1,1},\mathbb{S}_{3,1,1,1,1},\mathbb{S}_{1,1,1,2,1,1},\mathbb{S}_{1,1,2,1,1,1},$
$\displaystyle\mathbb{S}_{1,2,1,1,1,1},\mathbb{S}_{2,1,1,1,1,1},\mathbb{S}_{1,1,1,1,3},\mathbb{S}_{1,1,1,1,2,1},\mathbb{S}_{1,1,1,1,1,2},\mathbb{S}_{1,1,1,1,1,1,1}\Big{\\}}\,.$

We have $22$ constraints, which are described in Section 3, and construct from
these equations a matrix of dimension $22\times(64+1)$, where the last column
is multiplied by the factor $2^{-3}$, which we expect to be a common factor of
the desired expression. Then we append the transpose of this matrix,
multiplied by any large number, $8^{64}$ in our case, to the right of the unit
matrix of dimension $65\times 65$. The resulting matrix, which is of dimension
$65\times(65+22)$, is then translated into a form which can be read by the
fplll library [122], in which the LLL algorithm and further modifications of
it are implemented. We use the so-called Block– Korkine–Zolotarev (BKZ)
method,§§§See Refs. [127, 128, 129] for more details. with block size set to
$25$ using option -b and precision set to 555 using options -mpfr -p. The
library fplll produces a new matrix in which the lines are sorted according to
their Euclidean norm. Upon the execution of fplll, we have a new matrix, which
contains the solutions of the original non-uniform Diophantine system of
equations and also the solutions of the corresponding uniform system of
equations—the number of equations is less than the number of variables, and
there are combinations of variables which satisfy the system of uniform
equations. The first line of this matrix is the solution of the uniform
equation, with 0 in position 65,¶¶¶We suppress $22$ zeros at the end of the
line, as this is a general property of the solutions.

$\displaystyle\mathrm{fplll}_{1}$ $\displaystyle=$
$\displaystyle\big{\\{}0,0,-3,1,-6,-3,3,-9,-3,1,-2,2,2,8,1,-3,1,-5,-7,9,9,0,10,9,3,5,0,$
(49)
$\displaystyle-3,3,-1,3,0,1,7,-1,-12,0,5,-1,1,11,-12,-2,-10,-7,-7,1,-6,1,3,$
$\displaystyle-3,10,-11,-8,4,-1,2,5,5,0,0,0,0,0,0\big{\\}}\,,$

while the second line gives the desired solution of the non-uniform equation,
with 2 in position 65,

$\displaystyle\mathrm{fplll}_{2}$ $\displaystyle=$
$\displaystyle\big{\\{}0,0,0,0,2,-2,0,0,4,2,-2,-4,0,6,-2,-4,-12,6,2,-2,-6,12,-8,-4,8,$
(50) $\displaystyle
4,1,-3,-6,14,3,-9,3,1,0,0,-1,-3,8,8,-12,-4,-4,12,-4,4,-8,-8,$
$\displaystyle-2,14,-4,-4,-4,4,4,0,0,0,0,0,0,0,0,0,2\big{\\}}\,.$

The next possible solution of the non-uniform system is on line 20. So, there
is reason to assume that the solution in Eq. (50) is correct because

  * •

it is isolated from other solutions;

  * •

it contains many zeros;

  * •

it contains the same coefficients, up to a sign, for the most complicated
binomial harmonic sums in ansatz (48), i.e. those at the end of the ansatz;

  * •

it exhibits a rather uniform structure.

In principle, any line of the obtained matrix can be added to line 2
accommodating our conjectured solution. However, as can be seen by adding Eqs.
(49) and (50), we then loose the above-mentioned properties of the desired
result, which we expect from our observations made when computing anomalous
dimensions.

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Granular jamming gripper with integrated suction

granular jamming gripper with integrated suction

Angel Santarossa0000-0002-7898-2986,
Olfa D'Angelo0000-0002-7218-4596,
Achim Sack0000-0003-2536-2879,
and Thorsten Pöschel0000-0001-5913-1070
Angel Santarossa, Olfa D'Angelo, Achim Sack, and Thorsten Pöschel are with Institute for Multiscale Simulations,
Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 3, 91058 Erlangen, Germany

\begin@twocolumnfalse

Granular grippers can manipulate a wide variety of objects, but need to be pressed on the object to conform to it. If the object is placed on unstable ground, e.g., on sand or water, this step might cause the object to sink or move away from the gripper, hindering proper operation. We introduce a granular gripper with an integrated suction cup, where suction and jamming are controlled independently. We demonstrate the system's robust and enhanced gripping capabilities by comparing its grasping performance with a typical granular gripper design. We show that the proposed device can grip objects that are challenging for typical granular grippers, including those placed on unstable ground, as the suction cup stabilizes the object, allowing the gripper to conform.

Keywords: granular jamming, suction, granular gripper, soft robotics

\end@twocolumnfalse

[1]Angel Santarossa, Olfa D'Angelo, Achim Sack, and Thorsten Pöschel are with Institute for Multiscale Simulations,
Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 3, 91058 Erlangen, Germany

§ INTRODUCTION

Granular grippers are soft robotic effectors that exploit the effect of granular jamming to grip objects <cit.>. This type of gripper is highly adaptable and can easily manipulate objects of diverse geometry and surface properties, without the need for reconfiguration between gripping cycles<cit.>. The typical design of a granular gripper comprises a loosely packed granulate contained in a flexible, airtight membrane.
In this unjammed state, the granular material easily flows when deformed, akin to a fluid. When pressed against an object, the gripper deforms and conforms to the object's shape. To grip the object, the air is evacuated from within the gripper, which causes the gripper membrane to contract and compress the particles: the granular material jams and thus becomes mechanically stable by entering its solid-like state, characterized by a finite elastic modulus.
In this jammed state, the gripper exerts forces on the target object that can effectively grip and firmly hold it. To release the object, the pressure inside the gripper
is equalized to the ambient pressure, causing the membrane to relax and the granulate to return to a fluid-like state.

The holding force generated by a granular gripper results from the combination of three mechanisms <cit.>: friction due to tangential stress at the contact between the object and the membrane <cit.>, interlocking due to geometric constraints between the object and the jammed granulate <cit.>, and suction in regions where the membrane seals the surface of the object airtight <cit.>.

A key feature of granular jamming-based grippers is that they must be pressed against an object to conform to its shape <cit.>. The maximum pushing force the gripper applies to the object is the activation force. Deforming the gripper by compressing it onto an object works well for objects located on solid, stable ground. It works less reliably for objects in unstable positions such on rough surfaces, sand, wet soil, or floating on water <cit.>, where the object can sink or slide away from the gripper, see Fig. <ref>. In such situations, granular grippers tend to fail taking hold of objects or conform insufficiently to the object, resulting in low holding force.

Granular gripper reaching out to get hold of an object floating on a liquid. The gripper fails because it cannot exert sufficient force on the evading object.

In addition, typical granular grippers cannot reliably manipulate objects whose shape or size impedes the gripper to cover a sufficiently large area. This is frequently the case for large objects, e.g., glass panes. Also problematic are very soft objects (e.g., dough, flexible foil, a plastic bag) that deform when in contact with the gripper <cit.>.

We introduce a granular jamming gripper with integrated suction where suction- and jamming-based gripping can be activated independently. Granular gripper membranes equipped with suction cups have been proposed for grippers previously. In contrast to our device, however, either only suction was utilized for gripping <cit.>, or suction and granular jamming were activated simultaneously during gripping <cit.>.
We demonstrate the performance of our gripper when applied to objects of different geometries (notably a large flat plate), sizes, and surface properties. We also demonstrate its ability to grip objects lying on unstable grounds and floating on water.

§ GRANULAR GRIPPER WITH SUCTION: DESIGN

Our device is an ordinary granular gripper supplemented with a suction cup connected to an independent vacuum pump, as shown in Fig. <ref>.

(view1) at (0, 0) ;

[right=6mm of view1.6, anchor=west, text width=cm,] (vacuum)
[left=14mm of vacuum] (vacuumPt) ;

[-Circle[fill=none], thick](vacuum.west)–(vacuumPt);

[right=6mm  of view1.21, anchor=west,text width=cm] (vacjam)
holder (top) ;
[left=14mm of vacjam] (vacjamPt) ;

[-Circle[fill=none], thick](vacjam.west)–(vacjamPt);

[right=6mm  of view1.34, anchor=west,text width=cm] (vacjam)
vacuum for
jamming ;
[left=52mm of vacjam] (vacjamPt) ;

[-Circle[fill=none], thick](vacjam.west)–(vacjamPt);

[right=6mm  of view1.49, anchor=west,text width=cm] (vacjam)
vacuum for
suction ;
[left=32mm of vacjam] (vacjamPt) ;

[-Circle[fill=none], thick](vacjam.west)–(vacjamPt);

[right=6mm of view1.-16, anchor=west, text width=cm] (cupcon)
pipe to
suction cup;
[left=28.5mm of cupcon] (cupconPt) ;
[-Circle[fill=none], thick](cupcon.west)–(cupconPt);

[right=6mm of view1.-30, anchor=west, text width=cm] (membrane)
[left=10mm of membrane] (membranePt) ;
[-Circle[fill=none], thick](membrane.west)–(membranePt);

[right=6mm of view1.-40, anchor=west, text width=cm] (granu)
[left=15mm of granu] (granuPt) ;
[-Circle[fill=none], thick](granu.west)–(granuPt);

[right=6mm of view1.-48, anchor=west, text width=cm] (silcup)
silicon cup;
[left=26mm of silcup] (silcupPt) ;
[-Circle[fill=none], thick](silcup.west)–(silcupPt);

[right=6mm of view1.-53, anchor=west,text width=cm] (cup)
suction cup;
[left=26mm of cup] (cupPt) ;
[-Circle[fill=none], thick](cup.west)–(cupPt);

Cross section of the gripper with suction mechanism.

As for the typical gripper design, a holder maintains an elastic bag filled with particles. The gripper holder comprises two parts (blue and cyan in Fig. <ref>). The membrane is attached to the bottom part of the holder and fixed with a snap ring. A plastic filter on top of the membrane retains the particles during the evacuation. An O-ring between the bottom and top part of the holder ensures proper sealing.
At the side of the top part of the gripper holder, a flange allows loading the gripper's interior with positive air pressure or applying vacuum.

A suction cup is integrated into the lower center of the gripper membrane (Fig. <ref>a). The suction cup is placed in a silicon cup to prevent the particles from deforming the suction cup during conformation. A flexible pipe connects the suction cup to a vacuum pump.

Note that suction and jamming-based gripping can be activated individually, which is crucial to grip and hold objects on unsteady ground.

§ EXPERIMENTAL SETUP AND PROCEDURES

§.§ Gripping objects on stable grounds

The experimental setup consists of a fixed granular gripper and an object placed below, on a stage moving it along the $z$-direction, as sketched in Fig. <ref>.

[] (setup)  at (0, 0)  ;

[right=-2mm of setup.44, anchor=west, text width=2.5cm, text=black] (holder) gripper holder;
[left=20mm of holder] (holderPt) ;

[-Circle[fill=none], thick](holder.west)–(holderPt);

[right=-2mm of setup.25, anchor=west, text width=2.5cm, text=black] (gripper)
gripper filled
with particles;

[left=14mm of gripper] (gripperPt) ;

[-Circle[fill=none], thick](gripper)–(gripperPt);

[right=-2mm of setup.-11, anchor=west, text width=2.5cm, text=black] (object) object;
[left=23mm of object] (objectPt) ;

[-Circle[fill=none], thick](object.west)–(objectPt);

[right=-2mm of setup.-33, anchor=west, text width=2.5cm, text=black] (bore) bore to

[left=26mm of bore] (borePt) ;

[-Circle[fill=none], thick](bore.west)–(borePt);

[right=-2mm of setup.-48, anchor=west, text width=2.5cm, text=black] (loadcell) load cell;
[left=13mm of loadcell] (loadcellPt) ;

[-Circle[fill=none], thick](loadcell.west)–(loadcellPt);

[right=-2mm of setup.-54, anchor=west,text width=2.5cm, text=black] (zstage) $z$-stage;

[left=4.5mm of zstage] (zstagePtmid) ;

[-Circle[fill=none], thick](zstage.west)–(zstagePtmid);

Experimental setup for gripping objects from stable ground. The $z$-stage moves vertically. The object is attached to the $z$-stage via a load cell to measure the holding force.

The elastic, non-permeable, spherical-shaped bag (diameter $73.0 \pm \SI{0.5}{\mm}$) of both grippers is filled with expanded polystyrene (EPS) beads (diameter $4.2 \pm \SI{0.5}{\mm}$). We use EPS beads because soft particles produce significantly higher holding forces in granular grippers <cit.>. The particles are coated with an anti-static spray to minimize electrostatic interactions. The membrane is attached to the gripper holder.
The target object is mounted underneath on a $z$-stage. The $z$-stage position can be changed vertically using two stepper motors driven by a microcontroller. A load cell between the object and the $z$-stage measures the force acting on the target object throughout the gripping process.

In the following, we compare the granular gripper with suction to an identical device not equipped with a suction cup, referred to here as “ordinary gripper”. We study the gripping of six 3D-printed objects (three shapes, each with a smooth and rough surface):

* a small sphere (diameter $20.0 \pm \SI{0.5}{\mm}$, mass $17.5 \pm \SI{0.1}{\gram}$)

* a large sphere (diameter $40.0 \pm \SI{0.5}{\mm}$, mass $23.2 \pm \SI{0.1}{\gram}$ )

* a large square flat plate (surface $2500 \pm \SI{1}{\square\mm}$, thickness $6.0 \pm \SI{0.5}{\mm}$,  mass $26.2 \pm \SI{0.1}{\gram}$)

To obtain smooth surfaces, the objects are coated with lacquer (Lackspray matt Acryl) to facilitate suction between the membrane and the object. For rough surfaces, a mesh structure was printed on the surfaces of the objects to prevent suction. All objects have a borehole at the top, connected to a pressure sensor to monitor the pressure at the membrane-object interface (see Fig. <ref>). We studied (i) the ordinary granular gripper, (ii) our gripper with only suction, and (iii) our gripper with a combination of suction and jamming. Suction is active throughout each measurement. A pressure difference  $p_\text{suc}\approx \SI{90}{\kilo\pascal}$ between ambient and the suction cup is applied for suction.

Before each measurement, the interior of the gripper is pressurized at $\SI{5}{\kilo\pascal}$ to fluidize the enclosed granulate and erase the memory of previous cycles. An experimental measurement comprises the following steps:

* The $z$-stage displaces the object upwards, pushing it onto the gripper.

* When a predefined indentation depth is reached, the platform stops. Then the granulate relaxes for 20. In the first two steps, the gripper adapts to the shape of the object.

* The air is evacuated from the gripper. The pressure difference between the interior of the gripper and the surrounding atmosphere ($p_\text{vac}\approx \SI{90}{\kilo\pascal}$) causes the granulate to jam.

* The platform moves downwards until the object is no longer in contact with the gripper, which imitates the manipulation of an object.

§.§ Gripping on unstable grounds

To test the gripping performance of objects placed on unstable ground, the object previously fixed to a $z$-stage is replaced by an open-top container of length 56.0, width 70.0 and height 41.0, filled with water or sand, as shown in Fig. <ref>.
A spherical object is attached to the base of the box with a thread in such a way that the object can be moved by a distance that equals the height of the box. When the object is pulled to the maximum length of the thread, it is entirely outside of the water or sand.

As the object is pulled out but remains connected to the box and the force sensor, the maximum holding force is measured similarly to the previous setup (Fig. <ref>).

The spherical object consists of two parts: a lower smaller one and an upper larger one.
The upper part is hollow (lower density), and the lower one is solid (higher density), so the object floats on water. The box is filled up to 90% of its volume with water or sand, and the test object is placed on the surface.

[] (setup)  at (0, 0)  ;

[right=1mm of setup.47, anchor=west, text width=2.5cm, text=black] (holder) gripper holder;
[left=24mm of holder] (holderPt) ;

[-Circle[fill=none], thick](holder.west)–(holderPt);

[right=1mm of setup.25, anchor=west, text width=2.5cm, text=black] (gripper)
gripper filled
with particles;

[left=17mm of gripper] (gripperPt) ;

[-Circle[fill=none], thick](gripper)–(gripperPt);

[right=1mm of setup.-24, anchor=west, text width=2.5cm, text=black] (object)
hollow part;
[left=31mm of object] (objectPt) ;

[-Circle[fill=none], thick](object.west)–(objectPt);

[right=1mm of setup.-35, anchor=west, text width=2.5cm, text=black] (object)
solid part;
[left=31mm of object] (objectPt) ;

[-Circle[fill=none], thick](object.west)–(objectPt);

[right=1mm of setup.-43, anchor=west, text width=2.5cm, text=black] (bore) thread;

[left=12mm of bore] (borePt) ;

[-Circle[fill=none], thick](bore.west)–(borePt);

[right=1mm of setup.-47, anchor=west, text width=2.5cm, text=black] (water) water or sand;

[left=22.5mm of water] (waterPt) ;

[-Circle[fill=none], thick](water.west)–(waterPt);

[right=1mm of setup.-53, anchor=west, text width=2.5cm, text=black] (loadcell) load cell;
[left=14mm of loadcell] (loadcellPt) ;

[-Circle[fill=none], thick](loadcell.west)–(loadcellPt);

[right=1mm of setup.-57, anchor=west,text width=2.5cm, text=black] (zstage) $z$-stage;

[left=5.0mm of zstage] (zstagePtmid) ;

[-Circle[fill=none], thick](zstage.west)–(zstagePtmid);

Gripping objects from unstable grounds. An open box, placed on the $z$-stage, is filled with water or sand. The test object is placed on the unstable ground's surface and connected to the bottom of the box by a thread. The test object comprises two parts: a lower solid one and an upper hollow one. The difference in densities allows the object to float in water. The box is attached to a load cell to record the holding force.

§ RESULTS

§.§ Gripping from stable grounds

We first study the effect of adding suction to a granular gripper for manipulating objects of various shapes and surface properties placed on stable grounds. Figure <ref> shows the maximum holding force and maximum differential pressure (difference between the ambient pressure and the pressure at the interface between the membrane and the object), $\Delta p$, for the studied cases.

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Maximum holding force (a-c) and maximum differential pressure, $\Delta p$ (d-e) for the studied cases. Colors indicate the employed gripping mechanisms: (I) ordinary granular gripper (jamming only), (II) our gripper where only suction is active (no jamming), and (III) our gripper with both mechanism, suction and jamming, activated. All data are shown for smooth and rough surfaces.
Error bars indicate the standard deviation due to six independent measurements; the heights of the columns show the mean.

For a small sphere whose diameter is less than half the gripper's diameter (Fig. <ref>a, d),  the holding force is ca. 4 for both rough and smooth surface, when exploiting jamming only (case (I)), which is sufficient for reliable operation. Using only the suction cup (case (II)),
for the smooth surface, we reach up to $\approx$3.4, but on the rough surface, the gripper fails, evidenced by minimal holding force of $\approx$0.2.
For suction and jamming both activated (case (III)), we obtain the largest forces, ca. 4.5 for the rough sphere and ca. 9.2 for the smooth one which is an improvement by ca. the factor $2.5$ compared to the ordinary gripper.

For the large sphere whose diameter is more than half the gripper's diameter (Fig. <ref>b, e), the holding force of an ordinary gripper (case (I)) is low for both smooth and rough surface (ca. 1.5) since the gripper cannot embrace the sphere. No sealed cavities appear at the object-gripper interface, apparent from minimal pressure difference (Fig. <ref>e). When only suction is active (case (II)), the holding force is ca. 4.8 for a rough surface and ca. 7.7 for a smooth surface. Uneven surfaces lead to a smaller force, as roughness weakens the seal between the object and the suction cup.
For both mechanisms active, jamming and suction (case (III)), the largest holding forces are achieved, ca. 6.0 for a rough surface and 10.4 for a smooth one.

For a flat pane larger than the gripper (Fig. <ref>c, f), the ordinary gripper fails for both surfaces (force $<$0.2), which is expected: objects that the gripper cannot embrace due to their shape or size are challenging for granular grippers <cit.>. In contrast, significant holding forces (ca. 9) are achieved by the gripper with integrated suction, irrespective of the object's surface properties (rough or smooth). For both cases (II) and (III), i.e., suction only or suction and jamming, we obtain similar values for the holding forces and the differential pressure, which indicates that jamming does not make a significant contribution since the gripper fails to embrace the object.

For all measurements obtained for the ordinary granular gripper, no appreciable pressure difference, $\Delta p$, is measured at the membrane-object interface (case (I) in Fig. <ref>). This indicates that airtight cavities are not formed between the object and the gripper; therefore, in contrast to earlier speculations <cit.>, suction due to jamming only (i.e., not due to an independent suction cup) is not achieved.

For all objects tested, regardless of their surface state, jamming-based gripping in combination with an independent suction cup considerably increases the maximum holding force. For objects that are challenging for ordinary granular grippers (large sphere and flat plate), the ordinary gripper fails, while integrated suction effectively allows for proper operation.

§.§ Gripping from unstable grounds

To study the gripping of objects placed on unstable grounds, we perform the procedure sketched in Fig. <ref>:

[] (setup)  at (0, 0) ;

[right=-68mm of setup.south, anchor=north, text width=2.5cm, align=center] (suction)
suction is
[right=-23mm of setup.south, anchor=north, text width=2.5cm, align=center] (conforms)
[right=23mm of setup.south, anchor=north, text width=2.5cm, align=center] (jamming)
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[right=67mm of setup.south, anchor=north, text width=2.5cm, align=center] (manip)
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[-stealth, thick, black](suction)–(conforms);
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[-stealth, thick, black](jamming)–(manip);

[above=75mm of suction, anchor=south, align=center] (i)
[above=75mm of conforms, anchor=south, align=center] (ii)
[above=75mm of jamming, anchor=south, align=center] (iii)
[above=75mm of manip, anchor=south, align=center] (iv)

Sequence of steps for gripping a sphere (in red) floating on water with a granular gripper with integrated suction.

* Suction is applied to fix the object's position, preventing it from sinking or sliding away from the gripper.

* Air pressure is applied within the gripper to fluidize the granulate and inflate the gripper's membrane, which increases the contact surface between the gripper and the object.

* Applying vacuum enforces the grains to jam, causing the gripper to clamp the object.

* The object can then be handled safely.

Figure <ref> shows the maximum holding force when a sphere of diameter $20.0 \pm \SI{0.5}{\mm}$ is gripped from a sand surface or floating on water. Again, we study the ordinary granular gripper (case (I)) versus a granular gripper with integrated suction (case (III)).

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Maximum holding force for gripping a sphere from an unstable ground. (top) sphere is placed on a sand surface, (bottom) sphere floats on water. Data is shown for an ordinary granular gripper (case I) and a granular gripper with integrated suction (case III). Error bars show the standard deviation over six independent measurements; the height of the bar shows the mean.

When an ordinary gripper is pressed against the floating sphere to conform to its shape, the object sinks. Therefore, the gripper embraces only a small part of the object, resulting in a low holding force of ca. 0.1. Effectively, the gripper cannot grip the object, as it slips out of grip almost immediately.

For activated suction, the object is held in place during conformation. When jamming is enforced, the holding force reaches 7.8, corresponding to an increase of about one order of magnitude compared to the ordinary gripper.

Similarly, when the sphere is placed on a sand surface, it tends to sink into the ground when an ordinary granular gripper is pressed against its surface. Consequently, the conformation is hindered, and only a reduced holding force (ca. 1) is achieved, see Fig. <ref> (bottom). Using the gripper with activated suction (case (III)), a significantly larger holding force of 7.62 was found.

The results reveal that applying an independent suction mechanism can considerably enhance the performance of granular grippers. In particular, such grippers can grip objects located on unstable ground or even floating on water, which is impossible for ordinary granular grippers.

§ SUMMARY

Granular grippers are capable of gripping objects of different shapes and sizes. Despite their versatility, they cannot reliably grip flat, very soft, or deformable objects, objects larger than the gripper size, and objects resting on unstable ground. This paper presents a granular gripper with an integrated suction mechanism.

We show that this gripper performs better gripping and holding than ordinary granular grippers. For objects larger than the gripper and flat objects, adding an independent suction mechanism increases the holding force by about an order of magnitude. The improved gripper can also grip objects located on unstable grounds, such as water or sand, where ordinary granular grippers fail.

Granular grippers with additional suction devices can be used in demanding applications, e.g., picking up and placing objects of different geometries in factory automation or improving prosthetic systems. Due to its robustness and ability to grip objects on unstable surfaces, the presented gripper can be integrated into remote-controlled flying devices <cit.>.

§ ACKNOWLEDGEMENTS

The authors thank Walter Pucheanu for technical support. This work was supported by the Interdisciplinary Center for Nanostructured Films (IZNF), the Competence Unit for Scientific Computing (CSC), and the Interdisciplinary Center for Functional Particle Systems (FPS) at Friedrich-Alexander-Universität Erlangen-Nürnberg.

§ FUNDING STATEMENT

Angel Santarossa and Thorsten Pöschel gratefully acknowledge funding by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–Project Number

§ AUTHOR DISCLOSURE STATEMENT

The authors declare that they have no conflict of interest.