# The Power of Fragmentation: A Hierarchical Transformer Model for Structural
Segmentation in Symbolic Music Generation

Guowei Wu, Shipei Liu, Xiaoya Fan This paper was produced by the IEEE
Publication Technology Group. They are in Piscataway, NJ.Manuscript received
June 26, 2022; revised June 26, 2021.

###### Abstract

Symbolic Music Generation relies on the contextual representation capabilities
of the generative model, where the most prevalent approach is the Transformer-
based model. The learning of musical context is also related to the structural
elements in music, i.e. intro, verse, and chorus, which are currently
overlooked by the research community. In this paper, we propose a hierarchical
Transformer model to learn multi-scale contexts in music. In the encoding
phase, we first designed a Fragment Scope Localization layer to syncopate the
music into chords and sections. Then, we use a multi-scale attention mechanism
to learn note-, chord-, and section-level contexts. In the decoding phase, we
proposed a hierarchical Transformer model that uses fine-decoders to generate
sections in parallel and a coarse-decoder to decode the combined music. We
also designed a Music Style Normalization layer to achieve a consistent music
style between the generated sections. Our model is evaluated on two open MIDI
datasets, and experiments show that our model outperforms the best
contemporary music generative models. More excitingly, the visual evaluation
shows that our model is superior in melody reuse, resulting in more realistic
music.

###### Index Terms:

Symbolic music generation, Transformer-based model, structural segmentation,
multi-scale attention.

##  I Introduction

Symbolic Music Generation (SMG) refers to generating continuation from the
initial notes. It has received great attention with the prosperity of deep
learning [1]. Music can be seen as a sequence of notes in time. A musical
generative model should be able to refer to the context of note
representations, as required for natural language models. Hence, Language
Models (LMs), such as the auto-regression model, are prevalently used for
music generation. A representative model is PerformanceRNN [2], an LSTM-based
recurrent neural network designed to model polyphonic music with complex
dynamics. It performs well in generating short music ($\sim$ 30s). But
generating long music sequences ($\geq$ 4 minutes) is still a challenge
because errors accumulate as the length of the sequence increases. The output
length is often limited in current LMs, resulting in a restricted maximum
perception range. When the length of the generated sequence exceeds the
perception range, its performance degenerates quickly.

Many efforts have been devoted to maintaining long-term relevance with
reasonable computational complexity in SMG tasks. Especially the Transformer
model [3] have helped the advance state-of-the-art (SOTA) in symbolic music
generation. For instance, Google researchers [4] have proposed
MusicTransformer, which can deal with longer sequences ($\sim$ 4096 notes)
with optimized intermediate memory occupation. MuseNet [5] is a Transformer-
based model, using the same general-purpose unsupervised technology as GPT-2
[6], that generates 4-minute music and discovers patterns of harmony, rhythm,
and style in long-form music. Despite recent improvements in long-term music
generation, existing approaches fail to learn music structure effectively.
This may be attributed to their incompetence in segmenting structural
elements.

Most music is typically structural, such as intro, verse, chorus, etc.,
meaning that some attractive melodies are repeated throughout the song. In
traditional neural networks, hierarchical architectures have been proposed to
generate these structured melodies. MusicVAE [7] uses a hierarchical decoder
to model structural elements, such as the repetition and variation between
measures and sections of a piece of music. MuseGAN [8] consists of three
Generative Adversarial Networks (GANs) models, called jamming model, composer
model, and hybrid model, for generating multi-track music with their
respective temporal dynamics. Researchers have shown that the hierarchical
architecture enables the extraction of structural context, enhancing the long-
term relevance in the decoding phase. A similar idea has also been introduced
to a transformer-based model for text generation task [9] that references
paragraph-level context and achieves SOTA performance. Therefore, we argue
that structural segmentation and section-level contextual learning are the
primary challenges in generating realistic long-term music.

(a) Encoding (b) Decoding

Figure 1: An overview of the proposed framework. (a) The encoding phase: The
FSL layer is used to recognize chords and sections. It consists of two sub-
layers, a classification (cls) layer and a regression (reg) layer, for
recognizing sections and locating the location of each recognized section,
respectively. Then, these structural elements are fed into a Transformer
encoder that uses multi-scale attention to embed musical representations at
the note, chord and section levels. Fine-decoders of different lengths are
subsequently pre-trained by multi-scale representations. (b) The decoding
phase: At the bottom of the hierarchical architecture, fine-decoders are used
to generate output sections in parallel. We introduced an MSN layer to
normalize the music style through a variable $z$ and maintain consistency
between the generated sections. The normalized sections are combined by an
aggregation layer and fine-tuned by a followed coarse-decoder.

In this paper, we propose a hierarchical Transformer model to learn multi-
scale representations extracted by structural segmentation methods.
Concretely, chords were first recognized from note sequence and then fed to a
custom-designed Fragment Scope Localization (FSL) layer. The note sequence was
segmented into several structural sections by the region proposal method with
two sub-layers, namely the cls layer and the reg layer. Then, fine-decoders
were pre-trained with multi-scale attention through these structural corpora,
enabling chord-level and section-level contextual learning. Next, the first
few notes and labels are input as activation information to generate sections.
We propose a Music Style Normalization (MSN) layer to control the music style
of the generated sections. The followed aggregation layer and coarse-decoder
are used to combine the generated sections and fine-tune them at the global
scale, respectively. An illustrative diagram of the proposed framework is
shown in Fig. 1. Our main contributions are as follows:

  * •

We designed an FSL layer that divides the music into structural elements and
further analyzed its utility with different settings through ablation studies.

  * •

We proposed a multi-scale attention mechanism to learn music representations
at the note, chord, and section levels. Visual evaluation shows its
superiority in reusing melodies, such as chord progression at the chord-level.

  * •

We proposed a hierarchical architecture, using fine-decoders to generate
sections in parallel and fine-tune the combined music with a coarse-decoder.
Since the proposed model generates the music by section, it can alleviate the
problem of error accumulation. Experimental results show that it achieves SOTA
performance on two open datasets.

  * •

We designed an MSN layer to control music style and argued for the importance
of Mutual Information (MI) in this process. Thus, the proposed model showed
superiority in the style consistency of generated music compared to other
models.

##  II Related works

###  II-A Transformer baselines

The Transformer model has been the typical choice for sequence data modeling
due to its advantages in contextual representation capabilities. However,
applying the Transformer model to the SMG task is computationally prohibitive
since the computational complexity increases quadratically with the sequence
length [10]. Reformer [11] is an effective work, where the dot-product
attention is replaced by a locality-sensitive hashing-based calculation,
reducing the complexity from $O\left(L^{2}\right)$ to $O(L\sqrt{L})$. Many
other Transformer models have been devoted to resolving the problem of
building longer-term dependencies in sequence prediction tasks. For instance,
Transformer-XL [12] learns dependencies beyond a fixed length by a segment-
level recurrence mechanism and a positional encoding scheme. Some approaches
introduce additional modules to improve music generation. Transformer-GANs
[13] introduces a pre-trained discriminator that uses adversarial loss to
complement the negative log-likelihood objective, enabling improvement in
synthesizing minute-long compositions. These proposed baselines provide the
feasibility of solving SMG tasks by Transformer models.

###  II-B Music structure segmentation

Structure segmentation is commonly used for modeling long-form music, which
can be tackled with hierarchical architecture. In a music annotation work
[14], attentive convolution networks and recurrent networks were
hierarchically combined to solve the problem of audio music representation and
structure learning. McCallum [15] explores the use of Convolutional Neural
Networks (CNNs) for unsupervised training in music segmentation, aiming to
detect the boundaries of musical fragments in audio music. Dai [16] proposed a
hierarchical model to generate a full-length melody guided by long-term
repetitive structure and achieved near-human performance in melody generation
about half the time. These works suggest that it is beneficial to design a
structural segmentation module for SMG tasks.

###  II-C Music Style Control

Instance normalization methods are often used for style transfer. For
instance, Huang et al. [17] proposed an adaptive instance normalization
approach to achieve flexible style control. Ling et al. [18] used a region-
aware adaptive instance normalization module to formulate the visual style of
the background and transfer it to the foreground. MI is also commonly used for
style control. Chawla [19] proposed a model for formality style transfer,
which maximized the MI between original and target styles as the training
objective and achieved better performance. Inspired by these studies, we
utilized a MI-based strategy to maintain a consistent style in long-form
music.

##  III The proposed model

The key challenge for the SMG task is to generate long-term music with
structural relevance and consistent style. We begin with a description of note
sequence encoding and structural element fragmentation, followed by an
introduction of the hierarchical architecture and music style normalization. A
theoretical analysis of the coverage of attention patterns of the proposed and
baseline models was also performed.

###  III-A Encoding: Fragmentation scope localization and Multi-scale
attention mechanism

During the encoding phase, the FSL layer is used to recognize chords and
sections, which facilitates the extraction of multi-scale contexts in music.
Then, we propose multi-scale attention to learn musical representations at
different time scales and pre-train the fine-decoders. The detail of the FSL
layer and multi-scale attention is illustrated in Fig. 1 (a).

Chord recognition: A chord, in music, is a group (typically three or more) of
notes that sound together and serve as a basis of harmony. In tonal music
rules, a chord can also be a group of notes that appear dispersed in a bar.
Therefore, we extracted the chord feature, called chord profile, by counting
the notes of each bar. The notes (transferred to the same octave) was mapped
to a 12-dimensional vector, usually called twelve-tone equal temperament,
which is a set of pitch classes
$\left[\textbf{C}_{1},Db_{2},\textbf{D}_{3},Eb_{4},\textbf{E}_{5},\textbf{F}_{6},F\sharp_{7},\textbf{G}_{8},Ab_{9},\textbf{A}_{10},Bb_{11},\textbf{B}_{12}\right]$.

TABLE I: Description of usual chord profiles.

Id | Chord type | Chord profile | Id | Chord type | Chord profile
---|---|---|---|---|---
1 | $\left(empty\right)$ | [0, 4, 7] | 25 | 79$\sharp$ | [0, 3, 4, 7, 10]
2 | + | [0, 4, 8] | 26 | 79$\sharp$11$\sharp$ | [0, 3, 4, 6, 7, 10]
3 | +7 | [0, 4, 8, 10] | 27 | 79$\sharp$13 | [0, 3, 4, 5, 7, 9, 10]
4 | +79 | [0, 2, 4, 8, 10] | 28 | 7911 | [0, 2, 4, 5, 7, 10]
5 | +79$\sharp$ | [0, 3, 4, 8, 10] | 29 | 7911$\sharp$ | [0, 2, 4, 6, 7, 10]
6 | +7911$\sharp$ | [0, 2, 4, 6, 8, 10] | 30 | 7913 | [0, 2, 4, 5, 7, 9, 10]
7 | +79b | [0, 1, 4, 8, 10] | 31 | 7913b | [0, 2, 4, 5, 7, 8, 10]
8 | +j7 | [0, 4, 8, 11] | 32 | 79b | [0, 1, 4, 7, 10]
9 | - | [0, 3, 7] | 33 | 79b13 | [0, 1, 4, 5, 7, 9, 10]
10 | -6 | [0, 3, 7, 9] | 34 | 79b13b | [0, 1, 4, 5, 7, 8, 10]
11 | -69 | [0, 2, 3, 7, 9] | 35 | 7alt | [0, 1, 4, 6, 10]
12 | -7 | [0, 3, 7, 10] | 36 | j7 | [0, 4, 7, 11]
13 | -79 | [0, 2, 3, 7, 10] | 37 | j79 | [0, 2, 4, 7, 11]
14 | -7911 | [0, 2, 3, 5, 7, 10] | 38 | j79$\sharp$ | [0, 3, 4, 7, 11]
15 | -7913 | [0, 2, 3, 5, 7, 9, 10] | 39 | j79$\sharp$11$\sharp$ | [0, 3, 4, 6, 7, 11]
16 | -79b | [0, 1, 3, 7, 10] | 40 | j7911$\sharp$ | [0, 2, 4, 6, 7, 11]
17 | -j7 | [0, 3, 7, 11] | 41 | m7b5 | [0, 3, 6, 10]
18 | -j7911$\sharp$ | [0, 2, 3, 6, 7, 11] | 42 | o | [0, 3, 6]
19 | -j7913 | [0, 2, 3, 5, 7, 9, 11] | 43 | o7 | [0, 3, 6, 9]
20 | 6 | [0, 4, 7, 9] | 44 | sus | [0, 5, 7]
21 | 69 | [0, 2, 4, 7, 9] | 45 | sus7 | [0, 5, 7, 10]
22 | 6911$\sharp$ | [0, 2, 4, 6, 7, 9] | 46 | sus79 | [0, 2, 5, 7, 10]
23 | 7 | [0, 4, 7, 10] | 47 | sus7913 | [0, 2, 5, 7, 9, 10]
24 | 79 | [0, 2, 4, 7, 10] | 48 | Non-chord | [0]

We counted the 47 most common forms of chord composition (major triads, minor
sixth chords, minor-major seventh chords, suspension chords, etc.). Single
notes that cannot form a chord are denoted as type 48. For example, the $C$
triad chord consists of three notes: a root note $C$, intervals of a third
$E$, and a fifth above the root note $G$. Thus, the $C$ triad chord can be
represented by $\left[C,E,G\right]$, and in twelve-tone equal temperament by
$C_{chord}=C_{1}+Triad_{\left[0,4,7\right]}=\left[C_{1},E_{5},G_{8}\right]$.
The chord profile used in our model is shown in Tab. I.

In our model, chords are identified automatically from MIDI data in two steps:
1\. Template matching for complete chord; 2\. Sequence estimation for
unidentified bars. More specifically, we use the template matching method to
match the same chords as in the chord template, including the 47 common chord
profiles. These matching chords have neither missing notes nor other
ornamental notes. Then, we construct an n-gram sequence to represent
unidentified bars and use a nonlinear classifier to calculate correlation,
which can determine the probability of the chord category. The result of chord
recognition is demonstrated in Fig. 2.

Figure 2:  Example of chord recognition results. The first 20 music bars are
displayed in a visual MIDI diagram, which can easily exhibit the note
constitution of the chord between bar lines.

A series of chords is called a chord progression, and it has the
characteristic of being periodically reused in sections of the same label. For
example, in Fig. 2, the 19, 20, and 21 bars use the same chord as the 2, 3,
and 4 bars. The construction of these specific chord progressions is one of
the goals we hope to achieve with our model.

Fragment region proposal: We use the region proposal method to selectively
search for music sections. The candidate windows $\omega$ used to extract
section location and length, is applied to the note sequence
$chord\left(x\right)$ labeled by the chord profiles. Thus, the window size is
set to a multiple of the chord length (usually 16 semiquaver notes), with $K$
sizes available. Normally, sections are extracted through windows sliding from
left to right. The left-to-right (L2R) strategy can be written as:

$Cand\left[m\right]=\omega\left[k\right]\cdot
chord\left(Loc_{m}:Loc_{m}+\omega\left[k\right]\right)$ (1)

where $Cand\left[m\right]$ denotes the scope of $m$-th extracted fragment;
$\omega\left[k\right]$ denotes the candidate window with the appropriate size
$k$; The current location is calculated by
$Loc_{m}=\sum_{j=1}^{m-1}\omega_{j}$; The goal of L2R strategy is to select
suitable windows with the maximum classification probability
$\mathop{\arg\max}\limits\left(cls\left(\omega\cdot
chord\left(x\right)\right)\right)$ in order from left to right.

Obviously, the L2R strategy is prone to error accumulation of the window
location. To deal with this, we design a global strategy, where a random
location $\Delta Loc$ is added to the input, and the non-overlapping windows
are padded. The global strategy can be written as:

$Cand^{*}\left[m\right]=\omega\left[k\right]\cdot chord\left(\Delta
Loc_{m}\pm\frac{1}{2}\omega\left[k\right]\right)$ (2)

where $\Delta Loc_{m}$ denotes the center of the $m$-th random location; The
non-overlapping windows are padded by maximizing the average confidence,
calculated by
$\mathop{\arg\max}\limits\left(\frac{1}{M}\sum_{m=1}^{M}cls\left(\omega\cdot
chord\left(x\right)\right)\right)$. The comparison of the two alignment
strategies is shown in Fig. 3.

(a) Left-to-right strategy (b) Global strategy

Figure 3: Two alignment strategies for candidate window. (a) The candidate
windows slide over the note embeddings to produce proposal regions. The cls
layer identifies the section categories, and the reg layer locates the section
scopes to the ground truth. (b) The central locations of the candidate scopes
are randomly initialized on the note embedding. Then, the cls layer identifies
the section labels, and the reg layer regresses the candidate scopes.

Candidate classification: The cls layer produces category probabilities
$p_{\varphi}\in\mathbb{R}^{\phi}$ to predict the label $\varphi\in\phi$ of
candidate sections, such as intro, verse, chorus. The number of labels in the
training data is unbalance, so we use Gumbel Softmax [20] to optimize the cls
layer, which is defined as:

$L_{cls}=-\log\frac{\exp\left(\left(p_{\varphi}+\theta_{\varphi}\right)/\tau\right)}{\sum_{\tilde{\varphi}\sim\phi}\exp\left(\left(p_{\tilde{\varphi}}+\theta_{\tilde{\varphi}}\right)/\tau\right)}$
(3)

where $\tau$ is a non-negative coefficient of spread in the $Gumbel$
distribution; The smaller the coefficient, the closer the sample expectation
is to the $Argmax$ function; The larger the coefficient, the more average the
sample expectation is. $\theta$ are samples drawn from
$Gumbel\left(0,1\right)$. The $\tilde{\varphi}$ lists the all labels, and
$p_{\tilde{\varphi}}$ is the probabilities of candidate section in every
label. The value of $L_{cls}$ should be close to 0 if the candidate is
consistent with the ground-truth label.

Scope regression: To measure the overlap ratio between the candidate and
ground-truth sections, we check the percentage of shared and different notes
in these paired sections. Let
$D=\left[(\omega_{1},y_{1}),\cdots,(\omega_{M},y_{M})\right]$ be a training
set of instance pairs, $y$ are the ground-truth sections. We use the Jaccard
similarity coefficient to evaluate the similarity between the predicted and
ground-truth scopes, which is denoted as $Jac\left(a,b\right)=\left(a\cap
b\right)/\left(a\cup b\right)$. The regression loss function can be calculated
by:

$L_{reg}=-\log(Jac\left(\omega_{m},y_{m}\right))$ (4)

A low regression loss indicates high coincidence between the prediction and
the ground-truth. Accordingly, the loss function of our FSL layer can be
represented by:

$L_{FSL}=\frac{1}{N_{cls}}\sum_{m=1}^{N_{cls}}L_{cls}+\frac{1}{N_{reg}}\sum_{m=1}^{N_{reg}}L_{reg}$
(5)

where $L_{cls}$ and $L_{reg}$ are normalized by $N_{cls}$ and $N_{reg}$, which
are the sample number in these two layers, respectively.

After passing through the FLS layer, the note sequence is segmented into
structural sections. The fragmentation process is formulated as:

$Section_{m}\left(x\right)=reg\left(cls\left(Cand\left[m\right]\right)\right)$
(6)

where $cls$ and $reg$ stands for the classification and regression operation
of the $m$-th fragment, respectively;

Multi-scale attention and fine-decoders pre-training: In Transformer models,
the attention pattern determines the perceptual range. We designed a multi-
scale attention mechanism for musical representation, to enable learning of
chord-level and section-level contextual features, as shown in Fig. 4.

Figure 4:  A diagram of multi-scale attention. Each layer in the encoder
consists of three groups of attention heads, capable of learning music
representations at different scale.

Given a vector $X\in\mathbb{R}^{L\times D}$ of note sequence, where $L$ is the
length of section sequence and the $D$ is the dimension of the vector. The
output embedding $h\left(x\right)$ is obtained by a Transformer encoder, which
has $l$ hidden layers, denoted as:

$h_{l}=LN\left(h_{l-1}+FFN\left(MS\\_Attention\left(Q,K,V\right)\right)\right)$
(7)

where $Q$, $K$, $V$ represent the Query, Key, Value vector in the Transformer-
based model; $LN$ and $FFN$ represent the layer normalization operation and
feed-forward network, respectively; $MS\\_Attention$ represents the multi-
scale attention.

Multi-scale attention has a scale pattern $S=\left(S_{1},\cdots,S_{i}\right)$,
which we parameterize by default three scales $S_{note}$, $S_{chord}$, and
$S_{section}$, to control its working scope. The multi-scale attention can be
calculated as:

$\displaystyle
MS\\_Attention=\left[\begin{array}[]{c}head_{1}\left(S_{1}\right),\cdots,head_{N_{1}}\left(S_{1}\right)\\\
\cdots\\\
head_{1}\left(S_{i}\right),\cdots,head_{N_{i}}\left(S_{i}\right)\end{array}\right]$
(8)

where $N_{i}$ is the number of heads at each scale; The attention head is the
weighted sum of scaled dot-product of the input vectors, calculated by:

$head\left(S_{i}\right)=Softmax\left(\frac{Q_{S_{i}}K_{S_{i}}^{T}}{\sqrt{D}}\right)\cdot
V_{S_{i}}$ (9) $\displaystyle\left[\begin{array}[]{c}Q_{S_{i}}\\\ K_{S_{i}}\\\
V_{S_{i}}\end{array}\right]=\left[\begin{array}[]{c}W_{Q}\\\ W_{K}\\\
W_{V}\end{array}\right]\odot X\left(S_{i}\right)+\left[\begin{array}[]{c}0\\\
P_{S_{i}}\\\ P_{S_{i}}\end{array}\right]$ (10)

where $W_{Q},W_{K},W_{V}$ represents the weight matrix that transform the
input into the $Query$, $Key$, and $Value$ vector, respectively; $X(S_{i})$
defines the references at the $i$ scale, corresponding to $note(x)$,
$chord(x)$ and $section(x)$. $P_{S_{i}}$ represent the position embeddings
(PEs) in each scale, for example, use relative PEs (RPEs) in the note and
chord scales, and absolute PE (APEs) in the section scale.

Through the $MS\\_Attention$, we pre-trained several fine decoders, whose
maximum length is according to the section length. The training steps and
important parameters are detailed in Appendix.

###  III-B Decoding: hierarchical architecture with Music Style Normalization

The proposed hierarchical model, a bottom-up architecture, aims to generate
notes at scales from fine to coarse. The decoding phase is shown in Fig. 1
(b).

At the bottom of the hierarchical architecture, we use the fine decoders, pre-
trained with different maximum lengths, to generate music sections. Each
section is decoded in parallel with an activation information, which usually
uses the first few notes and a label:

$G_{m}=Decoder_{fine}\left(Section_{m}\left(x\right)\left[0:r\right];z_{m}\right)$
(11)

where $G_{m}$ represents the $m$-th generated section; Fine-decoders are
activated by start notes $Section_{m}\left(x\right)$ with default length $r$;
$z_{m}$ indicates the target label of the generated section, i.e., intro,
verse, and chorus, etc.

Specifically, given the start notes
$\left(h^{\prime}_{1},\cdots,h^{\prime}_{r}\right)$, the output embedding
$h^{\prime}_{r+1}$ is obtained by:

$h^{\prime}_{r+1}=LN\left(h^{\prime}_{r}+FFN\left(a_{r+1}\left(h^{\prime}_{1}:h^{\prime}_{r}\right)\right)\right)$
(12)

where $a_{n}$ is the self-attention block, which calculates a weight score
from previous notes.

Music Style Normalization: To enhance the style consistency between the
original and generated section, we designed an MSN layer. The musical style is
quantified by a variable $z$ that has the same dimensions as the note
embedding. Through the MSN layer, the style variable $z$, initialized by the
section label, is transferred to the generated sections, which can be
expressed as $z_{m}\to z_{m}^{\prime}$. For example, the MSN layer normalizes
the mean and standard deviation of note pitches of each original section and
maps them to the generated section:

$MSN\left(G,z\right)=\gamma_{z}\left(\frac{G_{m}-\mu(z_{m}^{\prime})}{\sigma(z_{m}^{\prime})}\right)+\beta_{z}$
(13)

where $\gamma_{z}$, $\beta_{z}$ are the scaling and translation parameters,
respectively, which are calculated independently for each category.
$\mu(z_{m}^{\prime})$ and $\sigma(z_{m}^{\prime})$ represent the mean and
standard deviation of note pitches in $i$-th generated section, respectively.

At the up of the hierarchical architecture, these sections are aggregated into
a long sequence by the weighted concatenation operation $v$. The coarse-
decoder used to learn the global contextual reference from the merged
sequence. The coarse decoding process can be represented by:

$G=Decoder_{coarse}\left(v\left(MSN\left(\sum_{m=1}^{M}G_{m}\right)\right)\right)$
(14)

where $G$ stands for the output sequence; The coarse-decoder $Decode_{coarse}$
uses a multi-scale attention with same fixed-length as the encoder.

Multi-Task Loss: The loss function we used in decoding phase consists of two
terms: (1) The $L_{MLM}$ for the note prediction in coarse decoder; (2) The
$L_{style}$ for music style normalization.

$L_{Decoding}=L_{MLM}\left(h_{\Pi}\mid h_{-\Pi}\right)+\lambda
L_{style}\left(x,G\left(x,z\right)\right)$ (15)

where hyperparameter $\lambda$ is used to balance the magnitude of loss terms.

Given a sequence $h_{1:n}$, decoders predicts $T$ masked notes among them. We
minimize the following MLM loss:

$L_{\mathrm{MLM}}\left(h_{\Pi}\mid
h_{-\Pi}\right)=-\frac{1}{T}\sum_{t=1}^{T}\log p\left(h_{\Pi_{t}}\mid
h_{-\Pi}\right)$ (16)

where $h_{\Pi}$ and $h_{-\Pi}$ denotes the masked and unmasked notes,
respectively.

To improve the stylistic similarity between the original and generated
sections, we maximize their MI in the optimizer. Mathematically, we can
examine the $KL$ divergence between the joint probability distribution and the
marginal probability distribution to determine whether the two variables are
approximately independent. Thus, we can use the variational information
maximization method [21] to estimate the $I\left(x;G\left(x,z\right)\right)$
by instantiating an intermediate tensor $Q\left(x\mid
G\left(x,z\right)\right)$:

$\begin{array}[]{c}I(x;G(x,z))=E_{p(x,G(x,z))}\left[\log\frac{Q(x\mid
G(x,z))}{P(x)}\right]\\\ +E_{p(G(x,z))}\left[D_{KL}(P(x\mid G(x,z))\|Q(x\mid
G(x,z)))\right]\end{array}$ (17)

where $P\left(x\right)$ is the distribution of training data;
$P\left(G\left(x,z\right)\right)$ is the distribution of generated samples; We
empirically deduce that $D_{KL}>0$, then the
$I\left(x;G\left(x,z\right)\right)$ can transform to:

$I(x;G(x,z))\geq E_{p(x,G(x,z))}\left[\log Q(x\mid
G(x,z))\right]+H\left(X\right)$ (18)

where $H\left(x\right)$ is the differential entropy of $X$. We can derive an
initial lower bound, which is tight when $Q\left(x\mid
G\left(x,z\right)\right)=P\left(x\mid G\left(x,z\right)\right)$. The musical
style loss $L_{style}\left(x,G\right)$ can calculate on
$I\left(x;G\left(x,z\right)\right)$:

$\begin{array}[]{c}L_{style}(x,G)=E_{p(x,G(x,z))}[\log Q\left(x\mid
G\left(x,z\right)\right)]+H(z)\\\ =E_{x\sim
P(x)}\left[E_{G^{\prime}\left(x,z\right)\sim P(G\left(x,z\right)\mid
x)}\left[\log Q\left(x\mid G^{\prime}\left(x,z\right)\right)\right]\right]\\\
+H(z)\leq I(x;G(x,z))\end{array}$ (19)

The $L_{style}\left(x,G\right)$ can be thought of as a negative reconstruction
error, in which the gradient of intermediate tensor $Q\left(x\mid G\right)$ is
tractable. Therefore, $L_{style}$ can be maximized via the re-parameterization
trick. This optimization conforms to the information-theoretic regularization:
MI is high when the two musical sections are similar. Hence, it is feasible to
add the $L_{style}$ loss to maintain consistency of generated sections.

Attention pattern analysis: At last, we analyze the advantage of the proposed
hierarchical model in attention patterns and compare it with others. We use a
16*16 image to compare attention mechanisms, where the differences between the
four patterns are revealed by the connection matrix of the output (rows) and
the input (columns).

As shown in Fig. 5 (a), local attention cannot exchange information with
distant notes. This situation is the opposite of dilated attention, see Fig. 5
(b). The drawback of sparse attention, in Fig. 5 (c), is that the dilated
stride is fixed and has no structure-related connection. In Fig. 5 (d), our
multi-scale attention enables both intra- and inter-section connectivity,
which has the advantage of enhancing intra-section integrity while learning
section-level dependencies. The range and location of section-level attention
are adaptive to the corresponding section in each music composition rather
than hard-coded in advance. Therefore, section segmentation can be viewed as a
dynamic assignment of full attention at a local scale, which has decreased
computational cost while maintaining the long perception range of full
attention.

(a) local (b) dilated (c) sparse (d) hierarchical

Figure 5: Illustration of four attention patterns: (a) Local attention (window
size 3); (b) Dilated attention (dispersion stride 1); (c) Sparse attention
(window size 3 and dispersion stride 3); (d) multi-scale attention (4 fine-
decoders for 4 sections of length $[3,7,3,6]$); The depth of blue represents
the weight of attention, while the white color block indicates the absence of
attention. The light blue, green, and orange block correspond to the attention
range of note-, chord- and section-level, respectively.  TABLE II: Comparison
of model performance with four other models, tested on J.S. Bach and Maestro
datasets. For each metric, the closer to the training set the values is, the
better. The values of the metrics are expected to be close to that for the
training set. The best performance is highlighted in bold.

Models | PPL | PCU | ISR | PRS | TUP | PR | APS | IOI | PCH | GS | CPI | SI
---|---|---|---|---|---|---|---|---|---|---|---|---
Training Set (J.S. Bach) | - | 8.407 | 0.835 | 0.422 | 25.45 | 34.32 | 11.67 | 0.064 | 2.994 | 0.930 | 0.985 | 0.011
PerformanceRNN | 1.96 | 7.5 | 0.853 | 0.446 | 9.3 | 28.4 | 3.901 | 0.05 | 2.665 | 0.999 | 0.999 | 0
Transformer-XL | 1.846 | 5.742 | 0.785 | 0.383 | 34.859 | 40.136 | 10.763 | 0.074 | 2.531 | 0.919 | 0.990 | 0.008
MusicTransformer | 1.833 | 5.768 | 0.792 | 0.386 | 36.069 | 42.542 | 10.257 | 0.075 | 2.535 | 0.916 | 0.993 | 0.009
Transformer-GANs | 1.789 | 5.9 | 0.807 | 0.423 | 34.817 | 40.41 | 10.993 | 0.072 | 2.537 | 0.921 | 0.987 | 0.009
Hierarchical model (Ours) | 1.716 | 7.061 | 0.819 | 0.412 | 25.194 | 34.597 | 10.982 | 0.124 | 2.944 | 0.872 | 0.994 | 0.01
Training Set (Maestro v3.0.0) | - | 6.665 | 0.815 | 0.429 | 66.01 | 68.15 | 11.59 | 0.083 | 2.886 | 0.901 | 0.958 | 0.208
PerformanceRNN | 1.403 | 3.99 | 0.682 | 0.268 | 5.52 | 10.71 | 3.98 | 0.099 | 1.912 | 0.880 | 0.322 | 0.014
Transformer-XL | 1.188 | 6.121 | 0.796 | 0.305 | 53.53 | 61.54 | 11.18 | 0.067 | 2.639 | 0.931 | 0.957 | 0.083
MusicTransformer | 1.167 | 6.16 | 0.833 | 0.499 | 55.55 | 62.72 | 11.69 | 0.071 | 2.539 | 0.919 | 0.990 | 0.095
Transformer-GANs | 1.145 | 6.136 | 0.814 | 0.298 | 56.17 | 63.93 | 11.95 | 0.091 | 3.51 | 0.919 | 0.997 | 0.129
Hierarchical model (Ours) | 1.063 | 6.716 | 0.815 | 0.429 | 57.26 | 67.62 | 11.35 | 0.087 | 2.985 | 0.894 | 0.987 | 0.191

##  IV Experiment Results

We used the same methods as MusicTransformer to process MIDI data. The
structural information of musical sections is automatically extracted from the
music score, which are obtained by the GuitarPro software, a music recording
tool. The annotation extraction method is detailed in Appendix.

Datasets: For pre-training the FSL layer, we collected a total of 300 scores
with 3,188 sections, called the GuitarPro dataset. After pre-training the FSL
layer, we trained and tested the proposed hierarchical Transformer based on
two canonical music corpora: (1) J.S. Bach Chorales [22] for short-term
generation; (2) Maestro v3.0.0 [23] for long-term generation. The sample
length of the Maestro dataset is significantly longer than that of the J.S.
Bach dataset. The split standard of these datasets abides by the rule of
80/10/10 proportion for train/validation/test. The detailed information of
datasets is shown in Tab. III.

TABLE III: Description of datasets.

Dataset | Scores | Sections | Notes | Average length
---|---|---|---|---
GuitarPro (Ours) | 300 | 3,188 | 355,411 |  $\sim$1185
J.S. Bach Chorales | 382 | - | 56,441 |  $\sim$148
Maestro v3.0.0 | 1276 | - | 7,040,164 |  $\sim$5517

Evaluation metrics: Several quantitative music metrics were used to evaluate
the generated music [24, 13]: PPL (Perplexity, measures the performance of
predict the next note); PCU (Unique pitch classes); ISR (Nonzero entries in C
major scale / Total nonzero entries); PRS (Time steps where the no. of pitches
$\geq$ 4 / Total time steps); TUP (Different pitches within a sample); PR
(Avg. difference of the highest and lowest pitch in semitones); APS (Avg.
semitone interval between two consecutive pitches); IOI (Time between two
consecutive notes); PCH (Pitch-Class Histogram Entropy, measures the
instability of pitch usage in shorter timescales); GS (Grooving Pattern
Similarity, measures consistency of rhythm across the entire piece); CPI
(Chord Progression Irregularity, measures consistency of harmony across the
entire piece); SI (Structural Indicator, detects presence of repeated
structures within a specified range of timescale);

Platform: All models were trained/tested on two Nvidia GeForce RTX 2080-Ti 12
GB GPU.

###  IV-A Symbolic Music Generation

We compared the performance of the present hierarchical model with other
outstanding music generation models that had achieved SOTA performance at the
moment. The experiments focused on assessing the pitch diversity (PCU, TUP,
PR, APS), non-blank rate (ISR, PRS), rhythm consistency (IOI, PCH, GS), and
musical rules (CPI, SI) of the generated samples. All models were retrained
with the same training sets. We ran the results 5 times and reported the
average values. The minimum unit of a visual token is a semiquaver note.

As shown in Tab. II, our hierarchical model surpasses other models in most
metrics. MusicTransformer and Transformer-XL use the similar relative
attention mechanism, and achieve similar results. Transformer-GANs achieved
the better performance in some metrics, especially in short-term music
generation (J.S. Bach dataset), duo to adversarial training that reduced the
distributional discrepancy between real and generated data. Our model performs
the best result (the value of metrics are the closest to the training set),
which can be attributed to the MSN layer that learns the musical style of the
training set. Meanwhile, our model achieved a higher ISR, TUP, PR, and PCH
compared to other models, suggesting better pitch diversity and rhythm
consistency in our generated samples. Most excitingly, we achieved the best
SI, indicating the superiority of our model in reusing structural melody.

TABLE IV: Ablation studies of FSL and MSN layers on J.S. Bach and Maestro
v3.0.0 datasets. For each metric, the closer to the training set the values
is, the better. The values of the metrics are expected to be close to that for
the training set. The best performance is highlighted in bold.

Baseline | Configuration | PPL | PCU | ISR | PRS | TUP | PR | APS | IOI | PCH | GS | CPI | SI
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Training Set (J.S. Bach) | - | - | 8.407 | 0.835 | 0.422 | 25.453 | 34.322 | 11.671 | 0.064 | 2.994 | 0.930 | 0.985 | 0.011
Local Attention | $\left[Global,MSN\right]$ | 1.872 | 6.979 | 0.815 | 0.417 | 24.812 | 34.832 | 11.153 | 0.125 | 2.92 | 0.871 | 0.99 | 0.009
Sparse attention | $\left[Global,MSN\right]$ | 1.724 | 7.123 | 0.817 | 0.419 | 24.859 | 34.699 | 11.237 | 0.124 | 2.934 | 0.922 | 0.993 | 0.013
Multi-scale Attention | $\left[L2R,None\right]$ | 1.917 | 5.565 | 0.802 | 0.391 | 22.178 | 41.955 | 6.983 | 0.084 | 2.566 | 0.879 | 1 | 0.005
Multi-scale Attention | $\left[L2R,MSN\right]$ | 1.729 | 6.933 | 0.815 | 0.415 | 24.68 | 34.704 | 11.32 | 0.127 | 2.915 | 0.874 | 0.997 | 0.011
Multi-scale Attention | $\left[Global,None\right]$ | 1.733 | 6.449 | 0.789 | 0.402 | 20.021 | 38.186 | 6.883 | 0.074 | 2.501 | 0.889 | 0.998 | 0.003
Multi-scale Attention | $\left[Global,MSN\right]$ | 1.716 | 7.061 | 0.819 | 0.412 | 25.194 | 34.597 | 10.982 | 0.124 | 2.944 | 0.872 | 0.994 | 0.01
Training Set (Maestro) | - | - | 6.665 | 0.815 | 0.429 | 66.002 | 68.146 | 11.598 | 0.083 | 2.886 | 0.901 | 0.958 | 0.208
Local Attention | $\left[Global,MSN\right]$ | 1.184 | 6.361 | 0.786 | 0.411 | 52.924 | 61.422 | 10.651 | 0.090 | 2.817 | 0.942 | 0.743 | 0.089
Sparse Attention | $\left[Global,MSN\right]$ | 1.086 | 6.626 | 0.834 | 0.436 | 73.965 | 76.011 | 10.767 | 0.082 | 2.863 | 0.892 | 0.998 | 0.171
Multi-scale Attention | $\left[L2R,None\right]$ | 1.108 | 6.516 | 0.825 | 0.44 | 76.0 | 77.7 | 11.4 | 0.087 | 2.949 | 0.882 | 0.997 | 0.093
Multi-scale Attention | $\left[L2R,MSN\right]$ | 1.042 | 6.759 | 0.832 | 0.435 | 74.17 | 76.93 | 11.09 | 0.083 | 2.987 | 0.887 | 0.992 | 0.156
Multi-scale Attention | $\left[Global,None\right]$ | 1.181 | 6.751 | 0.824 | 0.439 | 66.91 | 73.0 | 11.19 | 0.084 | 2.988 | 0.892 | 0.994 | 0.139
Multi-scale Attention | $\left[Global,MSN\right]$ | 1.063 | 6.716 | 0.815 | 0.429 | 57.26 | 67.62 | 11.35 | 0.087 | 2.985 | 0.894 | 0.987 | 0.191

Error accumulation Analysis: We further tested the generation performance
under different conditions, i.e., number of sections and length of the output,
using the PPL metric, which measures the underlying performance of the
language model. These tests were performed on the J.S. Bach and Maestro
datasets. As shown in Fig. 6, the PPL increases rapidly when the section
number exceeds 6 for MusicTransformer and Transformer-GANs, while staying
stable for our hierarchical model. Similarly, the PPL for MusicTransformer and
Transformer-GANs increases significantly when the output length exceeds 1500,
while the drop is much less for our hierarchical model. This indicates that
our model can better avoid the error accumulation problem compared to other
Transformer-based models.

(a) Impact of section number and output length in J.S. Bach dataset

(b) Impact of section number and output length in Maestro dataset

Figure 6: Results of the error accumulation analysis. Note that the PPL of
MusicTransformer and Transformer-GANs increases considerably with the length
and number of sections, even on the exponential scale. On the contrary, our
model was affected very slightly.

###  IV-B Ablation studies

The improvements in the different configurations of the FSL and MSN layers
were analyzed by ablation studies. We performed all the ablation studies on
the J.S. Bach and Maestro datasets, which were designed to test three aspects
of performance: (1) The influence of the baseline model choice; (2) The
optimal alignment strategy of candidate windows in the FSL layer; (3) The
effectiveness of the MSN layer; (4) The best configuration of our model. The
results are demonstrated in Tab. IV.

The results of ablation studies have suggested the effectiveness of the FSL
and MSN layers, while the sparse attention configured with
$\left[Global,MSN\right]$ parameters achieved the best performance on most
metrics. Almost all models that use the global strategy (compared to the L2R
strategy) to align the candidate windows lead to better performance of music
generation. The use of the MSN layer resulted in better pitch diversity in the
generated music (TUP, PR, and APS are great improvements), validating its
utility in musical style control. These results show the importance of
structural segmentation (FSL layer) and music style control (MSN layer) for
SMG tasks.

###  IV-C Visual Evaluation

In order to test the practical performance of our model, we specifically
analyzed the structure of the music in the generated samples. The sample
evaluation was demonstrated by the visual MIDI diagram. Fig. 7 shows an
example of a composition generated by our model, in which a steady rhythm is
maintained compared to the original music, and some strong melodies are
recreated (in the red boxes).

Figure 7: Illustration of visual comparison of the generated (upper in shallow
blue) and original music (lower in orange). Blue boxes indicate activation
notes; Red boxes are examples of reused melody. Figure 8: Visual comparison of
musical samples generated by four different models, i.e., Transformer-XL (top
left), MusicTransformer (top right), Transformer-GANs (bottom left), and our
hierarchical model (bottom right). The blue box shows the same activation
notes.

Further, samples generated by our model and other three comparison models were
also presented under the same experimental condition, as shown in Fig. 8. In
the sample generated by Transformer-XL, all notes are the minimum unit
(semiquaver note), making the composition sound monotonous and boring. The
sample generated by MusicTransformer is rhythmically inconsistent because
there is an in-equivalent note density in the sequence (see purple box),
making the generated music unrealistic. All notes in the samples generated by
Transformer-GANS are large in timescale (longer duration), in higher octaves
(higher pitches), and rigid in musical style. Our model generates samples with
lively tonal trends and stable rhythms. Furthermore, the samples reflect many
compositional techniques, such as the use of grace notes (bass tones) and
melodic repetition, which conform to the musical sense and are similar to
realistic music.

##  V Discussion and Conclusion

We propose a hierarchical Transformer model to generate music with structural
sections and reused chord progressions, which the advantage is learning
contextual representation at multi-scale. Results demonstrated that the
present model achieved SOTA performance on two opening datasets. Ablation
studies showed the effectiveness of the proposed FSL and MSN layers.

Another advantage of our model is that the decoding time of our model was
reduced to approximately 0.8 times compared to other Transformer models, after
using the hierarchical architecture. For example, the computational complexity
of the Sparse Transformer (the most common model) with a maximum length $L$ is
$O\left(L\log L\right)$. In our multi-scale attention, since chord-level and
section-level heads avoid connecting all sequence nodes. Benefiting from
parallel decoding, our hierarchical architecture reduces the computational
complexity by approximating
$O\left(\left(L/m\right)\log\left(L/m\right)\right)$. Tab. V shows the
computational complexity of contemporary canonical models.

TABLE V: L-related computation complexity. $B$ is the number of memory blocks
in MusicTransformer. $m$ is the number of section in the sample.

Model | Computational Complexity | Decoding Step
---|---|---
LSTM | $O\left(L\right)$ | $L$
Transformer | $O\left(L^{2}\right)$ | $L$
Sparse Transformer | $O\left(L\log L\right)$ | $L$
MusicTransformer | $O\left(\left(L/B\right)^{2}\right)$ | $L$
Transformer-GANs | $O\left(2L\log L\right)$ | $L$
Hierarchical Transformer | $O\left(\left(L/m\right)\log\left(L/m\right)\right)$ | $L/m$

However, we are also aware of some shortcomings that we have not addressed. In
some cases, the segmentation operation of the FSL layer may lose its
effectiveness for dealing with a composition with rhythmic variations, such as
in Beethoven’s Piano Sonata No.30, This problem could be improved by resizing
the windows according to the note density of the candidate scope.

In the future, we plan to explore other applications of the hierarchical
models, for instance, generating polyphony music that involves multiple
tracks. We can also try other strategies, such as adversarial learning and
curriculum learning, to improve the performance of music generation.

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Data Preprocessing: To pre-train the FSL layer in a supervised manner, the
location and length of structural section are needed, which can be
independently extracted from the GuitarPro (GTP) software. We proposed a
transfer method for automatically marking the section regions from GTP data.
The approach is as follows: (1) The GTP data was first converted into MIDI
data and the corresponding ASCII text was exported; (2) The ASCII symbols were
discretized onto a note grid and then serialized by iterating through all the
symbols within a time step; (3) The section labels can be marked by adding a
location coordinate since there is a direct correspondence between the
sequence location and the grid location. An example of a music section in GTP
data with the corresponding ASCII data and MIDI data is illustrated in Fig. 9.

(a) The GTP data

(b) The ASCII data

(c) The MIDI data

Figure 9:  An example of a music section in GTP data with the corresponding
ASCII data and MIDI data. The section label was extracted in the GTP data and
noted on the MIDI data through the ASCII code transfer method. Figure 10:
Illustration of the training details. Indigo boxes represent the starting
notes (16 semiquaver notes), which is used to activate the initial state in
each section; Green boxes and Gray boxes represent the input and output of the
decoder, respectively; Orange boxes represent section labels ($label_{i}$) and
style representation ($z$) of the generated section. Yellow boxes represent
start token and end token, which are used to change the state of the model.

In GTP data, the symbols “$\|:$” and “$:\|$” represent the beginning and the
end of the section, respectively, and are equivalent to the symbol “$\|$” in
ASCII data. These section labels can be synchronously marked in the MIDI data.
We have compiled 3188 sections from various genres of music, including blues,
flamenco, rock, classical, and other music. The diversity of music styles in
the music samples enables the FSL layer to generalize better and learn to
fragment musics with various styles. In the FSL layer, the number $K$ of
candidate windows was set to 8, and the window size was selected from
$[8,16,32,64,128,256,512,1024]$ to accommodate music sections with various
musical styles.

Training details: Applying LMs to downstream tasks by pre-training and fine-
tuning is a common strategy. Since the Transformer-based model has the problem
of error accumulation when dealing with musical sequences with thousands of
notes, it is also slow. We first pre-train fine-decoders to generate sections
at a fine scale and then fine-tune the combined sequence at a coarse scale.
The coarse-encoder and fine-decoders were trained with different default
hyper-parameters. While maintaining the overall design of our hierarchical
model, we set the maximum length of the coarse-decoder to be 4096, as in other
Transformer-based models in comparison. The maximum length of fine-decoders is
chosen from [256, 512, 768, 1024], using a carry-up strategy to fit the
section length (the smallest one that exceeds the section length).

Then, we fix the weights of fine-decoders and train the MSN layer and the
coarse decoder. The the MSN layer is optimized together with the coarse-
decoder. The MI-based loss $L_{style}$ is used to optimize the coarse-decoder,
which is calculated as the MI of the input and the renormalized music
sections. The training process of coarse-decoder is shown in Fig. 10.

In our experiments, both fine-decoders and coarse-decoder have 6 hidden
layers. We implemented the model in TensorFlow framework and the hyper-
parameters for training were as follows:

  * •

(a) 1e-03 initial learning rate minimized with 1e-04 weight decay;

  * •

(b) 100 epoch and 8 batch size;

  * •

(c) 0.2 dropout;

  * •

(d) dynamic position embedding;

  * •

(e) multiple GPU training and early stopping strategy.


# Higgs $p_{t}$ distribution in Higgs $+$ jet production at hadron colliders
within the noncommutative effective standard model

Mohamed El Arebi Gadja∗ and Lamine Khodja† Laboratoire de Rayonnement et
Plasmas et Physique de Surfaces,
Département de Physique, Faculté des Mathématiques et des Sciences de la
Matière,
Université Kasdi Merbah Ouargla, Ouargla 30000, Algeria
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>Yazid Delenda‡00footnotetext: ‡Corresponding
author. Laboratoire de Physique des Rayonnements et de leurs Interactions avec
la Matière,
Département de Physique, Faculté des Sciences de la Matière,
Université de Batna-1, Batna 05000, Algeria
<EMAIL_ADDRESS>
###### Abstract

We use the noncommutative Higgs effective standard model to make a
phenomenological prediction for the transverse momentum distribution of the
Higgs boson produced in association with a jet at hadron colliders. We
calculate at leading order in the noncommutative parameter $\Theta$ as well as
leading order in the strong coupling $\alpha_{s}$, the one-loop $p_{t}$
distribution of the Higgs boson. As in the standard model, the fixed-order
distribution suffers from large logarithms at small $p_{t}$ which require an
all-orders resummation. We find that the large-$p_{t}$ region of the
distribution is strongly affected by the non-commutativity, while
small-$p_{t}$ region is not. Following this observation, we propose a simple
matching method that allows us to compute a result that is also valid at small
$p_{t}$ obtained with standard-model parton showers such as Pythia 8. We also
compare our results with the next-to-leading order (NLO) and next-to-next-to-
leading order (NNLO) distributions in the standard model, in order to assess
the importance of higher-order effects in the search for non-commutativity at
colliders.

###### keywords:

Higgs effective standard model; noncommutative geometry; QCD.

PACS Nos.: 11.10.Nx, 14.80.Bn

##  1 Introduction

Following the discovery of the Higgs boson in 2012 [1, 2], the main focus of
Higgs physics has entered the era of precision phenomenology by measuring its
properties, specifically its kinematical behavior, couplings to Standard Model
particles, and new physics searches [3]. The Higgs production cross-section
and decay rate are used to study the coupling between the Higgs boson and
other particles. In this regard, the production of a high-$p_{t}$ jet
alongside the Higgs boson is a process of great interest both in making
precision measurements as well as the search for new-physics phenomenon at
hadron colliders.

The correlation between the Higgs particle and the jet will undoubtedly
provide important information and further reveal the electroweak coupling of
the Higgs boson [4, 5, 6, 7]. In this work, we study the process of production
of a Higgs plus jet at the LHC within the framework of the noncommutative
standard model. We specifically consider an important observable used
intensively to extract properties of the Higgs boson, namely the transverse
momentum ($p_{t}$) distribution of the produced Higgs.

The Higgs field is somehow related to the noncommutative nature of space-time,
an idea which emerged in the late 1980s and early 1990s [8, 9], and was
included in the description of the Standard Model in Refs.
chamseddine1997spectral and connes1996gravity. Several phenomenological works
have been performed in the literature that aim at studying the applications of
the non-commutativity of space-time in particle physics, such as Refs.
chaichian2003non,buric2007one,ohl2004testing,castro2015decays,haghighat2006photon,melic2005standard.

More precisely, noncommutative geometry [18] is a branch of mathematics
developed by Alain Connes that aims to generalize geometric ideas to spaces
where coordinates do not commute. Driven by quantum gravity, space-time is
assumed to be a noncommutative manifold at very high energy scales. Even if
the exact nature of this noncommutative manifold remains unknown, it seems
reasonable to assume that at an intermediate scale, say several orders of
magnitude lower than the Planck scale, the corresponding coordinate algebra is
only a slightly noncommutative matrix algebra [19].

In the canonical version of the noncommutative space-time one has

$[\hat{x}^{\mu},\hat{x}^{\nu}]=i\,\Theta^{\mu\nu}\,,$ (1)

where $\Theta^{\mu\nu}$ is a real constant anti-symmetric tensor, and
$\hat{x}$ denotes the noncommutative coordinates. The action in noncommutative
field theories is obtained by using the Moyal-star product and the Seiberg-
Witten maps [20], where the star product of two commutative fields $\psi$ and
$\phi$ is defined by [21, 22]

$\psi(x)\star\phi(x)=\psi(x)\exp\left[\frac{i}{2}\,\overleftarrow{\partial}_{\mu}\,\Theta^{\mu\nu}\,\overrightarrow{\partial}_{\nu}\right]\phi(x)\,,$
(2)

and the ordinary spinor ($\psi$) and gauge ($V_{\mu}$) fields transform via
the Seiberg-Witten maps as [20, 17, 23, 24]

$\displaystyle\hat{\psi}(x,\Theta)$
$\displaystyle=\psi(x)+\Theta\,\psi^{(1)}[V]+\mathcal{O}(\Theta^{2})\,,$ (3a)
$\displaystyle\hat{V}_{\mu}(x,\Theta)$
$\displaystyle=V_{\mu}+\Theta\,V_{\mu}^{(1)}[V]+\mathcal{O}(\Theta^{2})\,,$
(3b)

where to leading order in the $\Theta$ parameter[25, 26, 23]

$\displaystyle\Theta\,\psi^{(1)}[V]$
$\displaystyle=-\frac{1}{2}\,\Theta^{\mu\nu}\,V_{\mu}\,\partial_{\nu}\psi+\frac{i}{4}\,\Theta^{\mu\nu}\,V_{\mu}\,V_{\nu}\,\psi\,,$
(4a) $\displaystyle\Theta\,V^{(1)}_{\mu}[V]$
$\displaystyle=-\frac{1}{4}\Theta^{\alpha\beta}\\{V_{\alpha},\partial_{\beta}V_{\mu}+F_{\beta\mu}\\}\,,$
(4b)

where $F_{\beta\mu}$ is the QCD field strength tensor and
$\\{\cdots,\cdots\\}$ stands for the anti-commutator.

In this paper, we use the noncommutative Higgs effective field theory (NC-
HEFT) in order to make a phenomenological study of the Higgs transverse
momentum distribution. We calculate within this framework the production
cross-section of the Higgs boson, and estimate at leading order (LO) in the
strong coupling the transverse momentum distribution. The distribution at
fixed order suffers from large logarithms in the ratio $p_{t}/Q$, where $Q$ is
the hard scale of the process. To obtain a reliable result at low $p_{t}$,
where interesting physics are usually found, a resummation to all orders is
necessary. As it turns out, the distribution at small values of $p_{t}$ is not
affected by the non-commutativity of space-time, while the tail of the
distribution (high-$p_{t}$ region) is strongly affected by it. We therefore
propose a matching method that allows us to combine the two regions of $p_{t}$
in order to obtain a distribution that is valid in the entire spectrum of
$p_{t}$, and is suitable for phenomenology. Furthermore, we compare the
matched result with the next-to-leading order (NLO) distribution obtained with
MCFM [27] for the process under consideration[28], and with the next-to-next-
to-leading order (NNLO) result obtained from Ref. Boughezal:2015dra, and
discuss the possibility of distinguishing noncommutative effects from NLO and
NNLO corrections.

This paper is organised as follows. In Sec. 2, we present the HEFT in the
noncommutative geometry formalism, and calculate the cross-section of
production of a Higgs boson and a single jet at LO in $\alpha_{s}$ in $pp$
collisions at the LHC. We then, in Sec. 3, extract the Higgs $p_{t}$
distribution at LO. We use MadGraph5_aMC@NLO [30, 31] and MadAnalysis5 [32] in
order to perform the convolution with parton distribution functions (pdfs) and
include experimental cuts. Furthermore, we use Pythia8 [33, 34] parton shower
to produce a SM distribution, with which we match our NC distribution to
obtain a result valid for all values of $p_{t}$, and compare this distribution
with NLO and NNLO results. Finally, in the last section, we draw our
conclusions.

##  2 Higgs + Jet Production in the NC-HESM

As in the Higgs effective standard model (HESM), the Higgs + jet production
process in $pp$ collisions within the noncommutative Higss effective standard
model (NC-HESM) has the Feynman diagrams shown in Fig. 1.

Figure 1: Feynman diagrams contributing to the process $pp\to$ Higgs + jet.
The dot represents the effective gluon-Higgs coupling in the infinite top-
quark mass limit [4].

The interaction point of two gluons with the Higgs boson in ordinary Higgs
effective field theory has been calculated in Ref. dedes2017feynman. The
noncommutative geometry at first order in $\Theta$ results solely in a
correction term to the triple gluon vertex as well as the triple-gluon + Higgs
boson vertex, as shown in Fig 2.

$\displaystyle\begin{aligned}
&g_{s}\,f^{abc}\left[g_{\mu\nu}(k_{1}-k_{2})_{\rho}+g_{\nu\rho}(k_{2}-k_{3})_{\mu}-g_{\rho\mu}(k_{3}-k_{1})_{\nu}\right]\\\
&\qquad+\frac{1}{2}\,g_{s}\,d^{abc}\,\Theta_{3\,\mu\nu\rho}\end{aligned}$

$\displaystyle\begin{aligned}
&G_{H}\,g_{s}\,f^{abc}\left[g_{\mu\nu}(k_{1}-k_{2})_{\rho}+g_{\nu\rho}(k_{2}-k_{3})_{\mu}-g_{\rho\mu}(k_{3}-k_{1})_{\nu}\right]\\\
&\qquad+G_{h}\,g_{s}\,f^{abc}(k_{1}+k_{2}+k_{3})^{\alpha}\epsilon_{\mu\nu\rho\sigma}\\\
&\qquad-\frac{1}{2}G_{H}\,g_{s}d^{abc}\,\widetilde{\Theta}_{1\mu\nu\rho}-G_{h}\,g_{s}\,d^{abc}\,\widetilde{\Theta}_{2\,\mu\nu\rho}\end{aligned}$

Figure 2: Deformed vertices by noncommutative geometry.

In this figure, $g_{s}$ is the strong coupling, $f_{abc}$ and $d_{abc}$ are,
respectively, the SU(3) structure constants and $d$ symbols, $\epsilon$ is the
totally anti-symmetric Levi-Civita tensor, and $k_{i}$ represent the momenta
of the gluons. The Wilson coefficients are given by

$\displaystyle G_{H}$
$\displaystyle=-\frac{g_{s}^{2}}{12\pi^{2}v}\left(1+\frac{7\,m_{H}^{2}}{120\,m_{t}^{2}}+\frac{m_{H}^{4}}{168\,m_{t}^{4}}+\frac{13\,m_{H}^{6}}{16800\,m_{t}^{6}}\right),$
(5a) $\displaystyle G_{h}$
$\displaystyle=-\frac{g_{s}^{2}}{8\pi^{2}v}\left(1+\frac{m_{H}^{2}}{12\,m_{t}^{2}}+\frac{m_{H}^{4}}{90\,m_{t}^{4}}+\frac{m_{H}^{6}}{560\,m_{t}^{6}}\right),$
(5b)

with $m_{t}$ and $m_{H}$ being the masses of top quark and Higgs boson,
respectively, and $v$ is Higgs vacuum expectation value. In this figure,
$\Theta_{3}$ has been calculated in Ref. melic2005standard and is given by

$\displaystyle\Theta_{3}^{\mu\nu\rho}=\,$ $\displaystyle-(k_{1}\Theta
k_{2})\left[(k_{1}-k_{2})^{\rho}g^{\mu\nu}+(k_{2}-k_{3})^{\mu}g^{\nu\rho}+(k_{3}-k_{1})^{\nu}g^{\rho\mu}\right]$
$\displaystyle-\Theta^{\mu\nu}\left[k_{1}^{\rho}(k_{2}\cdot
k_{3})-k_{2}^{\rho}(k_{1}\cdot
k_{3})\right]-\Theta^{\nu\rho}\left[k_{2}^{\mu}(k_{3}\cdot
k_{1})-k_{3}^{\mu}(k_{2}\cdot k_{1})\right]$
$\displaystyle-\Theta^{\rho\mu}\left[k_{3}^{\nu}(k_{1}\cdot
k_{2})-k_{1}^{\nu}(k_{3}\cdot k_{2})\right]+(\Theta
k_{2})^{\mu}\left[g^{\nu\rho}k_{3}^{2}-k_{3}^{\nu}k_{3}^{\rho}\right]$
$\displaystyle+(\Theta
k_{3})^{\mu}\left[g^{\nu\rho}k_{2}^{2}-k_{3}^{\nu}k_{2}^{\rho}\right]+(\Theta
k_{3})^{\nu}\left[g^{\mu\rho}k_{1}^{2}-k_{1}^{\mu}k_{1}^{\rho}\right]$
$\displaystyle+(\Theta
k_{1})^{\nu}\left[g^{\mu\rho}k_{3}^{2}-k_{3}^{\mu}k_{3}^{\rho}\right]+(\Theta
k_{1})^{\rho}\left[g^{\mu\nu}k_{2}^{2}-k_{2}^{\mu}k_{2}^{\nu}\right]$
$\displaystyle+(\Theta
k_{2})^{\rho}\left[g^{\mu\nu}k_{1}^{2}-k_{1}^{\mu}k_{1}^{\nu}\right],$ (6)

where $(k_{i}\Theta k_{j})=k_{i\mu}\Theta^{\mu\nu}k_{j\nu}$, $(\Theta
k)^{\mu}=\Theta^{\mu\nu}k_{\nu}$, and $(\Theta k)^{2}=(\Theta k)^{\nu}(\Theta
k)_{\nu}$. In our kinematics, all these products vanish except for $(\Theta
k_{3})^{2}=\Theta^{2}k_{t}^{2}$. Furthermore, $\widetilde{\Theta}_{1}$ and
$\widetilde{\Theta}_{2}$ are computed in this work, in our choice of
noncommutative geometry parameter $\Theta$, to be

$\displaystyle\widetilde{\Theta}_{1}^{\mu\nu\rho}$ $\displaystyle=(\Theta
k_{3})^{\nu}k_{1}^{\rho}k_{2}^{\mu}+(\Theta
k_{3})^{\mu}k_{1}^{\nu}k_{2}^{\rho}+(\Theta
k_{3})^{\nu}k_{1}^{\rho}k_{3}^{\mu}$ $\displaystyle+(\Theta
k_{3})^{\mu}k_{2}^{\rho}k_{3}^{\nu}+\Theta^{\nu\rho}k_{3}^{\mu}\,k_{1}\cdot
k_{2}+\Theta^{\mu\rho}k_{3}^{\nu}\,k_{1}\cdot k_{2}$ $\displaystyle-(\Theta
k_{3})^{\nu}g^{\mu\rho}\left(k_{1}\cdot k_{2}+k_{1}\cdot k_{3}\right)-(\Theta
k_{3})^{\mu}g^{\nu\rho}\left(k_{1}\cdot k_{2}+k_{2}\cdot k_{3}\right)$
$\displaystyle-\left(\Theta^{\mu\rho}k_{1}^{\nu}+\Theta^{\mu\nu}k_{1}^{\rho}\right)k_{2}\cdot
k_{3}-\left(\Theta^{\nu\rho}k_{2}^{\mu}+\Theta^{\mu\nu}k_{2}^{\rho}\right)k_{1}\cdot
k_{3}\,,$ (7a) $\displaystyle\widetilde{\Theta}_{2}^{\mu\nu\rho}$
$\displaystyle=\Theta^{\nu\rho}\epsilon^{\mu\alpha\beta\gamma}k_{1\gamma}k_{2\beta}k_{3\alpha}+\Theta^{\mu\rho}\epsilon^{\nu\alpha\beta\gamma}k_{1\beta}k_{2\gamma}k_{3\alpha}$
$\displaystyle+\Theta^{\mu\nu}\epsilon^{\rho\alpha\beta\gamma}k_{1\beta}k_{2\alpha}k_{3\gamma}+2(\Theta
k_{3})^{\nu}\epsilon^{\mu\rho\alpha\beta}k_{1\alpha}k_{2\beta}$
$\displaystyle+2(\Theta
k_{3})^{\mu}\epsilon^{\nu\rho\alpha\beta}k_{1\alpha}k_{2\beta}\,.$ (7b)

In this paper we work with space-space non-commutativity, i.e.,
$\Theta^{i0}=0$, motivated by the unitary problem [36], and without loss of
generality we choose $\Theta^{21}=\Theta$ and $\Theta^{13}=\Theta^{23}=0$.
Then, the color and polarization-averaged/summed squared amplitude of the last
four Feynman diagrams in Fig. 1 is found to be

$\displaystyle\overline{\left|\mathcal{M}^{\mathrm{NC}}_{gg\to
hg}\right|^{2}}=\,$
$\displaystyle\overline{\left|\mathcal{M}^{\mathrm{SM}}_{gg\to
hg}\right|^{2}}+\Theta^{2}\,g_{s}^{2}\,\frac{\mathrm{C_{A}^{2}}-4}{128\,\mathrm{C_{A}^{2}C_{F}^{2}}}\,\Big{[}-\left(162G^{2}_{h}+47G^{2}_{H}\right)stu$
$\displaystyle+4G^{2}_{h}\,k_{t}^{2}\left(-3sm_{H}^{2}+48s^{2}+47s(t+u)+24t^{2}-2tu+24u^{2}\right)$
$\displaystyle+G^{2}_{H}\,k_{t}^{2}\left(5sm_{H}^{2}+21s^{2}+53st+57su+27t^{2}-3tu+35u^{2}\right)\Big{]}\,,$
(8)

where $\mathrm{C_{F}}=4/3$ and $\mathrm{C_{A}}=3$ are the Casimir scalars in
the fundamental and adjoint representations of SU(3), $s$, $t$, and $u$ are
the usual Mandelstam variables, and $k_{t}$ is transverse momentum of the
outgoing gluon. Here $\mathcal{M}^{\mathrm{SM}}_{gg\to hg}$ is the amplitude
of the process $gg\to hg$ within the HESM. Additionally, the
color/spin/polarization-averaged/summed squared amplitude of the first and
second diagrams in our choice of $\Theta$ is not affected by non-
commutativity, that is111Here the $q$ also implies processes with antiquarks.

$\displaystyle\overline{\left|\mathcal{M}^{\mathrm{NC}}_{gq\to
hq}\right|^{2}}$
$\displaystyle=\overline{\left|\mathcal{M}^{\mathrm{SM}}_{gq\to
hq}\right|^{2}}\,,$ (9a)
$\displaystyle\overline{\left|\mathcal{M}^{\mathrm{NC}}_{q\bar{q}\to
hg}\right|^{2}}$
$\displaystyle=\overline{\left|\mathcal{M}^{\mathrm{SM}}_{q\bar{q}\to
hg}\right|^{2}}\,.$ (9b)

Then, the cross-section for the process $pp\to h+j$ reads

$\sigma^{\mathrm{NC}}=\sum_{\delta}\int
dx_{a}dx_{b}f_{a}(x_{a},\mu_{\mathrm{F}}^{2})f_{b}(x_{b},\mu_{\mathrm{F}}^{2})\,\Xi\,\frac{\mathrm{d}p_{t}^{2}\mathrm{d}y}{16\pi^{2}s}\,\delta(s+t+u-m_{H}^{2})\overline{\left|\mathcal{M}^{\mathrm{NC}}_{\delta}\right|^{2}}\,,$
(10)

where the sum extends over all the partonic channels $\delta$, $x_{a}$ and
$x_{b}$ are the momentum fractions carried by the incoming partons, and
$f_{a}$ are the pdfs evaluated at the factorization scale $\mu_{\mathrm{F}}$,
and $\Xi$ denotes the experimental cuts. The $p_{t}$ and $y$ represent the
transverse momentum and rapidity of the Higgs boson.

##  3 Inclusive Cross-section and Transverse Momentum Distribution

Having obtained the squared amplitude in NC geometry, we now proceed to
calculate the total cross-section as well as the Higgs $p_{t}$ distribution.
To do so, a straightforward method is to use Monte Carlo event generators. For
this purpose, we use MadGraph5_aMC@NLO [30, 31] in order to generate a sample
of parton-level events for the process $pp\to Hj$ at LO within the SM-HEFT. To
incorporate NC geometry all we have to do is modify the weight of each event
by replacing the matrix element squared of the SM by the corresponding NC one.
Said differently, we may write the total cross-section in NC geometry as

$\sigma_{\mathrm{NC}}=\frac{\sigma_{\mathrm{SM}}}{N}\sum_{\mathrm{events}}\left(\overline{\left|\mathcal{M}^{\mathrm{NC}}_{\delta}\right|^{2}}/\overline{\left|\mathcal{M}^{\mathrm{SM}}_{\delta}\right|^{2}}\right),$
(11)

where the sum extends over all $N$ events in the generated sample, and
$\sigma_{\mathrm{SM}}$ is the SM-HEFT cross-section obtained with
MadGraph5_aMC@NLO, and $\delta$ is the corresponding channel. For each event
record we access the particle kinematics, using MadAnalysis5 [32], and compute
the corresponding weight as shown in Eq. (11).

To present our quantitative results, we consider $pp$ scattering at
$\sqrt{s}=14\,\mathrm{TeV}$ and fix the pole mass of the Higgs boson $m_{H}$
to the value $m_{H}=125\,\mathrm{GeV}$. We use NNPDF3.1 NNLO pdfs [37] through
LHAPDF [38], and impose the cut $\eta<5$ for the jet rapidity. To avoid the
soft/collinear singularities in the inclusive cross-section, we cut the jet
transverse momentum at 20 GeV. Further experimental cuts, for instance on the
decay products of the Higgs boson, can also be applied within
MadGraph5_aMC@NLO when comparing to experimental data.222The Higgs is set not
to decay in this work. For our analysis, we set the renormalization and
factorization scales at $\mu_{\mathrm{F}}=\mu_{\mathrm{R}}=m_{H}$. The SM
cross-section that we obtain is

$\displaystyle\sigma^{\mathrm{SM}}=(13.110\pm 0.001)\,\mathrm{pb}\,,$ (12)

and the results in the NC-HESM are shown in Table 3 for different values of
the $\Theta$ noncommutative parameter.

Cross-section for the production of the Higgs boson in association with a jet
in NC geometry for different choices of the $\Theta$ parameter.  NC parameter
$\Theta$ NC cross-section $\sigma$ ($\times 10^{-4}$ GeV-2) $(\mathrm{NC})$
(pb) $0.1$ $13.98$ $0.3$ $20.94$ $0.5$ $34.85$ $0.7$ $55.72$

We also calculate the Higgs $p_{t}$ distribution in a similar way. The results
are shown in Fig. 3 for different values of the non-commutativity parameter
$\Theta$. As we can see from the curves, the fixed-order (LO and NLO)
distribution is divergent for small values of $p_{t}$ both in the SM and NC-
SM. This divergence has a leading double logarithmic structure that goes like
$\alpha_{s}\ln^{2}(m_{H}^{2}/p_{t}^{2})$ as well as sub-leading single
logarithms $\alpha_{s}\ln(m_{H}^{2}/p_{t}^{2})$. These logarithms are
persistent at higher orders and may be resummed at next-to-leading logarithmic
(NLL) accuracy into an exponential form

$\sigma(p_{t})\sim\exp\left(Lg_{1}(\alpha_{s}L)+g_{2}\alpha_{s}(L)\right),$
(13)

with $L=\ln(m_{H}^{2}/p_{t}^{2})$ being the large logarithm, and $g_{1}$ and
$g_{2}$ being resummation functions of $\alpha_{s}L$.

Figure 3: Higgs $p_{t}$ distribution.

We notice that the $\mathcal{O}(\Theta^{2})$ noncommutative correction to the
squared matrix element does not manifest a soft/collinear divergence when
$k_{t}\to 0$. The source of the above-mentioned large logarithms is purely the
SM contribution to the squared amplitude. For this reason, we can perform the
resummation for the SM-like part of the distribution, and include the
$\mathcal{O}(\Theta^{2})$ corrections due to non-commutativity in the form of
a matched distribution given by

$\frac{\mathrm{d}\sigma_{\mathrm{matched}}^{\mathrm{NC}}}{\mathrm{d}p_{t}}=\frac{\mathrm{d}\sigma^{\mathrm{SM}}_{\mathrm{resummed}}}{\mathrm{d}p_{t}}-\frac{\mathrm{d}\sigma_{\mathrm{LO}}^{\mathrm{SM}}}{\mathrm{d}p_{t}}+\frac{\mathrm{d}\sigma_{\mathrm{LO}}^{\mathrm{NC}}}{\mathrm{d}p_{t}}\,.$
(14)

This way, the expansion of the matched NC distribution at first order in the
strong coupling exactly reproduces the NC LO distribution (with the correct
$\mathcal{O}(\Theta^{2})$ terms). This matched distribution also remains valid
for all values of $p_{t}$ owing to the resummation of the large logarithms in
the low-$p_{t}$ region in the SM part.

We do not explicitly perform the resummation in this paper. However, we can
use Monte Carlo Parton Shower results to represent a resummed distribution and
perform the matching between the fixed-order result and the parton shower
distribution. The results for this are shown in Fig. 3. Here we observe the
reasonable behaviour of the Higgs $p_{t}$ distribution at low $p_{t}$ form the
SM parton shower Pythia8, and we see that the matched NC distribution
coincides with this distribution in this region. At large values of $p_{t}$,
the matched distribution approaches the LO distribution and has a noticeable
deviation from the SM prediction. This indicates that the signature of
noncommutative geometry may be hinted from the tail of this distribution.

It is therefore important to assess the impact of noncommutative geometry on
the large-$p_{t}$ region of the distribution compared to other QCD effects
such non-perturbative dynamics and higher-order contributions, namely NLO and
NNLO corrections. Luckily, this region is not strongly affected by the former
(non-perturbative effects), and is usually accurately estimated using just
fixed-order calculations (up to NNLO). Thus, in this work, we use the fixed-
order Monte Carlo program MCFM [27] to compute the Higgs $p_{t}$ distribution
at NLO within the SM-HEFT. Scale uncertainties are also estimated by varying
the renormalization and factorization scales by factors of $2$ and $1/2$
around their central values. The results are shown in Fig. 3. We notice that
NLO corrections at the tail of the distribution have comparable size to
noncommutative corrections for values of $\Theta\sim 0.3\times
10^{-4}\,\mathrm{GeV}^{-2}$. For larger values, $\Theta\gtrsim 0.5\times
10^{-4}\,\mathrm{GeV}^{-2}$, the noncommutative corrections are noticeably
higher than the NLO corrections with scale uncertainties.

Given the large size of NLO corrections, it is important to also consider the
comparison of our results to the NNLO distribution. To achieve this, we use
the result presented in Ref. Boughezal:2015dra for the NNLO Higgs $p_{t}$
distribution in which the $m_{t}\to\infty$ approximation is employed. The
results are displayed in Fig. 4 for $\sqrt{s}=8\,\mathrm{TeV}$, and where we
use the NNPDF2.3 [39] pdf set and impose the cut on the jets $p_{t}>30$ GeV.
We observe that the NNLO scale-uncertainty band almost entirely overlaps with
the NLO one at the tail of the distribution, indicating that NNLO corrections
are very small in this large-$p_{t}$ region. The scale uncertainties at NNLO
are a lot smaller than those at NLO. This means that the values of
$\Theta\gtrsim 0.5\times 10^{-4}\,\mathrm{GeV}^{-2}$ still lead to effects on
the distribution that are clearly distinguishable from fixed-order
corrections. This indicates that such large values of $\Theta$ can easily be
confirmed or ruled out when compared to real experimental data of the same
distribution. Additionally, an upper bound on the $\Theta$ parameter may be
obtained from comparison to experimental data on this distribution — an
investigation that we leave to our forthcoming work.

Figure 4: Comparison between Higgs $p_{t}$ distribution in the NC-SM and
fixed-order results up to NNLO. The fixed-order results are extracted from
Ref. [29], and assume the infinite-top-mass limit.

##  4 Conclusions

In this work, we used the NC-HESM in order to perform detailed calculations
that are relevant for Higgs phenomenology at the LHC. Specifically we
calculated the noncommutative corrections to the SM squared amplitudes for the
process of production of a Higgs boson in association with a jet at hadron
colliders. We found that only the gluon-gluon fusion channel is affected by NC
geometry at leading order in $\Theta$, while the other channels (which are
actually suppressed in the SM) are unaffected.

Following this, we calculated the total cross-section for the production of
the Higgs boson and the jet in the NC-HESM using MadGraph5_aMC@NLO by
modifying the weights of the events to include noncommutative corrections to
the squared amplitudes. Furthermore, we calculated the differential
distribution in the Higgs transverse momentum at leading order in the strong
coupling. We also proposed a simple matching formula that allowed us to extend
the range of validity of the distribution to low values of $p_{t}$, by
showering (or resumming) the distribution in the SM and including the
noncommutative corrections at leading order. Finally, we compared the size of
noncommutative corrections for different values of $\Theta$ to those obtained
in QCD at NLO using the Monte Carlo program MCFM and at NNLO using the results
of Ref. Boughezal:2015dra, and discussed how it is possible to confirm or rule
out the impact of noncommutative geometry when comparing to experimental data
of this distribution.

## Acknowledgments

This work is supported by PRFU research project B00L02UN050120230003. The
authors wish to thank the Algerian Ministry of Higher Education and Scientific
Research and DGRSDT for financial support.

Some of the numerical calculations presented here have been performed in the
HPC cluster at the University of Batna 2.

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