# LLMCarbon: Modeling the End-To-End Carbon Footprint of Large Language Models

Ahmad Faiz, Sotaro Kaneda, Rita Osi∗, Ruhan Wang, Parteek Sharma, Fan Chen,
Lei Jiang
Indiana University ∗Jackson State University
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

The carbon footprint of large language models (LLMs) is substantial, stemming
from their training, inference, experimentation, and storage processes,
encompassing both operational and embodied carbon emissions. Precisely
assessing the carbon impact of emerging LLMs before their actual training,
which involves substantial GPU usage, is crucial. Although many previous
studies have reported the carbon footprint of LLM training, only one prior
tool, mlco2, can predict the carbon footprint of new neural networks before
their physical training. However, mlco2 exhibits several limitations. Firstly,
it cannot extend its carbon footprint estimation to include dense or mixture-
of-experts (MoE) LLMs. Secondly, mlco2 disregards essential architectural
parameters of networks, such as parameter counts, leading to inflated
projections. Thirdly, mlco2 focuses solely on GPUs, excluding TPUs and
assuming uniform peak computing throughput across GPUs, resulting in imprecise
carbon footprint estimations. Lastly, mlco2 cannot model the embodied carbon
footprint of an LLM. To address these gaps, we present an end-to-end carbon
footprint projection model, LLMCarbon, designed for both dense and MoE LLMs.
Compared to mlco2, LLMCarbon greatly improves the estimation accuracy of the
carbon footprint of various LLMs.

##  1 Introduction

Large language models (LLMs) have become the dominant solution for solving a
wide range of natural language processing applications Brown:NIPS2020. The
triumph of prior LLMs has incited a trend towards even larger model
architectures Sanderson:Nature2023,Anil:TECH2023, coupled with expansive
datasets, thus embedding LLM inferences into numerous facets of daily life
Campello:IQ2023. However, it is crucial to acknowledge that the adoption of
LLMs has aggravated the already significant environmental implications of
machine learning (ML) Thompson:Spectrum2021. For example, creating a
transformer with 213M parameters via neural architecture search has been
estimated to produce a carbon dioxide equivalent (CO2eq) comparable to that
emitted by 5 cars over their lifespan Strubell:ACL2019.

In light of the potential ecological ramifications associated with LLMs, it
becomes imperative for both cloud service providers and ordinary users to
acquire a comprehensive understanding of the carbon footprint of new LLMs.
This awareness is particularly important before starting expensive training
endeavors that involve thousands of GPUs. During the initial design phase,
critical parameters like LLM’s parameter count, hardware configurations, and
hosting data center’s efficiency are incorporated into a carbon footprint
projection model. This model should be able to swiftly and accurately assess
the carbon footprint, including both operational and embodied carbon
emissions. Additionally, it should offer valuable insights into factors like
test loss, training duration, and inference latency of the LLM. The
availability of such a carbon footprint projection model empowers cloud
providers to adroitly explore the trade-off between test loss and carbon
footprint of their new LLM designs. Furthermore, it encourages everyday users
to adopt strategies that alleviate LLM carbon footprint by enabling
quantitative comparisons across diverse LLM configurations.

However, there is currently a near-complete absence of an end-to-end carbon
footprint projection model designed for LLMs. Most prior endeavors
Henderson:JMLR2020,Wu:MLS2022,Anthony:ARXIV2020,Schwartz:ACM2020,Patterson:Carbon2021,Dodge:CFAT2022,Strubell:ACL2019,Lakim:BIG2022
are limited to recording and reporting the carbon footprint associated with
the actual training phase of ML models. To the best of our knowledge, only one
carbon footprint model, mlco2 Lacoste:ARXIV2019, can predict the carbon
footprint of an ML task based on factors like GPU usage, training duration,
and data center efficiency. However, mlco2 exhibits several limitations.
Firstly, mlco2 is confined to convolutional neural networks (CNNs) and cannot
estimate the carbon footprint of LLMs. Secondly, mlco2 overlooks crucial
architectural specifications of ML models, such as parameter count, leading to
inflated predictions. Thirdly, mlco2 considers only GPUs and disregards
specialized ML hardware like TPUs Jouppi:ISCA2017 in its predictions, assuming
that all GPUs consistently operate at peak computing throughput, which leads
to inaccurate carbon footprint estimations. Lastly, although the embodied
carbon footprint of an ML task holds equal significance to its operational
carbon footprint Wu:MLS2022, mlco2 is incapable of modeling the embodied
carbon footprint of an LLM based on its hardware resources.

In this paper, we propose an end-to-end carbon footprint projection model,
LLMCarbon, which can accurately predict the carbon footprint of both dense and
MoE LLMs during their training, inference, experimentation, and storage
phases. LLMCarbon incorporates critical LLM, hardware, and data center
parameters, such as LLM parameter count, hardware type, system power, chip
area, and data center efficiency, to model both operational and embodied
carbon footprints of an LLM. When validated against Google’s published LLM
carbon footprints, the results generated by LLMCarbon exhibit differences of
only $\leq 8.2\%$, and are more accurate than those of mlco2.

##  2 Background

Carbon Footprint of an LLM. The carbon footprint of an LLM can be divided into
two components Gupta:MICRO2022: the operational footprint, which encompasses
carbon emissions originating from hardware energy consumption, and the
embodied footprint, which encapsulates carbon emissions arising from hardware
manufacturing. While prior investigations
Henderson:JMLR2020,Wu:MLS2022,Anthony:ARXIV2020,Schwartz:ACM2020,patterson2022carbon,Dodge:CFAT2022,Strubell:ACL2019
have predominantly focused on recording and reporting the operational carbon
footprint of various ML tasks, only Wu:MLS2022 has studied the embodied carbon
footprint of several ML tasks. This study Wu:MLS2022 indicates that the
embodied carbon footprint of an LLM deployed within a Meta data center
accounts for $\sim 50\%$ of its operational carbon footprint.

Neural Scaling Law. The neural scaling law Kaplan:ARXIV2020 elucidates a
power-law relationship between the test loss of an LLM and each of three
pivotal factors: the quantity of model parameters, the scale of the training
dataset, and the computational resources utilized during training. This
relationship remains consistent across different architectures and downstream
ML tasks, considering zero-shot, prompted, and fine-tuned settings
Ethan:ICLR2023.

Mitigating the Carbon Footprint of LLMs. Prior work has focused on reducing
the carbon footprint of LLMs through progress in four domains. Firstly, the
proposal of sparse MoE architectures Fedus:JMLR2022,du2022glam aims to enhance
LLM performance by increasing LLM parameters while maintaining a similar
computational load. Secondly, the adoption of specialized ML hardware like
TPUs Jouppi:ISCA2017 has emerged as a replacement for power-hungry GPUs,
aiming to process LLMs more efficiently. Thirdly, data centers devoted to ML
tasks have streamlined their facilities into large-scale systems, optimizing
power usage effectiveness (PUE) by reducing cooling and infrastructure
overhead liu2020energy. Lastly, these data centers are working towards
mitigating LLM operational carbon footprints by transitioning to renewable
energy sources such as solar and wind power acun2023carbon. Nevertheless,
recent years have seen a significant fourfold Wu:MLS2022 expansion in the
deployment of ML-specific hardware within these data centers, catering to the
diverse demands of various ML tasks. Consequently, the gap between the
operational and embodied carbon footprints of LLMs is predicted to widen in
the near future.

Parallelism in LLM Training and Inference. To process LLMs effectively,
numerous computing devices like GPUs or TPUs are used, due to the high LLM
parameter count. Four types of parallelism, i.e., data, tensor, pipeline
Narayanan:SC2021, and expert Kim:ARXIV2021, are commonly considered to enhance
hardware efficiency, which gauges actual throughput relative to peak
throughput.

  * •

Data Parallelism. Data parallelism Xing:TBD2015 involves distributing a full
model copy to each computing device, while the input dataset is partitioned
among these devices. Periodic gradient aggregation ensures that all devices
maintain a consistent version of the model weights.

  * •

Tensor Parallelism. When implementing tensor parallelism Narayanan:SC2021, an
LLM’s layers are distributed across multiple devices. A transformer layer
consists of a self-attention block and a two-layer MLP. The self-attention
block partitions key, query, and value matrices through column-parallel
division. The output linear layer directly handles the attention operation’s
partitioned output, with the weight matrix partitioned by rows. In the two-
layer MLP, the first layer is divided along columns, and the second along
rows. Both the self-attention block and the MLP layer utilize two all-reduce
operations in forward and backward passes to efficiently coordinate data among
different partitions on distinct devices.

  * •

Pipeline Parallelism. Pipeline parallelism Narayanan:SC2021 partitions an
LLM’s layers across multiple devices. For LLMs with repeated transformer
blocks, each device handles an equal number of layers. Microbatches split a
batch for pipelined execution. Ensuring consistent weight versions for
synchronous weight updates requires well-defined pipelining. Optimizer
integrity is maintained with periodic pipeline flushes synchronizing steps
across devices, albeit inducing idle time at batch starts and ends–termed
“pipeline bubbles”. Minimizing such bubbles is crucial for enhancing the
efficiency of pipeline model parallelism.

  * •

Expert Parallelism. Expert parallelism Kim:ARXIV2021 is tailored for
parallelizing the training of MoE LLMs. This approach involves distributing
distinct experts across various devices, enabling parallel execution. However,
due to the separation of experts across multiple computing devices, explicit
communication using all-to-all primitives becomes essential.

Table 1: The comparison of LLMCarbon against prior work. scheme | predictive | MoE | architectural | specialized | operational | embodied
---|---|---|---|---|---|---
modeling | support | parameters | hardware | carbon | carbon
mlco2 | ✓ | ✗ | ✗ | ✗ |  | ✗
others | ✗ | ✗ | ✗ | ✗ | ✓ | ✓
LLMCarbon | ✓ | ✓ | ✓ | ✓ | ✓ | ✓

##  3 Related Work

Figure 1: The overview of LLMCarbon.

Table 1 presents a comparative analysis between LLMCarbon and prior research
efforts. The majority of existing studies
Henderson:JMLR2020,Wu:MLS2022,Anthony:ARXIV2020,Schwartz:ACM2020,Dodge:CFAT2022,Strubell:ACL2019
focus on recording and reporting the carbon footprint stemming from the actual
training phase of ML models, denoted as “others” in the table. However, only
one prior model, mlco2 Lacoste:ARXIV2019, has the capability to predict the
carbon footprint of an ML task based on metrics like GPU utilization, training
duration, and data center efficiency. Nonetheless, mlco2 encounters four
noteworthy limitations. Firstly, it cannot estimate the carbon footprint of
LLMs, particularly sparse MoE LLMs. Secondly, mlco2 overlooks pivotal
architectural attributes of LLMs, e.g., LLM parameter count, resulting in
inflated projections. Thirdly, mlco2 exclusively considers GPUs and neglects
specialized ML hardware such as TPUs Jouppi:ISCA2017 in its predictions,
assuming uniform peak computing throughput across all GPUs, thereby yielding
imprecise carbon footprint estimations. Lastly, mlco2 cannot model the
embodied carbon footprint of an LLM based on its hardware configuration.

##  4 LLMCarbon

###  4.1 Overview

Figure 1 presents an overview of LLMCarbon for predicting the carbon footprint
of an LLM. The inputs to LLMCarbon encompass the LLM’s architectural
description, data center specification, and hardware configuration. The output
of LLMCarbon is the LLM’s carbon footprint expressed in CO2eq units. To
generate this output, LLMCarbon employs a series of models, each processing
distinct input details. LLMCarbon can use the parameter model to calculate the
LLM’s parameter count based on its architectural attributes, or directly use
the LLM’s parameter count as the input. Leveraging the neural scaling law
Kaplan:ARXIV2020, LLMCarbon determines the test loss by employing the LLM’s
parameter count and training token count. Moreover, the FLOP model estimates
the volume of FLOPs required for LLM training or inference, using its
architectural description. Through the parameter count, LLMCarbon generates
the optimal data, tensor, pipeline, and expert parallelism setting. Taking
into account the parallelism setting and hardware configuration, LLMCarbon’s
hardware efficiency model computes the hardware efficiency, representing the
real computing throughput divided by the peak computing throughput. Utilizing
data center details, hardware efficiency, and FLOP count, LLMCarbon applies
the operational carbon model to derive the LLM’s operational carbon footprint.
Similarly, by considering the hardware configuration, LLMCarbon’s embodied
carbon model yields the LLM’s embodied carbon footprint. The total carbon
footprint of the LLM is calculated by summing both the operational and
embodied carbon footprints.

###  4.2 Parameter Model

Among all architectural factors of an LLM, the number of LLM parameters has a
large impact on its accuracy Kaplan:ARXIV2020. To reduce projection errors,
LLMCarbon can take the parameter count as a direct input. Optionally, the
parameter count of an LLM can be also estimated by the parameter model of
LLMCarbon. The parameter model’s input comprises the LLM’s architectural
information including the hidden size ($h$), the number of layers ($l$), the
vocabulary size ($V$), and the number of experts ($N_{e}$). For a dense LLM,
we calculate its parameter count ($P_{d}$) by Equation 1 Narayanan:SC2021. An
MoE LLM Rajbhandari:ICML2022 replaces $\rho$ ($\rho\leq 1$) feed-forward (FF)
layers in its counterpart dense LLM with MoE layers. An MoE layer’s parameter
count is the sum of the expert parameter count per layer
($P_{exp}=8h^{2}N_{e}$) and the self-attention parameter count
($P_{att}=4h^{2}$) in the layer. So the parameter count ($P_{e}$) of an MoE
LLM can be computed using Equation 2. The parameter model of LLMs adopting an
encoder-decoder architecture can be viewed in Appendix A.
$P_{d}\approx 12lh^{2}+Vh$ (1)
$P_{e}\approx(1-\rho)P_{d}+\rho(8h^{2}N_{e}+4h^{2})l$ (2)

###  4.3 Neural Scaling Law

The neural scaling law Kaplan:ARXIV2020 serves to predict an LLM’s test loss
based on its parameter count $P$ and the training dataset size $D$. For
ensuring the comparability of test losses across various models, sizes, and
datasets, we adopt the Chinchilla scaling law hoffmann2022training, which is
formulated as Equation 3, where $A$, $B$, $\alpha$, $\beta$, and $E$ are
fitting constants. The projected loss $L$ is presented as the summation of an
irreducible term $E$ and a reducible term that diminishes through the scaling
of $P$ and $D$.
$L(P,D)=\frac{A}{P^{\alpha}}+\frac{B}{D^{\beta}}+E$ (3) $TC\approx 6PD$ (4)
$IC\approx 2PD$ (5)

###  4.4 FLOP Model

The FLOP model receives two inputs: the count of parameters ($P$) and the
number of tokens ($D$) processed by the LLM during training or inference. The
primary component of FLOPs is the multiply-accumulate operations involving LLM
weights and intermediate results. Within our FLOP model, the FLOP count
necessary for training a dense LLM ($TC$) is estimated following Equation 4.
In the context of dense LLM inference, the FLOP count ($IC$) is approximated
as per Equation 5. To compute the FLOP count for the training and inference
processes of an MoE LLM, we input the parameter number of the dense base model
Rajbhandari:ICML2022 of the MoE LLM into Equations 4 and 5, respectively.

###  4.5 Hardware Efficiency Model

To efficiently process an LLM, it is critical to attain the highest hardware
efficiency, defined as actual computing throughput divided by peak computing
throughput, through an optimal configuration of data, tensor, pipeline, and
expert parallelism, utilizing a certain number of computing devices
Narayanan:SC2021. Insufficient or excessive devices, or improperly-configured
parallelism setups, result in diminished hardware efficiency, due to frequent
swaps between host and device memories, as well as intensive communications
among various devices. For instance, achieving the optimal parallelism
configuration for training a GPT-3 model with 175 billion parameters demands
1.5K V100 GPUs, resulting in a hardware efficiency of 47% Narayanan:SC2021.
Conversely, training the same model with an unoptimized parallelism
configuration utilizing 10K V100 GPUs yields a substantially lower hardware
efficiency of only 19.7% Patterson:Carbon2021.

Figure 2: The parallelism setting for processing dense LLMs.

Figure 3: The parallelism setting for processing MoE LLMs.

Figure 4: The computing device number for processing LLMs.

Figure 5: The hardware efficiency for processing LLMs.

Optimal Parallelism Setting. We denote the optimal parallelism configuration
as $(p,t,d,e)$, where $p$, $t$, $d$, and $e$ represent degrees of pipeline,
tensor, data, and expert parallelism, respectively. For processing dense LLMs,
we use optimal settings from Narayanan:SC2021, depicted in Figure 5, where
$e=1$ is omitted. To heighten processing parallelism, we initially raise $t$
up to $z$ (e.g., $z=8$) when employing $z$-device servers Narayanan:SC2021,
each featuring $z$ interconnected computing devices. This increment in $t$ is
confined to not exceed $z$ to avoid surpassing communication bandwidth limits
essential for all-reduce operations during both forward and backward phases.
Once $z$ is reached, scaling up computations of larger LLMs across multiple
servers involves augmenting $p$ Narayanan:SC2021. However, the model-parallel
size $t\cdot p$ must not exceed a threshold to ensure that parameters and
intermediate data of the LLM can fit into device memory. As illustrated in
Figure 5, the number of computing devices for attaining the optimal hardware
efficiency of dense LLM processing is computed as $n=t\cdot p\cdot d$
Narayanan:SC2021. Leveraging the optimal hardware efficiency data from
Narayanan:SC2021, a degree-2 polynomial regression model is proposed to
predict the optimal hardware efficiency for dense LLM processing, as shown in
Figure 5. For MoE LLM processing, as Figure 5 shows, the optimal parallelism
settings are adopted from Chen:ARXIV2023. Assuming 64 experts within an MoE
LLM, the expert parallelism size is always set to 64, intertwining $d$ and $e$
for a uniform expert distribution. Token embeddings are assigned to and
returned from each expert, consolidating into the original data parallel rank,
and resulting in two inter-device all-to-all communications during forward
passes Chen:ARXIV2023. Therefore, $d$ is fixed at 1 to reduce inter-device
all-to-all communications. Scaling the parallelism of MoE LLMs is facilitated
by elevating $p$. The device number for attaining the optimal hardware
efficiency of MoE LLM processing can also be computed as $n=t\cdot p\cdot d$,
as displayed in Figure 5. MoE LLMs necessitate fewer computing devices
compared to dense LLMs with equivalent parameter counts, given their
diminished total computational overhead. The optimal hardware efficiency
during MoE LLM processing is depicted in Figure 5. MoE LLMs achieve $80\%\sim
90\%$ Chen:ARXIV2023 of the optimal hardware efficiency of their dense base
models, attributed to extra memory swaps between host and computing devices
due to considerably larger parameter counts.
$\mathit{eff}_{re}=\begin{cases}\gamma_{0}\cdot\frac{re}{n}\cdot\mathit{eff}_{n}&\text{$re<n$}\\\
\gamma_{1}\cdot\frac{n}{re}\cdot\mathit{eff}_{n}+\gamma_{2}&\text{$re>n$}\end{cases}$
(6)
$t_{\mathit{dev}}=\frac{\mathit{TFLOP}}{n_{dev}\cdot\mathit{FLOP}_{\mathit{peak}}\cdot\mathit{eff}}$
(7)

Fewer or More Computing Devices. For a given LLM parameter count, if we have
$re$ ($re\neq t\cdot p\cdot d$) computing devices, the hardware efficiency of
LLM processing would decrease. Specifically, the hardware efficiency
($\mathit{eff}_{re}$) when processing the LLM with $re$ devices can be
computed as Equation 6, where $\gamma_{0}\sim\gamma_{2}$ are fitting
constants; $\mathit{eff}_{n}$ is the highest hardware efficiency; and $n$ is
the number of devices that can obtain $\mathit{eff}_{n}$.

###  4.6 Operational Carbon Model

Using the FLOP count ($\mathit{TFLOP}$) associated with the LLM processing,
the hardware efficiency ($\mathit{eff}$), and the number of computing devices
($n_{dev}$), we can determine the execution time of a device through Equation
7, where $\mathit{FLOP}_{\mathit{peak}}$ stands for the peak throughput of the
computing device. The total energy ($\mathit{energy}_{\mathit{hard}}$)
consumed by all hardware units can be calculated using Equation 8, where
$P_{i}$ denotes the peak power of hardware unit $i$; $\mathit{eff}_{i}$
represents the hardware efficiency of hardware unit $i$; $n_{i}$ indicates the
count of hardware unit $i$; and $t_{i}$ means the execution time of hardware
unit $i$. The hardware units in Equation 8 include CPUs, LLM computing
devices, memories, SSDs and the other components. For hardware units like SSDs
or network interfaces, the power consumption parameter $P_{i}$ in the equation
is typically expressed in Watts per TB, while the data storage or transfer
capacity $n_{i}$ is expressed in TB.
$\mathit{energy}_{\mathit{hard}}=\sum_{i\in
hardware\\_set}(P_{i}\cdot\mathit{eff}_{i}\cdot n_{i}\cdot t_{i})$ (8)
$\mathit{energy}_{\mathit{oper}}=\mathit{energy}_{\mathit{hard}}\cdot\mathit{PUE}$
(9)

PUE. Power Usage Effectiveness (PUE) Henderson:JMLR2020 serves as the industry
standard metric for assessing the energy efficiency of a data center. It is
defined as the ratio of the total energy consumption of the data center,
including all auxiliary components like cooling, to the energy consumed solely
by the computing hardware within the data center. The operational energy
($\mathit{energy}_{\mathit{oper}}$) associated with the LLM processing can be
calculated using Equation 9, where $\mathit{energy}_{\mathit{hard}}$ denotes
the energy used by the computing hardware within a data center, and
$\mathit{PUE}$ indicates the PUE of the specific data center.
$\mathit{CO2eq}_{\mathit{oper}}=\mathit{energy}_{\mathit{oper}}\cdot\mathit{carb\\_inten}$
(10) $\mathit{CO2eq}_{\mathit{chip}}=\mathit{area}\cdot\mathit{CPA}$ (11)

Table 2: The data center efficiency. data | carbon | carbon
---|---|---
center | free | intensity
name | energy | $\mathit{gCO2eq/kWh}$
asia-east2 | 28% | 360
europe-north1 | 91% | 127
us-central1 | 97% | 394
us-south1 | 40% | 296

Table 3: The comparison of embodied carbon footprints. hardware | description | unit | CPA
---|---|---|---
CPU | TSMC 16nm | 147 $mm^{2}$ | 1 $\mathit{kgCO2/cm^{2}}$
DRAM | Micron 18nm | 256 GB | 0.024 $\mathit{kgCO2/GB}$
SSD | Samsung 20nm | 32 TB | 0.4 $\mathit{kgCO2/GB}$
TPUv3 | TSMC 16nm | 700 $mm^{2}$ | 1 $\mathit{kgCO2/cm^{2}}$
TPUv4 | TSMC 7nm | 400 $mm^{2}$ | 1.6 $\mathit{kgCO2/cm^{2}}$
V100 | TSMC 12nm | 815 $mm^{2}$ | 1.2 $\mathit{kgCO2/cm^{2}}$
H100 | TSMC 4nm | 814 $mm^{2}$ | 1.8 $\mathit{kgCO2/cm^{2}}$

Carbon Intensity. Carbon intensity is a metric that gauges the environmental
friendliness of a data center’s energy consumption, while carbon free energy
(CFE) denotes the proportion of renewable, carbon-free energy used within a
data center. A data center that increases its utilization of renewable energy
will witness a rise in its CFE and a corresponding drop in its carbon
intensity. Table 3 provides insight into the carbon intensity and CFE values
for some data centers. The operational carbon footprint
($\mathit{CO2eq}_{\mathit{oper}}$) attributed to the LLM processing is
computed by Equation 10, where $\mathit{energy}_{\mathit{oper}}$ indicates the
operational energy for the LLM processing, and $\mathit{carb\\_inten}$ denotes
the carbon intensity of the specific data center.
$\mathit{CO2eq}_{\mathit{emb}}=\sum_{i\in
hardware\\_set}\frac{\mathit{t}_{i}\cdot\mathit{CO2eq}_{\mathit{chip}_{i}}}{\mathit{lifetime}_{i}}$
(12)
$\mathit{CO2eq}=\mathit{CO2eq}_{\mathit{oper}}+\mathit{CO2eq}_{\mathit{emb}}$
(13)

###  4.7 Embodied Carbon Model

To quantify the chip’s embodied carbon footprint
($\mathit{CO2eq}_{\mathit{chip}}$) within a specific hardware unit, Equation
11 is employed, where $\mathit{area}$ represents the chip’s area. The Carbon
emitted Per unit Area ($\mathit{CPA}$) is contingent on various semiconductor
fabrication parameters, including yield, energy consumption per unit area
during manufacturing, emissions from chemicals utilized in hardware
production, and emissions associated with raw material sourcing for
fabrication. Specific values for area and CPA for distinct hardware units are
elaborated in Table 3, where area values for CPU, DRAM, SSD, TPU, and GPU are
drawn from sources such as [singh20202], [choe2021memory], [wiki:TPU], and
[wiki:GPU]. CPA values for Micron, Samsung, and TSMC are extracted from
[Garcia:IEDM2020], and [TSMC:SUS2019]. The total embodied carbon footprint
($\mathit{CO2eq}_{\mathit{emb}}$) originating from all hardware units involved
in the LLM processing is assessed using Equation 12, where
$\mathit{CO2eq}_{\mathit{chip}_{i}}$ denotes the chip’s embodied carbon
footprint for hardware unit $i$, $\mathit{lifetime}_{i}$ means the lifespan of
hardware unit $i$, and $t_{i}$ represents the execution duration of hardware
unit $i$. The hardware units mentioned in Equation 12 include CPUs, LLM
computing devices, memories, SSDs, and other components. Notably, Meta’s data
centers achieve an average utilization rate of $60\%$ throughout the 5-year
lifespan of hardware units Wu:MLS2022.

###  4.8 Total Carbon Footprint

The total carbon footprint ($\mathit{CO2eq}$) resulting from the LLM
processing is determined using Equation 13, where
$\mathit{CO2eq}_{\mathit{oper}}$ indicates the operational carbon footprint of
the LLM, and $\mathit{CO2eq}_{\mathit{emb}}$ denotes the embodied carbon
footprint of the LLM.

##  5 Validation

We employ LLMCarbon to compute the operational footprints of five LLMs,
including dense and MoE architectures, developed by Google, OpenAI, and Meta
during their training phases. We also compute the operational footprint of
another LLM, Noor Lakim:BIG2022, during its storage phase. To validate the
predictions of LLMCarbon, we compare our calculated operational footprint
values with the previously published data for these LLMs. Moreover, we utilize
LLMCarbon to predict the embodied footprint of an LLM developed by Meta and
validate the result by comparing it with the actual embodied footprint data.

Table 4: The validation on the operational carbon footprints of various LLMs. LLM | T5 | GPT3 | GShard | Switch | XLM
---|---|---|---|---|---
reference |  Patterson:Carbon2021 |  Wu:MLS2022
developer | Google | OpenAI | Google | Google | Meta
type | dense | dense | MoE | MoE | dense
parameter # (B) | 11 | 175 | 619 | 1500 | 0.55
base model param. # (B) | - | - | 2.3 | 7.41 | -
token # (B) | 500 | 300 | 1K | 2K | 7K
$\mathit{CO}_{2}\mathit{eq/KWh}$ | 0.545 | 0.429 | 0.177 | 0.33 | 0.413
PUE | 1.12 | 1.1 | 1.09 | 1.1 | 1.1
computing device | TPUv3 | V100 | TPUv3 | TPUv3 | V100
device TPD (W) | 450 | 300 | 450 | 450 | 300
avg. system power (W) | 310 | 330 | 288 | 245 | 342
peak TFLOPs/s | 123 | 125 | 123 | 123 | 125
achieved TFLOPs/s | 45.6 | 24.6 | 48 | 34.4 | 26.5
hardware efficiency | 37% | 19.7% | 39% | 28% | 21.2%
device # | 512 | 10K | 1K | 1K | 512
total zettaFLOPs | 40.5 | 314 | 13.3 | 82.2 | 23.9
training days | 20 | 14.8 | 3.1 | 27 | 20.4
actual $\mathit{tCO}_{2}\mathit{eq}$ | 46.7 | 552.1 | 4.3 | 59.1 | 39
mlco2 predicted $\mathit{tCO}_{2}\mathit{eq}$ | 89.4 | 955.2 | 8.4 | 137.3 | 66.96
mlco2 $\Delta$ | $+91.3\%$ | $+73\%$ | $+95.3\%$ | $+132\%$ | $+69\%$
LLMCarbon predicted $\mathbf{\mathit{tCO}_{2}\mathit{eq}}$ | 45.66 | 553.87 | 4.46 | 63.9 | 37.6
$\mathbf{LLMCarbon~{}\Delta}$ | $\mathbf{-2.22\%}$ | $\mathbf{+0.32\%}$ | $\mathbf{+3.8\%}$ | $\mathbf{+8.2\%}$ | $\mathbf{-3.54\%}$

###  5.1 Operational Carbon Footprint Validation

Training Phase. Table 4 presents the validation results of LLMCarbon’s
predictions on the training operational carbon footprint. To validate the
training operational carbon footprint estimations yielded by LLMCarbon, we
selected five LLMs: T5 Colin:JMR2020, GPT-3 Brown:NIPS2020, GShard
Dmitry:ICLR2021, Switch Fedus:JMLR2022, and XLM Conneau:ACL2020. We list the
inputs and outputs of LLMCarbon in Table 4. Within the table, “device TPD (W)”
indicates the Chip Thermal Design Power of a computing device, while “avg.
system power (W)” conveys the average system power per computing device,
including TPU/GPU, host CPU, DRAM, and network interface. The inputs on the
parameters of LLMs, hardware, and data centers, and the actual training
operational carbon footprint values of these LLMs were collected from
Patterson:Carbon2021 and Wu:MLS2022. Since the parameter count of an LLM is
considered as an architectural parameter of the LLM in Patterson:Carbon2021
and Wu:MLS2022, we skipped the parameter model and directly used the parameter
count as an input to LLMCarbon. The validation of the parameter model of
LLMCarbon can be found in Appendix B. Owing to the adoption of suboptimal
parallelism settings, the hardware efficiencies for training these LLMs hover
within the range of $39\%$ to $19.7\%$, lower than the efficiencies achieved
with optimal parallelism configurations. Comparing the predicted operational
carbon footprints to actual data, LLMCarbon’s projections display disparities
of $\leq 8.2\%$. When predicting the operational carbon footprint during
training for MoE LLMs, LLMCarbon incurs a higher margin of error, due to the
intricacy of their architectures. On the contrary, when compared to actual
data, the training operational carbon footprint estimations made by mlco2
Lacoste:ARXIV2019 suffer from huge disparities of more than $69\%$, because
mlco2 assumes all computing devices consistently operate at peak computing
throughput and consume peak power.

Storage Phase. The typical power consumption of cloud storage is reported as
11.3W/TB Posani:ARXIV2018, while the power consumption for data transfer
within a data center is around 1.48W/TB Baliga:IEEE2011. Over a six-month
storage phase, the Noor LLM Lakim:BIG2022 encompasses 32.7TB of storage data,
comprising curated data, bulk data, and the model. Additionally, it transfers
a data volume of 277.4TB. Based on LLMCarbon’s estimations, the storage data
energy is predicted as 1.596MWh (compared to the actual 1.69MWh
Lakim:BIG2022), while the energy consumption attributed to data transfer is
projected to be 1.77MWh (compared to 1.8MWh Lakim:BIG2022). Notably, the
projection accuracy of LLMCarbon regarding the operational energy during the
storage phase showcases an error margin of less than 3.6%.

###  5.2 Embodied Carbon Footprint Validation

Table 5: The embodied carbon footprint validation against Meta XLM. hardware | number | $\mathit{CO2eq}_{\mathit{chip}}$ | $\frac{time}{lifetime}$ | $\mathit{CO2eq}_{\mathit{emb}}$
---|---|---|---|---
($\mathbf{\mathit{kgCO}_{2}\mathit{eq}}$) | ($\mathbf{\mathit{tCO}_{2}\mathit{eq}}$)
GPU | 512 | 9.78 | 1.12% | 0.056
CPU | 64 | 1.47 | 1.12% | 0.0018
SSD | 64 | 576 | 1.12% | 0.412
DRAM | 64 | 102.4 | 1.12% | 0.073
others | 64 | 148.2 | 1.12% | 0.096
predicted sum |  |  | 0.64
actual 0.66 $\mathbf{\mathit{tCO}_{2}\mathit{eq}}$, $\mathbf{\Delta}$
$\mathbf{-3.05\%}$

Table 5 presents the validation results of the embodied carbon footprint
estimated by LLMCarbon in comparison to the published data of XLM Wu:MLS2022.
This is the only publicly available data regarding the embodied carbon
footprint of a LLM training hardware infrastructure to our best knowledge. The
setup consists of 512 V100 GPUs organized into 64 8-GPU servers, each equipped
with a CPU, a 32TB SSD disk, and a 256GB DRAM main memory system. Using the
unit and CPA data from Table 3, we computed the values of
$\mathit{CO2eq}_{\mathit{chip}}$ presented in Table 5. The training duration
of XLM is 20.4 days, and Wu:MLS2022 assumed a hardware unit lifetime of 5
years. Consequently, the $\frac{time}{lifetime}$ values for all hardware units
were determined to be $1.12\%$. Apart from CPU, GPU, SSD, and DRAM, other
hardware components (others) such as the motherboard, chassis, and PSU
collectively contribute to $15\%$ Tannu:WSCSDI2022 of the anticipated total
embodied carbon footprint. In contrast to the reported embodied carbon
footprint of XLM Wu:MLS2022, the predictions produced by LLMCarbon reveal a
disparity of $-3.05\%$.

##  6 Case Studies Using LLMCarbon

We used LLMCarbon to demonstrate the following case studies.

Large Embodied Carbon Footprint. The embodied carbon footprint throughout the
life-cycle of an LLM is significant. Even when no computing activities occur,
the LLM still incurs embodied carbon overhead due to the idle hardware
allocated to the LLM. As illustrated in Figure 7, the embodied carbon
footprint of an LLM across its entire life-cycle contributes to approximately
$24\%\sim 35\%$ of the overall carbon footprint (including embodied, training,
inference, experimentation, and storage carbon footprints) of the LLM. We
adopted the ratio between training, inference, and experimentation activities
from Wu:MLS2022. Furthermore, as data centers progressively shift towards
adopting renewable energy sources, the embodied carbon footprint of an LLM
will dominate the entire life-cycle carbon footprint of the LLM in the near
future. For instance, 97% of the operational energy in a Meta data center
Wu:MLS2022 is provided by renewable sources. The embodied carbon footprints of
diverse LLMs operating within this data center constitute $92\%\sim 95\%$ of
their entire life-cycle carbon footprints. This underscores the pivotal role
of accounting for embodied carbon in the sustainability evaluation of LLMs.

Optimal Parallelism Setting. As discussed in Section 5.1, the training
processes of the LLMs used in our validation lacked optimized parallelism
settings. By using LLMCarbon, we pinpoint the optimal configurations for data,
tensor, pipeline, and expert parallelism pertaining to these three LLMs. As
illustrated in Figure 7, the adoption of these optimal parallelism settings
leads to a noteworthy decrease (i.e., $16\%\sim 39\%$) in their operational
carbon footprints.

New Accelerators. When employing distinctive computing devices for the LLM
processing, the operational carbon footprints of an LLM tend to differ, while
the embodied carbon footprints remain similar. Figure 7 showcases the outcomes
derived from training, inferring, and experimenting with three LLMs utilizing
V100 GPU, H100 GPU, TPUv3, and TPUv4. Their embodied carbon footprints exhibit
similarity, as the embodied carbon emissions of SSD and DRAM dominate their
total embodied carbon footprints. However, compared to V100 GPUs, the
operational carbon footprints of these LLMs are notably curtailed by 71% and
41% when employing H100 and TPUv4 accelerators, respectively. Embracing novel
computing devices for LLMs presents a pragmatic path to mitigate their
operational carbon footprints.

Figure 6: The carbon footprint of three LLMs in case studies.

Figure 7: The carbon footprint of GPT3 trained by different computing devices.

Figure 8: The trade-off between training carbon footprint and test loss.

Training Carbon Footprint Scaling. In addition to the LLMs (i.e., T5, GPT3,
GShard, Switch, XLM, and Noor) we used in validations, we also included other
LLMs in our analysis, such as PaLM Chowdhery:ARXIV2022, Gopher rae2021scaling,
Chinchilla hoffmann2022training, LaMDA thoppilan2022lamda, Jurassic-1
lieber2021jurassic, MT-NLG smith2022using, Bloom scao2022bloom, YaLM
yalm_100b, GLM Aohan:ICLR2023, GLaM du2022glam, FacebookMoE Artetxe:ARIV2021,
ST-MoE zoph2022st, and PR-MoE Rajbhandari:ICML2022. Among these LLMs, GShard,
Switch, GLaM, FacebookMoE, ST-MoE, and PR-MoE use sparse MoE architectures,
while the other LLMs adopt dense architectures. We do not aim to directly
compare the accuracy and carbon emissions of these original LLMs, since they
were trained by different datasets and in different data centers. Instead, we
study the test losses and training operational carbon footprints of some new
LLM designs adopting the same architectures as these LLMs. We assume these new
LLM designs are trained using the same dataset and the same hardware
infrastructure in the same data center. We present the test losses and
training operational carbon footprints of these LLMs in Figure 8. To compute
the test loss, we adopt the fitting constants including $\alpha=0.34$,
$\beta=0.28$, $A=406.4$, $B=410.7$, and $E=1.69$ for Equation 3 from
hoffmann2022training. Since the test loss of an MoE LLM with $P$ parameters is
similar to that of its dense counterpart with only $P/8$ parameters
Rajbhandari:ICML2022, we decreased the $P$ of MoE LLMs to $P/8$ in Equation 3.
The training processes of all LLMs use their optimal parallelism settings and
the corresponding numbers of V100 GPUs hosted by a data center where PUE is
1.1 and $\mathit{CO}_{2}\mathit{eq/KWh}$ is 0.431. Overall, an LLM with a
larger number of parameters and trained on more tokens achieves a lower test
loss but also consumes a larger training operational carbon footprint.
Compared to dense LLMs, the Pareto front of MoE LLMs is closer to the origin
point, indicating that an MoE LLM can obtain a lower test loss by the same
training carbon footprint.

##  7 Conclusion

In this paper, we propose LLMCarbon, an end-to-end carbon footprint modeling
tool for dense and MoE LLMs, which contribute significantly to carbon
emissions during training, inference, experimentation, and storage processes.
LLMCarbon can accurately assess the operational and embodied carbon footprints
of an LLM, enabling efficient exploration of the design space by considering
the trade-off between carbon footprint and test loss. It also promotes the
adoption of carbon footprint reduction measures by facilitating quantitative
comparisons among various LLM configurations.

Table 6: The architectural description of the dense LLMs used in validations and explorations. LLM | Param. | Vocab. | Hidden | FF | Layer | Predicted | Diff.
---|---|---|---|---|---|---|---
Name | (B) | Size ($V$) | Size ($h$) | Size ($f$) | Num. ($l$) | Param. (B) | $\Delta$
T5 Colin:JMR2020 | 11 | 32K | 1024 | 65536 | 24 | 11.3 | +2.79%
GPT3 Brown:NIPS2020 | 175 | 51.2K | 12288 | 49152 | 96 | 174.58 | -0.24%
XLM lample2019cross | 0.55 | 250K | 1024 | 4096 | 12 | 0.557 | +1.45%
Noor Lakim:BIG2022 | 13 | - | - | - | - | - | -
PaLM Chowdhery:ARXIV2022 | 540 | 256K | 18432 | 73728 | 118 | 534.5 | -1.01%
Gopher rae2021scaling | 280 | 32K | 16384 | 65536 | 80 | 258.5 | -7.65%
Chinchilla hoffmann2022training | 70 | 32K | 8192 | 32768 | 80 | 64.84 | -7.36%
LaMDA thoppilan2022lamda | 137 | 32K | 8192 | 65536 | 64 | 137.9 | +0.63%
Jurassic-1 lieber2021jurassic | 178 | 256K | 13824 | 55296 | 76 | 175 | -1.68%
MT-NLG smith2022using | 530 | 250K | 20480 | 81920 | 105 | 529.6 | -0.09%
Bloom scao2022bloom | 176 | 250K | 14336 | 57344 | 70 | 173.4 | -1.49%
YaLM yalm_100b | 100 | - | - | - | - | - | -
GLM Aohan:ICLR2023 | 130 | 150K | 12288 | 49152 | 70 | 127.5 | -1.95%

##  Appendix A More on the Parameter Model of LLMs

GPT3-like Dense LLMs: The parameter count for most dense LLMs structured on a
GPT3-like architecture Brown:NIPS2020 can be determined using Equation 1. In
each layer of these dense LLMs, there exists a self-attention layer and a
feed-forward layer. The $W_{q}$, $W_{k}$, and $W_{v}$ matrices of the self-
attention layer possess dimensions of $hN_{head}d_{head}$, with $h$
representing the hidden size, $N_{head}$ indicating the number of heads, and
$d_{head}$ denoting the head dimension. The $W_{o}$ matrix that links the
self-attention layer to the feed-forward layer also has a dimension of
$hN_{head}d_{head}$. In the feed-forward layer, two $hd_{ff}$ weight matrices
are used, where $d_{ff}$ signifies the dimension of the feed-forward layer. In
a conventional LLM architecture, we have $n_{head}d_{head}=h$ and $d_{ff}=4h$.
Consequently, the parameter count for a single dense LLM layer can be
calculated as $4hN_{head}d_{head}+2hd_{ff}=12h^{2}$. Additionally, a dense LLM
possesses $Vh$ token embedding parameters, where $V$ denotes the vocabulary
size. In total, a dense LLM utilizing a GPT3-like architecture incorporates
$12h^{2}l+Vh$ parameters, where $l$ stands for the number of layers.

Encoder-Decoder Dense LLMs: Certain dense LLMs from Google, such as T5
Colin:JMR2020, employ an encoder-decoder transformer architecture. Within a
single layer of these LLMs, there exist both an encoder and a decoder. The
encoder comprises a self-attention layer and a feed-forward layer, while the
decoder includes two self-attention layers and a feed-forward layer. The
parameter count for the encoder is $4hN_{head}d_{head}+2hd_{ff}$, whereas the
parameter count for the decoder is $8hN_{head}d_{head}+2hd_{ff}$. Therefore,
the total parameter count for a single LLM resembling T5 becomes
$12hN_{head}d_{head}+4hd_{ff}$. In the case of a T5-like LLM, where
$n_{head}d_{head}\neq h$ and $d_{ff}\neq 4h$, we cannot derive a further
simplified equation. The overall parameter count for a T5-like LLM can be
estimated as:

$P_{d}\approx(12hN_{head}d_{head}+4hd_{ff})l+Vh.$ (14)

For some dense LLMs like LaMDA thoppilan2022lamda, which consist of only a
decoder in each layer, the total parameter count for a LaMDA-like LLM is:

$P_{d}\approx(8hN_{head}d_{head}+2hd_{ff})l+Vh.$ (15)

MoE LLMs: In the case of certain MoE LLMs, especially those developed by
Google, we also encounter scenarios where $n_{head}d_{head}\neq h$ and
$d_{ff}\neq 4h$. Consequently, within an MoE layer, we can compute the expert
parameter count as $P_{exp}=2hd_{ff}N_{e}$, while the self-attention parameter
count can be determined as $P_{att}=4hn_{head}d_{head}$. The overall parameter
count for such an MoE LLM can be estimated as follows:

$P_{e}\approx(1-\rho)P_{d}+\rho(2hd_{ff}N_{e}+4hn_{head}d_{head})l$ (16)

Table 7: The architectural description of the MoE LLMs used in validations and explorations. LLM | Param. | Base | Hidden | MoE | Layer | Experts | Predicted | Diff.
---|---|---|---|---|---|---|---|---
Name | (B) | Param. | Size | ($\rho$) | Num. | Num. | Param. (B) | $\Delta$
Gshard Dmitry:ICLR2021 | 600 | 2 | 1024 | 0.5 | 36 | 2048 | 606.9 | +3.28%
Switch Fedus:JMLR2022 | 1571 | - | 2080 | 1 | 15 | 2048 | 1546.4 | -1.56%
GLaM du2022glam | 1200 | 95 | 8192 | 0.5 | 64 | 64 | 1133.9 | -5.51%
FacebookMoE Artetxe:ARIV2021 | 1100 | 2.3 | 4096 | 0.5 | 32 | 512 | 1103.8 | +0.34%
ST-MoE zoph2022st | 269 | 32 | 5120 | 0.25 | 27 | 64 | 264.1 | -1.83%
PR-MoE Rajbhandari:ICML2022 | 31 | 1.3 | 2048 | 0.5 | 24 | 128 | 30.5 | -1.65%

##  Appendix B Parameter Model Validation

Validation of Dense LLMs: We present the architectural parameters of dense
LLMs in Table 6. It’s worth noting that while Noor was utilized in the
validation of training operational energy and YaLM in LLM scaling, their
original papers Lakim:BIG2022,yalm_100b do not provide architectural
specifications, thus preventing us from determining their parameter count
using LLMCarbon. In Table 6, we apply Equation 1 to calculate the parameter
count for models such as GPT3, XLM, Gopher, Chinachilla, Jurassic-1, MT-NLG,
Bloom, and GLM. Additionally, we use Equation 14 to estimate the parameter
count for T5, and Equation 15 for PaLM and LaMDA. Among all dense LLMs, Gopher
and Chinchilla exhibit the most substantial disparities between the predicted
and actual parameter numbers. This deviation is primarily attributed to the
usage of positional encoding mechanisms in these LLMs, with the weights
employed in their relative positional encodings not included in our
calculations. For instance, Gopher incorporates 21.5 billion weights in its
relative positional encoding, contributing to this observed difference.

Validation of MoE LLMs: We present the architectural parameters of MoE LLMs in
Table 7. To compute the parameter count for GLaM, FacebookMoE, and PR-MoE, we
utilize Equation 2. For Gshard, Switch, and ST-MoE, we apply Equation 16, with
specific values: in Gshard, we set $n_{head}=16$, $d_{head}=128$, and
$d_{ff}=8192$; for Switch, we have $n_{head}=32$, $d_{head}=64$, and
$d_{ff}=6144$; and in ST-MoE, the parameters are $n_{head}=64$,
$d_{head}=128$, and $d_{ff}=20480$. It’s noteworthy that PR-MoE and ST-MoE
incorporate additional feed-forward layers. The most substantial differences
are observed in GShard and GLaM due to unaccounted parameters in their routing
networks.


# Centrality dependence and isospin effect on ${\rm W}^{\pm}$ and ${\rm
Z}^{0}$ productions
in nucleus-nucleus collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV

Dai-Mei Zhou1<EMAIL_ADDRESS>Yu-Liang Yan2<EMAIL_ADDRESS>Liang Zheng3, Ming-Rui Zhao2, Xiao-Mei Li2, Xiao-Ming Zhang1, Gang Chen3, An-
Ke Lei1, Ya-Qian Zhu1, Xu Cai1, and Ben-Hao Sa1,2<EMAIL_ADDRESS>1 Key
Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle
Physics, Central China Normal University, Wuhan 430079, China.
2 China Institute of Atomic Energy, P. O. Box 275 (10), Beijing, 102413 China.
3 School of Mathematics and Physics, China University of Geosciences (Wuhan),
Wuhan 430074, China.

###### Abstract

In this paper, the centrality dependent ${\rm Z}^{0}$ and ${\rm W}^{\pm}$
production and the isospin effect in ${\rm W}^{\pm}$ production are
investigated with a parton and hadron cascade model PACIAE in Pb–Pb collisions
at $\sqrt{s_{\rm NN}}=5.02$ TeV. ALICE data of ${\rm Z}^{0}$ production in
Pb–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV are found to be reproduced
fairly well. The prediction on ${\rm W}^{\pm}$ production in the same
collision system is given as well. An interesting isospin effect is observed
in exploring the charge asymmetry between ${\rm W}^{+}$ and ${\rm W}^{-}$ as a
function of the asymmetry between number of valence $u$\- and $d$-quarks
varied from small to large collision systems at center-of-mass energy 5.02
TeV. The results serve as a important benchmarks for understanding the initial
conditions of heavy-ion collisions.

##  I Introduction

${\rm W}^{\pm}$ and ${\rm Z}^{0}$ vector bosons are heavy particles with
masses of $m_{{\rm W}^{\pm}}=80.39$ GeV/$c^{2}$ and $m_{{\rm Z}^{0}}=91.19$
GeV/$c^{2}$ pdg . They are mainly produced in the hard partonic scattering
processes with large momentum transfer at the early stage of the
(ultra-)relativistic nucleus–nucleus collisions. Their main production
processes are

$u\overline{d}\rightarrow{\rm W}^{+},\hskip
14.22636ptd\overline{u}\rightarrow{\rm W}^{-}$

and

$u\overline{u}\rightarrow{\rm Z}^{0},\hskip
14.22636ptd\overline{d}\rightarrow{\rm Z}^{0}$

in the leading order approximation martin . With the involvement of the
valence $u$\- and $d$-quarks inside the beam particles, the isospin difference
between protons and neutrons is important on the production of ${\rm W}^{+}$
and ${\rm W}^{-}$. It is therefore expected that the charge asymmetry between
${\rm W}^{+}$ and ${\rm W}^{-}$ is related to the relative abundance of
protons or neutrons of the colliding nuclei, irrespective of the collision
system size.

In comparison with the evolution time of the heavy-ion collision system, $10$
to $100$ fm/$c$ for instance, decay time of ${\rm W}^{\pm}/{\rm Z}^{0}$, which
can be estimated with the full decay width $\Gamma$ pdg ,

$\displaystyle t=\frac{\hbar}{\Gamma},\hskip 11.38092ptt_{{\rm
W}^{\pm}}=0.0922~{}{\rm fm/}c,\hskip 11.38092ptt_{{\rm Z}^{0}}=0.0791~{}{\rm
fm/}c,$

is very short. The ${\rm W}^{\pm}/{\rm Z}^{0}$ leptonic decays

${\rm W}^{+}\rightarrow l^{+}\nu_{l},\hskip 14.22636pt{\rm Z}^{0}\rightarrow
l^{+}l^{-},~{}(l{\rm:}~{}e,\mu,\tau)$

are nearly instantaneous. As the produced leptons weakly interact with the
partonic and hadronic matter created in heavy ion collisions, ${\rm W}^{\pm}$
and ${\rm Z}^{0}$, similar as the prompt direct photons, are powerful probes
for investigating the properties of the initial stage of the evolving system
and the partonic structure of the colliding nuclei.

The measurement of ${\rm W}^{\pm}/{\rm Z}^{0}$ production is very important to
calibrate the collision geometry information of nuclear collisions and of
great experimental interests. Many observables in the heavy ion studies
require the knowledge of geometric quantities such as the nuclear thickness
function $\left\langle T_{\rm AA}\right\rangle$, the number of participant
nucleons $\left\langle N_{\rm part}\right\rangle$, and the number of binary
collisions $\left\langle N_{\rm coll}\right\rangle$, which are usually
obtained using the Glauber model shor ; abel ; abel1 ; misk . Without
participating the final state interactions, ${\rm W}^{\pm}/{\rm Z}^{0}$ bosons
are ideal probes to test the validity of collision geometries applied in the
experimental analysis.

The CMS and ATLAS Collaborations have first measured ${\rm W}^{\pm}$ and ${\rm
Z}^{0}$ production in Pb–Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV cms1 ;
atlas1 ; cms2 ; atlas2 . Recently, the ALICE and ATLAS Collaborations
published the measurements of ${\rm Z}^{0}$ production at forward rapidities
alice1  and ${\rm W}^{\pm}/{\rm Z}^{0}$ production at mid-rapidity atlas3 ;
atlas4 , in Pb–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV, respectively.
The ${\rm W}^{\pm}$ and ${\rm Z}^{0}$ production cross sections are also
measured in p–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ by the ALICE and CMS
collaborations alice2 ; cms3 . All those measurements are declared to be well
reproduced by the leading-order (LO) and/or next-to-leading-order (NLO)
perturbative Quantum Chromo Dynamics (pQCD) calculations ct14 ; eps09 ; cteq15
; epps  using the CT14 Parton Distribution Function (PDF) set ct14  with and
without the parameterized nuclear modified PDF (nPDF) like EPPS16 epps . As
the experimental data analysis relies strongly on the template calculated with
NLO pQCD powheg , comparing experimental data to the LO or NLO pQCD
predictions only is incomplete. The study of ${\rm W}^{\pm}/{\rm Z}^{0}$
production in heavy-ion collisions with Monte-Carlo simulation may provide
more differential understandings into the microscopic transport properties of
the collision system.

##  II Model

A parton and hadron cascade model PACIAE sa1  is employed to simulate ${\rm
Z}^{0}$ production in pp and Pb–Pb collisions at centre-of-mass energy $5.02$
TeV. The results are compared with the ALICE measurements alice1 . The ${\rm
W}^{\pm}$ production is predicted in Pb–Pb collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV. A systematic analysis of ${\rm W}^{\pm}$ charge asymmetry in
nucleon–nucleon (NN): pp, pn, np, nn collisions and in nucleus–nucleus (AA):
Cu–Cu, Au–Au, Pb–Pb and U–U collisions is also studied.

Table 1: Per-event multiplicity of ${\rm Z}^{0}$ and ${\rm W}^{\pm}$ bosons in
rapidity interval $2.5<y<4.0$ for three different centrality classes
($0$–$(\%$20), $20$–$90\%$ and $0$–$90\%$) in Pb–Pb collision at $\sqrt{s_{\rm
NN}}=5.02$ TeV.

|  |  |
---|---|---|---
Centrality |  $0$–$20\%$ |  $0$–$90\%$ |  $20$–$90\%$
|  |  |
${\rm d}N_{{\rm Z}^{0}}/{\rm d}y|_{exp.}$ |  $0.996\times 10^{-7}$ |  $0.379\times 10^{-7}$ |  $0.198\times 10^{-7}$
|  |  |
Rescaled ${\rm d}N_{{\rm Z}^{0}}/{\rm d}y|_{exp.}^{**}$ |  $5.03$ |  $1.91$ |  $1.00$
|  |  |
${\rm d}N_{{\rm Z}^{0}}/{\rm d}y|_{paciae}$ |  $9.92$ |  $4.42$ |  $2.84$
|  |  |
Rescaled ${\rm d}N_{{\rm Z}^{0}}/{\rm d}y|_{paciae}^{**}$ |  $3.50$ |  $1.56$ |  $1.00$
|  |  |
${\rm d}N_{{\rm W}^{+}}/{\rm d}y|_{paciae}$ |  $4.97$ |  $2.23$ |  $1.43$
|  |  |
Rescaled ${\rm d}N_{{\rm W}^{+}}/{\rm d}y|_{paciae}^{**}$ |  $3.49$ |  $1.56$ |  $1.00$
|  |  |
${\rm d}N_{{\rm W}^{-}}/{\rm d}y|_{paciae}$ |  $5.30$ |  $2.37$ |  $1.50$
|  |  |
Rescaled ${\rm d}N_{{\rm W}^{-}}/{\rm d}y|_{paciae}^{**}$ |  $3.54$ |  $1.58$ |  $1.00$
|  |  |
* Taken from ALICE alice1
** Rescale to the digital in 20-90% centrality.

Figure 1: Centrality dependance on ${\rm Z}^{0}$ production in Pb–Pb
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV: panel (a) for ${\rm d}N/{\rm d}y$
/ $\left\langle T_{\rm AA}\right\rangle$, (b) for $R_{\rm AA}$, and (c) for
${\rm d}N/{\rm d}y$.

The PACIAE model is based on Pythia event generator (version $64.28$) soj1 .
For NN collisions, with respect to Pythia, the partonic and hadronic
rescatterings are introduced before string formation and after the
hadronization, respectively. The final hadronic state is developed from the
initial partonic hard scattering and parton showers, followed by parton
rescattering, string fragmentation, and hadron rescattering stages. Thus, the
PACIAE model provides a multi-stage transport description on the evolution of
the collision system.

Figure 2: Centrality dependance on ${\rm W}^{\pm}$ productions in Pb–Pb
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV: panel (a) for ${\rm d}N/{\rm d}y$
/ $\left\langle T_{\rm AA}\right\rangle$, (b) for $R_{\rm AA}$, and (c) for
${\rm d}N/{\rm d}y$.

For AA collisions, the initial positions of nucleons in the colliding nucleus
are sampled according to the Woods-Saxon distribution. Together with the
initial momentum setup of $p_{x}=p_{y}=0$ and $p_{z}=p_{\rm beam}$ for each
nucleon, a list containing the initial state of all nucleons in a given AA
collision is constructed. A collision happened between two nucleons from
different nuclei if their relative transverse distance is less than or equal
to the minimum approaching distance: $D\leq\sqrt{\sigma_{\rm NN}^{\rm
tot}/\pi}$. The collision time is calculated with the assumption of straight-
line trajectories. All such nucleon pairs compose a NN collision time list. A
NN collision with least collision time is selected from the list and executed
by Pythia (PYEVNW subroutine) with the hadronization temporarily turned-off,
as well as the strings and diquarks broken-up. The nucleon list and NN
collision time list are then updated. A new NN collision with least collision
time is selected from the updated NN collision time list and executed with
repeating the aforementioned steps till the NN collision list is empty.

With those procedures, the initial partonic state for a AA collision is
constructed. Then, the partonic rescatterings are performed, where the LO-pQCD
parton-parton cross section ranft ; field  is employed. After partonic
rescatterings, the string is recovered and then hadronized with the Lund
string fragmentation scheme resulting in an intermediate hadronic state.
Finally, the system proceeds into the hadronic rescattering stage and produces
the final hadronic state observed in the experiments.

The per-event ${\rm W}^{\pm}/{\rm Z}^{0}$ production yield is very low, e.g.,
${\rm d}N_{{\rm Z}^{0}}/{\rm d}y\sim 10^{-7}$ at forward-rapidity in the most
$20\%$ central Pb–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV, as shown in
Tab. 1. In this study, the relevant production channels are activated in a
user controlled approach by setting ${\rm MSEL}=0$ in Pythia together with the
following subprocesses switched on:

$\displaystyle f_{i}\overline{f}_{j}$ $\displaystyle\rightarrow$
$\displaystyle{\rm W}^{+}/{\rm W}^{-}$ $\displaystyle f_{i}\overline{f}_{j}$
$\displaystyle\rightarrow$ $\displaystyle g{\rm W}^{+}/{\rm W}^{-}$
$\displaystyle f_{i}\overline{f}_{j}$ $\displaystyle\rightarrow$
$\displaystyle\gamma{\rm W}^{+}/{\rm W}^{-}$ $\displaystyle f_{i}g$
$\displaystyle\rightarrow$ $\displaystyle f_{k}{\rm W}^{+}/{\rm W}^{-}$
$\displaystyle f_{i}\overline{f}_{j}$ $\displaystyle\rightarrow$
$\displaystyle{\rm Z}^{0}{\rm W}^{+}/{\rm W}^{-}$ $\displaystyle
f_{i}\overline{f}_{i}$ $\displaystyle\rightarrow$ $\displaystyle{\rm
W}^{+}{\rm W}^{-}$

for ${\rm W}^{\pm}$ production, and

$\displaystyle f_{i}\overline{f}_{i}$ $\displaystyle\rightarrow$
$\displaystyle\gamma^{*}/{\rm Z}^{0}$ $\displaystyle f_{i}\overline{f}_{i}$
$\displaystyle\rightarrow$ $\displaystyle g(\gamma^{*}/{\rm Z}^{0})$
$\displaystyle f_{i}\overline{f}_{i}$ $\displaystyle\rightarrow$
$\displaystyle\gamma(\gamma^{*}/{\rm Z}^{0})$ $\displaystyle f_{i}g$
$\displaystyle\rightarrow$ $\displaystyle f_{i}(\gamma^{*}/{\rm Z}^{0})$
$\displaystyle f_{i}\overline{f}_{i}$ $\displaystyle\rightarrow$
$\displaystyle(\gamma^{*}/{\rm Z}^{0})(\gamma^{*}/{\rm Z}^{0})$ $\displaystyle
f_{i}\overline{f}_{j}$ $\displaystyle\rightarrow$ $\displaystyle{\rm
Z}^{0}{\rm W}^{+}/{\rm W}^{-}$

for ${\rm Z}^{0}$ production. In aforementioned equations $f$ refers to
fermions (quarks) and its subscript stands for flavor code.

The production of the ${\rm W}^{\pm}/{\rm Z}^{0}$ boson is performed using a
triggered simulation approach. A normalization factor is needed to account for
the trigger bias effect. Unfortunately, this normalization factor is not able
obtaining from default Pythia run where the ${\rm W}^{\pm}/{\rm Z}^{0}$
production subprocesses were not active. The comparison between results using
bias sampling technique and experimental data is available investigating trend
of the observable only. Therefore we apply a self-normalized rescaling
technique to both of the experimental observables and the model calculations
with respect to a reference value at a given argument (such as $N_{\rm part}$
in Fig. 1), as indicated in Tab. 1 and Fig. 1.

As the Quark Gluon Matter and Hadronic Matter are both nearly transparent to
the ${\rm W}^{\pm}/{\rm Z}^{0}$ bosons, the rescattering of ${\rm
W}^{\pm}/{\rm Z}^{0}$ bosons are not considered in the partonic and hardonic
evolutions in the PACIAE simulations. Thus the results of ${\rm W}^{\pm}/{\rm
Z}^{0}$ productions from PACIAE simulations are nearly the same as the ones in
pQCD for pp collisions.

Figure 3: The correlation between boson (${\rm W}^{\pm}$) and fermion (valence
quark) charge asymmetry in the elementary nuclear-nuclear collisions: n–n
(triangle-up); n–p (triangle-left); p–n (triangle-right); p-p (triangle-up) at
$\sqrt{s_{NN}}$=5.02 TeV in three symmetrical y intervals: $|y|<1$,
$2.5<|y|<4.0$ and full $y$ phase space. The linear fitting are shown for those
$y$ intervals as well. Figure 4: The correlation between boson (${\rm
W}^{\pm}$) and fermion (valence quark) charge asymmetry in symmetrical
nucleus-nucleus collisions: U–U (full circle), Pb–Pb (full square), Au–Au
(star), Cu–Cu (full diamond) at $\sqrt{s_{NN}}$ = 5.02 TeV in three
symmetrical y intervals: $|y|<1$, $2.5<|y|<4.0$ and full $y$ phase space. The
linear fitting are shown for those $y$ intervals as well.

##  III Results and Discussions

The comparison of the rescaled ${\rm Z}^{0}$ rapidity-differential density
${\rm d}N_{{\rm Z}^{0}}/{\rm d}y~{}/\left\langle T_{\rm AA}\right\rangle$ as a
function of ${N_{\rm part}}$ between PACIAE simulations and the ALICE
measurements is shown on the Fig. 1 panel (a) in Pb–Pb collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV. In this figure the $0$–$20\%$ centrality class
is the reference point. The panel (a) shows that ALICE measurements alice1
are well reproduced by the PACIAE simulations within uncertainties and the
centrality dependence is not strong. Similarly, Fig. 1 panel (b) gives the
centrality dependence of $R_{\rm AA}({\rm Z}^{0})$

$R_{\rm AA}=\frac{1}{\left\langle N_{\rm coll}\right\rangle}\frac{{\rm
d}N/{\rm d}y|_{\rm PbPb}}{{\rm d}N/{\rm d}y|_{\rm pp}}.$ (1)

Again, PACIAE simulations agree with the ALICE measurements within errors. It
is interesting that the centrality dependence here is nearly the same as the
one in panel (a). Finally, Fig. 1 panel (c) is the centrality dependence of
${\rm d}N_{{\rm Z}^{0}}/{\rm d}y$. Here the centrality dependence is stronger
than the one in panel (a) or (b), as it should be.

Figure 5: The correlation between boson (${\rm W}^{\pm}$) and fermion (valence
quark)charge asymmetry in the elementary nuclear-nuclear collisions and
symmetrical nucleus-nucleus collisions.

A similar model prediction for ${\rm W}^{\pm}$ production in Pb–Pb collisions
at $\sqrt{s_{\rm NN}}=5.02$ TeV is shown in Fig. 2. It is found that the
centrality dependence shown in Fig. 2 is quite close to the one in Fig. 1.

The asymmetry between ${\rm W}^{+}$ and ${\rm W}^{-}$ productions stemming
from the isospin effect can be studied with the ${\rm W}^{\pm}$ charge
asymmetry

$A_{\rm{\rm W}^{\pm}}=\frac{N_{{\rm W}^{-}}-N_{{\rm W}^{+}}}{N_{{\rm
W}^{-}}+N_{{\rm W}^{+}}}.$ (2)

as a function of colliding system valence quark charge asymmetry

$A_{v}=\frac{N_{u}-N_{d}}{N_{u}+N_{d}},$ (3)

where $N_{u}$ ($N_{d}$) stands for the total number of valence $u$ ($d$) quark
in the colliding system.

The correlation between ${\rm W}^{\pm}$ and valence quark charge asymmetry in
the elementary nucleon–nucleon collisions: n–n (triangle-up), n–p (triangle-
left), p–n (triangle-right) and p–p (triangle-down) at $\sqrt{s}=5.02$ TeV at
different $y$ intervals: $|y|<1$, $2.5<|y|<4.0$, and full y phase space are
shown in Fig. 3. All those results in three $y$ intervals can be fitted by a
linear function with different slopes. We can also find a feature that the
positive correlation flips into negative correlation from n–n to n–p to p–n
and to p–p in these three different rapidity intervals. It shows that the
slope obtained at mid-rapidity is gentler than that at forward rapidities.
This is due to that the isospin effect on ${\rm W}^{\pm}$ production is more
pronounced at forward rapidities zconesa .

In Fig. 4 the correlation between ${\rm W}^{\pm}$ and valence quark charge
asymmetries in minimum bias symmetrical nucleus-nucleus: U–U (full circle),
Pb–Pb (full square), Au–Au (star), Cu–Cu (full diamond) collisions at
$\sqrt{s}=5.02$ TeV in three different $y$ intervals:$|y|<1$, $2.5<|y|<4.0$,
and full y phase space are given. The linear fitted results are shown as well.
Also different $y$ interval has different fitted slope. The valence quark
charge asymmetry factor $A_{v}$ are all negative, while the ${\rm W}^{\pm}$
charge asymmetry factor $A_{\rm{\rm W}^{\pm}}$ are all positive, that is to
say the correlation have no sign-change form U–U to Pb–Pb to Au–Au and to
Cu–Cu collision systems. If we put the data from Fig. 3 and Fig. 4 together,
as shown in Fig. 5, we can get very good linear correlations for all collision
systems at different y intervals.

##  IV Summary

The parton and hadron cascade model PACIAE is employed to simulate the
dynamical production of ${\rm W}^{\pm}$ and ${\rm Z}^{0}$ bosons in
nucleon–nucleon (NN) and nucleus–nucleus (AA) collisions for the first time.
The rescaled ${\rm d}N/{\rm d}y/\left\langle T_{\rm AA}\right\rangle$ and
$R_{\rm AA}$ for ${\rm Z}^{0}$ bosons measured by ALICE in p–p and Pb–Pb
collisions at centre-of-mass energy $5.02$ TeV alice1  are reproduced. The
correlations between ${\rm W}^{\pm}$ charge asymmetry ($A_{{\rm W}^{\pm}}$)
and valence $u$\- and $d$-quark number asymmetry ($A_{\nu}$) are studied in
three rapidity intervals (mid-rapidity $\left|y\right|<1$, forward rapidities
$2.5<\left|y\right|<4$, and the full rapidity region). In each rapidity
interval, a linear scaling behavior between $A_{{\rm W}^{\pm}}$ and $A_{\nu}$
is observed by combing the results in different elementary NN (p–p, p–n, n–p
and n–n) collision systems. Results obtained from various AA (Cu–Cu, Au–Au,
Pb–Pb and U–U) collision systems well follow the scaling established by NN
collisions. The nuclear-modified PDF (nPDF) is not adopted in the AA collision
setup in this analysis. And the dynamic shadowing on nucleon implemented in
PACIAE simulations may affect only the probability of binary collisions in a
given AA collision, but not the kinematics of the hard-scattering out-going
particles (e.g., ${\rm W}^{\pm}/{\rm Z}^{0}$). The universal scaling behavior
between $A_{{\rm W}^{\pm}}$ and $A_{\nu}$ observed in both NN and AA collision
systems hence may provide a vacuum (without the nuclear modification) baseline
of the isospin effect on ${\rm W}^{\pm}$ production. The nuclear modification
on the vacuum nucleon PDF may change the in-going parton kinematics for the
hard processes and consequently may impact the $p_{\rm T}$\- and
$y$-distribution of the out-going particles. This effect may result in a
breaking of the $A_{{\rm W}^{\pm}}$–$A_{\nu}$ scaling established in vacuum if
looking into a given ${\rm W}^{\pm}$ $p_{\rm T}$–$y$ phase space. Therefore,
by measuring $p_{\rm T}$ and $y$ differentially, the deviation of the $A_{{\rm
W}^{\pm}}$–$A_{\nu}$ correlations between NN and AA systems should provide a
new tool to constrain the nPDF in a wide Bjorken-$x$ region. This study serves
as a benchmark for the research on the initial conditions of heavy-ion
collisions and also sheds light on the understanding of medium-induced higher-
order effects on ${\rm W}^{\pm}/{\rm Z}^{0}$ production in future works.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(11775094, 11805079, 11905188, 11775313), the Continuous Basic Scientific
Research Project (No.WDJC-2019-16) in CIAE, National Key Research and
Development Project (2018YFE0104800) and by the 111 project of the foreign
expert bureau of China.

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