# Cryptoasset Competition and Market Concentration in the Presence of Network
Effects

Konstantinos Stylianou  University of LeedsWoodhouse LaneLeedsUnited Kingdom
<EMAIL_ADDRESS>,  Leonhard Spiegelberg  Brown University115 Waterman
StProvidenceRhode IslandUSA<EMAIL_ADDRESS>,  Maurice Herlihy  Brown
University115 Waterman StProvidenceRhode IslandUSA<EMAIL_ADDRESS>and
Nic Carter  Coin MetricsBostonMassachusettsUSA<EMAIL_ADDRESS>
(2020)

###### Abstract.

When network products and services become more valuable as their userbase
grows (network effects), this tendency can become a major determinant of how
they compete with each other in the market and how the market is structured.
Network effects are traditionally linked to high market concentration, early-
mover advantages, and entry barriers, and in the cryptoasset market they have
been used as a valuation tool too. The recent resurgence of Bitcoin has been
partly attributed to network effects too. We study the existence of network
effects in six cryptoassets from their inception to obtain a high-level
overview of the application of network effects in the cryptoasset market. We
show that contrary to the usual implications of network effects, they do not
serve to concentrate the cryptoasset market, nor do they accord any one
cryptoasset a definitive competitive advantage, nor are they consistent enough
to be reliable valuation tools. Therefore, while network effects do occur in
cryptoasset networks, they are not a defining feature of the cryptoasset
market as a whole.

network effects, Metcalfe, Metcalfe’s Law, concentration, monopolization,
monopoly

††journalyear: 2020††copyright: rightsretained††conference: ; ††doi: ;

##  1\. Introduction

The rapid appreciation and popularization of cryptoassets over the past few
years has incited a large body of scholarship on understanding their behavior
and their positioning in the market, particularly financial markets. As
cryptoassets gradually became a household investment and transaction medium,
they began to invite greater regulatory and investor scrutiny, which created
the need to better understand their function as a market of their own and as
market that forms part of the greater economy. While early analyses focused on
simple economic illustrations of the functioning of cryptoasset networks in
isolation, later work started exploring market-wide phenomena, including the
dominance patterns of some cryptoassets over others.

Since cryptoassets are based on blockchain networks and are therefore network
markets, one important parameter that reflects and determines their behaviour
is the relationship between their userbase and their value. This relationship
has a long history in network markets under the theory of network effects.
Network effects theory states that the value of a product or service $V$ is
co-determined by its userbase $u$. Then, for products or services that obey
network effects, one can derive the value of the network, and therefore their
relative value to each other too, for a given userbase assuming that the
relationship between $V$ and $u$ is known, for example $V\propto nlog(u)$,
$V\propto u^{2}$, $V\propto 2^{u}$ etc.

Initially, this insight attracted attention because of its predictive
potential of cryptoasset valuation. Indeed a number of studies attempted to
develop valuation models based on network effects that could be used by
investors to predict the future value of their assets and the value of the
market as a whole. However, the implications of network effects go far beyond
valuation and, understood properly, they inform also the structure and
competitiveness of the market making them a key input into policy-making and
regulatory decisions. Most notably, markets that are characterized by network
effects are commonly thought to be winner-take-all markets, where first mover
advantage is key, entry barriers are high, networks hit tipping points of no
return, and contestable monopolies or high concentration can be the natural
state of the market. This is for two reasons: firstly, because the value of
joining a network is increasing in the number of other network adopters,
because the bigger the number of existing adopters the greater the utility
every new adopter derives from it (pure network effects), and secondly,
because for every new adopter joining the network, existing adopters also
benefit (network externalities). In both cases bigger equals better
(everything else equal), creating an incentive for users to join the network
where the value will grow larger both for new and for existing users, which
creates a snowball effect. This kind of power concentration in networks that
exhibit network effects usually makes regulators uneasy, and therefore, if
cryptoassets exhibit network effects, they would (and should) attract higher
regulatory and investor scrutiny.

Extant literature on network effects in cryptoassets is limited and has
focused almost exclusively on confirming or rejecting, usually for Bitcoin
only, a specific application of network effects, namely Metcalfe’s law, which
states that the value of a network is proportional to the square of its users
($V\propto u^{2}$), and, if confirmed, it would be a useful valuation tool.
However, this line of literature presents only a binary distinction between
the existence or not of a specific type of network effects, focuses only on
valuation, uses sub-optimal data, and has also been temporally limited to the
period before the recent resurgence in mid 2019, or excludes periods,
therefore missing key parts in the cryptoasset market evolution.

By contrast, our analysis takes a more comprehensive view of network effects
in cryptoassets, and, while it confirms that network effects occur in
cryptoassets, it shows that they do not have the usual implications associated
with them in terms of according competitive advantages, resulting in market
concentration, or serving as a reliable valuation tool. Firstly, we define
network effects to occur when the value of a cryptoasset network changes
supra- or infra-proportionately to changes in its userbase, thereby showing
both positive and reverse network effects, while not being constrained by a
specific version of network effects. We also use two proxies for value and
userbase each to better capture what users perceive as the value of the
network and how the network size (userbase) should be measured, and we base
our results on cleaner vetted data. Moreover, we examine multiple cryptoassets
to get a broader view of the industry, as opposed to previous works which
focused on Bitcoin. Lastly, our analysis covers the entire history of the
studied cryptoassets, which includes the valuation spikes and subsequent
declines in 2014, 2017 and 2019. The spike in 2019 and the preceding decline
from the heights of 2017 are particularly valuable because they help us show
that the results obtained in previous studies which sampled only up to early
2018 do not hold based on more recent history.

Figure 1. Price and userbase development since 2016.

##  2\. Background, Motivation and Implications

Network effects were first studied in the 1970s to more accurately capture the
value and growth of telecommunications networks (Rohlfs, 1974). The intuition
was that when the nature of a product or service is such that it relies on
linking users together, the value of the product $V$ is co-determined by its
userbase $u$. More specifically, for every user added to the userbase of a
product, value is created not just for the joining user but for existing users
as well. As a result, each new user derives value from joining a network that
is relative to the size of the network (pure network effects) and creates an
externality in the form of value that is captured by the network of existing
users (network externality). Conversely, for every exiting user, value is lost
both for the exiting user and for existing users. This type of network effects
was called direct network effects to distinguish it from later extensions to
the theory, which accounted for the effects changes in the network’s userbase
have on complementary products and services developed for that network (Church
et al., 2008). This latter type was called indirect network effects, and it is
not the kind that will concern us here.

The powerful implication of (direct) network effects is the increasing returns
to the userbase and ultimately to the product exhibiting network effects.
Because for products that exhibit network effects every new adopter makes the
product more valuable relative to existing size of its network, it creates
incentives for other adopters to adopt the product with the bigger network
over its competitors. Consequently, the more the userbase grows the more it
invites further growth rendering the product increasingly more valuable and
competitive. The exact relationship between value and userbase can vary; While
one can say that in the most basic version of network effects the value of a
product grows linearly with the number of users added to its userbase
($V\propto u$) (Swann, 2002), most commonly network effects are used to
describe relationships that are logarithmic ($V\propto nlog(u)$) (Briscoe et
al., 2006), quadratic ($V\propto u^{2}$) (Metcalfe, 2013) or other (e.g.
$V\propto 2^{u}$).

Network effects have found application in numerous industries and business
models ranging from telecommunications (Birke and Swann, 2004; Gallaugher and
Wang, 2002), to web servers, PC software (Gandal, 1995), airline reservation
systems, ATMs (Economides and Salop, 1992), and platform systems (Church and
Gandal, 2005). Indeed, the intuition and implications of network effects have
been so pervasive that they have been invoked in any industry where the
consumption or use of a product by consumers makes the product more valuable
for others (for a collection of relevant literature see (Garcia-Swartz and
Garcia-Vicente, 2015). It is no surprise that cryptoassets have also been
hypothesized to exhibit network effects. The combination of the inherent
network nature, the meteoric rise in popularity (read: userbase), and the
substantial price volatility (read: value) has suggested a strong-if elusive-
relationship.

The particular motivation behind the study of network effects in cryptoassets
has so far been to discover a valuation formula: if we know the function
between userbase and value, then with informed guesses on the network’s growth
we can predict future prices (Peterson, 2018; Van Vliet, 2018; Shanaev et al.,
2019). But valuation formulas reduce network effects down to a binary
distinction represented by a single function. While useful as prediction tools
and high-level descriptors of cryptoasset trends, valuation formulas provide
little granularity.

Our motivation and goal is, instead, to provide more high-level view of how
network effects influence the cryptoasset market as a whole, and particularly
what they say about the potential for concentration in the market and about
competitive (dis)advantages of one cryptoasset over others. These are the most
impactful implications of network effects, and they are desirable for those
networks that can exploit them, but undesirable for their competitors or for
regulators who have to deal with concentrated markets. We work with numerous
cryptoassets so that we can obtain a market-wide overview (limited by how big
and representative our sample is), and we study them from their inception
until early 2020 which allows us to capture all historically important phases,
including the resurgence in 2019, which extant literature has not had a chance
to consider. This type of approach allows us to draw insights about the
structure and competitive dynamics of the cryptoasset market. It goes back to
the early wave of ”Bitcoin maximalism”, which stood for the idea that the
optimal number of currencies as alternatives to the mainstream financial
system is one, and altcoins will eventually be rendered obsolete as more and
more users gravitate toward the biggest, most stable, most widely accepted
cryptocurrency, namely Bitcoin. At the time, Bitcoin maximalism was rejected
by Vitalik Buterin, the creator of Ethereum, correctly pointing out that the
cryptoasset universe is not a homogeneous thing, and that therefore there is
no one single ”network” around which network effects would form (Buterin,
2014). We expand on that thinking.

Looking at network effects to study the competitive dynamics of the
cryptoasset market and its potential to concentrate around one or a small
number of cryptoassets can provide useful insights for industrial policy.
Normally, a showing that cryptoassets exhibit network effects would suggest
that early cryptoassets have a first-mover advantage and may lock the market
in (Economides, 1996; Katz and Shapiro, 1985; Gandal and Halaburda, 2016),
even if they are intrinsically inferior to other comparable cryptoassets
(Farrell and Saloner, 1985; Hagiu and Rothman, 2016; Briscoe et al., 2006).
While, the market seems to have moved away from that danger, network effects
theory also suggests that, assuming homogeneity, once a cryptoasset hits a
tipping point, it may fully prevail because new users will always prefer the
cryptoasset with the larger userbase (the so called ”winner-take-all” markets,
which Bitcoin maximalism relied on) (Economides, 1996; Katz and Shapiro,
1985). Homogeneity is, of course, a matter of degree, and it is still likely
that, if a cryptoasset exhibits stronger network effects than its peers, it
can prevail at least within a sub-segment of the market. The flip side of
network effects can also be observed, whereby the loss of a user results in a
supra-proportionate loss of value (i.e. more value than the user intrinsically
contributed individually), which incites further losses and so on. This means
that rapid depreciation is more likely in cryptoassets characterized by
network effects. The rapid appreciation and depreciation cycles coupled with
the winner-take-all characteristic can in turn result in cryptoasset markets
that are successively dominated by a new winner in every era (successive
contestable monopolies). Then, if this is the natural state of the market,
artificially forcing more competition may not be optimal.

These insights are well-applicable in financial markets. For instance, the
influential ”Cruickshank report”, an independent report on banking services in
the United Kingdom prepared for the UK Treasury, which has in turn influenced
regulatory and legal decisions (noa, 2007, 2011), warned about the far
reaching implications of network effects: ”Network effects also have profound
implications for competition, efficiency and innovation in markets where they
arise. Establishing critical mass is the first hurdle, as the benefits to
customers and businesses of a network arise only gradually with increasing
use. It is possible to imagine a world in which electronic cash is widely held
and used, for example, but much harder to see how to get there. Once a network
is well established, it can be extremely difficult to create a new network in
direct competition. … Where network effects are strong, the number of
competing networks is likely to be small and the entry barriers facing new
networks will be high” (Cruickshank, 2000). As the fintech industry is heating
up, network effects have also been cited there as a strong factor in
entrenching existing market power of financial services (see e.g. the recent
proposed acquisition of Plaid by Visa (Cyphers, 2020)), and such risks have
also been highlighted in the cryptoasset market, with models showing that
certain conditions can allow cryptoasset markets to become oligopolies and
market players entrench their position in the market (Arnosti and Weinberg,
2018; Cong et al., 2020).

##  3\. Prior Literature and Contribution

A number of papers have investigated aspects of the application of network
effects in cryptoasset networks. The focus has been to determine whether the
value of cryptoassets (and mainly Bitcoin) complies with network effects, and
in particular on whether it follows Metcalfe’s law, which is the most popular
iteration of network effects and stipulates that the value of a network grows
at a rate proportional to the square of the number of users ($V\propto
u^{2}$).

The early influential analysis by Peterson (Peterson, 2018) remains the point
of reference. Peterson developed a valuation model for Bitcoin’s price based
on Metcalfe’s law for the period 2009-2017, using wallets as a proxy for
users, Bitcoin prices as the proxy for value, and a Gompertz function to
account for growth. He found that the price of Bitcoin follows Metcalfe’s law
with R-square of 85 percent. In a revised version of the original paper that
extends through 2019, Peterson re-confirms the application of Metcalfe’s law
to Bitcoin (Peterson, 2019). However, he excludes significant periods of time
on the grounds of price manipulation, during which the value of the Bitcoin
network, as measured by the USD price of Bitcoin, lies well outside of
Peterson’s model predictions. Van Vliet (Van Vliet, 2018) enhanced Peterson’s
model by incorporating Rogers’ diffusion of innovation models to better
capture population parameters and growth rates. By doing so, van Vliet raised
R-squared to 99 percent. Shanaev et al. (Shanaev et al., 2019) acknowledge the
utility of Peterson’s and van Vliet’s analyses but depart from them in that
their model does not rely on historical data for the estimation of the
coefficient of proportionality, which raises an endogeneity problem. They
still use Metcalfe’s law but only as only as one of the building blocks of
their model. Civitarese (Civitarese, 2018) rejects the applicability of
Metcalfe’s law to the value of the Bitcoin network by running a cointegration
test between price and an adjusted number of wallets’ connections.

Gandal and Halaburda (Gandal and Halaburda, 2016) use a completely different
approach to examine the existence of network effects in cryptoasset networks.
They define network effects as the reinforcement effects the price of a
cryptoasset has on the price of another cryptoasset. With Bitcoin as the base
cryptoasset, the idea is that, if network effects are in place, as Bitcoin
becomes more popular (price increase), more people will believe that it will
win the winner-take-all race against other cryptoassets resulting in further
demand and higher prices. Therefore, network effects would manifest themselves
as an inverse (negative) correlation between the prices of the sampled
cryptoassets. For the period May 2013 - July 2014, their results showed signs
of network effects after April 2014.

Our analysis complements and differs from prior literature in several ways.
Firstly, we do not focus on a specific network effects formula; we rather look
at when, to what degree, in which cryptoassets, and for what proxies of value
and userbase network effects are observable (defined as supra-proportional
change in value relative to userbase) regardless of which particular
curve/function they follow. Secondly, we go beyond Bitcoin to examine six
cryptoassets that we have selected as representative of different features and
characteristics to better be able to observe potential industry-wide trends.
This helps us notice whether one cryptoasset has the potential to dominate the
market or multiple cryptoassets benefit from the same network effect forces.
Thirdly, we use different parameters as proxies for value and userbase to more
fully capture the functionality and usage of cryptoassets in the market.
Importantly, we do not rely on the total number of users as a proxy for
userbase like extant literature, because many of those addresses are dormant
or permanently inaccessible and therefore economically irrelevant. Fourthly,
we study the full history of cryptoassets from their inception to today which
allows us to observe their different phases, including the price collapse in
2018 and the resurgence in mid-2019, which dramatically change the picture of
network effects and which have been missed by previous studies. Lastly, we
work with data sets that have been meticulously cleaned to filter out spurious
or manipulative activity, which improves the accuracy of our results compared
to data-sets that are pulled unfiltered from the network. Our analysis
confirms the existence of network effects, but also that they do not have the
results usually associated with them on the market.

##  4\. Methodology and Development

We study the application of network effects in Bitcoin (BTC), Dogecoin (DOGE),
Ethereum (ETH), Litecoin (LTC), XRP and Tezos (XTZ). The selection of these
cryptoassets was made on the basis of diversity and feasibility. We aimed to
study cryptoassets that exhibited different attributes in terms of age, market
capitalization and any special features that make them stand out from other
competing cryptoassets in order to build a representative sample of the
crypto-economy (Irresberger et al., 2020). We also limited the study to
cryptoassets for which we could get reliable, standardized time-series data
from the cryptoassets’ initial release to the time of the study (Metrics,
2020). The unreliability of the prices reported by exchanges in the early days
of the industry led us to consider Bitcoin from July 2010, Litecoin from March
2013, and XRP from August 2014—the rest from their beginning. Table 1
summarizes the attributes of each chosen cryptoasset.

| Age | Market cap (2020) |  Features
---|---|---|---
Bitcoin (BTC) | Old (2009) | V. Large ($170B) |  Popularity, first cryptocurrency, UTXO based
Dogecoin (DOGE) | Old (2013) | V. Small ($0.3B) |  ”Joke cryptocurrency”, early BTC contender , UTXO based
Ethereum (ETH) | Medium (2015) | Medium ($25B) |  Turing complete, programmable, account based
Litecoin (LTC) | Old (2011) | Small ($2.6B) |  First major BTC fork, UTXO based
XRP | Old (2012) | Small ($8B) |  Consensus, fintech-orientated, account based
Tezos (XTZ) | New (2018) | Small ($1.7B) |  Centralized PoS, on chain governance, account based
Table 1. List of studied cryptoassets, chosen to cover different
characteristics.

We first define network effects. Network effects occur where the value of the
network $V$ grows supra-proportionately to the number of users $n$ that
participate in the network. Reverse network effects occur where the value $V$
drops supra-proportionately to the number of users $n$ that leave the network.
Unless there is a reason to distinguish between positive and reverse network
effects, we collectively refer to them as network effects. Therefore, we
define network effects to occur in cryptoassets when a positive value change
$\Delta V>0$ is larger than a positive userbase change $\Delta u>0$, or when a
negative value change $\Delta V<0$ is smaller than a negative userbase change
$\Delta u<0$. Notice that we do not consider that network effects apply when
value and userbase move in different directions, e.g. when the value increases
while the userbase decreases, regardless of which increases or decreases more.

Thus, network effects occur if

$\Delta V>\Delta u\geq 0\ \lor\Delta V<\Delta u\leq 0$

In our analysis we define change at time $t$ similar to log returns, i.e.

(1) $\displaystyle\Delta V$ $\displaystyle:=\ln\frac{V_{t+1}}{V_{t}}$ (2)
$\displaystyle\Delta u$ $\displaystyle:=\ln\frac{u_{t+1}}{u_{t}}$

Then, we identify appropriate proxies to represent value $V$ and userbase $u$.
To represent $V$ we use two proxies: (a) token price and (b) transaction
value. The two proxies represent different aspects of the value users assign
to cryptoassets. In theory, even one proxy applied to one cryptoasset would be
enough to demonstrate (or not) network effects (as has, for example, been done
in previous literature that relied only on token price), assuming the proxy
and cryptoasset are representative. However, because cryptoassets are
differentiated resulting in diversified usage patterns, and because the chosen
proxies express different ways by which users perceive the value of the
network, a multitude of cryptoassets and proxies was used in an effort to
better represent the industry.

Token Price (PriceUSD): The first parameter we use is token price, which is
the fixed closing price of the asset as of 00:00 UTC the following day (i.e.,
midnight UTC of the current day) denominated in USD (for a detailed
explanation of Coin Metric’s methodology on toke price see (Metrics, 2020)).
Token price expresses value in terms of market forces, namely the point at
which supply meets demand. It is the value that users as market participants
collectively assign to a given cryptoasset by deciding to buy and sell at that
price level. We assume that the studied cryptoassets trade under normal market
conditions; any acknowledgement of price manipulation that may have occurred
at times has been accounted for in the cleaning of data by Coin Metrics
(Metrics, 2020).

Transaction Value (TxTfrValAdjUSD): The second proxy of choice is transaction
value, which expresses the USD value of the sum of native units transferred
between distinct addresses per day removing noise and certain artifacts to
better reflect the real economically relevant value circulating in the
network. The assumption is that as the network becomes more valuable to users,
they will use it more frequently and/or to transfer greater value among them.
Therefore, transaction value as a proxy sees cryptoassets as means of
transaction. We considered and rejected transaction count as an appropriate
proxy, because on some networks a large number of recorded transactions are
unrelated to value transfer, but rather to the operation of the network, e.g.
consensus formation on Tezos (Perez et al., 2020). One could retort that even
these non-value-carrying transactions reflect engagement with the network and
that therefore are an indication of the value of the network to users. Even
so, lumping together value-carrying and operational transactions would taint
the comparison across cryptoassets, since on some cryptoassets the majority of
transactions are operational (e.g. Tezos, see (Perez et al., 2020)), while on
others value-carrying (e.g. Bitcoin).

Next, to represent $u$ we select the following proxies: (a) addresses with
non-zero balance (b) trailing 6-month active addresses and . Different ways to
represent userbase more fully captures the relationship between value and
userbase. We considered and rejected counting userbase based on total number
of addresses (like all previous literature), because of the large number of
inactive addresses. Contrary to other industries where network effects have
been studied and where inactive users are eventually purged from the network
(e.g. mobile phone subscriptions, social networks), so that total user count
may still be a good approximation of the economically meaningful userbase,
this is not the case with cryptoassets. Instead we opted for two variants of
addresses with non-zero balance, as defined below.

Addresses with Non-Zero Balance (AdrBalCnt): This proxy represents the sum
count of unique addresses holding any amount of native units as of the end of
that day. Only native units are considered (e.g., a 0 ETH balance address with
ERC-20 tokens would not be considered). The utility of this proxy lies in that
it excludes all non-economically active addresses, the assumption being that
addresses with zero balance are dormant (similar to bank accounts with zero
balance). This choice responds to criticism that has been raised with regard
to extant literature that tended to use all addresses or wallets as a proxy
for users. Despite our choice of improved metric, it still remains a fact that
there is no one-to-one mapping between addresses and actual users, which is a
common problem to any network or service, e.g. the same person may have
multiple bank accounts. While there are methods to de-cluster actual users
from wallets and addresses, these are not sufficiently precise and are
unavailable or inapplicable across cryptoassets (Harrigan and Fretter, 2016).
We also acknowledge that on networks with lower transaction fees it is easier
to generate and/or maintain addresses with balance, and to counter that we
could raise the amount of native units the counted addresses should have, but
this would introduce a subjectivity question without even fully eradicating
the initial problem of spurious addresses.

Trailing 6-Month Active Addresses (6MAdrActCnt): This proxy counts all unique
addresses that have been active at least once over the trailing 6-month period
from the time of measurement. Repeat activity is not double-counted.
Traditionally, most userbase measurements are taken in time frames that range
from one month to one year. Given that cryptoassets are of relatively young
age, which may suggest that their userbase is expected to interact with them
less frequently, and that part of their utility involves simply owning them,
which does not generate any activity, we decided that a 6-month time frame
sufficiently captures active userbase.

Before we derive network effects, we first calculate the Pearson correlation
between value $V$ and users $u$ which is informative in terms of their overall
relationship. Next, we obtain relevant measurements of network effects. We
rely predominantly on the PriceUSD-AdrBalCnt pair of proxies for value and
userbase, but additional measurements are in the Appendix. To see how
prevalent network effects are in the studied cryptoassets we calculate the
ratio of total days to the days where network effects were observed
(separately for positive and reverse) for each cryptoasset. To see how strong
network effects are we calculate the ratio of total days to the sum of the
network effects observations over the days they occurred for each cryptoasset
(separately for positive and reverse). To see how strong network effects are
in cryptoassets relative to each other we reduce to a 100 day period. The
results are presented in Part 5 and the analysis of the results in Part 6.

Metric abbr | Metric meaning
---|---
PriceUSD | Token price
TxTfrValAdjUSD | Transaction value
AdrBalCnt | Addresses with non-zero balance
6MAdrActCnt | Trailing 6-month active addresses
NFX | Network effects
Table 2. Legend of metrics in use.

##  5\. Results

We are looking for network effects in the relationship between value $V$ and
users $u$ of various cryptoassets as represented by the proxies defined
previously. Four pairs (2x2 proxies) are possible:

  * •

`Token Price - Addresses with Non-Zero Balance`: This pair demonstrates
network effects expressed as the change of monetary value of a cryptoasset
relative to the users that hold any amount of that cryptoasset. By counting
only accounts with non-zero balance, we filter out economically dormant users.

  * •

`Token Price - Trailing 6-month Active Addresses`: This pair demonstrates
network effects expressed as the change of monetary value of a cryptoasset
relative to the users that have been active at least once in the trailing
6-month period on that cryptoasset’s network. Counting all active users over a
recent time segment (usually 1, 6 or 12 months) is a common measurement of
network or platform userbase and less conservative than daily active users.

  * •

`Transaction Value - Addresses with Non-Zero Balance`: This pair demonstrates
network effects expressed as the change of transaction value of a cryptoasset
relative to the users that hold any amount of that cryptoasset.

  * •

`Transaction Value - Trailing 6-month Active Addresses`: This pair
demonstrates network effects expressed as the change of transaction value of a
cryptoasset relative to the users that have been active at least once in the
trailing 6-month period on that cryptoasset’s network.

Before we derive network effects, we calculate, based on the above pairs, the
Pearson correlation between value $V$ and users $u$ which tells us whether, as
a general matter, cryptoasset value and userbase are moving in the same
direction. This already provides an indication of whether cryptoassets become
more valuable as their adoption increases.

|  | User proxies
---|---|---
Cryptoasset | Value proxy | AdrBalCnt | 6MAdrActCnt
BTC | PriceUSD | 0.878760 | 0.800890
BTC | TxTfrValAdjUSD | 0.771601 | 0.734617
DOGE | PriceUSD | 0.532856 | 0.255025
DOGE | TxTfrValAdjUSD | 0.258791 | 0.141790
ETH | PriceUSD | 0.256837 | 0.475199
ETH | TxTfrValAdjUSD | 0.048093 | 0.214427
LTC | PriceUSD | 0.646814 | 0.844012
LTC | TxTfrValAdjUSD | 0.258648 | 0.431706
XRP | PriceUSD | 0.551157 | 0.803027
XRP | TxTfrValAdjUSD | 0.189622 | 0.278429
XTZ | PriceUSD | -0.477943 | -0.681394
XTZ | TxTfrValAdjUSD | -0.169407 | -0.240346
Table 3. Pearson correlation between value and user proxies

It is evident that only BTC shows a strong correlation between value and
userbase, at least when userbase is measured by our main proxy of total
addresses with non-zero balance (AdrBalCnt), with LTC showing the next highest
correlation, which is, however, average and only holds when value is measured
as value in fiat currency (PriceUSD). Correlations when userbase is measured
as addresses that have been active in the trailing 6-month period
(6MAdrActCnt) tend to be higher although still not consistently so. Higher
correlation using 6MAdrActCnt might be explained on the grounds that user
activity picks up during phases of large price movements. Overall, the
mediocre and inconsistent correlations between value and userbase provide a
first indication that a blanket conclusion that the cryptoasset market is
characterized or not by network effects is unwarranted.

Next, we obtain relevant measurements based on the PriceUSD-AdrBalCnt pair of
proxies for value and userbase as presented in Table 4 (additional
measurements for other pairs are in the Appendix). As explained in the
methodology, we believe these are the most appropriate proxies. Column 5 of
Table 4 shows prevalence of network effects for each cryptoasset as calculated
by the ratio of total days to the days where network effects were observed
(separately for positive and reverse). Column 6 of Table 4 shows strength of
network effects as calculated by the ratio of total days to the sum of the
network effects observations over the days they occurred for each cryptoasset
(separately for positive and reverse). Column 7 of Table 4 shows relative
strength of network effects across cryptoassets by reducing to a 100 day
period. This allows us to compare how strong network effects are across
cryptoassets regardless of how prevalent they are across them.

Crypto | Total days |  Days of NFX (pos-reverse) |  Sum (strength) of NFX (pos-reverse) |  Ratio of total days/NFX days (pos-reverse) |  Relative strength of NFX (pos-reverse)
---|---|---|---|---|---
Bitcoin | 3461 |  1434 |  47.1 |  0.400 |  3.28
|  |  243 |  9.2 |  0.070 |  3.78
Doge | 2175 |  695 |  30.7 |  0.310 |  4.40
|  |  295 |  11.1 |  0.130 |  3.70
Ethereum | 1614 |  707 |  33.5 |  0.430 |  4.70
|  |  12 |  0.2 |  0.007 |  1.66
Litecoin | 2473 |  722 |  34.5 |  0.290 |  4.77
|  |  354 |  11.2 |  0.140 |  3.16
XRP | 1973 |  901 |  41 |  0.450 |  4.55
|  |  - |  - |  - |  -
Tezos | 558 |  244 |  10.8 |  0.430 |  4.40
|  |  - |  - |  - |  -
|  |  |  |  |
Table 4. Network effects measurements based on the Token price - Addresses
with non zero balance proxy pair

(a) BTC

(b) DOGE

(c) ETH

(d) LTC

(e) XRP

(f) XTZ

Figure 2. Network effect observations and distribution (blue: positive NFX,
red: reverse NFX, white: no NFX); userbase measured by total addresses with
non-zero balance, value measured by USD token price.

##  6\. Analysis

Our results are useful in reaching a number of conclusions on how network
effects inform the structure and evolution of the cryptoasset market.

### (1) Network effects do not provide precise valuation predictions:

The most common application of network effects theory has been to draw
insights into future cryptoasset pricing based on the evolution of their
userbase. Our results indicate that network effect observations in
cryptoassets are frequent but inconsistent and therefore they cannot be relied
on, generally, as a valuation tool as previous literature suggests (Figures 2
and 3). They are most frequent in XRP (45 percent of time in the pair Token
Price-Addresses with Non-Zero Balance) and least frequent in LTC (29 percent
of time in the same pair). While they appear more consistent in ETH and XRP,
their results can be somewhat misleading at first glance: ETH’s and XRP’s
userbase (AdrBalCnt) was constantly increasing and so any supra-proportionate
increase in price registered as a (positive) network effect observation (blue
lines in (c) and (e) in Figure 2). However, the positive network effect
observations are frequently punctuated by days/periods of no network effect
observations during which the price either does not rise supra-proportionately
to userbase or drops. In cryptoassets such as BTC and LTC, where userbase
fluctuates, it is easier to notice the changes in network effects trends (blue
and red lines in (a) and (d) in Figures 2 and 3), even through network effect
frequency is comparable to ETH and XRP. Therefore, it is hard to conclude that
in any cryptoasset network effects exhibit constant patterns that, if extended
into the future, can hold predictive value. This does not mean that we do not
acknowledge the exponential long-term price increase of some cryptoassets
(Figure 1), but we note that this is not linked consistently to their userbase
growth, which is what network effects theory suggests. One explanation of why
our results do not support the conclusions of previous studies can relate to
the different time frames. Most previous studies’ datasets end around the
valuation peak of January 2018, missing the precipitous fall in 2018 and the
subsequent rise in 2019, which upend the relatively smoother network effect
curves of valuations up until the end of 2017. Another explanation relates to
methodology. For example, Peterson’s revised study, which covers up to 2020
and confirms the finding of the paper’s previous popular version that
Bitcoin’s valuation follows Metcalfe’s law, excludes certain sizeable time
periods, which, if accounted for, show a poor(er) fit (Peterson, 2019). A
third explanation relates to the proxies used. Some previous studies rely on
wallets (total addresses) as the proxy for userbase, which is a more crude
measurement than our preferred addresses with non-zero balance, as the latter
show only economically active users and are therefore a better approximation
of relevant userbase.

### (2) Reverse network effects are also noticeable meaning that cryptoassets
are vulnerable to rapid decline, not just conducive to rapid growth:

While network effects have mostly been used to describe growth patterns, they
are equally applicable in describing decline. Reverse network effects reflect
situations where a decrease in users is linked to a larger decrease in value.
Such observations are important, because they show that each user loss incurs
a greater loss of value and therefore expose the potential for a rapid decline
of the network once user exodus begins. Reverse network effects therefore
highlight the precariousness of success (as measured by proxies of value).
Most cryptoassets exhibited at least one prolonged period where reverse
network effects were dominant, during which phases their value contracted
disproportionately to the contraction of their userbase ending up less
valuable than their userbase size would otherwise suggest or mandate during
that period. This is noticeable both when userbase is measured by addresses
with non zero balance, but it is even more pronounced when userbase is
measured as trailing 6-month active addresses (Figure 3). This makes sense
since the users active in the trailing 6-month period are more likely to be
responsive to price fluctuations compared to users who simply hold some
balance on their account. From Figure 3 it is also evident that user
disengagement is almost consistently observed after every price crash (as
manifested through the reverse network effects that begin 6 months after many
of the crashes), and the fact that price continues to decrease supra-
proportionately to userbase, as measured by active users in the trailing
6-month period, 6 months after the crash, may be indicative of the lasting
effects user exodus has on the value of cryptoasset networks. Generally,
however, while reverse network effects serve as a cautionary note that rapid
decline of value can be triggered by user exit, they are weaker in magnitude
than positive network effects (Table 4). So, overall, positive network effects
(albeit inconsistent) still seem to characterize cryptoasset networks.

### (3) Cryptoassets do not seem to be a winner-take-all market:

A common corollary of network effects is that they eventually cause the market
to gravitate toward oligopolistic structure, since, everything else equal,
users prefer to join the network where the value from their joining will be
maximized. This causes a ”rich-get-richer” effect where the most valuable
network continues to become even more valuable as users prefer to join that
over others. Such markets tend to become oligopolistic, with the usual
downsides of such industry structure (higher prices, reduced output, entry
barriers; lower variety and innovation), and can therefore be a cause for
concern. For this to be more likely to happen the various networks
(=cryptoassets) must be undifferentiated and switching among and multi-homing
across networks must be rare or costly (Schmalensee, 2011). These features do
not seem to characterize the cryptoasset market, which accordingly appears
less susceptible to a winner-take-all trend, at least on account of network
effects. Indeed, of the thousands of available cryptoassets many serve
different purposes, and users can own multiple cryptoassets at the same time
and enter and exit their networks without friction. As evidenced by our
results, the fact that the various cryptoassets we studied exhibit network
effects of comparable relative strength (Column 7 in Table 4), and that they
retain their userbase and valuation cycles (Figure 1) seems to suggest that
the underlying market features, including network effects, do not lead it
toward an oligopolistic structure.

### (4) Network effects strength across cryptoassets is comparable and
therefore network effects do not accord a single cryptoasset a strong
comparative advantage over its peers, undermining fears of concentration:

Besides frequency and duration, i.e. what period of a cryptoasset’s lifetime
is dominated by network effects, another useful parameter of network effect
observations in cryptoassets is their strength, i.e. the magnitude of the
impact of a userbase change to value change (Shankar and Bayus, 2003). Strong
network effects can be indicative of higher homogeneity or cohesion within the
network, where the addition of each new user (e.g. investor) affects existing
users of that closely-knit network more than if it was a different looser
network. In turn, this is reflected in the value of the network, or they may
be indicative of stronger reputational effects, where the addition of each new
user signals major changes for the network, which are then reflected in its
value. Our results show that the comparative strength of network effects
across the studied cryptoassets is similar (Table 4). This leads us to believe
that no single cryptoasset benefits from network effects significantly more
than its peers and therefore that no cryptoasset enjoys an overwhelming
competitive advantage over its peers on account of network effects. A
necessary corollary observation is that network effects accrue at similar
levels to the studied cryptoassets, which means that network effects as a
phenomenon, characterizes the cryptoasset industry as a whole (at least based
on our sample), not just Bitcoin, which has been the main subject of many of
extant studies in the area. This is not a surprising finding, but it is worth
highlighting that it lends support to the previous point that the structure of
the cryptoasset market does not seem to be such where network effects lead it
to concentration around a small number of cryptoassets or that it helps
cryptoassets overtake their peers on account of network effects. This is most
likely because cryptoassets are differentiated and multi-homing and switching
are pervasive.

### (5) Network effects are not consistently observed during the early days of
cryptoassets and therefore it is doubtful that they can be relied on as a tool
to bootstrap a new cryptoasset:

A common business model when launching new products or services in digital
markets is to exploit network effects to quickly establish a growing foothold.
Particularly if the product or service is also the first of its kind to hit
the market, network effects can dramatically augment the first mover
advantage, everything else equal. Our results indicate that network effects
are not consistently observed in the studied cryptoassets during their early
days (the first year of data); in particular, DOGE, XTZ and LTC do not exhibit
consistent positive network effects neither by token price (PriceUSD) nor by
transaction value (TxTfrValAdjUSD) as proxies for value (Figures 2 and 5). The
lack of consistency is even more pronounced when userbase is measured by
active addresses in the trailing 6-month period, which is an instructive
measure here, because it tracks recent user activity which is the driver of
early adoption. In Figure 3 only BTC and ETH have a claim to positive early
network effects and in ETH they are sparser. This suggests that new
cryptoassets cannot necessarily hope that network effects will assist in their
initial uptake. It is useful to dispel this hypothesis because investors are
looking for patterns in events that may trigger valuation changes (e.g. the
hypothesis that cryptoasset value as measured in monetary terms increases once
the cryptoasset is listed on a major crypto-exchange).

### (6) Comparison between network effects on price and transaction value
reveals sensitivity to price, which can be a competitive disadvantage:

Extant literature has relied exclusively on token price as the proxy for
network value. Using transaction value too helps us draw useful comparisons.
For this, it is most instructive to rely on trailing 6-month active addresses
as the proxy for userbase, because this proxy is more responsive to value
fluctuations. Then, a comparison between the strength of network effects
measured by token price (PriceUSD) and by transaction value (TxTfrValAdjUSD)
reveals that some cryptoassets experience greater fluctuations in their
transaction value relative to their token price. During upturns, network
effects tell us that token price and transaction value increase more than the
userbase increases, and during downturns, reverse network effects show the
opposite. By comparing the ratios among cryptoassets of the sum of network
effects when value is measured by token price and the sum of network effects
when value is measured by transaction value one can observe differences in how
transaction value is affected among cryptoassets. Specifically, the ratios for
BTC, DOGE, ETH and LTC are similar ranging from 0.12 to 0.14 for positive
network effects and 0.07 to 0.09 for reverse network effects, whereas XRP’s is
0.07 and for XTZ’s is 0.06 for positive network effects and 0.04 and 0.03 for
reverse network effects (compare sum ratios in Figure 3 and Figure
LABEL:fig:stemplots_fx__6m). This means that during periods of positive
network effects, XRP’s and XTZ’s transaction value grows more than their token
price grows relative to their userbase, and that during periods of reverse
network effects, XRP’s and XTZ’s transaction value drops more than their token
price drops relative to their userbase. This kind of increased volatility may
be generally undesirable, but it is particularly problematic during downturns
(reverse network effects) because it shows that activity on XRP and XTZ
networks is more drastically affected making them more sensitive and less
resilient, which is a competitive disadvantage. Our results hold too when we
look exclusively at 2017 and 2018 as the years with the most sustained price
increase and decrease respectively.

##  7\. Conclusion

Network effects can be among the most common and influential factors shaping
market dynamics in industries where products and services are built around
networks. It is no wonder that they have been cited as a determinant in how
cryptoassets grow in value and compete. Our analysis show that while network
effects do characterize cryptoassets, they do not result in the usual
concentration and competitive advantage implications usually associated with
them. Our work also invites further research to determine the exact scope and
conditions under which network effects apply. More precise proxies for
userbase and value and accounting for exogenous effects are steps in the right
direction.

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##  8\. Appendix

(a) BTC

(b) DOGE

(c) ETH

(d) LTC

(e) XRP

(f) XTZ

Figure 3. Network effect observations and distribution (blue: positive NFX,
red: reverse NFX, white: no NFX); userbase measured by trailing 6 month
addresses, value measured by USD token price.

(a) BTC

(b) DOGE

(c) ETH

(d) LTC

(e) XRP

(f) XTZ

Figure 4. Network effect observations and distribution (blue: positive NFX,
red: reverse NFX, white: no NFX); userbase measured by trailing 6 month
addresses, value measured by transaction value.

(a) BTC

(b) DOGE

(c) ETH

(d) LTC

(e) XRP

(f) XTZ

Figure 5. Network effect observations (blue: positive NE, red: reverse NE,
white: no NE); userbase measured by total addresses with non-zero balance,
value measured by transaction value.


#
Reduction of Open String Amplitudes
by Mostly BRST Exact Operators

Shigenori<EMAIL_ADDRESS>
Osaka City University Advanced Mathematical Institute (OCAMI),
3-3-138, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

and

Tomohiko<EMAIL_ADDRESS>
Department of Physics, Nara Women’s University,
Nara 630-8506, Japan

We study two and three-point tree-level amplitudes of open strings. These
amplitudes are reduced from higher-point correlation functions by using mostly
BRST exact operators for gauge fixing. For simplicity, we focus on an open
string tachyon. The two-point amplitude of open string tachyons is reduced
from a three-point function of two tachyon vertex operators and one mostly
BRST exact operator. Similarly the three-point amplitude is from a four-point
function of three tachyon vertex operators and one mostly BRST exact operator.
One can also obtain the two-point amplitude from the four-point function of
two tachyon vertex operators and two mostly BRST exact operators. In these
derivation from four-point functions, moduli integrals are significant. We
discuss the overall signs of amplitudes which are indefinite in this
formalism.

12 August 2021

###### Contents

  1. 1 Introduction
  2. 2 Reduction by one ${\mathpzc E}$
    1. 2.1 Two-point amplitude from three-point function
    2. 2.2 Three-point amplitude from four-point function
    3. 2.3 Overall sign of amplitudes
  3. 3 Reduction by two ${\mathpzc E}$’s
    1. 3.1 Two-point amplitude from four-point function
  4. 4 Concluding remarks

##  1 Introduction

String scattering amplitudes is one of important origins of string theories.
It has been studied in detail for a long time and even now provides unexpected
and interesting subjects for us. In bosonic open string theory, the Veneziano
amplitude given by the Euler beta function was discovered to explain stringy
nature such as duality, and in a modern formulation, more generic string
amplitudes are given as correlation functions of vertex operators in the
Polyakov path integral, where the measure has to be divided by the volume of
diffeomorphism and Weyl symmetry. For open string tree amplitudes,
correlarators has to be divided by the volume of $PSL(2,\mathbb{R})$ on an
upper-half plane and so the one and two-point amplitudes were thought to be
vanished due to the infinite volume of the residual symmetry. However, this
view has been overturned by [1] for two-point amplitudes. They have shown that
the infinite volume and the divergence of energy-conserving delta function
$\delta(0)$ are canceled each other for the two-point amplitudes and then the
standard free particle expression can be obtained.

Motivated by their work, the present authors introduced the operator which
fixes a part of the $PSL(2,\mathbb{R})$ gauge symmetry in the BRST operator
formalism. The operator is defined by

$\displaystyle{\mathpzc
E}(z)\equiv\frac{1}{\pi}\int_{-\infty}^{\infty}dq\,\Bigl{(}c\partial
X^{0}e^{-iqX^{0}}-i\alpha^{\prime}q(\partial c)e^{-iqX^{0}}\Bigr{)}\,,$ (1.1)

where $X^{0}(z,\bar{z})$ is the 0-th string coordinate and $c(z)$ is the ghost
field, and $z$ is a point on the boundary.111 In this paper, we multiply the
operator ${\cal V}_{0}$ in [2] by the factor $i\alpha^{\prime}$ to define the
mostly BRST operator ${\mathpzc E}$. This is a BRST invariant operator and
moreover it is written by using the BRST charge $Q_{\rm B}$:

$\displaystyle{\mathpzc
E}(z)=\frac{1}{\pi}\int_{-\infty}^{\infty}dq\,\frac{i}{q}\,\delta_{B}e^{-iqX^{0}}\,,\quad\delta_{B}e^{-iqX^{0}}=[\,Q_{\rm
B},\,e^{-iqX^{0}}\,]\,.$ (1.2)

We call it a mostly BRST exact operator, since the integrand for $q\neq 0$ is
indeed BRST exact, but it is not so at the singular point $q=0$. In [2], the
two-point amplitude is reproduced by calculating the three point function in
which two arbitrary on-shell open string vertex operators and the operator
(1.1) are inserted on an upper-half plane.

The concept of “mostly BRST exactness” is very important. The correlation
function including the integrand of (1.1) should behave as a kind of
distributions supported at $q=0$, since it becomes zero for $q\neq 0$ due to
BRST exactness. Then, performing the $q$-integration, we should have a non-
zero result, which is applicable to two-point amplitudes. Mostly BRST
exactness also assures that the operator satisfies Lorentz and conformal
invariance, although depending only on the time-direction. Moreover, the
concept can be widely applied in the BRST formalism. Indeed, a mostly BRST
exact operator is constructed in the pure spinor formalism of superstring
theory [3], and then two-point superstring amplitudes have been found to agree
with the corresponding expression in the field theories.

An insertion of the mostly BRST exact operator fixes one degree of freedom of
$PSL(2,\mathbb{R})$, since the operator is a BRST extension of the gauge
fixing delta functional and the Fadeev-Popov determinant discussed in [1].
Then, it is natural to ask whether general $n$-point amplitudes can be
obtained by fixing a part of $PSL(2,\mathbb{R})$ in terms of the mostly BRST
exact operator, as mentioned in the last part of [2]. The main purpose of this
paper is to report on the results of the $n$-point amplitudes by inserting
this gauge fixing operator. Consequently, it is confirmed that the mostly BRST
exact operator works well for gauge fixing of $PSL(2,\mathbb{R})$.

In the derivation of the two-point amplitudes in [2], the energy-conserving
delta function becomes $\delta(q)$, if two states have the same mass and one
of them is an incoming state and the other outgoing. After the
$q$-integration, we can find the correct expression of two-point amplitudes
not involving the delta function for energy. However, if we attempt to
evaluate three-point amplitudes by an additional insertion of the gauge fixing
operator, the positions of the two matter vertex operators is fixed, but the
position of the remaining vertex has to be integrated. Compared with the
conventional three-point amplitudes, this expression includes, as extra
components, the moduli and $q$ integrations. In this paper, we indicate that
the Feynman $i\varepsilon$ plays a key role in elimination of these integrals
to derive the correct amplitudes.

In the Veneziano amplitude, there are poles for the Mandelstam variables and,
precisely, the poles are defined by the $i\varepsilon$ prescription as well as
Minkowski amplitudes[4]. This prescription is related to the fact that the
string worldsheet hides Lorentzian nature although it is treated as Euclidean
space generically[5]. Similarly, the moduli integral of our amplitudes
generates poles which include the delta function of the energy zero mode $q$.
Then, we will find that, after the $q$-integration, the conventional
expression of three-point amplitudes is derived from the additional insertion
of the gauge fixing vertex.

We should comment on the indefiniteness of the overall sign of the amplitude
given by our gauge fixing operators. First, we remind that even for
conventional amplitudes in the BRST formalism, the overall sign is indefinite
and it is fixed by referring to the Jacobian factor in the Fadeev-Popov
procedure.222The sign can also be determined by deriving the amplitude from
string field theories based on the BRST formalism. This indefiniteness arises
from removing the absolute value from the Fadeev-Popov determinant to
introduce the ghost fields. The same extract of the absolute value causes
uncertainty in the amplitudes with our gauge fixing operator. Then, we will
see the signature factor depending on momenta of open string states even in
the two-point amplitude.

More precisely, we will show that conventional open string amplitudes can be
reproduced by using our gauge fixing operator for $PSL(2,\mathbb{R})$, except
the overall sign which can be positive, negative or zero. The sign factor is
expressed by using momenta of external strings and we may attempt to interpret
this expression in terms of a signed intersection number associated with the
gauge fixing condition. This interpretation is essentially same as the
discussion of a subtlety relating to a gauge choice in [6], which is inspired
by the result of [1].

We begin in the section 2 by illustrating the derivation of the two and three-
point amplitudes fixed a part of $PSL(2,\mathbb{R})$ by the insertion of the
mostly BRST exact operator. For simplicity, we choose open string tachyons as
external states. We then verify in the subsection 2.3 the overall sign factor
by interpreting it as a signed intersection number. In the section 3, we fixes
two degrees of freedom in $PSL(2,\mathbb{R})$ by the mostly BRST exact
operators to derive two-point amplitudes for open string tachyons. This result
justifies our gauge fixing procedure of $PSL(2,\mathbb{R})$ and the
interpretation of the overall sign in terms of the intersection. Finally, we
provide concluding remarks in the section 4.

##  2 Reduction by one ${\mathpzc E}$

###  2.1 Two-point amplitude from three-point function

In [2], it is shown that the mostly BRST exact operator leads to correct two-
point amplitudes for any open string vertex operators. Here, we illustrate how
to derive a two-point amplitude in terms of open string tachyons in the 26
dimensional flat Minkowski spacetime.

We consider an upper-half plane as the world-sheet and the real axis is
regarded as the open string boundary. If a part of $PSL(2,\mathbb{R})$ is
fixed by the insertion of ${\mathpzc E}$, the amplitude for two tachyon vertex
operators is given by

$\displaystyle{\cal A}_{2}=ig_{\rm o}^{2}C_{D_{2}}\left<0\right|{\mathpzc
E}(y_{0})\,cV_{1}(y_{1})\,cV_{2}(y_{2})\left|0\right>.$ (2.1)

where the tachyon vertex operator $V_{i}$ is defined as

$\displaystyle V_{i}(y)\equiv e^{ip_{i}\cdot X}(y)\,,$ (2.2)

and $C_{D_{2}}$ is a normalization factor for disk amplitudes:
$C_{D_{2}}=1/(\alpha^{\prime}g_{\rm o}^{2})$ [4]. The positions $y_{i}$ are
along the real axis and we set $y_{0}<y_{1}<y_{2}$. The tachyon vertex
operators include a factor of the open string coupling $g_{\rm o}$, but
$g_{\rm o}$ is not assigned to the operator ${\mathpzc E}$, because ${\mathpzc
E}$ does not add an extra string to the process. $\left|0\right>$ is the
$SL(2,\mathbb{R})$ invariant vacuum normalized as $\langle
0|0\rangle=(2\pi)^{26}\delta^{26}(0)$.

Calculating the correlation function, we obtain

$\displaystyle{\cal A}_{2}$ $\displaystyle=ig_{\rm
o}^{2}C_{D_{2}}{\alpha^{\prime}(p_{1}^{0}-p_{2}^{0})\over
i}(2\pi)^{25}\delta^{25}({\bm{p}}_{1}+{\bm{p}}_{2})\int_{-\infty}^{\infty}dq\,\delta(q+p_{1}^{0}+p_{2}^{0})|y_{01}|^{-2\alpha^{\prime}qp_{1}^{0}}|y_{02}|^{-2\alpha^{\prime}qp_{2}^{0}}|y_{12}|^{\alpha^{\prime}q^{2}}\,.$
(2.3)

where $y_{ij}$ is defined as $y_{ij}\equiv y_{i}-y_{j}$. The integrand with
the exponent including $q$ implies that the operator in the $q$-integration of
${\mathpzc E}$ apparently breaks conformal invariance for $q\neq 0$, however
it will be found immediately that the resulting amplitude has the invariance
and no dependence of the position of the vertex operators.

By the on-shell condition $\alpha^{\prime}(p_{i})^{2}=1$, we have
$p_{i}^{0}=\pm\sqrt{({\bm{p}}_{i})^{2}+(-1/\alpha^{\prime})}$ and so the
amplitude vanishes if $p_{1}^{0}$ and $p_{2}^{0}$ have the same signature due
to the factor $p_{1}^{0}-p_{2}^{0}$. In general, this vanishing property is
followed from the BRST invariance, since only the $q=0$ part is not BRST exact
in ${\mathpzc E}$ and so energies for two tachyons should be conserved in the
non-zero amplitude. Actually, if $p_{1}^{0}$ and $p_{2}^{0}$ have opposite
sign, we find that the integral depends only on the contribution from $q=0$,
namely including $\delta(q)$, and then the amplitude becomes non-zero. In this
case, one tachyon corresponds to an incoming state and another is outgoing.

The resulting amplitude is given by

$\displaystyle{\cal A}_{2}=\frac{p_{1}^{0}}{|p_{1}^{0}|}\times
2|p_{1}^{0}|(2\pi)^{25}\delta^{25}({\bm{p}}_{1}+{\bm{p}}_{2})\,.$ (2.4)

Thus, we can obtain the two-point amplitude in the standard free particle
expression, however it should be noted that the signature is not determined by
this gauge fixing procedure, as mentioned in the introduction. This sign
factor is depend on a momentum of external states.

###  2.2 Three-point amplitude from four-point function

Let us consider a three open tachyon amplitude by fixing the two vertex
operators to positions $y_{1},\ y_{2}$ on the real axis. To fix the residual
gauge symmetry of $PSL(2,\mathbb{R})$, we insert the operator ${\mathpzc E}$
to a position $y_{0}$.

$\displaystyle{\cal A}_{3}=ig_{\rm o}^{3}C_{D_{2}}\left<0\right|{\mathpzc
E}(y_{0})\,cV_{1}(y_{1})\,cV_{2}(y_{2})\int_{-\infty}^{\infty}dy_{3}\,V_{3}(y_{3})\left|0\right>.$
(2.5)

Although it is a three point amplitude, the position $y_{3}$ of the third
vertex operator is integrated as a result of the insertion of ${\mathpzc E}$.
Since $y_{3}$ varies from $-\infty$ to $\infty$, this expression includes two
ordering of the three tachyons and so it is not needed to add an extra
contribution from the exchange, $p_{1}\leftrightarrow p_{2}$, unlike a
conventional case in [4].

Here, for convenience, introducing a covariant expression for the momentum $q$
in $\mathpzc E$ defined by (1.2): $p_{0}=(q,0,\dots,0)$, we rewrite
$\delta_{\rm B}e^{-iqX^{0}}$ as $\delta_{\rm B}e^{ip_{0}\cdot X}$. We fix the
positions as $y_{0}<y_{1}<y_{2}$, since the ordering of $y_{0}$, $y_{1}$ and
$y_{2}$ is related only to the overall sign. The correlation function in the
amplitude (2.5) can be calculated as

$\displaystyle{\cal F}_{1}(q)$
$\displaystyle=\left<0\right|\biggl{(}{i\over\pi q}\delta_{\rm
B}e^{ip_{0}\cdot
X}(y_{0})\biggr{)}\,cV_{1}(y_{1})\,cV_{2}(y_{2})\int_{-\infty}^{\infty}dy_{3}\,V_{3}(y_{3})\left|0\right>$
$\displaystyle=\biggl{|}\frac{y_{01}y_{02}}{y_{12}}\biggr{|}^{-\alpha^{\prime}(p_{0})^{2}}(2\pi)^{26}\delta^{26}(\sum_{j=0}^{3}p_{j})\,I\,,$
$\displaystyle I$ $\displaystyle={i\over\pi
q}\int_{-\infty}^{\infty}dx\,\frac{1}{2}\biggl{(}\alpha^{\prime}(p_{0})^{2}+2\alpha^{\prime}p_{0}\cdot
p_{1}-2\alpha^{\prime}p_{0}\cdot
p_{3}\frac{x}{1-x}\biggr{)}|x|^{2\alpha^{\prime}p_{2}\cdot
p_{3}}|1-x|^{2\alpha^{\prime}p_{0}\cdot p_{3}}\,,$ (2.6)

where $x$ denotes a moduli parameter given by the cross ratio,

$\displaystyle x=\frac{y_{01}y_{23}}{y_{02}y_{13}}\,.$ (2.7)

The prefactor with the exponent
$-\alpha^{\prime}(p_{0})^{2}=\alpha^{\prime}q^{2}$ implies that ${\mathpzc E}$
apparently breaks $PSL(2,\mathbb{R})$ invariance for $q\neq 0$, but it is not
problematic since the final result should depend only on the $q=0$ part.

The amplitude is written in terms of the Mandelstam variables;

$\displaystyle s=-(p_{0}+p_{1})^{2},\quad t=-(p_{0}+p_{2})^{2},\quad
u=-(p_{0}+p_{3})^{2}.$ (2.8)

However, unlike the Veneziano amplitude, these are independent since $p_{0}$
does not satisfy an on-shell condition, i.e., $s+t+u=q^{2}-3/\alpha^{\prime}$.
The moduli integral (2.6) is rewritten as

$\displaystyle I(q)={-i\over 2\pi
q}\int_{-\infty}^{\infty}dx\biggl{(}(\alpha^{\prime}s+1){1\over
1-x}+(\alpha^{\prime}t+1){x\over
1-x}\biggr{)}|x|^{-\alpha^{\prime}s-2}|1-x|^{\alpha^{\prime}(s+t)+2}\,.$ (2.9)

Noting that the moduli integral $I$ splits into three ranges, i.e.,
$I=I_{\rm(i)}+I_{\rm(ii)}+I_{\rm(iii)}$, for

$\displaystyle({\rm i})\ -\infty<y_{3}<y_{0},\ y_{2}<y_{3}<\infty\qquad({\rm
ii})\ y_{0}<y_{3}<y_{1}\qquad({\rm iii})\ y_{1}<y_{3}<y_{2}\,,$ (2.10)

or equivalently (i) $0<x<1$, (ii) $1<x<\infty$, (iii) $-\infty<x<0$, the
amplitude becomes

$\displaystyle{\cal A}_{3}=ig_{\rm
o}^{3}C_{D_{2}}\int_{-\infty}^{\infty}dq\,{\cal F}_{1}=ig_{\rm
o}^{3}C_{D_{2}}\int_{-\infty}^{\infty}dq\,\biggl{|}{y_{01}y_{02}\over
y_{12}}\biggr{|}^{-\alpha^{\prime}(p_{0})^{2}}(2\pi)^{26}\delta^{26}(\sum_{j=0}^{3}p_{j})(I_{\rm(i)}+I_{\rm(ii)}+I_{\rm(iii)})\,,$
(2.11)

where

$\displaystyle I_{\rm(i)}$ $\displaystyle={-i\over 2\pi
q}\\{(\alpha^{\prime}s+1)B(-\alpha^{\prime}s-1,\alpha^{\prime}(s+t)+2)+(\alpha^{\prime}t+1)B(-\alpha^{\prime}s,\alpha^{\prime}(s+t)+2)\\}\,,$
(2.12) $\displaystyle I_{\rm(ii)}$ $\displaystyle={-i\over 2\pi
q}\\{-(\alpha^{\prime}s+1)B(-\alpha^{\prime}t,\alpha^{\prime}(s+t)+2)-(\alpha^{\prime}t+1)B(-\alpha^{\prime}t-1,\alpha^{\prime}(s+t)+2)\\}\,,$
(2.13) $\displaystyle I_{\rm(iii)}$ $\displaystyle={-i\over 2\pi
q}\\{(\alpha^{\prime}s+1)B(-\alpha^{\prime}s-1,-\alpha^{\prime}t)-(\alpha^{\prime}t+1)B(-\alpha^{\prime}s,-\alpha^{\prime}t-1)\\}\,.$
(2.14)

The Euler beta function is defined by

$\displaystyle B(u,v)=\int_{0}^{1}dx\,x^{u-1}(1-x)^{v-1}\,.$ (2.15)

If the momentum variables are in the convergence region of the integral $B$,
we find that $I_{\rm(i)}$, $I_{\rm(ii)}$ and $I_{\rm(iii)}$ vanish. This
reflects the mostly BRST exactness of ${\mathpzc E}$, however, it does not
necessarily imply that the amplitude becomes trivially zero. To derive the
physical result, we should study the $q=0$ part of the amplitude, which is
given as the divergence of the integral.

To study the divergence, we have to introduce a convergence factor which
corresponds to the Feynman $i\varepsilon$. For instance, one of the integrals
in $I_{\rm(i)}$ has the convergence factor $\exp(-i\varepsilon(\log
x+\log(1-x)))$ [5] and it should be precisely defined as

$\displaystyle
B(-\alpha^{\prime}s-1,\alpha^{\prime}(s+t)+2)=\int_{0}^{1}dxx^{-\alpha^{\prime}s-2-i\varepsilon}(1-x)^{\alpha^{\prime}(s+t)+1-i\varepsilon}\,.$
(2.16)

To extract the singularity at $q=0$, we have only to expand it in rational
fractions and to use the formula,

$\displaystyle\frac{1}{x-i\varepsilon}={\rm P}\frac{1}{x}+\pi i\delta(x)\,,$
(2.17)

where P denotes the principal value. In this example, the singularity can be
evaluated as

$\displaystyle B(-\alpha^{\prime}s-1,\alpha^{\prime}(s+t)+2)$
$\displaystyle=\frac{1}{-\alpha^{\prime}s-1-i\varepsilon}+\frac{1}{\alpha^{\prime}(s+t)+2-i\varepsilon}+\cdots$
$\displaystyle=\frac{\pi
i}{2\alpha^{\prime}}\biggl{(}\frac{1}{|p_{1}^{0}|}+\frac{1}{|p_{3}^{0}|}\biggr{)}\delta(q)+\cdots\,.$
(2.18)

These terms are generated from the singular configuration of the worldsheet
with $y_{3}$ approaching to $y_{0}$ or $y_{2}$.

Similarly, by extracting $\delta(q)$, the moduli integration gives the
following results for each integration range:

$\displaystyle I_{\rm(i)}\sim{1\over
2}\biggl{(}\frac{p_{1}^{0}}{|p_{1}^{0}|}-\frac{p_{3}^{0}}{|p_{3}^{0}|}\biggr{)}\delta(q)\,,\quad
I_{\rm(ii)}\sim{1\over
2}\biggl{(}-\frac{p_{2}^{0}}{|p_{2}^{0}|}+\frac{p_{3}^{0}}{|p_{3}^{0}|}\biggr{)}\delta(q)\,,\quad
I_{\rm(iii)}\sim{1\over
2}\biggl{(}\frac{p_{1}^{0}}{|p_{1}^{0}|}-\frac{p_{2}^{0}}{|p_{2}^{0}|}\biggr{)}\delta(q)\,,$
(2.19)

where $\sim$ stands for equal up to analytic terms. By substituting these
results into (2.11) and performing the $q$-integration, we obtain the final
expression of the amplitude:

$\displaystyle{\cal A}_{3}={1\over
2}\biggl{(}\frac{p_{1}^{0}}{|p_{1}^{0}|}-\frac{p_{2}^{0}}{|p_{2}^{0}|}\biggr{)}\times\frac{2ig_{\rm
o}}{\alpha^{\prime}}(2\pi)^{26}\delta^{26}(p_{1}+p_{2}+p_{3})\,.$ (2.20)

This result agrees with the correct three string amplitude except the
prefactor taking the value $\pm 1$ or $0$, which can not be determined in this
gauge fixing as mentioned above.

###  2.3 Overall sign of amplitudes

The resulting three-point amplitude includes the sign factor

$\displaystyle\frac{1}{2}\biggl{(}\frac{p_{1}^{0}}{|p_{1}^{0}|}-\frac{p_{2}^{0}}{|p_{2}^{0}|}\biggr{)}\,,$
(2.21)

which is $\pm 1$ if $p_{1}^{0}$ and $p_{2}^{0}$ have opposite signs each
other, and becomes zero if they are the same signs. Let us verify the validity
of this factor.

The operator ${\mathpzc E}$ corresponds to the gauge fixing $X^{0}(y_{0})=0$,
where the point $y_{0}$ is on the segment of the boundary of the disk. In the
case $y_{0}<y_{1}<y_{2}$, the segment consists of the regions,
$-\infty<y<y_{1}$ and $y_{2}<y<\infty$, noting that $\pm\infty$ are
identified. The tachyon vertex operators are inserted at the points $y_{1}$
and $y_{2}$.

From the path integral formulation, we integrate all over the function
$X^{0}(y)$ on the boundary to derive the amplitude. However, $X^{0}(y)$ should
approach $+\infty$ at the inserted points of the vertex operators if the open
string corresponds to outgoing states, and $-\infty$ for incomming states,
because $X^{0}$ is the time coordinate of the target space.

Suppose that the operator at $y_{2}$ is incoming and the one at $y_{1}$ is
outgoing, which implies $p_{2}^{0}>0$ and $p_{1}^{0}<0$. In this case,
$X^{0}(y_{2})=-\infty$ and $X^{0}(y_{1})=+\infty$. Since the equation
$X^{0}(y)=0$ has necessarily roots, it is possible to impose the gauge fixing
condition $X^{0}(y_{0})=0$.

Figure 1: The function $X^{0}(y)$ with $y$ in the region, $-\infty<y<y_{1}$ or
$y_{2}<y<\infty$, for the case that the total number of roots is three.

In Fig. 1, we show the function $X^{0}(y)$ satisfying the gauge choice
$X^{0}(y_{0})=0$ for the case that the total number of roots is three. For the
three roots, there are three possibilities how to choose intersections at
$y_{0}$, namely, I, II and III in Fig. 1. As mentioned in the introduction,
the operator ${\mathpzc E}$ originates in the Fadeev-Popov determinant for the
gauge fixing, but we have to define ${\mathpzc E}$ by using the singed
determinant. Therefore, for the three possibilities, $\partial X^{0}(y_{0})$
has different signatures: positive for I and III and negative for II. Then,
the amplitude becomes positive for I and III and negative for II.
Consequently, after summing over all $X^{0}$ in the case of the three
intersection, the amplitude is given as a positive result if I, II and III
lead to the same absolute value. So, we expect that the sign of the amplitude
is given as the signed intersection number of the graph $u=X^{0}(y)$ with
$u=0$ for any number of the roots.

The signed intersection number is $+1$ for $p_{2}^{0}>0$ and $p_{1}^{0}<0$,
and it is $-1$ for $p_{2}^{0}<0$ and $p_{1}^{0}>0$. If $p_{2}^{0}$ and
$p_{1}^{0}$ have the same sign, both of the tachyon states are incomming or
outgoing and so the signed intersection number becomes zero. These results
agree with the signature of (2.21).

There is the case where the point $y_{3}$ enters the same segment of $y_{0}$,
but the signed intersection number is unchanged regardless of whether
$X^{0}(y_{3})$ is infinity or minus infinity.

The two point amplitude is given by (2.4), which is given by the gauge choice
$X^{0}(y_{0})=0$ and the vertex insertions at $y_{1}$ and $y_{2}$. So, we
apply the same discussion as the above to the signature of (2.4). The only
difference is that the sum of $p_{1}^{0}$ and $p_{2}^{0}$ is zero for the two
point amplitude, and so the signature (2.21) equals to
$p_{1}^{0}/|p_{1}^{0}|$, which agrees with the sign of (2.4).

Thus, we can verify the validity of the signature of the amplitude from the
point of view of the signed intersection number.

##  3 Reduction by two ${\mathpzc E}$’s

###  3.1 Two-point amplitude from four-point function

Let us consider two open tachyon amplitudes by inserting the mostly BRST exact
operators at two points, $y_{0}$ and $y_{1}$. To fix $PSL(2,\mathbb{R})$ gauge
symmetry, we fix the one tachyon vertex operator at the position $y_{2}$ and
so the position $y_{3}$ of another vertex operator is integrated. The
amplitude is given by

$\displaystyle{\cal A}_{2}$ $\displaystyle=\frac{1}{2}\times ig_{\rm
o}^{2}C_{D_{2}}\left<0\right|{\mathpzc E}(y_{0}){\mathpzc
E}(y_{1})cV_{2}(y_{2})\int_{-\infty}^{\infty}dy_{3}V_{3}(y_{3})\left|0\right>.$
(3.1)

Here, the two mostly BRST exact operators at $y_{0}$ and $y_{1}$ are
indistinguishable each other and so we should divide the amplitude by the
statistical factor $2$ from the perspective of the Feynman diagram of four-
point amplitudes. This factor can be also understood in world-sheet language.
The metric on the upper-half plane is given by

$\displaystyle ds^{2}=\frac{dzd\bar{z}}{|z-\bar{z}|^{2}}\,,$ (3.2)

which is invariant under $PSL(2,\mathbb{R})$ transformation:
$z\rightarrow(az+b)/(cz+d)$, $ad-bc=1$. In addition, the metric is invariant
under the discrete transformation $z\rightarrow-\bar{z}$. These symmetries
generate the conformal Killing group (CKG). In the usual gauge fixing, the
positions of three different vertex operators are fixed and so any conformal
Killing invariance is not remained. However, if the two positions are fixed by
the indistinguishable operators, a $\mathbb{Z}_{2}$ invariance as a part of
the CKG is unfixed. More explicitly, the $\mathbb{Z}_{2}$ transformation is
given by

$\displaystyle z\rightarrow
f(z)=\frac{-(y_{0}y_{1}-y_{2}^{2})\bar{z}+y_{2}(2y_{0}y_{1}-y_{0}y_{2}-y_{1}y_{2})}{-(y_{0}+y_{1}-2y_{2})\bar{z}+(y_{0}y_{1}-y_{2}^{2})}\,.$
(3.3)

This mapping function satisfies $f(y_{0})=y_{1}$, $f(y_{1})=y_{0}$,
$f(y_{2})=y_{2}$ and $f(f(z))=z$. Therefore, the $\mathbb{Z}_{2}$ symmetry
should be fixed by restricting the integration range of $y_{3}$ to half of the
boundary, or by dividing the factor 2 while the $\mathbb{Z}_{2}$ symmetry
remains unfixed. In the expression (3.1), we adopt the latter procedure to fix
the CKG.333For the Veneziano amplitude, this $\mathbb{Z}_{2}$ symmetry is
remained if two momenta are identical, however the identical case has measure
zero and so this factor is irrelavent to the amplitude at generic momenta.

Now, let us evaluate the correlation function in (3.1) according to the
discussion in the previous section. After tedious calculation, we obtain

$\displaystyle{\cal F}_{2}(q,q^{\prime})$
$\displaystyle=\left<0\right|\biggl{(}{i\over\pi q}\delta_{\rm
B}e^{ip_{0}\cdot X}(y_{0})\biggr{)}\biggl{(}{i\over\pi q^{\prime}}\delta_{\rm
B}e^{ip_{1}\cdot X}(y_{1})\biggr{)}ce^{ip_{2}\cdot
X}(y_{2})\int_{-\infty}^{\infty}dy_{3}e^{ip_{3}\cdot X}(y_{3})\left|0\right>$
$\displaystyle=\Big{|}\frac{y_{12}}{y_{01}y_{02}}\Big{|}^{\alpha^{\prime}(p_{0})^{2}}\Big{|}\frac{y_{02}}{y_{01}y_{12}}\Big{|}^{\alpha^{\prime}(p_{1})^{2}}(2\pi)^{26}\delta^{26}(\sum_{j=0}^{3}p_{j})\,J\,,$
$\displaystyle J$
$\displaystyle={1\over\pi^{2}qq^{\prime}}\int_{-\infty}^{\infty}dx\,\frac{1}{2}\biggl{(}\alpha^{\prime}p_{0}\cdot
p_{1}-(\alpha^{\prime})^{2}(p_{0}+p_{1})^{2}(p_{0}\cdot p_{1})$
$\displaystyle\qquad-(\alpha^{\prime})^{2}\bigl{(}(p_{0})^{2}+2p_{0}\cdot
p_{1}\bigr{)}(p_{1}\cdot
p_{3})x+(\alpha^{\prime})^{2}\bigl{(}(p_{1})^{2}+2p_{0}\cdot
p_{1}\bigr{)}(p_{0}\cdot p_{3})\frac{x}{1-x}$
$\displaystyle\qquad+2(\alpha^{\prime})^{2}(p_{0}\cdot p_{3})(p_{1}\cdot
p_{3})\frac{x^{2}}{1-x}\biggr{)}|x|^{2\alpha^{\prime}p_{2}\cdot
p_{3}}|1-x|^{2\alpha^{\prime}p_{0}\cdot p_{3}}\,,$ (3.4)

where $p_{0}=(q,0,\cdots,0)$, $p_{1}=(q^{\prime},0,\cdots,0)$. $q$ and
$q^{\prime}$ are integrated later in ${\mathpzc E}(y_{0})$ and ${\mathpzc
E}(y_{1})$, respectively. Due to the BRST invariance, the $q\neq 0$ part does
not contribute to the amplitude and so it can be easily seen that the
$x$-integration of (3.4) vanishes in the convergence region. Then we have only
to extract the singularity of $q=0$ as the previous case.

We should comment on a different type of singularity proportional to
$\delta((q+q^{\prime})^{2})$, which seems to arise from (3.4), since there are
some terms in the moduli integration which have the pole at
$2\alpha^{\prime}p_{2}\cdot p_{3}+2=-\alpha^{\prime}(q+q^{\prime})^{2}=0$.
This pole corresponds to a massless open string state as the intermediate one
of the two tachyon collision. However, this pole should vanish since the
amplitude has twist symmetry for two tachyons. Indeed, gathering together
factors of this pole, using on-shell condition and momentum conservation, we
find that they totally cancel each other and then the dangerous distribution
does not appear in the amplitude.

Now, we show the final expression of the amplitude by extracting $\delta(q)$
and $\delta(q^{\prime})$. Taking $y_{0}<y_{1}<y_{2}$, the integration (3.4)
for three regions of (2.10) leads to $J=J_{\rm(i)}+J_{\rm(ii)}+J_{\rm(iii)}$
with

$\displaystyle
J_{\rm(i)}\sim\frac{i\alpha^{\prime}}{2\pi}|p_{3}^{0}|\,\delta(q)\,,\quad
J_{\rm(ii)}\sim\frac{i\alpha^{\prime}}{2\pi}|p_{3}^{0}|\,\bigl{(}\delta(q)+\delta(q^{\prime})\bigr{)}\,,\quad
J_{\rm(iii)}\sim\frac{i\alpha^{\prime}}{2\pi}|p_{3}^{0}|\,\delta(q^{\prime})\,.$
(3.5)

Unless $p_{2}^{0}+p_{3}^{0}=0$, the amplitude vanishes as well as the two-
point amplitude reduced from the three-point function. Finally, in the case of
$p_{2}^{0}+p_{3}^{0}=0$, the resulting amplitude is given by

$\displaystyle{\cal A}_{2}={ig_{\rm o}^{2}C_{D_{2}}\over
2}\int_{-\infty}^{\infty}dq\int_{-\infty}^{\infty}dq^{\prime}\,{\cal
F}_{2}=-1\times
2|p_{3}^{0}|(2\pi)^{25}\delta^{25}({\bm{p}}_{2}+{\bm{p}}_{3})\,.$ (3.6)

Similar to the previous cases, the signature of the amplitude can not be
determined in this gauge fixing. However, contrastingly, the signature of
(3.6) does not depend on momenta of the vertex operators. This can be also
understood in terms of the signed intersection. For simplicity, suppose that
$p_{2}^{0}>0$ and $p_{3}^{0}<0$, and so $X^{0}(y_{2})=-\infty$ and
$X^{0}(y_{3})=+\infty$. We set $y_{0}<y_{1}<y_{2}$. If $y_{1}<y_{3}$ or
$y_{3}<y_{0}$, the intersection number of $X^{0}$ on the segment in which
$y_{0}$ and $y_{1}$ are located becomes an odd number. By assigning these
intersections to $y_{0}$ and $y_{1}$ and adding the contributions of all
combination, the corresponding amplitude is expected to be zero since
$\sum{\rm sgn}(\partial X^{0}(y_{0})){\rm sgn}(\partial X^{0}(y_{1}))=0$ in
this case. If $y_{0}<y_{3}<y_{1}$, the intersection number on each segment for
$y_{0}$ and $y_{1}$ is odd and moreover we find that $\sum{\rm sgn}(\partial
X^{0}(y_{0}))=+1$ and $\sum{\rm sgn}(\partial X^{0}(y_{1}))=-1$. Then, the
signature factor is $+1\times(-1)=-1$ in this case. It can be easily seen that
the same signature is obtained for $p_{2}^{0}<0$ and $p_{3}^{0}>0$, and
therefore the signature is independent of momenta in the two-point amplitude
reduced from the four-point function.

##  4 Concluding remarks

We have shown that the two and three-point tree-level amplitudes of open
string tachyons are reduced from the three and four-point correlation
functions by inserting the mostly BRST exact operators ${\mathpzc E}$. Firstly
we have obtained the two-point amplitude by calculating the three-point
correlation function with one ${\mathpzc E}$: $\langle{\mathpzc
E}\,cV_{1}\,cV_{2}\rangle$. This two-point amplitude has the same expression
as the standard free particle, but its overall sign depends on the sign of the
momentum of external state.

Similarly we have shown that the three-point amplitude is reduced from the
four-point function of three tachyon vertex operators and one ${\mathpzc E}$:
$\langle{\mathpzc E}(y_{0})\,cV_{1}(y_{1})\,cV_{2}(y_{2})\,\int
dy_{3}V_{3}(y_{3})\rangle$. In this computation, the moduli integration plays
an important role, so that $\delta(q)$ appears from the singularities of the
Euler beta functions. Combining it with the $q$-integration, the four-point
function leads to the correct three-point amplitude up to the overall sign.
The operator ${\mathpzc E}$ implies the gauge fixing, $X^{0}(y_{0})=0$.
$X^{0}(y)$ at $y_{1}$ and $y_{2}$, where tachyon vertex operators are
inserted, are equal to $\pm\infty$, of which sign depends on the choice of
incoming or outgoing for each external state. The configuration of $X^{0}(y)$
with $X^{0}(y_{0})=0$ and $X^{0}(y_{1,2})=\pm\infty$ fixes the total signed
intersection number of the graph $u=X^{0}(y)$ with $u=0$. One can recognize
that the overall sign of the two-point amplitude is given by this intersection
number.

We have also studied the two insertions of ${\mathpzc E}$. The four-point
function $\langle{\mathpzc E}\,{\mathpzc E}\,cV_{1}\,\int V_{2}\rangle$ is
reduced to the two-point amplitude. The two ${\mathpzc E}$’s provide $q$ and
$q^{\prime}$-integrals. In the similar way as the computation for the
reduction from four-point to three, the moduli integration yields $\delta(q)$
and $\delta(q^{\prime})$. Due to the $q$ and $q^{\prime}$-integration with
these delta functions, we can obtain the correct two-point amplitude. There
again appear a problem of the overall sign, but in this case the signature
does not depend on the external momenta.

In this paper, by focusing only on open string tachyons, we have explicitly
evaluated the correlation functions, and, as a result, we have found they are
reduced to the lower-point amplitudes. In the case of two-point amplitudes of
general vertex operators, Refs. [2, 1] have shown that the two-point
amplitudes can be derived from the three-point function. However, for the case
of higher-point amplitudes, we should understand more how the operator
${\mathpzc E}$ acts in the higher-point functions of general vertex operators.

It is not straightforward to apply our formalism using the operator ${\mathpzc
E}$ for closed string amplitudes. For instance, if we extend ${\mathpzc E}$ to
${\mathpzc E}_{\rm closed}\sim\int{dq\over q}[\,Q^{\rm c}_{\rm
B},\,e^{-iqX^{0}(z,{\bar{z}})}\,]$, where $Q^{\rm c}_{\rm B}$ is the BRST
charge of closed string, it causes a problem of ghost number. Therefore, we
need to find a new operator, which plays a same role as ${\mathpzc E}$ in the
closed string theory [7].

## Acknowledgments

The authors would like to thank Isao Kishimoto for valuable comments. This
work was supported in part by JSPS Grant-in-Aid for Scientific Research (C)
#20K03972. S. S. was supported in part by MEXT Joint Usage/Research Center on
Mathematics and Theoretical Physics at OCAMI and by JSPS Grant-in-Aid for
Scientific Research (C) #17K05421.

## References

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