11institutetext: University of the Aegean, 83200 Karlovasi, Greece 11email:
{efchatzoglou, tzisis<EMAIL_ADDRESS>22institutetext: Norwegian University
of Science and Technology, 2802 Gjøvik, Norway 22email:
<EMAIL_ADDRESS>33institutetext: European Commission, Joint Research
Centre (JRC), Ispra, Italy 33email<EMAIL_ADDRESS>
# Keep your memory dump shut: Unveiling data leaks in password managers

Efstratios Chatzoglou 11 0000-0001-6507-5052 Vyron Kampourakis 22
0000-0003-4492-5104 Zisis Tsiatsikas 11 0000-0002-9481-0906 Georgios
Karopoulos envelope 33 0000-0002-0142-7503 Georgios Kambourakis 11
0000-0001-6348-5031

###### Abstract

Password management has long been a persistently challenging task. This led to
the introduction of password management software, which has been around for at
least 25 years in various forms, including desktop and browser-based
applications. This work assesses the ability of two dozen password managers,
12 desktop applications, and 12 browser-plugins, to effectively protect the
confidentiality of secret credentials in six representative scenarios. Our
analysis focuses on the period during which a Password Manager (PM) resides in
the RAM. Despite the sensitive nature of these applications, our results show
that across all scenarios, only three desktop PM applications and two browser
plugins do not store plaintext passwords in the system memory. Oddly enough,
at the time of writing, only two vendors recognized the exploit as a
vulnerability, reserving CVE-2023-23349, while the rest chose to disregard or
underrate the issue.

###### Keywords:

Password Managers Security Data leaks Vulnerability

##  1 Introduction

Twenty years after Bill Gates predicted the death of passwords [14], here we
are still using them. What is more, there are still major data breaches that
are due to weak or insecure passwords, such as in the recent cases of
SolarWinds [10] and Colonial Pipeline [30]. Passwords come in different forms,
such as personal identification numbers (PINs), passphrases, and frequently as
one of the components of multifactor authentication. In between secure but
impractical solutions, such as remembering all passwords, and practical but
insecure ones, such as using the same password everywhere, one of the most
popular ways for protecting credentials is password managers (PMs). Moreover,
even when using PMs, usability clearly affects the security and
trustworthiness of such applications [7]. Thus, the main question behind our
study is: 25 years after the development of the first PM [18], how secure are
our passwords when stored in such applications?

Looking first at how software developers and vendors handle PM security, as
software applications, PMs capitalize on relevant principles, methodologies
and tools, as well as standards and best practices for secure software
development. Nevertheless, absolute security cannot be guaranteed, even when
using all the above means. Moreover, one of the main building blocks of a
secure software development lifecycle is risk management. During this process,
however, software vendors need to decide which security issues to tackle,
wittingly ignoring in several cases those that they consider of minor
importance or extravagant to address. In fact, during our study, out of the
seven PM vendors contacted, only two recognized the issues reported in this
paper and took action to remediate them.

Moving to how PM end-users perceive the value of security, the work in [26]
suggests that users of standalone password managers may prioritize security,
whereas users of built-in managers seem to value convenience more. The work in
[9] compared users of password managers with non-users of such applications,
finding that the main motivation of password manager users was convenience,
instead of security, and the impeding factor for non-users was security
concerns. These findings suggest that PM users do not always put security as
their first priority when adopting such applications.

There are frequent mentions of PMs in the media in the form of advertisements,
comparisons or security analyses [31], attempting to help the user select the
most suitable PM. However, as absolute security cannot be guaranteed, it turns
out that sporadically even popular PMs, which are recognized for the level of
security they provide, can suffer from security issues [25]. One such example
is a recent critical vulnerability affecting KeePass, which allowed the
attackers to recover the master key [29] safeguarding the PM’s vault. In this
context, this paper presents a security analysis of 12 desktop and 12 browser
plugin PMs to understand the level of private information they potentially
leak. Our study is focused on the period during which each of these
applications is loaded on the system memory.

The contributions of this work can be summarized as follows:

  * •

We examine which modern PMs allow the extraction of plaintext credentials from
the RAM, offering exploits that can streamline the retrieval process.

  * •

We investigate to which extent the leaking information reveals repetitive
patterns that augment the attack surface.

  * •

We examine PM vendor conformity to OWASP secure development guidelines with
respect to the exposure of private information.

  * •

Following responsible disclosure, we notified the respective PM vendors about
the identified vulnerabilities and succinctly discuss our interaction with
them.

The rest of the paper is structured as follows. The next section details the
threat model. Section 3 presents our methodology, the derived results, and
ways of exploitation in the context of red teaming. Section 4 elaborates on
best practices, including possible mitigation strategies. The related work is
given in Section 5. The last section concludes and offers directions for
future work.

##  2 Threat model

Generally, data leakage attacks directly violate the first two properties of
the Confidentiality, Integrity, and Availability (CIA) triad. For PMs,
however, availability is also indirectly affected in case an attacker obtains
the PM’s master password and subsequently changes it, locking the legitimate
user outside the PM’s vault. Similarly, a user can be locked outside a service
account if an attacker changes the user’s password. Essentially, in the case
of PMs, where the stored information is considered critical, maintaining these
properties is paramount. Nevertheless, as this work proves through an
experimental evaluation of several renowned PMs in Section 3, data leakage is
still feasible under certain circumstances. This section establishes the
threat model under which an outsider attacker operates.

We concentrate on client-side PMs assuming that the targeted machine is
situated within a local network, typically protected by a firewall, say, the
one provided by default by MS Windows. In this common setting, the adversary
needs to somehow penetrate the network and gain access to the victim’s
machine. This can be done in a plethora of ways, which however are outside the
scope of the current work; for instance, the attacker might lure the victim to
download and execute a piece of malware, often the result of a phishing
attack.

After that, the attacker needs to dump the PM’s processes; typically, these
are more than one. With reference to Section 3, for all the 12 standalone PMs
but 1Password this is straightforward; no special privileges are required. On
the other hand, 1Password’s processes run with a high integrity level. This
means that even if the targeted user has administrative privileges, due to the
User Account Control (UAC) Windows security feature, the malware (or generally
the attacker’s process) will often default to medium integrity, obstructing
access to resources with higher integrity levels. In such a case, the attacker
needs to elevate the integrity level silently, without spawning a UAC pop-up
to the user. This can be accomplished through a UAC bypass method, including
registry key manipulation, DLL hijacking, and elevated COM interface.
Indicatively, a well-utilized DLL hijack-based method is given in [13]. Last
but not least, the attacker must devise a way to efficiently search each one
of the acquired dump (.DMP) files, say, through some script implementing
pattern matching as detailed in subsection 3.4.

##  3 Experiments

The current section details our methodology and findings. Namely, after
presenting the utilized testbed, we report on the different test scenarios and
discuss the derived results.

###  3.1 Testbed and test scenarios

Each of the examined PM applications was installed on a separate clean Virtual
Machine (VM) instance running MS Windows 10; the last OS update was performed
in Jan 2024. Every VM was equipped with 5GB of RAM and a quad-core CPU. For
testing each PM, we compiled six distinct, representative of use scenarios, S1
to S6, as presented in Table 1. Overall, these test scenarios correspond to
the different states a PM application may be in relation to the actions of the
end-user.

With reference to the leftmost column of Table 2(a), the first round of
experiments involved 12 popular PM desktop applications [31]. In a subsequent
round, as shown in the leftmost column of Table 3, we repeated all six
scenarios against 12 frequently encountered browser PM plugins [8]. The latter
checks have been carried out on Chrome v120.0.6099.216. By comparing the
above-mentioned table columns, it is obvious that half of the PM products are
common in both Tables 2 and 3. All the experiments were carried out in Jan.
2024 using the latest at that time version of each PM or browser plugin.

Scenario |  Description
---|---
S1 |  Enter the master password and dump the relevant processes.
S2 |  Manually lock the PM and dump the relevant processes.
S3 |  After a certain amount of idle time, the PM is locked automatically; dump the relevant processes.
S4 |  After creating a new entry password, dump the PM’s processes.
S5 |  While the PM is unlocked, click on a random entry in the corresponding list and dump the relevant processes.
S6 |  Kill the relevant processes through the task manager, rerun the application without entering the master password and dump the relevant processes.

Table 1: Utilized scenarios per PM application. The term “relevant processes”
in the scenarios refers to either the PM’s or the browser’s processes in case
of a plugin.

###  3.2 Results

Every dump (.DMP) file created after executing each of the scenarios of Table
1 was examined for possibly containing secret information in cleartext.
Specifically, we look for user credentials, either the PM’s master password or
entry passwords; usernames are also of interest, but for the needs of this
subsection the concentration is on passwords. Each PM entry corresponds to the
user’s sign-in credentials to some service, say, Facebook, as they are
contained in the PM’s vault. Typically, an entry comprises a username,
password, and the URL of the corresponding service. No less important,
normally, a PM (or browser in the case of a plugin) spawns more than one
processes, therefore a memory dump event as those in Table 1 may comprise
numerous .DMP files. Such a file contains arbitrary information and its
average size may be several megabytes, which renders its manual analysis with
a hex editor cumbersome. Instead, as detailed in subsection 3.4, this can be
relatively easily automated.

The results of applying all six scenarios to each standalone PM are
recapitulated in Table 2(a), where the ✓ and the ✗ symbols indicate either a
positive or negative result regarding password leakage, respectively.
Specifically, for each PM and scenario, the table shows if one or more of the
respective dump files contain either or both the end-user’s master and entry
passwords. Note that each scenario is independent of the others, i.e., before
executing each scenario, the PM application (or the browser in the case of
plugins) was terminated.

With reference to Table 2(a), it is observed that only three PMs do not expose
any plaintext password across all six scenarios. This does not mean that the
password(s) are indeed not present in the memory (dump); contrastingly, they
may exist in the dump, but they are in some secret obfuscated form that only
the PM can parse. Also on the positive side, no PM leaks any plaintext
password in S6, which however was somewhat expected. On the downside, four PMs
expose either or both types of passwords in all scenarios but S6. In addition,
three more PMs leak both types of passwords in at least three different
scenarios. On top of that, half of the PMs leak the master password in S1,
which is expected to be executed much more frequently than the rest of the
scenarios. The latter outcome is not much better for S2 to S5, where three,
four, five, and six PMs leak the master password. With reference to entry
passwords, the situation is not markedly better given that in S1 to S5 at
least two PMs expose entry passwords; the worst results are obtained in S4 and
S5, where nine and seven PMs leak this type of password, respectively.
Overall, across all the scenarios and all the PMs, 50 leaks are perceived, 24
and 26 for the master and entry passwords, respectively. Naturally, this
figure cannot be seen as positive, summoning vendors up for corrective
actions.

An additional important remark that emerged during our experiments with the
standalone PM applications relates to the number of times a password appears
in plaintext in the corresponding memory dump file. This information per
scenario per PM is summarized in Table 2(b). It should be clarified that for
S4 and S5, the table contains the number of times the newly added (S4) or
clicked on (S5) entry password is found in the respective .DMP file. From a
bird’s eye view, as expected, the results between the two subtables of Table 2
coincide. Concentrating on Table 2(b), the worst results for the entry
password(s) are observed for S1, S4, and S5, where two, six, and four PMs
expose at least four instances of the same password, respectively. This number
even reaches double digits in S4 (for three PMs) and S5 (for one PM).
Continuing from the previous result, S4 yields the worst score, demonstrating
six PMs exposing at least four instances of the same newly added entry
password. On the other hand, the situation regarding the master password seems
better, with the poorer and best scenario in terms of security being S1 and
S2, respectively. That is, in S1 three PMs expose at least four instances of
the master password, while in S2 no PM exposes more than two instances. As a
general comment, with special reference to S4, and to a lesser degree to S5
and S1, a significant number of PMs expose user’s entry passwords too many
times, which in turn increases the attacker’s chances of discerning the
password in the dump file, even if the latter is inspected visually.

The corresponding results referring to the 12 browser plugins are given in
Table 3. As perceived, only two plugins expose zero plaintext leaks across all
scenarios. Notably, however, six plugins leak a password in at least two
different scenarios. Significant also is that 9 out of 12 or $\approx$75% of
the plugins leak a password in S1, which is considered the most common from a
user’s perspective. Altogether, across all the scenarios and all the plugins,
23 leaks are observed, which are split into 3 and 20 for the master and entry
passwords, respectively. Although somewhat better vis-à-vis the results of
Table 2(a), this figure reveals again a rather unfavorable average result in
terms of security engineering for this type of software.

For better appreciating the results of Table 3, the following observations are
noteworthy. First, some plugins, including 1Password and Ironvest require
prior interaction between the browser and themselves for leaking an entry
password. Second, certain plugins, including ⋆PM, Kaspersky, and Sticky
Password mandate the installation of the homonymous desktop PM; therefore,
when the desktop PM application is locked, the plugin is also locked
automatically. Third, a minority of plugins, e.g., Roboform, need prior
browser interaction with any saved entry URL or the user to click on the
plugin’s dashboard for loading all entries. Fourth, in some cases, e.g.,
1Password, the leak is visible after several seconds, possibly for the browser
to finish loading the respective plugin. Lastly, most plugins do not sweep PM-
loaded user credentials away from the browser’s process after some time has
elapsed. Bitdefender and Bitwarden are the only exceptions to this situation,
clearing the browser’s process after 5 and 10 min respectively. Comparing the
common six PMs in Tables 2(a) and 3 it is extrapolated that, as expected due
to the second point above, there are two zero-leak PMs in both their
standalone and plugin version, while for the remaining ones the situation is
more or less better in their plugin version.

PM application | S1 | S2 | S3 | S4 | S5 | S6
---|---|---|---|---|---|---
1Password | ✓ | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✓ | ✓ | ✗ | ✗ | ✗
Bitwarden | ✓ | ✓ | ✗ | ✗ | ✗ | ✗ | ✓ | ✓ | ✓ | ✓ | ✗ | ✗
⋆PM | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ | ✗ | ✓ | ✗ | ✗
Kaspersky | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
KeePass 2 | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
KeePassXC | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗
Keeper | ✓ | ✓ | ✗ | ✗ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✗ | ✗
Nordpass | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ | ✓ | ✓ | ✗ | ✗
Passwarden | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✗ | ✗
PasswordBoss | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✗ | ✗
RoboForm | ✓ | ✓ | ✓ | ✗ | ✓ | ✗ | ✓ | ✓ | ✓ | ✓ | ✗ | ✗
StickyPassword | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
(a) Leaks per type of password per scenario

PM application | S1 | S2 | S3 | S4 | S5 | S6
---|---|---|---|---|---|---
1Password | 10 | - | 2 | - | 2 | - | 2 | 4 | 1 | - | -
Bitwarden | 8 | 2 | - | - | - | - | 1 | 7 | 1 | 3 | -
⋆PM | - | - | - | - | - | - | - | 1 | - | 1 | -
Kaspersky | - | - | - | - | - | - | - | - | - | - | -
KeePass 2 | - | - | - | - | - | - | - | - | - | - | -
KeePassXC | - | - | - | - | - | - | - | 1 | - | - | -
Keeper | 4 | 4 | - | - | 4 | 4 | 4 | 4 | 4 | 4 | -
Nordpass | 2 | - | - | - | - | - | - | 24 | 4 | 19 | -
Passwarden | 2 | 3 | 2 | 3 | 2 | 3 | 2 | 12 | 2 | 8 | -
PasswordBoss | - | 8 | - | 4 | - | 1 | - | 11 | - | 7 | -
RoboForm | 1 | 1 | 1 | - | 2 | - | 1 | 2 | 1 | 2 | -
StickyPassword | - | - | - | - | - | - | - | - | - | - | -
(b) Password instances per scenario

Table 2: Results for standalone PMs. Master and entry cleartext password leaks
are given in white and light-gray background, respectively (✓: password leak,
✗: no password leak). The vendor preceded with a star ($\star)$ requested to
remain undisclosed until the vulnerability has been patched.

Browser PM plugin | S1 | S2 | S3 | S4 | S5 | S6
---|---|---|---|---|---|---
1Password | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✗ | ✗ | ✗ | ✓ | ✗ | ✗
Avira | ✗ | ✓ | ✗ | ✓ | n/a | ✗ | ✗ | n/a | ✗ | ✓
Bitdefender | ✓ | ✗ | ✓ | ✗ | n/a | ✗ | ✗ | ✗ | ✓ | ✗ | ✗
Bitwarden | ✗ | ✓ | ✗ | ✗ | n/a | ✗ | ✗ | n/a | ✗ | ✗
Dashlane | ✓ | ✓ | ✗ | ✗ | n/a | ✗ | ✗ | n/a | ✗ | ✗
⋆PM | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗
Ironvest | ✗ | ✓ | ✗ | ✓ | n/a | ✗ | ✗ | n/a | ✗ | ✗
Kaspersky | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | n/a | ✗ | ✗ | ✗ | ✗
LastPass | ✗ | ✓ | ✗ | ✓ | n/a | ✗ | ✗ | n/a | ✗ | ✓
Norton | ✗ | ✓ | ✗ | ✓ | n/a | ✗ | ✗ | n/a | ✗ | ✓
RoboForm | ✗ | ✓ | ✗ | ✗ | n/a | ✗ | ✗ | n/a | ✗ | ✗
StickyPassword | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗

Table 3: Results for PM browser plugins. Master and entry cleartext password
leaks are given in white and light-gray background, respectively (✓: password
leak, ✗: no password leak). n/a (not applicable) means that either the plugin
does not have an automatic lock feature (S3), or the plugin does not offer an
“add new entry” feature (S4), or the plugin loads all entries automatically
when the browser is started (S5).

###  3.3 Discussion

Abiding by a Coordinated Vulnerability Disclosure (CVD) process, the results
have been promptly communicated to most of the affected PM vendors along with
the corresponding exploit(s). Two vendors did acknowledge the issue and
reserved a Common Vulnerabilities and Exposures (CVE) ID, namely
CVE-2023-23349, pledging to also provide a respective patch promptly. The rest
of them provided arguments in two directions. The first implies that the
vendors are already aware of the problem, but they largely tend to underrate
it. Namely, their basic claim is that performing such an attack would require
an already compromised system and a malicious actor with escalated privileges.
On the one hand, this assertion is half true; based on our results in
subsection 3.2, there do exist PM products that do not leak secret information
in plaintext, and this stands true for both the standalone PMs (three
instances) and the respective plugins (two instances). On the other hand, as
detailed in Section 2, all PMs except one do not even require elevated
privileges for dumping their processes. In this respect, the attacker does not
even need to bypass UAC, which generally is cumbersome.

In relation to the previous argument, others may reason that a keylogger
malware may achieve the same goal, without collecting and inspecting a PM
memory dump file. The counterargument here is that a keylogger needs to
stealthily operate round-the-clock and opportunistically wait for the victim
to type a password. Even more, the attacker can save the relevant .DMP file
and analyze it offline, let alone that if the respective PM divulges the
master password, then the attacker instantly gains access to every entry the
victim keeps in the PM’s vault.

The second line of arguments, even not so common, suggests that some of the PM
vendors are unsure about how to address the issue effectively. That is, if
they apply some form of obfuscation to the passwords loaded in memory, they
may provide a false sense of security; this is a form of security by
obscurity, which generally, as well as in this case, is only effective as long
as the (secret) obfuscation pattern remains confidential. On the flip side,
unauthorized access to the PM will typically lead to multiple breaches (this
equals the number of different entry passwords stored in the PM). In the
presence of no other viable and stronger defense, obfuscation seems, at least
for the moment, the only way forward. To enhance the robustness of the
obfuscation procedure, a variable, frequently changing pattern should be used,
ideally paired with other techniques, e.g., the system-protected process
security model, also known as Protected Process Light (PPL) technology [20].
For more information on this issue, the reader is referred to Section 4.

###  3.4 Red teaming and exploits

To propel research and advance red teaming methodologies and responsible
disclosure in this field, we developed an open-source tool called Pandora,
which is publicly available in a GitHub repository [4]. Pandora assists in
harvesting user credentials, both usernames and passwords, possibly present in
cleartext in the processes spawned by several well-known PMs. The tool
supports MS Windows 10 standalone PMs and browser plugins; at the time of
writing, it supports 14 PMs, with 18 different implementations, e.g., it can
dump end-user credentials either from the desktop application or the browser
plugin of the same product. The tool requires the PM to be running and
unlocked. For additional and up-to-date information on the tool, the
interested reader is referred to [4].

Moreover, for reasons of completeness, Algorithm 1 offers a generic exploit in
pseudocode. As seen in line 3, after opening the respective .DMP file, the
algorithm applies a search pattern. The latter is based on the common
technique exploited by anti-malware services; a file is flagged as possibly
malicious when a sequence of hex bytes, i.e., a signature, is present within
the file. In this work, we exploit the same logic to identify the patterns
that hold the user’s credentials in the corresponding PM’s process memory
dump. To identify a possible pattern, first, one needs to configure each PM
with different credentials, namely, master and entry passwords along with
their matching usernames, if required by the PM depending on the case. Then,
using a hex editor, each dump file is manually investigated for containing one
or more of these entries. If some entry is found, that entry’s preceding and
succeeding bytes are kept as possible search patterns. Then, the investigator
checks if the same bytes (pattern) stay identical across different PM states
with the same or other entries. If yes, then this pattern can be used to
produce an exploit for the specific PM. In Algorithm 1, given the pattern of
line 3, lines 5-21 gather the next or previous n bytes, and convert them to
UTF-8 for revealing the password (and username) in plaintext. The number of
bytes n is defined empirically, and it may be different per PM; for Algorithm
1 this parameter is set to 300 bytes in line 12.

Algorithm 1 getCreds

1:function getCreds

2: Open file “app.dmp” in binary mode

3: $searchPattern\leftarrow$ [0x80, 0x00, 0x04 … 0x00]

4: $foundData\leftarrow$ empty list

5: while not end of file do

6: Read a byte from file and store it in variable $c$

7: if $c=$ next element in $searchPattern$ then

8: Add $c$ to $foundData$

9: if length of $foundData=$ length of $searchPattern$ then

10: $extractedData\leftarrow$ empty list

11: $dataCount\leftarrow 0$

12: while $dataCount<300$ and not end of file do

13: Read a byte from file and add it to $extractedData$

14: Increment $dataCount$

15: $utf8ExtractedData\leftarrow$ convert $extractedData$ to UTF-8 string

16: Print “Pattern Data:” concatenated with $utf8ExtractedData$

17: Save into file using saveFile function with parameter

18: $utf8ExtractedData$

19: Clear $foundData$

20: else

21: Clear $foundData$

22: Close file

##  4 Best practices

The current section offers a brief description of the best practices that
should be employed for sensitive information management by a PM application.
As discussed in Section 3, the success of the attack hinges on the fact that
essential information is written in an unprotected (plaintext) format by the
PM application. Nevertheless, there exist numerous guidelines [6, 23, 22] that
dictate the proper way for handling private information by such applications.

Primarily, as delineated by the Open Worldwide Application Security Project
(OWASP), the relevant guidelines concentrate on the proper employment of
cryptographic primitives [23, 22]. In this respect, encryption and hashing
algorithms are considered the mainstream tools in the quiver of the PM
developers to protect private information, even though key management remains
an important matter in the whole process. Along the same line, obfuscation
techniques can be used to hide essential information when loaded in memory;
note that the obfuscation pattern must remain secret, which indeed is a form
of security by obscurity. Moreover, the pattern should be frequently changed,
which also entails management operations, and naturally, there is no silver
bullet that can ensure key management security horizontally. No less
important, OWASP suggests that private data should be stored only when
necessary during the application life cycle [22] and remain the minimum
possible time in the RAM [24]. This can be achieved through spawning garbage
collection requests soon after every critical operation, or by overwriting the
content of a critical object once the object is no longer needed.

Another commonly used technique when it comes to the application development
stage is to eliminate variable copies [6]. Precisely, modern programming
languages afford several operations on variables, which essentially create
copies of the same information in the memory stack; each variable assignment
creates a copy of the same information. This might be the case with the
scrutinized PMs, as Table 2(b) confirms that both the master and entry
passwords were identified multiple times in the respective memory dumps. In
turn, this facilitates an attacker to monitor the repeating piece of data,
subsequently revealing repeating private information and the associated byte
patterns. Another relevant point is that some PMs, including Keeper do provide
an option for wiping secret data after the PM is automatically locked.
Nevertheless, this option is provided as opt-in and it is disabled by default,
which does not abide by the well-known principle of establishing security
defaults, namely establish default secure settings and possibly provide opt-in
and opt-out. Even more, a minority of PMs support Two-Factor Authentication
(2FA), which is positively appraised. Nevertheless, in some of them, the 2FA
token created during user sign-in with the PM leaks in the memory as well.

Last but not least, as already pointed out in Sections 2 and 3.3, developers
should also take advantage of the UAC mandatory access control enforcement
feature, which limits the ability of malicious code to execute with
administrator privileges. By means of UAC, every application that needs the
administrator access token must prompt the end-user for approval, the so-
called “elevation prompt”. Typically, when a user runs a process, that process
executes with their token and can do anything the user can do; in short, a
token says who you are and what you are allowed to do. In this context, every
time a Windows thread wishes to access a protected object, the operating
system conducts a security check. With reference to subsection 3.3, vendors
can potentially capitalize on the so-called Protected anti-malware services,
also known as PPL [20]. This enables specially-signed programs to run in a way
that they are invulnerable to tampering and termination, even by
administrative users. On the downside, running a PM with administrative rights
to enable PPL, may open the door for privilege escalation attacks through DLL
sideloading, e.g., as given in CVE-2023-48861.

##  5 Related Work

This section elaborates on the related work on PM security, concentrating on
publications from the past decade. The first set of related works focuses on
applications that expose sensitive data while running on different operating
systems, namely Windows, Linux and Android. [2] investigated the potential
leakage of private data from Android applications residing in RAM. According
to the findings, 29 out of 30 applications were found to disclose information
that facilitated the retrieval of user passwords. A similar research in [15]
presented an analysis of Android applications, including PMs, regarding the
persistence of passwords in system memory even after they are not needed.
Based on their results, one of the main components responsible for leaking
private information is UI widgets. To address this, they proposed an
enhancement to the Android platform by introducing the SecureTextView, a
secure version of Android TextView, which employs the zeroization of specific
buffers. In a related research [19], the authors explored zeroization
techniques across MS Windows and Linux environments. They concluded that only
the MS Windows operating system provides a reliable function that can be used
for this purpose. Finally, they showcased credential extraction from memory
allocated to browsers, highlighting potential security vulnerabilities.

Extending the analysis to web-based PMs that offer standalone applications
and/or browser extensions, the authors of [21] conducted a security evaluation
of seven web-based PMs. Their evaluation covered both the standalone
application and browser extension-based versions, encompassing the entire PM
lifecycle, from password generation and storage to autofill functions. Despite
the improvement of the scrutinized PMs in terms of security compared to prior
evaluations, persistent issues remain. These include unencrypted metadata,
insecure defaults, and vulnerabilities to clickjacking attacks. The authors
explored the methods that each of the PMs utilizes to securely store client
data in the respective database files. However, they overlooked the manner in
which the very same data are treated by the host’s OS and how they are stored
in RAM.

In the same vein, the authors in [16] performed a security analysis of five
web-based PMs, uncovering varying susceptibilities across all the examined
PMs. Among the identified vulnerabilities were issues related to logic and
authorization, misinterpretation of the web security model, and sensitivity to
CSRF/XSS attacks. Nevertheless, the authors specifically focused on
vulnerabilities stemming from the web-based nature of PMs. In contrast, our
study takes a different angle by delving into the local security posture of
PMs, with a particular emphasis on examining how sensitive data are stored
within the users’ RAM. Another study [3] presented flaws in the LastPass
browser extension. In this work, the authors performed a password extraction
attack directly from the extension’s memory while the extension was active. In
their research, Zhao, Yue, and Sun [32] provided a vulnerability analysis of
LastPass and Roboform, which both provide browser extensions for operating the
software and cloud storage of the passwords. They identified potential
threats, including the storage of credentials in plaintext locally and/or on
cloud servers. Additionally, they provided recommendations to enhance data
protection measures, improving the overall security level of these products.

A second set of works focused on data leakage during the autofill function
that modern PMs offer. The authors of [28] studied how different PMs handle
automatic filling of passwords. Their work encompassed three categories of
password managers: (a) built into browsers, (b) used in mobile devices, and
(c) 3rd party applications. Their findings suggested that the various autofill
policies followed by these tools can expose users to serious risks, such as a
remote attacker stealing multiple passwords from their PM without their
knowledge. Similarly, [11] presented a technique, called AutoSpill, to leak
users’ saved credentials during an autofill operation on a webpage loaded into
an application’s WebView.

As a third tendency, some authors have followed forensic techniques to study
the security of PM applications. In [12], three desktop PMs were analyzed,
namely KeePass, Password Safe, and RoboForm, following a systematic forensic
analysis approach. The findings of this study showed that unencrypted
passwords and sensitive data were found in the clipboard, Temp folders, Page
files, or the Recycle Bin. Moreover, the authors of [27] followed a digital
forensic procedure to analyze the storage security of two Android PM
applications. Their focus is, however, on protecting long-term data stored in
Android devices, leaving out attacks targeting encryption keys in main system
memory.

In a similar direction to our work, [17] scrutinized the security of three
open-source PMs against various types of attacks, both new and already known.
The PMs they studied were Passbolt, Padlock, and Encryptr; among other
vulnerabilities, the authors found that all three applications were
susceptible to keyloggers and clipboard attacks. Additionally, the authors of
[5] analyzed five popular PMs against already disclosed and newly discovered
vulnerabilities. Notably, their findings indicated that four of the five
studied PMs, namely Dashlane, LastPass, Keeper, and RoboForm, were prone to
clipboard attacks.

When it comes to tools for password recovery from PMs, LaZagne [1] is an open-
source application capable of retrieving locally stored computer passwords. It
supports various methods that applications use to store their passwords, such
as plaintext, APIs, custom algorithms, and databases. Among the supported
applications are popular browsers, mail clients, chat apps, databases, and
memory dumps from password managers such as Keepass. Pandora [4], the tool we
have developed for our experiments, centers specifically on PMs and delves
deeper than LaZagne. It uncovers passwords that are inadequately managed by 18
different implementations of PMs, spanning Windows apps, browsers, and browser
plugins.

##  6 Conclusions

Password credentials are still the dominant way to perform user
authentication. Nevertheless, with the numerous accounts that the average user
manages today, it is challenging to create secure passwords while at the same
time remembering them by heart. PMs come with the pledge to fill this gap by
proposing strong passwords and storing them securely on desktop and mobile
devices. But leaving aside the convenience provided by such applications and
the widely held opinion that they are safe to use, can they really be trusted
to fully deliver what they promise?

This paper provides an up-to-date security analysis of the main desktop and
browser plugin PMs. This is done under the strong but viable assumption that
the adversary has somehow gained access to the local system, being also
capable of dumping the respective processes. Our findings suggest that, across
six representative from an end-user’s viewpoint scenarios, 75% of desktop PMs
and 83% of browser PM plugins store passwords in plaintext in the system
memory while in use. Among others, the key issue with these security-sensitive
applications is that this “storing” is done in a way that creates patterns,
which can be easily discerned by the adversary. Even worse, this secret
information not only remains in the memory for a significant time after
serving its purpose, but it is also stored in multiple copies, making the
attacker’s job easier. A possible direction for future work is to expand this
research to mobile platforms and to several other popular applications of
different kinds, including financial/banking, team collaboration, smart home,
etc.

## Acknowledgements

This work is supported by the Research Council of Norway through the SFI
Norwegian Centre for Cybersecurity in Critical Sectors (NORCICS) project no.
310105 and by the European Union through the Horizon 2020 project PERSEUS
(Grant No. 101034240).

## References

  * [1] AlessandroZ: LaZagne, https://github.com/AlessandroZ/LaZagne, accessed 14/01/2024
  * [2] Apostolopoulos, D., Marinakis, G., Ntantogian, C., Xenakis, C.: Discovering authentication credentials in volatile memory of android mobile devices. In: Conference on e-Business, e-Services and e-Society. pp. 178–185 (2013)
  * [3] Barten, D.: Client-side attacks on the lastpass browser extension (2019), https://api.semanticscholar.org/CorpusID:153315475
  * [4] C., E.: Pandora, https://github.com/efchatz/pandora, accessed 14/01/2024
  * [5] Carr, M., Shahandashti, S.F.: Revisiting security vulnerabilities in commercial password managers. In: ICT Systems Security and Privacy Protection. pp. 265–279 (2020)
  * [6] Chapman, R.: Sanitizing sensitive data: How to get it right (or at least less wrong…). In: Reliable Software Technologies – Ada-Europe 2017. pp. 37–52 (2017)
  * [7] Chaudhary, S., Schafeitel-Tähtinen, T., Helenius, M., Berki, E.: Usability, security and trust in password managers: A quest for user-centric properties and features. Computer Science Review 33, 69–90 (2019)
  * [8] Cybernews: Chrome password managers: Best extensions in 2023. https://cybernews.com/best-password-managers/password-managers-for-chrome, accessed 14/01/2024
  * [9] Fagan, M., Albayram, Y., Khan, M.M.H., Buck, R.: An investigation into users’ considerations towards using password managers. Human-centric Computing and Information Sciences 7(1), 12 (Mar 2017)
  * [10] Fung, B., Sands, G.: Former SolarWinds CEO blames intern for ‘solarwinds123’ password leak, https://edition.cnn.com/2021/02/26/politics/solarwinds123-password-intern/index.html, accessed 14/01/2024
  * [11] Gangwal, A., Singh, S., Srivastava, A.: Autospill: Credential leakage from mobile password managers. In: Proceedings of the Thirteenth ACM Conference on Data and Application Security and Privacy. pp. 39–47 (2023)
  * [12] Gray, J., Franqueira, V.N.L., Yu, Y.: Forensically-sound analysis of security risks of using local password managers. In: 2016 IEEE 24th (REW). pp. 114–121 (2016)
  * [13] hfiref0x: Defeating windows user account control, https://github.com/hfiref0x/UACME#readme, accessed 14/01/2024
  * [14] Kotadia, M.: Gates predicts death of the password, https://www.cnet.com/news/privacy/gates-predicts-death-of-the-password, accessed 14/01/2024
  * [15] Lee, J., Chen, A., Wallach, D.S.: Total recall: Persistence of passwords in android. Proceedings 2019 Network and Distributed System Security Symposium (2019), https://api.semanticscholar.org/CorpusID:142503488
  * [16] Li, Z., He, W., Akhawe, D., Song, D.: The $\\{$Emperor’s$\\}$ new password manager: Security analysis of web-based password managers. In: 23rd USENIX. pp. 465–479 (2014)
  * [17] Luevanos, C., Elizarraras, J., Hirschi, K., Yeh, J.h.: Analysis on the security and use of password managers. In: 2017 18th (PDCAT). pp. 17–24 (2017)
  * [18] Machine, W.: Counterpane systems brings the security of blowfish to a password database, https://web.archive.org/web/19980119124510/http://www.counterpane.com/passsafe.html, accessed 14/01/2024
  * [19] Malliaros, S., Ntantogian, C., Xenakis, C.: Protecting sensitive information in the volatile memory from disclosure attacks. In: 2016 11th International Conference on Availability, Reliability and Security (ARES). pp. 687–693 (2016)
  * [20] Microsoft: Protecting anti-malware services, https://learn.microsoft.com/en-us/windows/win32/services/protecting-anti-malware-services-, accessed 14/01/2024
  * [21] Oesch, S., Ruoti, S.: That was then, this is now: A security evaluation of password generation, storage, and autofill in browser-based password managers. In: Proceedings of the 29th USENIX. pp. 2165–2182 (2020)
  * [22] OWASP: Cryptographic storage, https://cheatsheetseries.owasp.org/cheatsheets/Cryptographic_Storage_Cheat_Sheet.html#minimise-the-storage-of-sensitive-information, accessed 14/01/2024
  * [23] OWASP: Secure product design, https://cheatsheetseries.owasp.org/cheatsheets/Secure_Product_Design_Cheat_Sheet.html, accessed 14/01/2024
  * [24] OWASP: Testing memory for sensitive data. https://mas.owasp.org/MASTG/tests/android/MASVS-STORAGE/MASTG-TEST-0011/, accessed 14/01/2024
  * [25] PASCU, L.: 29 vulnerabilities found in android password managers, https://www.bitdefender.com/blog/hotforsecurity/29-vulnerabilities-found-in-top-rated-password-managers-for-android/, accessed 14/01/2024
  * [26] Pearman, S., Zhang, S.A., Bauer, L., Christin, N., Cranor, L.F.: Why people (don’t) use password managers effectively. In: SOUPS 2019. pp. 319–338 (2019)
  * [27] Sabev, P., Petrov, M.: Android password managers and vault applications: Data storage security issues identification. J. Inf. Secur. Appl. 67(C) (2022)
  * [28] Silver, D., Jana, S., Boneh, D., Chen, E., Jackson, C.: Password managers: Attacks and defenses. In: 23rd USENIX. pp. 449–464 (2014)
  * [29] sysdig: Keepass cve-2023-32784: Detection of processes memory dump. https://sysdig.com/blog/keepass-cve-2023-32784-detection/, accessed 14/01/2024
  * [30] Turton, W., Mehrotra, K.: Colonial Pipeline, https://www.governing.com/security/bad-password-gave-colonial-pipeline-hackers-access, accessed 14/01/2024
  * [31] Wired: The best password managers to secure your digital life. https://www.wired.com/story/best-password-managers/, accessed 14/01/2024
  * [32] Zhao, R., Yue, C., Sun, K.: Vulnerability and risk analysis of two commercial browser and cloud based password managers. Academy of Science and Engineering Science Journal 1, 183–197 (2013)


Gradient Flow and Holography from a Local Wilsonian Cutoff

Ulrich Ellwanger
University Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France

We consider the vacuum partition function of a 4d scalar QFT in a curved
background as function of bare marginal and relevant couplings. A local UV
cutoff $\Lambda(x)$ transforming under Weyl rescalings allows to construct
Weyl invariant kinetic terms including Wilsonian cutoff functions. The local
cutoff can be absorbed completely by a rescaling of the metric and the bare
couplings. The vacuum partition function satisfies consistency conditions
which follow from the Abelian nature of local redefinitions of the cutoff, and
which differ from Weyl rescalings. These imply a gradient flow for beta
functions describing the cutoff dependence of rescaled bare couplings. The
consistency conditions allow to satisfy all but one Hamiltonian constraints
required for a holographic description of the flow of bare couplings with the
cutoff.

##  1 Introduction

In quantum field theories (QFTs) coupling constants including masses can be
promoted to local functions of space-time in which case they become sources
for corresponding operators. If QFTs are considered in curved space-time,
local Weyl transformations can be defined. These are sensitive to the
ultraviolet regularization and renormalization of QFTs, i.e. to counter terms
and beta functions. The Abelian nature of local Weyl transformations implies
consistency conditions on coefficients of the vacuum partition function and
beta functions. This program was initiated by Osborn and Jack and Osborn [JO]
in [1, 2, 3]. Implied relations among coefficients of $\beta$ functions in
dimensional regularization have been studied in [4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14] for the Standard Model and others.

In dimensional regularization classical Weyl invariance is broken by the scale
$\mu$ introduced for otherwise dimensionless couplings. Here we consider a
regularization by a Wilsonian cutoff $\Lambda$, by which we understand a
modification of kinetic terms in the action such that modes with momenta
$p^{2}\gg\Lambda^{2}$ are suppressed, i.e. the kinetic terms become very (e.g.
exponentially) large for $p^{2}\gg\Lambda^{2}$. We consider a local cutoff
$\Lambda(x)$ transforming under local Weyl transformations which allows to
define local renormalization group equations (RGEs). A local RGE was
considered as early as 1987 in [15] in two-dimensional curved spacetime sigma
models in order to derive consistency conditions on allowed backgrounds in
string theory. A local Wilsonian cutoff was considered e.g. in [16, 17], with
the aim to study exact (functional) RGEs in 2 and 4 dimensions.

In the present paper we focus on the dependence of bare couplings on the
cutoff, and on the bare vacuum partition function before adding counter terms.
The dependence of bare couplings on the cutoff is also described by beta
functions, which are functions of bare couplings and of the cutoff in case of
relevant couplings (such as masses) as considered here. Via subdivergences,
the bare vacuum partition function is also sensitive to the cutoff dependence
of bare couplings. (A Wilsonian cutoff underlines the hierarchy problem, i.e.
renormalized and bare masses are naturally of the order of the UV cutoff. For
renormalized masses this might be solved by fine tuning or by Supersymmetry;
the subsequent results do not depend on whether or how this issue is
solved.111We remark that we will not employ the (exact) Wilsonian
renormalization group.)

The introduction of a local Wilsonian cutoff transforming under Weyl
transformations allows to construct Weyl invariant kinetic terms for scalars
in 4d [18]; here the local component of the cutoff plays the role of a
compensator. Assuming massive fields only, one finds that the local cutoff
dependence of the bare vacuum partition function can be absorbed completely by
rescalings of the space-time metric and bare relevant couplings, a particular
feature of a Wilsonian cutoff. This allows to express the dependence of the
bare vacuum partition function on the local cutoff in terms of its dependence
on the rescaled space-time metric and, via corresponding beta functions, on
bare marginal and rescaled relevant couplings.

The possibility to absorb the local cutoff by rescalings of the space-time
metric and bare relevant couplings does not imply the absence of anomalies.
Assuming massive fields only, the cutoff appears in the form of logarithms
$\log(\Lambda^{2}/m_{0}^{2})$, i.e. bare masses play the role of the
renormalization scale $\mu$ in dimensional regularization. After a rescaling,
bare masses squared $m_{0}^{2}$ become $\Lambda^{2}\hat{m}_{0}^{2}$ with
$\hat{m}_{0}^{2}$ dimensionless, hence logarithms
$\log(\Lambda^{2}/m_{0}^{2})$ become $-\log(\hat{m}_{0}^{2})$. In the bare
vacuum partition function such logarithms, multiplying e.g. the Euler density
which is not Weyl invariant, indicate the presence of Weyl anomalies as
studied (including their dependence on dimensionless couplings) by [JO].

Local transformations of the cutoff are Abelian transformations similar to
(but different from) local Weyl transformations. Accordingly they lead to
consistency conditions similar to (but different from) local Weyl
transformations. These consistency conditions imply a gradient flow for the
beta functions describing the cutoff dependence of bare marginal and rescaled
relevant couplings. In an expansion to first non-trivial order in derivatives
the role of the potential of the gradient flow is played by the coefficient of
the Riemann scalar $R$ in the bare vacuum partition function.

Subsequently we turn to a possible holographic description of the flow of bare
couplings with the cutoff. Soon after the birth of the AdS/CFT correspondence
it became clear that a cutoff on the extra coordinate $\rho$ for $\rho\to 0$
of an asymptotic AdS space can be related to a UV cutoff or a RG scale of a
conformal field theory (CFT) (see [19] for an early review). A finite result
for the CFT action on the boundary is obtained only if the induced metric and
the fields are rescaled by appropriate powers of $\rho$. In addition, extra
counter terms are required to cancel logarithmic divergences multiplying e.g.
the Euler density. Requiring these counter terms to be local and covariant,
Henningson and Skenderis have shown in [20] that the so obtained finite action
on the boundary still varies under conformal transformations of the boundary
metric, and that the corresponding Weyl anomalies match the ones of the CFT.
In the present approach counter terms are omitted, and the Weyl anomalies
appear directly as non-invariance under conformal transformations of
corresponding terms in the bare action on the boundary. However, within our
expansion to second order in derivatives these Weyl anomalies are not yet
visible for a 4d QFT on the boundary of a 5d bulk for which Weyl anomalies are
of fourth order in derivatives.

It was frequently underlined, however, that a better understanding of the
precise relation between a UV cutoff on a QFT and on the extra coordinate of
an asymptotic AdS space would be desirable (see e.g. [21, 22, 23]). It is
clear that dimensional regularization of a QFT is of little help here as the
notion of a UV cutoff does not exist.

In the framework of classical general relativity in a 5d bulk, the flow with
the fifth coordinate of fields living on a 4d boundary can be described within
the Hamilton-Jacobi formalism [24] (see also [25]). The Hamiltonian constraint
implies constraints on the coefficients of the action on the 4d boundary, and
on the beta functions once the fields on the boundary are identified with
running bare couplings of a 4d QFT. For these beta functions one of the
Hamiltonian constraints implies a gradient flow. Based on this feature a
condition for a holographic duality was proposed in [26]. (An interpretation
of the renormalization group flow as a Hamiltonian vector flow was already
proposed by B. Dolan in 1994 [27].)

Only few studies concern the question whether the RG flows of generic QFTs
(non-CFTs) admit a holographic description [28]. Confining oneself to
classical general relativity in the 5d bulk, mostly two approaches were
persued in order to establish a link between 4d QFTs (conformal or non-CFTs)
and holography: One approach uses the formal similarity between the Hamilton-
Jacobi formalism (quadratic in functional derivatives of the action on the
boundary with respect to fields) and Wilson’s (or Polchinski’s [29]) exact RG,
see e.g. [30, 31, 32, 33]. Alternatively one can focus on the link between
Weyl transformations acting on the 4d QFT/CFT vacuum partition function on the
boundary and diffeomorphisms in an AdS5 bulk [34], see e.g. [35, 36, 37, 38,
39, 40] for corresponding studies. An approach towards quantum gravity in the
bulk from a matrix field theory via quantum RG has been persued in [41].

A known issue is whether a description of a QFT on an (inner) boundary of AdS
in terms of a local Lagrangian in the bulk is possible at all [42] (see also
[43] and refs. therein). Here we assume this to be the case, which corresponds
to the assumption that an expansion of a local covariant 5d Lagrangian in
derivatives exists allowing to obtain connected N point functions of fields at
different points. For points on the boundary, these must match the N point
functions of the QFT after suitable rescalings.

Here we address the question whether the RG flow of bare marginal and relevant
couplings of a generic massive (non-CFT) scalar QFT admits a holographic dual.
The number of bare marginal and relevant couplings will be denoted by $n_{g}$.
We review shortly the Hamilton-Jacobi formalism leading to flow equations for
fields and the Hamiltonian constraint on the action on the boundary of a 5d
bulk. Given a 5d Lagrangian in the bulk, the Hamiltonian constraint leads to
constraints on coefficients of the action on the boundary. Assuming this
action to be given by the vacuum partition function of a generic QFT expanded
to second order in derivatives, its coefficients seem to be heavily over-
determined.

However, we find that this is not so if the following dictionnary is applied:
1) The fields on the boundary of the 5d bulk are identified with 4d bare
marginal or rescaled relevant couplings, and the boundary action with the 4d
vacuum partition function.
2) The 4d metric on the boundary of the 5d bulk satisfies a field dependent
flow equation. Nevertheless, the 4d metric on the boundary of the 5d bulk is
identified with the 4d QFT metric after its rescaling by the local cutoff
$\Lambda(x)$. This implies that, implicitely, the fifth bulk coordinate on the
boundary is related in a field dependent way to the local QFT cutoff
$\Lambda$. (As expected, the 5d bulk metric is AdS up to field dependent
corrections.)
3) Then $n_{g}$ consistency conditions on coefficients of the QFT originating
from the Abelian nature of local cutoff transformations coincide exactly with
$n_{g}$ Hamiltonian constraints which seem to over-determine the vacuum
partition function of a generic QFT. The remaining Hamiltonian constraints
allow to reconstruct a suitable 5d bulk Lagrangian for a generic (massive) 4d
QFT provided one additional condition (within the considered order of an
expansion in derivatives) is satisfied. This is the main result of the second
part of this paper.

In Section 2 we construct Weyl invariant kinetic terms including a Wilsonian
cutoff as well as interaction terms following [18]. We introduce a rescaled
metric $\hat{\gamma}_{\mu\nu}$ and rescaled bare couplings $\hat{g}_{0}^{a}$
in terms of which the explicit dependence of the vacuum partition function
$\hat{S}_{QFT}$ on the local cutoff disappears. A consistency condition to be
satisfied by $\hat{S}_{QFT}$, expanded to second order in derivatives,
following from the Abelian nature of local variations of the cutoff is
derived. This consistency condition assumes the form of an exact gradient flow
for the beta functions.

In Section 3 we review the Hamilton-Jacobi formalism following to a large
extend refs. [24, 25]. Parts of the derivation are given in Appendix A. The
corresponding flow equations and Hamiltonian constraints are derived. In
Section 4 we show that, within the expansion up to second order in
derivatives, all Hamiltonian constraints up to one can be satisfied by an
arbitrary 4d scalar QFT thanks to the consistency condition following from the
local Wilsonian cutoff. A summary and outlook is given in Section 5.

##  2 Gradient Flow From Weyl Invariant Wilsonian Cutoff Functions

In this Section we construct a Weyl invariant kinetic term, including a
Wilsonian UV cutoff, for a real scalar field $\varphi$ in $d$ dimensions. The
UV cutoff $\Lambda$ is assumed to be local, i.e. $\Lambda(x)$, and to
transform under Weyl rescalings together with the background metric
$\gamma_{\mu\nu}$ and the scalar field:

$\displaystyle\delta_{\sigma}\gamma_{\mu\nu}=$
$\displaystyle-2\sigma\gamma_{\mu\nu}\;,$ (2.1)
$\displaystyle\delta_{\sigma}\varphi=$
$\displaystyle\sigma\left(\frac{d}{2}-1\right)\varphi\;,$
$\displaystyle\delta_{\sigma}\Lambda=$ $\displaystyle\sigma\Lambda\;.$

One may write the local cutoff $\Lambda(x)$ in the form

$\Lambda(x)=\overline{\Lambda}e^{\lambda(x)}\;,\quad\delta_{\sigma}\overline{\Lambda}=0\;,\quad\delta_{\sigma}\lambda(x)=\sigma(x)\;.$
(2.2)

$\lambda(x)$ plays the role of a compensator: Due to the presence of
$\lambda(x)$ the effective action satisfies a local RGE (see eq. (2.21)
below), but in the “gauge” $\lambda(x)=0$ this hold no longer.

For the construction of a kinetic term including a local Wilsonian cutoff we
start with the known expression for a Weyl covariant generalization of the
covariant Laplacian

$\nabla^{2}-\xi
R\;,\quad\nabla^{2}=\frac{1}{\sqrt{\gamma}}\partial_{\mu}\sqrt{\gamma}\gamma^{\mu\nu}\partial_{\nu}\;,\quad\gamma=\det({\gamma_{\mu\nu}})\;,\quad\xi=\frac{d-2}{4(d-1)}\;,$
(2.3)

which satisfies

$\delta_{\sigma}\left[(\nabla^{2}-\xi R){\cal
O}\right]=\left(\frac{d}{2}+1\right)\sigma\left[(\nabla^{2}-\xi R){\cal
O}\right]$ (2.4)

provided the operator $\cal{O}$ satisfies $\delta_{\sigma}{\cal
O}=\sigma\left(\frac{d}{2}-1\right){\cal O}$ such that, for ${\cal
O}=\varphi$, $\delta_{\sigma}(\sqrt{\gamma}\varphi(\nabla^{2}-\xi
R)\varphi)=0$ (using $\delta_{\sigma}\sqrt{\gamma}=-d\,\sigma\sqrt{\gamma}$).
A Weyl invariant Wilsonian cutoff can be constructed with help of the operator

$D_{\Lambda}=\Lambda^{-2}(\nabla^{2}-\xi R)\;.$ (2.5)

Using $\delta_{\sigma}\Lambda^{-2}=-2\sigma\Lambda^{-2}$, any expression of
the form

$F(D_{\Lambda})$ (2.6)

satisfies

$\delta_{\sigma}\left(F(D_{\Lambda})\varphi\right)=\left(\frac{d}{2}-1\right)\sigma
F(D_{\Lambda})\varphi\;.$ (2.7)

The possibility to construct $F(D_{\Lambda})$ with this property under local
Weyl transformations requires the use of the $x$ dependent cutoff $\Lambda$
transforming as in eq. (2.1). An example is

$F(D_{\Lambda})=e^{-D_{\Lambda}}$ (2.8)

which leads to an exponentially suppressed propagator for large $p^{2}$ in
momentum space. The kinetic part $S_{k}$ of the action reads then

$S_{k}(\gamma,\varphi,\Lambda)=\frac{1}{2}\int\sqrt{\gamma}d^{d}x\,\varphi(-\nabla^{2}+\xi
R)\,F(D_{\Lambda})\varphi$ (2.9)

and satisfies

$\sigma(x)\left\\{-2\gamma_{\mu\nu}(x)\frac{\delta}{\delta\gamma_{\mu\nu}(x)}+\varphi(x)\frac{\delta}{\delta\varphi(x)}+\Lambda(x)\frac{\delta}{\delta\Lambda(x)}\right\\}S_{k}(\gamma,\varphi,\Lambda)=0\;.$
(2.10)

In terms of a rescaled metric

$\hat{\gamma}_{\mu\nu}=\gamma_{\mu\nu}\Lambda^{2}$ (2.11)

eq.(2.10) becomes

$\sigma(x)\left\\{\varphi(x)\frac{\delta}{\delta\varphi(x)}+\Lambda(x)\frac{\delta}{\delta\Lambda(x)}\right\\}S_{k}(\hat{\gamma},\varphi,\Lambda)=0\;.$
(2.12)

As interactions we consider operators ${\cal O}^{a}(\varphi,\nabla)\equiv{\cal
O}^{a}(\varphi,\gamma_{\mu\nu})$ which satisfy

$\left\\{\int\sqrt{\gamma}d^{d}z\,\sigma\left(-2\gamma_{\mu\nu}\frac{\delta}{\delta\gamma_{\mu\nu}}+\varphi\frac{\delta}{\delta\varphi}\right)\right\\}\sqrt{\gamma(x)}\,{\cal
O}^{a}(x)=-d^{a}_{0}\sigma(x)\sqrt{\gamma(x)}\,{\cal O}^{a}(x)\;.$ (2.13)

A typical case is ${\cal O}^{a}=\varphi^{n}$ for which $d^{a}_{0}=d-n$. The
interaction part $S_{int}$ involving real scalar fields
$\varphi\equiv\left\\{\varphi_{k}\right\\}$ (indices of fields will be
suppressed for simplicity) reads

$S_{int}(\gamma,\varphi,g_{0},\Lambda)=\int\sqrt{\gamma}d^{d}x\sum_{a}g^{a}_{0}(\Lambda,g)\,{\cal
O}^{a}(x)$ (2.14)

where $g^{a}_{0}(\Lambda,g)$ are bare marginal or relevant couplings of
canonical dimension $d^{a}_{0}$. They include counter terms depending on
$\Lambda$ and on renormalized couplings $g^{a}$. (These counter terms include
polynomials in the renormalized couplings and, for relevant couplings,
derivatives of $g^{a}$ as well as of $\gamma$ in the form of suitable
contractions of the curvature tensor.) In the presence of massless fields
$\varphi$ a renormalization scale $\mu$ has to be introduced; otherwise a
$\mu$-independent on-shell renormalization scheme can be used.

A Weyl rescaling of $S_{int}(\gamma,\varphi,g_{0},\Lambda)$ gives

$\displaystyle\sigma(x)\left(-2\gamma_{\mu\nu}(x)\frac{\delta}{\delta\gamma_{\mu\nu}(x)}+\varphi(x)\frac{\delta}{\delta\varphi(x)}+\Lambda(x)\frac{\delta}{\delta\Lambda(x)}\right)$
$\displaystyle S_{int}(\gamma,\varphi,g_{0},\Lambda)$ $\displaystyle=$
$\displaystyle-\sigma(x)d_{a}^{0}g^{a}_{0}\frac{\delta}{\delta
g^{a}_{0}}S_{int}(\gamma,\varphi,g_{0},\Lambda)\;.$ (2.15)

In terms of rescaled bare couplings

$\hat{g}_{0}^{a}=\Lambda^{-d_{0}^{a}}g_{0}^{a}$ (2.16)

and $\hat{\gamma}_{\mu\nu}$ from (2.11),
$\hat{S}_{int}(\hat{\gamma},\varphi,\hat{g}_{0},\Lambda)$ satisfies

$\sigma(x)\left(\varphi(x)\frac{\delta}{\delta\varphi(x)}+\Lambda(x)\frac{\delta}{\delta\Lambda(x)}\right)\hat{S}_{int}(\hat{\gamma},\varphi,\hat{g}_{0},\Lambda)=0\;.$
(2.17)

The rescaled couplings are still nontrivial functions of renormalized
couplings $g$ and the cutoff. After the rescaling the beta functions for
relevant couplings (see below) contain tree level terms $\sim-d_{0}^{a}$.

The full action $S(\gamma,g_{0},\Lambda,\varphi)=S_{k}+S_{int}$ satisfies a
local RGE

$\sigma(x)\left\\{-2\gamma_{\mu\nu}(x)\frac{\delta}{\delta\gamma_{\mu\nu}(x)}+\varphi(x)\frac{\delta}{\delta\varphi(x)}+\Lambda(x)\frac{\delta}{\delta\Lambda(x)}+d_{a}^{0}g^{a}_{0}\frac{\delta}{\delta
g^{a}_{0}(x)}\right\\}S(\gamma,g_{0},\Lambda,\varphi)=0\;,$ (2.18)

whereas $\hat{S}(\hat{\gamma},\hat{g}_{0},\Lambda,\varphi)$ satisfies

$\sigma(x)\left\\{\varphi(x)\frac{\delta}{\delta\varphi(x)}+\Lambda(x)\frac{\delta}{\delta\Lambda(x)}\right\\}\hat{S}(\hat{\gamma},\hat{g}_{0},\Lambda,\varphi)=0\;.$
(2.19)

We consider the vacuum partition function $S^{0}_{QFT}(\gamma,g_{0},\Lambda)$
given by

$e^{-{S^{0}_{QFT}(\gamma,g_{0},\Lambda)}}=\frac{1}{\cal N}\int{\cal
D}\varphi\,e^{-S(\gamma,g_{0},\Lambda,\varphi)}\;.$ (2.20)

Since $\varphi$ are dummy variables on the right hand side of eq. (2.20), eq.
(2.18) implies a local RGE for ${S^{0}}_{QFT}(\gamma,g_{0},\Lambda)$

$\sigma(x)\left\\{-2\gamma_{\mu\nu}(x)\frac{\delta}{\delta\gamma_{\mu\nu}(x)}+\Lambda(x)\frac{\delta}{\delta\Lambda(x)}+d^{a}_{0}g^{a}_{0}(x)\frac{\delta}{\delta
g^{a}_{0}(x)}\right\\}{S^{0}}_{QFT}(\gamma,g_{0},\Lambda)=0\;.$ (2.21)

For ${\Lambda}\to\infty$, the vacuum partition function
${S^{0}}_{QFT}(\gamma,g_{0},\Lambda)$ suffers from subdivergences as well as
from superficial divergences. Subdivergences are removed if one expresses the
bare couplings $g_{0}(g,\Lambda)$ explicitely in terms of $\Lambda$ and
renormalized cutoff independent couplings $g$, and
$S^{0}_{QFT}(\gamma,g_{0}(g,\Lambda),\Lambda)$ becomes
${S^{0}}^{\prime}_{QFT}(\gamma,g,\Lambda)$. The remaining dependence of
${S^{0}}^{\prime}_{QFT}(\gamma,g,\Lambda)$ on ${\Lambda}$ concerns superficial
divergences which require extra counter terms
${S^{CT}}_{QFT}(\gamma,g,\Lambda,\mu)$; again a renormalization scale $\mu$ is
required in the presence of massless fields. (Anomalies for massless CFTs
originate from violations of the local RGE (2.21) by
${S^{CT}}_{QFT}(\gamma,g,\Lambda,\mu)$ and hence by the finite renormalized
vacuum partition function.) In the following we will not add
${S^{CT}}_{QFT}(\gamma,g,\Lambda,\mu)$ but consider
${S^{0}}_{QFT}(\gamma,g_{0},\Lambda)$ for finite $\Lambda$.

From now on we consider $d=4$, and assume all fields to be massive. In an
expansion of $S^{0}_{QFT}(\gamma,g_{0},\Lambda)$ up to quadratic order in
derivatives acting on $\gamma_{\mu\nu}$, $g_{0}$ and $\Lambda$, $S^{0}_{QFT}$
contains terms of the form

$\displaystyle S^{0}_{QFT}$ $\displaystyle(\gamma,g_{0},\Lambda)=\int
d^{4}x\sqrt{\gamma}\Bigg{(}V^{0}(g_{0},{\Lambda})-L^{0}(g_{0},{\Lambda})R_{4}$
$\displaystyle+\gamma^{\mu\nu}\left\\{\frac{1}{2}K^{0}_{ab}(g_{0},{\Lambda})\partial_{\mu}g_{0}^{a}\partial_{\nu}g_{0}^{b}+K^{0}_{a\lambda}(g_{0},{\Lambda})\partial_{\mu}g_{0}^{a}\partial_{\nu}\Lambda+\frac{1}{2}K^{0}_{\lambda\lambda}(g_{0},{\Lambda})\partial_{\mu}\Lambda\partial_{\nu}\Lambda\right\\}\Bigg{)}$
(2.22)

where $R_{4}$ denotes the curvature scalar constructed from $\gamma_{\mu\nu}$.
The origin of derivatives $\partial_{\mu}\Lambda(x)$ are vertices involving
$\Lambda(x)$ in the field theory action in the kinetic action $S_{k}$ via the
cutoff functions, and cannot be dropped for the computation of $S^{0}_{QFT}$
since invariance under local Weyl variations $\sim\sigma(x)$ would not hold.

On the other hand, in terms of $\hat{g}_{0}^{a}$ and $\hat{\gamma}_{\mu\nu}$,
$\hat{S}_{QFT}(\hat{\gamma},\hat{g}_{0})$ satisfies

$\Lambda(x)\frac{\delta}{\delta\Lambda(x)}\hat{S}_{QFT}(\hat{\gamma},\hat{g}_{0})=0\;,$
(2.23)

i.e. $\hat{S}_{QFT}(\hat{\gamma},\hat{g}_{0})$ does not depend explicitely on
$\Lambda(x)$ or its derivatives. For a constant cutoff eq. (2.23) corresponds
just to naive power counting involving the dimensionful rescaled metric
$\hat{\gamma}$ and derivatives. However, the absence of derivatives acting on
$\Lambda$ is not trivial. The expansion of
$\hat{S}_{QFT}(\hat{\gamma},\hat{g}_{0})$ to second order in derivatives reads

$\displaystyle\hat{S}_{QFT}(\hat{\gamma},\hat{g}_{0})=\int
d^{4}x\sqrt{\hat{\gamma}}\Big{(}\hat{V}(\hat{g}_{0})-\hat{L}(\hat{g}_{0})R_{4}(\hat{\gamma})+\hat{\gamma}^{\mu\nu}\frac{1}{2}\hat{K}_{ab}(\hat{g}_{0})\partial_{\mu}\hat{g}_{0}^{a}\partial_{\nu}\hat{g}_{0}^{b}\Big{)}\;.$
(2.24)

It is straightforward to replace $\hat{\gamma}$ by $\gamma$ and
$\hat{g}^{a}_{0}$ by $g^{a}_{0}$ in $\hat{S}_{QFT}(\hat{\gamma},\hat{g}_{0})$.
This generates terms of the form
$K^{0}_{a\lambda}\partial_{\mu}g_{0}^{a}\partial_{\nu}\Lambda$ and
$K^{0}_{\lambda\lambda}\partial_{\mu}\Lambda\partial_{\nu}\Lambda$, and a
$\Lambda$ dependence of $V^{0}$, $L^{0}$ and all $K^{0}$. Given the local eq.
(2.21), these terms have to match the ones in
$S^{0}_{QFT}(\gamma,g_{0},\Lambda)$.

Next we introduce transformations characterized by a local parameter $\tau(x)$
which describe the dependence of $\hat{\gamma}$ and $\hat{g}^{a}_{0}$ on the
local cutoff $\Lambda(x)$ (and which differ from Weyl transformations since
$\gamma_{\mu\nu}$ remains inert):

$\delta_{\tau}\Lambda=\tau\Lambda\;,\quad\delta_{\tau}\hat{\gamma}_{\mu\nu}=2\tau\hat{\gamma}_{\mu\nu}\;,\quad\delta_{\tau}\hat{g}_{0}^{a}=\tau\beta^{a}_{\tau}\;,\quad\beta^{a}_{\tau}=\Lambda\frac{\delta\hat{g}_{0}^{a}}{\delta\Lambda}\;.$
(2.25)

For rescaled relevant couplings $\hat{g}_{0}^{a}$, $\beta^{a}_{\tau}$ contains
a tree level term $-d_{0}^{a}$ as well as possible derivative terms. Recursion
relations ensure that $\beta^{a}_{\tau}$ can again be expressed in terms of
bare couplings.

The $\tau$ variation of $\hat{S}_{QFT}(\hat{\gamma},\hat{g}_{0})$ is given by

$\delta_{\tau}\hat{S}_{QFT}(\hat{\gamma},\hat{g}_{0})=\left(\Delta_{\tau}^{\gamma}+\Delta_{\tau}^{\beta}\right)\hat{S}_{QFT}(\hat{\gamma},\hat{g}_{0})$
(2.26)

where

$\Delta_{\tau}^{\gamma}=2\tau\hat{\gamma}_{\mu\nu}\frac{\delta}{\delta\hat{\gamma}_{\mu\nu}}\;,\quad\Delta_{\tau}^{\beta}=\tau\beta^{a}_{\tau}\frac{\delta}{\delta\hat{g}^{a}_{0}}\;.$
(2.27)

As in the case of local Weyl transformations considered by JO, the fact that
local $\tau$ transformations are Abelian imposes constraints on the
coefficients of the vacuum partition function. (In [1, 2, 3] local Weyl
transformations are applied to the vacuum partition function as function of
the renormalized couplings in dimensional regularization, and after adding
counter terms ${S^{CT}}_{QFT}$.) In the present case these consistency
conditions originate from

$\left[(\Delta_{\tau^{\prime}}^{\gamma}+\Delta_{\tau^{\prime}}^{\beta}),(\Delta_{\tau}^{\gamma}+\Delta_{\tau}^{\beta})\right]\hat{S}_{QFT}(\hat{\gamma},\hat{g}_{0})\overset{!}{=}0\;.$
(2.28)

Inserting $\hat{S}_{QFT}(\hat{\gamma},\hat{g}_{0})$ from (2.24) into (2.28)
one obtains from terms proportional to
$\left(\tau^{\prime}\partial_{\mu}\tau-\tau\partial_{\mu}\tau^{\prime}\right)\nabla^{\mu}\hat{g}^{a}_{0}$:

$6\,\partial_{a}\hat{L}-\beta_{\tau}^{b}\hat{K}_{ba}\overset{!}{=}0$ (2.29)

where $\partial_{a}$ denotes $\partial/\partial\hat{g}^{a}_{0}$. Eq. (2.29)
corresponds to a gradient flow for $\beta_{\tau}^{b}$, with $\hat{K}_{ba}$ a
metric in the space of couplings. Within the considered order in derivatives
this result holds only for the terms without derivatives in
$\beta_{\tau}^{b}$.

Some comments on the cutoff dependences are in order: First, since $\hat{L}$
does not depend explicitely on $\Lambda$, eq. (2.29) implies

$\Lambda\frac{d}{d\Lambda}\hat{L}=\frac{1}{6}\beta_{\tau}^{a}\hat{K}_{ab}\beta_{\tau}^{b},$
(2.30)

i.e. an “l-theorem”

$\Lambda\frac{d}{d\Lambda}\hat{L}>0\ \text{iff the metric}\ \hat{K}_{ab}\
\text{is positive.}$ (2.31)

Second, the coefficients $\hat{V}(\hat{g}_{0})$, $\hat{L}(\hat{g}_{0})$ and
$\hat{K}_{ab}(\hat{g}_{0})$ in $\hat{S}_{QFT}$ are all dimensionless. After
the replacement of $\hat{\gamma}$ and $\hat{g}_{0}$ as in eqs. (2.11) and
(2.16), $\sqrt{\hat{\gamma}}$ scales as $\Lambda^{4}$. Rescaled bare masses
$\hat{m}_{0}^{2}$ scale as $m_{0}^{2}/\Lambda^{2}$. Subsequently the
coefficients $V^{0}(g_{0},{\Lambda})$, $L^{0}(g_{0},{\Lambda})$ and
$K^{0}_{ab}(g_{0},{\Lambda})$ in $S^{0}_{QFT}$ assume their canonical
dimensions which determine their leading powers in $\Lambda$ modified by
powers of $\ln(\hat{m}_{0}^{2})=\ln(m_{0}^{2}/\Lambda^{2})$. These divergences
in $V^{0}$, $L^{0}$ and $K^{0}$ include both subdivergences and superficial
divergences, i.e. the information on subdivergences is present via the
$\Lambda$ dependence of these coefficients. This information can be extracted
once the beta functions are computed via the consistency condition (2.29). The
beta functions for relevant couplings like masses are cutoff dependent in
general. Particular conditions on the UV limit of couplings including relevant
couplings do not follow from the present formalism.

##  3 Towards a Holograpic Dual

In the first part of this Section we review the construction of a 4d action on
the boundary $\partial{\cal M}$ of a 5d bulk ${\cal M}$ following, to a large
extend, the Hamilton-Jacobi approach used in [24, 25]. With ansätze for the 5d
Lagrangian ${\cal L}_{5}$ and the action $\bar{S}$ on the boundary, both
expanded to second order in derivatives, we establish relations between the
corresponding parameters which follow from the Hamiltonian constraint ${\cal
H}_{5}=0$.

For the metric in the bulk we take

$ds^{2}=\frac{l^{2}}{4}\frac{d\rho^{2}}{\rho^{2}}+{G}_{4ij}(x,\rho)dx^{i}dx^{j}\;.$
(3.1)

From now on indices $\mu,\nu$ refer to indices in 5 dimensions. The 5d metric
will be denoted by $G_{5\mu\nu}$, and $i,j$ will be indices in 4 dimensions.
The 5d Lagrangian ${\cal L}_{5}$ includes scalars $\Phi^{a}$ and reads,
expanded to second order in derivatives222A non-Einstein factor Z in front of
$R_{5}$ can be removed by a Weyl rescaling of $G_{5}$ and a subsequent
diffeomorphism such that no additional degree of freedom in the final boundary
action remains.,

${\cal
L}_{5}=-R_{5}+\frac{1}{2}G_{5}^{\mu\nu}L_{ab}(\Phi)\partial_{\mu}\Phi^{a}\partial_{\nu}\Phi^{b}+V(\Phi)\;.$
(3.2)

Next we consider the 5d action regularized for $\rho\to 0$ by
$\rho\geqq\bar{\rho}$,

$2\kappa_{5}^{2}S=\int d^{5}x\sqrt{G_{5}}{\cal L}_{5}+S_{GH}=\int
d^{4}x\sqrt{{G_{4}}}\int_{\bar{\rho}}d\rho\frac{l}{2\rho}{\cal
L}_{5}+S_{GH}\;,$ (3.3)

where $S_{GH}$ denotes the Gibbons-Hawking action on the boundary
$\partial{\cal M}$ at $\rho=\bar{\rho}$:

$S_{GH}=-\frac{2\bar{\rho}}{l}\int
d^{4}x\sqrt{\bar{g}}\bar{g}^{ij}\dot{\bar{g}}_{ij}$ (3.4)

where $\bar{g}_{ij}$ denotes ${G}_{4ij}(x,\rho=\bar{\rho})$. Here and below a
dot denotes a derivative $\partial/\partial\rho$.

In the Hamilton-Jacobi formalism, the bulk fields ${G}_{4ij}$ and $\Phi^{a}$
are replaced by their solutions of the equations of motion with $\bar{g}_{ij}$
and $\bar{\phi}^{a}$ as boundary conditions on $\partial{\cal M}$
corresponding to $\rho=\bar{\rho}$. This allows to express $S$ as function of
$\bar{g}_{ij}$ and $\bar{\phi}^{a}$, denoted by
$\bar{S}(\bar{g}_{ij},\bar{\phi}^{a})$. For completeness, details of this
procedure are given in Appendix A. Hamiltonian constraints will relate
$\bar{S}(\bar{g}_{ij},\bar{\phi}^{a})$ to $\bar{{\cal L}}_{5}$, the bulk
Lagrangian at $\rho=\bar{\rho}$ as function of $\bar{g}_{ij}$ and
$\bar{\phi}^{a}$.

For $\bar{S}$ expanded to second order in derivatives we make the ansatz
($g_{ij}$ and $\phi^{a}$ denote $\bar{g}_{ij}$ and $\bar{\phi}^{a}$ to lowest
order in derivatives, respectively)

$\bar{S}=\int
d^{4}x\sqrt{{g}}\left(W({\phi})-L({\phi})R_{4}({g})+\frac{1}{2}K_{ab}({\phi})\nabla^{i}{\phi}^{a}\nabla_{i}{\phi}^{b}\right)\;.$
(3.5)

The flow equations for the leading terms of ${g}_{ij}$ and ${\phi}^{a}$ in an
expansion in derivatives become (see Appendix A and [24])

$\displaystyle\dot{{g}}_{ij}$ $\displaystyle=\frac{l}{6\bar{\rho}}g_{ij}W\;,$
(3.6) $\displaystyle\dot{{\phi}}^{a}$
$\displaystyle=-\frac{l}{2\bar{\rho}}L^{ab}\partial_{b}W\;.$ (3.7)

The Hamiltonian constraint ${\cal H}_{5}=0$ imposes the following relations
between the coefficients in $\bar{{\cal L}}_{5}$ and $\bar{S}$ which are
derived in Appendix A:

$\displaystyle V=$
$\displaystyle-\frac{1}{3}W^{2}+\frac{1}{2}\partial_{a}WL^{ab}\partial_{b}W\;,$
(3.8) $\displaystyle 0=$ $\displaystyle
1+\frac{1}{3}WL-\partial_{a}WL^{ab}\partial_{b}L\;,$ (3.9) $\displaystyle 0=$
$\displaystyle
L_{ab}+W\left(\frac{1}{3}K_{ab}+2\partial_{a}\partial_{b}L\right)-\partial_{c}WL^{cd}(\partial_{d}K_{ab}-\partial_{a}K_{bd}-\partial_{b}K_{ad})\;,$
(3.10) $\displaystyle 0=$ $\displaystyle
W\partial_{a}L+\partial_{b}WL^{bc}K_{ca}\;.$ (3.11)

These agree with [24, 25].

We write $g_{ij}$ in the form

$g_{ij}=a({\phi},\bar{\rho})\tilde{\gamma}_{ij}$ (3.12)

with $\tilde{\gamma}_{ij}$ independent from $\bar{\rho}$. From eq. (3.6) the
flow equation for $a({\phi},\bar{\rho})$ reads

$\dot{a}=a\frac{l}{6\bar{\rho}}W\;.$ (3.13)

Next we consider local transformations by $\tau(x)$. To $g_{ij}$ resp. $a$ we
attribute $\tau$ transformation laws

$\delta_{\tau}g_{ij}=2\tau g_{ij}\;,\quad\delta_{\tau}a=2\tau a$ (3.14)

such that we can identify $g_{ij}$ with the QFT metric $\hat{\gamma}_{ij}$,
and $a$ to the QFT cutoff $\Lambda^{2}$ which varies under Weyl
transformations as in eq. (2.1).

Let us recall the role of the parameter $\bar{\rho}$ for the boundary action
$\bar{S}$: $\bar{\rho}$ does not appear explicitely in $\bar{S}$, only
implicitely in $\phi(\bar{\rho})$ and $a({\phi},\bar{\rho})$. As argued in
[24] one can perform a change of variables and consider $a$ instead of
$\bar{\rho}$ as independent variable. The flow equation for $\phi(a)$ reads
now

$a\frac{\delta{\phi}^{a}}{\delta a}=-\frac{3}{W}L^{ab}\partial_{b}W\;.$ (3.15)

For $\tau$ transformations of $\phi$ we assume that these are induced by the
dependence of $\phi$ on $a$:

$\delta\phi^{a}=\tau\beta_{\tau}^{a}\;,\quad\beta_{\tau}^{a}=2a\frac{\delta{\phi}^{a}}{\delta
a}=-\frac{6}{W}L^{ab}\partial_{b}W\;.$ (3.16)

For the local $\tau$ transformations of $\bar{S}(g,\phi)$ one obtains

$\delta_{\tau}\bar{S}(g,\phi)=(\Delta_{\tau}^{g}+\Delta_{\tau}^{\beta})\bar{S}(g,\phi)$
(3.17)

with

$\Delta_{\tau}^{g}=2\tau g_{ij}\frac{\delta}{\delta
g_{ij}}\;,\quad\Delta_{\tau}^{\beta}=\tau\beta_{\tau}^{a}\frac{\delta}{\delta\phi^{a}}\;.$
(3.18)

As in the case of $\hat{S}_{QFT}(\hat{\gamma},\hat{g}_{0})$, the local $\tau$
transformations of $\bar{S}(g,\phi)$ in (3.17) imply consistency conditions
like in (2.29). In terms of the coefficients of $\bar{S}$ they read

$6\,\partial_{a}L-\beta_{\tau}^{b}K_{ba}\overset{!}{=}0$ (3.19)

With $\beta_{\tau}^{a}$ from (3.16) eq. (3.19) becomes

$W\partial_{a}L+K_{ab}L^{bc}\partial_{c}W\overset{!}{=}0\;.$ (3.20)

Eq. (3.20) corresponds to the Hamiltonian constraint (3.11), which will play
an important role in the next Section.

##  4 Matching the Bulk to the QFT

The starting point of most explicit applications of the holographic
renormalization group is a 5d Lagrangian ${\cal L}_{5}$, and resulting
properties of the effective action and constraints on $\beta$ functions of a
corresponding 4d QFT. Here we ask the opposite question: Given a set of
$\beta$ functions describing the running of bare (marginal or relevant)
couplings $g_{0}^{a}$ of a 4d QFT, can this RG flow always be related to the
flow of fields $\phi^{a}$ living on the boundary of a 5d manifold? Or, can the
vacuum partition function of a 4d QFT in a gravitational background as
function of local bare couplings $g_{0}^{a}(x)$ always be related to an action
for fields $\phi^{a}$ on the boundary of a 5d manifold?

As a first step in this direction we consider both functionals to first
nontrivial order in an expansion in 4d derivatives. To this order the vacuum
partition function $\hat{S}_{QFT}$ of a 4d QFT as function of the rescaled
metric $\hat{\gamma}$ and rescaled local couplings $\hat{g}_{0}$ is specified
by the three coefficients $\hat{V}(\hat{g}_{0})$, $\hat{L}(\hat{g}_{0})$ and
$\hat{K}_{ab}(\hat{g}_{0})$ in eq. (2.24). In turn, the boundary action
$\bar{S}$ as function boundary fields $\phi$ is specified by three
coefficients $W(\phi)$, $L(\phi)$ and $K_{ab}(\phi)$ in eq. (3.5).

We assume that the rescaled QFT metric $\hat{\gamma}_{ij}$ can be identified
with the boundary metric $g_{ij}$, the bare rescaled QFT couplings
$\hat{g}_{0}^{a}$ with the boundary fields $\phi^{a}$, and the QFT cutoff
$\Lambda^{2}$ with the coefficient $a$ in eq. (3.12) (and not with the
boundary coordinate $\bar{\rho}$ unless $lW=6$). Note that both
$\hat{\gamma}_{ij}$ and $g_{ij}$ diverge identically for
$\Lambda^{2}=a\to\infty$.

Given the coefficients $\hat{V}(\hat{g}_{0})$, $\hat{L}(\hat{g}_{0})$ and
$\hat{K}_{ab}(\hat{g}_{0})$ in $\hat{S}_{QFT}$, a correspondence

$\displaystyle W(\bar{\phi}^{a})$
$\displaystyle=\hat{V}(\hat{g}_{0}^{a}=\phi^{a})$ (4.1) $\displaystyle
L(\bar{\phi}^{a})$ $\displaystyle=\hat{L}(\hat{g}_{0}^{a}=\phi^{a})$ (4.2)
$\displaystyle K_{ab}(\bar{\phi}^{a})$
$\displaystyle=\hat{K}_{ab}(\hat{g}_{0}^{a}=\phi^{a})$ (4.3)

faces the following potential obstruction: If interpreted as coefficients of a
boundary action $\bar{S}$ resulting from the classical equations of motion
based on a 5d Lagrangian ${\cal L}_{5}$ including gravity, $W$, $L$ and
$K_{ab}$ must satisfy the Hamiltonian constraints (3.8)-(3.11).

In the present approach these have to be interpreted as equations which
determine the coefficients $V(\Phi)$ and $L_{ab}(\Phi)$ in ${\cal L}_{5}$ in
(3.2). Eq. (3.8) determines $V(\Phi)$, and the Hamiltonian constraint (3.10)
allows, in principle, to compute $L_{ab}(\Phi)$ in terms of $W$, $L$ and
$K_{ab}$. For $n_{g}$ bare couplings, the Hamiltonian constraint (3.11) seems
to impose $n_{g}$ additional conditions. At this point the consistency
conditions originating from local $\tau$ transformations – satisfied by both
$\hat{S}_{QFT}$ and $\bar{S}$ – come into play: First, recall that the $\beta$
functions $\beta_{\tau}^{a}$ describe the flow of both $g_{0}^{a}$ in the QFT
with the cutoff $\Lambda$, and of the boundary fields $\phi^{a}$ with
$a=\Lambda^{2}$, respectively. Second, the consistency conditions imply the
same gradient flows eq. (2.29) and eq. (3.19), hence the $\beta$ functions for
the QFT and the boundary fields coincide. Third, the consistency condition in
the form (3.20) implies the validity of the $n_{g}$ Hamiltonian constraints
(3.11) without further assumptions. Hence it remains only the Hamiltonian
constraint (3.9) which, with $L_{ab}(\Phi)$ determined by (3.10), imposes one
extra condition on the coefficients of $\hat{S}_{QFT}$.

Finally a successful QFT/bulk correspondence implies another relation: As
shown in [44], the weaker energy condition (or null energy condition) in the
5d bulk implies a c-theorem, i.e. the existence of a c-function which
decreases towards the infrared. Following [25] adapted to our conventions,
this c-function is given by

$c(\phi)=-\frac{1}{W^{3}}$ (4.4)

and satisfies

$2a\frac{d}{da}c(\phi)\equiv\beta_{\tau}^{a}\frac{d}{d\phi^{a}}c(\phi)\geq
0\;.$ (4.5)

With $c(\phi)$ as in eq. (4.4) and $\beta_{\tau}^{a}$ from eq. (3.16), the
c-theorem is equivalent to

$-\beta_{\tau}^{a}\frac{1}{W^{3}}L_{ab}\beta_{\tau}^{b}\geq 0\;.$ (4.6)

Accordingly the metric in field space

$-\frac{1}{W^{3}}L_{ab}$ (4.7)

must be semi-positive. If this condition (and the Hamiltonian constraint
(3.9)) are satisfied, the c-theorem applies also to the 4d QFT, but with
respect to the Wilsonian UV cutoff $\Lambda$: Using the relation (4.1), the
expression

$c_{QFT}(\hat{g}_{0})=-\frac{1}{\hat{V}^{3}(\hat{g}_{0})}$ (4.8)

increases for increasing $\Lambda$, i.e. $\hat{V}(\hat{g}_{0})$ increases for
increasing $\Lambda$ due to the running of $\hat{g}_{0}^{a}$ with $\Lambda$.

We recall that the origin of the Hamiltonian constraint in general relativity
can be traced back to the (explicit) independence of the boundary action
$\bar{S}$ on the coordinate of the boundary. The corresponding feature of the
QFT action in the form of $\hat{S}_{QFT}$ is that $\hat{S}_{QFT}$ satisfies
eq. (2.23) implying its (explicit) independence on the cutoff $\Lambda$. These
are the central results of the present approach.

##  5 Summary and Outlook

The introduction of a local ultraviolet cutoff in the kinetic terms of a
generic (here: multi-flavour scalar) QFT together with local couplings in a
curved background allows to gain new insight into properties of the bare
vacuum partition function as function of bare couplings.

First, it allows to realize local Weyl transformations under which the local
cutoff transforms; here the local component of the cutoff plays the role of a
compensator allowing for a local RGE satisfied by the bare vacuum partition
function. This local RGE can be solved in terms of a rescaled space-time
metric, and rescaled bare couplings such that they are dimensionless. (For
marginal bare couplings, this rescaling is trivial.) Next one can study the
variation of the bare vacuum partition function as function of the rescaled
space-time metric and rescaled bare couplings under local variations of the
cutoff. Since these variations are Abelian they imply consistency conditions
similar to the Weyl consistency conditions introduced by JO. In an expansion
to first nontrivial order in derivatives we studied the consequences of these
consistency conditions and found that they imply a gradient flow for the beta
functions describing the cutoff dependence of the rescaled bare couplings.

Second, we reviewed the holographic description of the renormalization group
flow which implies Hamiltonian constraints on the action on the boundary of a
5d manifold, following to a large extend [24]. At first sight the number of
these constraints does not allow to identify an arbitrary 4d vacuum partition
function of local bare couplings with an action on the boundary of a 5d
manifold, identifying scalar fields on the boundary with local rescaled bare
couplings in the 4d QFT. It turns out, however, that the consistency condition
to be satisfied by coefficients of the 4d vacuum partition function correspond
precisely to one of the Hamiltonian constraints such that only one additional
condition on the 4d vacuum partition function remains. (An open question is
whether a particular choice of the cutoff function of the 4d QFT allows to
satisfy this remaining condition.)

We underline that this consistency condition coincides with the one of the QFT
only if the holographic beta functions describe the flow of bulk fields with
the field dependent scale factor $a(\phi,\bar{\rho})$ multiplying the induced
metric on the boundary and not, as it is frequently assumed, with $\bar{\rho}$
directly. This is plausible, given that fields in the bulk modify the metric
in the bulk via the Einstein equations, and physical distances in the bulk
depend on the metric in the bulk.

Although this result was obtained only to first nontrivial order in an
expansion in derivatives it nourishes the hope that generic non-conformal
quantum field theories may allow for a description in terms of fields on the
boundary of a 5d bulk subject to equations of motion including general
relativity. Of course, to establish such a correspondence beyond an expansion
in derivatives is a more ambitious task.

Independently thereof, the present approach may also be helpful for the study
of the ultraviolet behaviour of 4d QFTs such as the search for UV fix points
including relevant couplings, related to extrema of the coefficient
$\hat{L}^{0}(\hat{g}_{0})$ in (2.24). The generalisation to QFTs including
Fermions and/or gauge fields remains a task for the future.

##  Appendix A Appendix

## The Hamiltonian Constraint

In this Appendix we sketch the Hamilton-Jacobi formalism relating an action
$\bar{S}$, a functional of fields $\bar{g},\bar{\phi}$ on the boundary
$\partial{\cal M}$, to the bulk Lagrangian ${\cal L}_{5}$ evaluated on the
boundary. It leads to flow equations for the fields $\bar{g},\bar{\phi}$ with
respect to the extra coordinate. The Hamiltonian constraint implies relations
between the parameters of $\bar{S}$ and the bulk Lagrangian ${\cal L}_{5}$. We
assume ${\cal L}_{5}$ quadratic in derivatives.

For the bulk metric we take

$ds^{2}=\frac{l^{2}}{4}\frac{d\rho^{2}}{\rho^{2}}+{G}_{4ij}dx^{i}dx^{j}\;,\qquad\sqrt{G_{5}}=b\sqrt{{G_{4}}}\;,\qquad
b=\frac{l}{2\rho}\;.$ (A.1)

For ${\cal L}_{5}$ we take

${\cal
L}_{5}=-R_{5}+V(\Phi)+\frac{1}{2}G_{5}^{\mu\nu}L_{ab}(\Phi)\partial_{\mu}\Phi^{a}\partial_{\nu}\Phi^{b}\;.$
(A.2)

The regularized bulk action is

$2\kappa_{5}^{2}S=\int d^{5}x\sqrt{G_{5}}{\cal L}_{5}+S_{GH}=\int
d^{4}x\int^{\infty}_{\bar{\rho}}d\rho\sqrt{{G_{4}}}b{\cal L}_{5}+S_{GH}$ (A.3)

where $S_{GH}$ denotes the Gibbons-Hawking action on the boundary
$\bar{\rho}$:

$S_{GH}=-\frac{1}{\bar{b}}\int d^{4}x\sqrt{{g}}{g}^{ij}\dot{{g}}_{ij}$ (A.4)

where a dot denotes $\partial/\partial\bar{\rho}$ and
$\bar{b}=\frac{l}{2\bar{\rho}}$.

Solving the equations of motion for ${G}_{4ij}$ and $\Phi$, $S$ becomes a
function of the corresponding values ${g},{\phi}$ on the boundary $\bar{\rho}$
and will be denoted by $\bar{S}({g},{\phi})$. ${\cal L}_{5}$ evaluated on the
boundary will be denoted by $\bar{{\cal L}}_{5}$. Separating 4d derivatives
from derivatives w.r.t. $\rho$ and after a partial integration w.r.t. $\rho$,
$\bar{{\cal L}}_{5}$ becomes

$\bar{{\cal
L}}_{5}=-R_{4}({g})+\frac{1}{4\bar{b}^{2}}(\dot{{g}}_{ij}{g}^{ik}{g}^{jl}\dot{{g}}_{kl}-({g}^{ij}\dot{{g}}_{ij})^{2})+\frac{1}{2}L_{ab}({\phi})\big{(}\frac{1}{\bar{b}^{2}}\dot{{\phi}}^{a}{\dot{\phi}}^{b}+{g}^{ij}\partial_{i}{\phi}^{a}\partial_{j}{\phi}^{b}\big{)}+V({\phi})\;.$
(A.5)

In $\bar{{\cal L}}_{5}$, contributions from partial integration of second
derivatives w.r.t. $\rho$ have been cancelled by the Gibbons-Hawking action.

In order to relate $\bar{S}({g},{\phi})$ to $\bar{{\cal L}}_{5}({g},{\phi})$
via the solutions of the equations of motion for ${G}_{4ij}$ and $\Phi^{a}$
inserted into $\sqrt{G_{5}}{\cal L}_{5}$ we combine the bulk fields
${G}_{4ij}$ and $\Phi^{a}$ into fields $q_{\alpha}$ were $\alpha$ denote the
corresponding indices:

$q_{\alpha}=\left\\{{G}_{4ij},\Phi^{a}\right\\}\;.$ (A.6)

It is convenient to introduce

$\tilde{\cal L}_{5}=\sqrt{G_{5}}{\cal L}_{5}=b\sqrt{{G_{4}}}{\cal L}_{5}\;.$
(A.7)

The equations of motion are

$\frac{\delta\tilde{\cal L}_{5}}{\delta
q_{\alpha}}-\partial_{i}\frac{\delta\tilde{\cal
L}_{5}}{\delta\partial_{i}q_{\alpha}}-\partial_{\rho}\frac{\delta\tilde{\cal
L}_{5}}{\delta\dot{q}_{\alpha}}=0\;.$ (A.8)

The variations of $S$ with $\delta q_{\alpha}$ and its derivatives are

$\frac{2\kappa_{5}^{2}}{\sqrt{g}}\delta S=\int
d^{4}x\int_{\bar{\rho}(x)}^{\infty}d\rho\left(\frac{\delta\tilde{\cal
L}_{5}}{\delta q_{\alpha}}\delta q_{\alpha}+\frac{\delta\tilde{\cal
L}_{5}}{\delta\partial_{i}q_{\alpha}}\delta\partial_{i}q_{\alpha}+\frac{\delta\tilde{\cal
L}_{5}}{\delta\dot{q}_{\alpha}}\delta\dot{q}_{\alpha}\right)\;.$ (A.9)

Using the equations of motion and partial integration under the $d\rho$
integral one obtains

$\frac{2\kappa_{5}^{2}}{\sqrt{g}}\delta S=\left[\int
d^{4}x\frac{\delta\tilde{\cal L}_{5}}{\delta\dot{q}_{\alpha}}\delta
q_{\alpha}\right]_{\infty}-\left[\int d^{4}x\frac{\delta\tilde{\cal
L}_{5}}{\delta\dot{q}_{\alpha}}\delta q_{\alpha}\right]_{\partial{\cal M}}\;.$
(A.10)

With $\bar{q}_{\alpha}=q_{\alpha}|_{\partial{\cal M}}$ and
$\bar{S}={2\kappa_{5}^{2}}S(q_{\alpha}=\bar{q}_{\alpha})$ one obtains

$\frac{1}{\sqrt{g}}\frac{\delta\bar{S}(\bar{q})}{\delta\bar{q}_{\alpha}}=-\frac{1}{\sqrt{{g}}}\frac{\delta\tilde{\cal
L}_{5}}{\delta\dot{\bar{q}}_{\alpha}}=-\bar{b}\frac{\delta\bar{\cal
L}_{5}}{\delta\dot{\bar{q}}_{\alpha}}$ (A.11)

or, returning from $\bar{q}_{\alpha}$ to ${g}_{ij},{\bar{\phi}^{a}}$,

$\frac{\delta\bar{\cal
L}_{5}}{\delta\dot{{g}}_{ij}}=-\frac{1}{\bar{b}\sqrt{{g}}}\frac{\delta\bar{S}}{\delta{g}_{ij}}\;,\qquad\frac{\delta\bar{\cal
L}_{5}}{\delta\dot{{\phi}}^{a}}=-\frac{1}{\bar{b}\sqrt{{g}}}\frac{\delta\bar{S}}{\delta{\phi}^{a}}\;.$
(A.12)

(Conventionally $\bar{b}\frac{\delta\bar{\cal L}_{5}}{\delta\dot{{g}}_{ij}}$
and $\bar{b}\frac{\delta\bar{\cal L}_{5}}{\delta\dot{{\phi}}^{a}}$ are denoted
by canonical momentum variables $\pi_{ij}$ and $\pi_{a}$, respectively [24,
25].)

With ${\cal L}_{5}$ and hence $\bar{\cal L}_{5}$ quadratic in dotted
derivatives it is straightforward to express $\frac{\delta\bar{\cal
L}_{5}}{\delta\dot{{g}}_{ij}}$ and $\frac{\delta\bar{\cal
L}_{5}}{\delta\dot{{\phi}}^{a}}$ in (A.12) in terms of $\dot{{g}}_{ij}$ and
$\dot{{\phi}}^{a}$, and to obtain flow equations for $\dot{{g}}_{ij}$ and
$\dot{{\phi}}^{a}$:

$\displaystyle\dot{{g}}_{ij}$
$\displaystyle=-2\bar{b}\left({g}_{ik}{g}_{jl}-\frac{1}{3}{g}_{ij}{g}_{kl}\right)\frac{1}{\sqrt{{g}}}\frac{\delta\bar{S}}{\delta{g}_{kl}}\;,$
(A.13) $\displaystyle\dot{{\phi}}^{a}$
$\displaystyle=-\bar{b}L^{ab}\frac{1}{\sqrt{{g}}}\frac{\delta\bar{S}}{\delta{\phi}^{b}}\;.$
(A.14)

The Hamilton density ${\cal H}_{5}$ is given by

${\cal H}_{5}=-\dot{g}_{ij}\frac{\delta{\cal
L}_{5}}{\delta\dot{g}_{ij}}-\dot{\Phi}^{a}\frac{\delta{\cal
L}_{5}}{\delta\dot{\Phi}^{a}}+{\cal L}_{5}\;.$ (A.15)

Replacing the bulk fields by the solutions of the equations of motion, ${\cal
H}_{5}$ satisfies the Hamiltonian constraint333Without gauge fixing of the
bulk metric, i.e. introducing lapse and shift as in the ADM decomposition, the
Hamiltonian constraint follows from the variation w.r.t. the lapse.

${\cal H}_{5}=0$ (A.16)

which (evaluated on the boundary $\rho=\bar{\rho}$) is equivalent to
$\partial\bar{S}/\partial\bar{\rho}=0$ corresponding to the only dependence of
$\bar{S}$ on the coordinate $\bar{\rho}$ via arguments of the metric
${g(\bar{\rho})}$ and the fields ${\phi}(\bar{\rho})$. In terms of $\bar{S}$,
the Hamiltonian constraint ${\cal H}_{5}=0$ evaluated on the boundary becomes

$\frac{1}{\sqrt{{g}}}\frac{\delta\bar{S}}{\delta{g}_{ij}}\left({g}_{ik}{g}_{jl}-\frac{1}{3}{g}_{ij}{g}_{kl}\right)\frac{1}{\sqrt{{g}}}\frac{\delta\bar{S}}{\delta{g}_{kl}}+\frac{1}{2}\frac{1}{\sqrt{{g}}}\frac{\delta\bar{S}}{\delta{\phi}^{a}}L^{ab}\frac{1}{\sqrt{{g}}}\frac{\delta\bar{S}}{\delta{\phi}^{b}}={\cal
L}_{4}$ (A.17)

where

${\cal
L}_{4}=-R_{4}+\frac{1}{2}L_{ab}({\phi}){g}^{ij}\,\partial_{i}{\phi}^{a}\partial_{j}{\phi}^{b}+V({\phi})\;.$
(A.18)

Eq. (A.17) coincides with the Hamilton-Jacobi constraint in [24, 25].

Next we consider $\bar{S}$ expanded to second order in derivatives,

$\bar{S}=\int
d^{4}x\sqrt{{g}}\left(W({\phi})-L({\phi})R_{4}({g})+\frac{1}{2}K_{ab}({\phi})\nabla^{i}{\phi}^{a}\nabla_{i}{\phi}^{b}\right)\;.$
(A.19)

First, following [24], we expand the flow equations (A.13) and (A.14) in
derivatives. To lowest order one obtains

$\displaystyle\dot{{g}}_{ij}$
$\displaystyle=\frac{1}{3}g_{ij}\bar{b}W=\frac{l}{6\bar{\rho}}Wg_{ij}\;,$
(A.20) $\displaystyle\dot{{\phi}}^{a}$
$\displaystyle=-\bar{b}L^{ab}\partial_{b}W=-\frac{l}{2\bar{\rho}}L^{ab}\partial_{b}W\;.$
(A.21)

(In a slight abuse of notation, the solutions $\hat{g}_{ij}$ and
$\hat{\phi}^{a}$ of the flow equations (A.13) and (A.14) can be expanded as
$\hat{g}_{ij}=g_{ij}+\text{derivatives}$,
$\hat{\phi}^{a}=\phi^{a}+\text{derivatives}$. Inserted into
$\bar{S}(\hat{g}_{ij},\hat{\phi}^{a})$ and expanding $\bar{S}$ as in (A.19),
the derivative terms in $\hat{g}_{ij}$ and $\hat{\phi}^{a}$ induce flows for
the coefficients $L$ and $K_{ab}$ which have to be consistent with the
Hamiltonian constraints below.)

Using ${\cal L}_{4}$ from (A.18) the Hamiltonian constraint (A.17) implies
relations between the coefficients in $\bar{S}$ and ${\cal L}_{4}$. From the
terms without derivatives $\partial_{i}$ on both sides of (A.17) one obtains

$V=-\frac{1}{3}W^{2}+\frac{1}{2}\partial_{a}WL^{ab}\partial_{b}W\;.$ (A.22)

From terms $\sim R_{4}$ one obtains

$1+\frac{1}{3}WL-\partial_{a}WL^{ab}\partial_{b}L=0\;.$ (A.23)

From terms $\sim\nabla^{i}{\phi}^{a}\nabla_{i}{\phi}^{b}$ one obtains

$L_{ab}+W\left(\frac{1}{3}K_{ab}+2\partial_{a}\partial_{b}L\right)-\partial_{c}WL^{cd}(\partial_{d}K_{ab}-\partial_{a}K_{bd}-\partial_{b}K_{ad})=0\;.$
(A.24)

Finally terms $\sim\nabla^{i}\nabla_{i}{\phi}^{a}$ require

$W\partial_{a}L+\partial_{b}WL^{bc}K_{ca}=0\;.$ (A.25)

## References

  * [1] H. Osborn, Phys. Lett. B 222 (1989) 97.
  * [2] I. Jack and H. Osborn, Nucl. Phys. B 343 (1990) 647.
  * [3] H. Osborn, Nucl. Phys. B 363 (1991) 486.
  * [4] O. Antipin, M. Gillioz, J. Krog, E. Mølgaard and F. Sannino, JHEP 1308 (2013) 034 [arXiv:1306.3234 [hep-ph]].
  * [5] I. Jack and H. Osborn, Nucl. Phys. B 883 (2014) 425 [arXiv:1312.0428 [hep-th]].
  * [6] F. Baume, B. Keren-Zur, R. Rattazzi and L. Vitale, JHEP 08 (2014), 152 [arXiv:1401.5983 [hep-th]].
  * [7] I. Jack and C. Poole, JHEP 01 (2015), 138 [arXiv:1411.1301 [hep-th]].
  * [8] I. Jack, D. R. T. Jones and C. Poole, JHEP 09 (2015), 061 [arXiv:1505.05400 [hep-th]].
  * [9] O. Antipin, N. A. Dondi, F. Sannino and A. E. Thomsen, Phys. Rev. D 99 (2019) no.2, 025004 [arXiv:1808.00482 [hep-th]].
  * [10] C. Poole and A. E. Thomsen, Phys. Rev. Lett. 123 (2019) no.4, 041602 [arXiv:1901.02749 [hep-th]].
  * [11] C. Poole and A. E. Thomsen, JHEP 09 (2019), 055 [arXiv:1906.04625 [hep-th]].
  * [12] J. Davies, F. Herren, C. Poole, M. Steinhauser and A. E. Thomsen, Phys. Rev. Lett. 124 (2020) no.7, 071803 [arXiv:1912.07624 [hep-ph]].
  * [13] L. Sartore, Phys. Rev. D 102 (2020) no.7, 076002 [arXiv:2006.12307 [hep-ph]].
  * [14] T. Steudtner, JHEP 12 (2020), 012 [arXiv:2007.06591 [hep-th]].
  * [15] G. M. Shore, Nucl. Phys. B 286 (1987), 349-377
  * [16] A. Codello, G. D’Odorico, C. Pagani and R. Percacci, Class. Quant. Grav. 30 (2013), 115015 [arXiv:1210.3284 [hep-th]].
  * [17] A. Codello, G. D’Odorico and C. Pagani, Phys. Rev. D 91 (2015) no.12, 125016
[arXiv:1502.02439 [hep-th]].

  * [18] U. Ellwanger, Eur. Phys. J. C 81 (2021) no.5, 422 [arXiv:2101.07615 [hep-th]].
  * [19] K. Skenderis, Class. Quant. Grav. 19 (2002), 5849-5876 [arXiv:hep-th/0209067 [hep-th]].
  * [20] M. Henningson and K. Skenderis, JHEP 07 (1998), 023 [arXiv:hep-th/9806087 [hep-th]].
  * [21] I. Heemskerk and J. Polchinski, JHEP 06 (2011), 031 [arXiv:1010.1264 [hep-th]].
  * [22] Y. Nakayama, JHEP 06 (2015), 092 [arXiv:1502.07049 [hep-th]].
  * [23] T. Hartman, J. Kruthoff, E. Shaghoulian and A. Tajdini, JHEP 03 (2019), 004
[arXiv:1807.11401 [hep-th]].

  * [24] J. de Boer, E. P. Verlinde and H. L. Verlinde, JHEP 08 (2000), 003 [arXiv:hep-th/9912012 [hep-th]].
  * [25] M. Fukuma, S. Matsuura and T. Sakai, Prog. Theor. Phys. 109 (2003), 489-562 [arXiv:hep-th/0212314 [hep-th]].
  * [26] V. Balasubramanian, E. G. Gimon and D. Minic, JHEP 05 (2000), 014 [arXiv:hep-th/0003147 [hep-th]].
  * [27] B. P. Dolan, Int. J. Mod. Phys. A 10 (1995), 2703-2732 [arXiv:hep-th/9406061 [hep-th]].
  * [28] E. Kiritsis, W. Li and F. Nitti, Fortsch. Phys. 62 (2014), 389-454 [arXiv:1401.0888 [hep-th]].
  * [29] J. Polchinski, Nucl. Phys. B 231 (1984), 269-295
  * [30] M. Li, Nucl. Phys. B 579 (2000), 525-532 [arXiv:hep-th/0001193 [hep-th]].
  * [31] E. T. Akhmedov and E. T. Musaev, Phys. Rev. D 81 (2010), 085010 [arXiv:1001.4067 [hep-th]].
  * [32] R. G. Leigh, O. Parrikar and A. B. Weiss, Phys. Rev. D 91 (2015) no.2, 026002
[arXiv:1407.4574 [hep-th]].

  * [33] J. M. Lizana, T. R. Morris and M. Perez-Victoria, JHEP 03 (2016), 198 [arXiv:1511.04432 [hep-th]].
  * [34] C. Fefferman and R. Graham, “Conformal Invariants”, Astérisque, hors série, 1995, p.95.
  * [35] C. Imbimbo, A. Schwimmer, S. Theisen and S. Yankielowicz, Class. Quant. Grav. 17 (2000), 1129-1138 [arXiv:hep-th/9910267 [hep-th]].
  * [36] J. Erdmenger, Phys. Rev. D 64 (2001), 085012 [arXiv:hep-th/0103219 [hep-th]].
  * [37] Y. Nakayama, Phys. Rept. 569 (2015), 1-93 [arXiv:1302.0884 [hep-th]].
  * [38] S. Rajagopal, A. Stergiou and Y. Zhu, JHEP 11 (2015), 216 [arXiv:1508.01210 [hep-th]].
  * [39] K. Kikuchi and T. Sakai, PTEP 2016 (2016) no.3, 033B02 [arXiv:1511.00403 [hep-th]].
  * [40] V. Shyam, JHEP 03 (2018), 171 [arXiv:1712.07955 [hep-th]].
  * [41] S. S. Lee, JHEP 01 (2014), 076 [arXiv:1305.3908 [hep-th]].
  * [42] T. Banks, M. R. Douglas, G. T. Horowitz and E. J. Martinec, [arXiv:hep-th/9808016 [hep-th]].
  * [43] S. El-Showk and K. Papadodimas, JHEP 10 (2012), 106 [arXiv:1101.4163 [hep-th]].
  * [44] D. Z. Freedman, S. S. Gubser, K. Pilch and N. P. Warner, Adv. Theor. Math. Phys. 3 (1999), 363-417 [arXiv:hep-th/9904017 [hep-th]].