diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzgeyo" "b/data_all_eng_slimpj/shuffled/split2/finalzzgeyo" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzgeyo" @@ -0,0 +1,5 @@ +{"text":"\n\n\\subsection{Experimental plan}\n\nIn this work, we present the analysis of OpenStack version 3.12.1 (release \\textit{Pike}), which was the latest version of OpenStack when we started this work. We injected bugs into the most fundamental services of OpenStack \\cite{denton2015learning,solberg2017openstack}: (i) the \\textbf{Nova} subsystem, which provides services for provisioning instances (VMs) and handling their life cycle; (ii) the \\textbf{Cinder} subsystem, which provides services for managing block storage for instances; and (iii) the \\textbf{Neutron} subsystem, which provides services for provisioning virtual networks for instances, including resources such as \\emph{floating IPs}, \\emph{ports} and \\emph{subnets}. Each subsystem includes several components (e.g., the Nova sub-system includes \\emph{nova-api}, \\emph{nova-compute}, etc.), which interact through message queues internally to OpenStack. The Nova, Cinder, and Neutron sub-systems provide external REST API interfaces to cloud users.\n\n \n\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n \\begin{centering}\n \\includegraphics[width=0.75\\columnwidth]{testbed_architecture_v2.pdf}\n \n \\end{centering}\n \\vspace*{-4.5mm}\n \\caption{OpenStack testbed architecture.}\n \\label{fig:testbed_architecture}\n\\vspace{-0.8cm}\n\\end{figure}\n\n\\figurename{}~\\ref{fig:testbed_architecture} shows the testbed used for the experimental analysis of OpenStack. We adopted an all-in-one virtualized deployment of OpenStack, in which the OpenStack services run on the same VM, for the following reasons: (1) to prevent interferences on the tests from transient issues in the physical network (e.g., sporadic network faults, network delays caused by other user traffic in our local data center, etc.); (2) to parallelize a high number of tests on several physical machines, by using the \\emph{Packstack} installation utility \\cite{packstack} to have a reproducible installation of OpenStack across the VMs; (3) to efficiently revert any persistent effect of a fault injection test on the OpenStack deployment (e.g., file system issues), in order to assure independence among the tests. Moreover, the all-in-one virtualized deployment is a common solution for performing tests on OpenStack \\cite{evaluating_openstack,Markelov2016}. The hardware and VM configuration for the testbed includes: 8 virtual Intel Xeon CPUs (E5-2630L v3 @ 1.80GHz); 16GB RAM; 150 GB storage; Linux CentOS v7.0.\n\n\n\n\n\n\n\nIn addition to the core services of OpenStack (e.g., Nova, Neutron, Cinder, etc.), the testbed also includes our own components to automate fault injection tests. \nThe \\emph{Injector Agent} is the component that analyzes and instruments the source code of OpenStack. The \\emph{Injector Agent} can: (i) scan the source code to identify injectable locations (i.e., source-code statements where the bug types discussed in \\S{}~\\ref{subsec:fault_injection} can be applied); (ii) instrument the source code by introducing logging statements in every injectable location, in order to get a profile of which locations are covered during the execution of the workload (\\textbf{coverage analysis}); (iii) instrument the source code to introduce a bug into an individual injectable location.\n\nThe \\emph{Controller} orchestrates the experimental workflow.\nIt first commands the \\emph{Injector Agent} to perform a preliminary coverage analysis, by instrumenting the source code with logging statements, restarting the OpenStack services, and launching the \\emph{Workload Generator}, but without injecting any fault. \nThe \\emph{Workload Generator} issues a sequence of API calls in order to stimulate OpenStack services.\nThe \\emph{Controller} retrieves the list of injectable locations and their coverage from the \\emph{Injector Agent}. Then, it iterates over the list of injectable locations that are covered, and issues commands for the \\emph{Injector Agent} to perform fault injection tests. For each test, the \\emph{Injector Agent} introduces an individual bug by mutating the source code, restarts the OpenStack services, starts the workload, and triggers the injected bug as discussed in \\S{}~\\ref{subsec:fault_injection}. The \\emph{Injector Agent} collects the logs files from all OpenStack subsystems and from the \\emph{Workload Generator}, which are sent to the \\emph{Controller} for later analysis (\\S{}~\\ref{subsec:failure_analysis}).\n\n\n\n\n\n\n\nWe performed a full scan of injectable locations in the source code of Nova, Cinder, and Neutron, for a total of \\numprint{2016} analyzed source code files. We identified \\numprint{911} injectable faults that were covered by the workload. \\figurename{}~\\ref{fig:fault_injection_tests} shows the number of faults per sub-system and per type of fault. \nThe number of faults for each type and sub-system depends on the number of calls to the target functions, and on their input and output parameters, as discussed in \\S{}~\\ref{subsec:fault_injection}.\nWe executed one the test per injectable location, by injecting one fault at a time. \n\n\n\\begin{figure}[t]\n \\begin{centering}\n \\includegraphics[width=0.72\\columnwidth]{fault_injection_tests.png}\n \\end{centering}\n \\vspace*{-5mm} \n \\caption{Number of fault injection tests.}\n \\label{fig:fault_injection_tests}\n \\vspace{-5.3mm}\n\\end{figure}\n\n\\begin{figure}[t]\n \\begin{centering}\n \\includegraphics[width=0.72\\columnwidth]{failure_types_distribution.png}\n \\vspace{-0.5cm}\n \\end{centering}\n \n \\caption{Distribution of OpenStack failures.}\n \\label{fig:all_component_failure_types}\n \\vspace{-0.75cm}\n\\end{figure}\n\n\nAfter executing the tests, we found failures respectively in 52.6\\% (231 out of 439 tests), 46.4\\% (125 out of 269 tests), and 61\\% (124 out of 203 tests) of tests in Nova, Cinder, and Neutron, for a total of \\numprint{480}.\nIn the remaining 47.3\\% of the tests (431 out of 911 tests), instead, there were neither an API error nor assertion failures: in these cases, the fault was not activated (even if the faulty code was covered by the workload), or there was no error propagation to the component interface. The occurrence of tests not causing failures is a typical phenomenon that occurs with code mutations, which may not infect the state even when the faulty code is executed \\cite{christmansson1996generation,lanzaro2014empirical}. Yet, the injections provided us a large and diverse set of failures for our analysis.\n\n\n\n\n\n\n\n\n\\subsection{Does OpenStack show a fail-stop behavior?}\n\\label{subsec:rq1}\n\n\n\nWe first analyze the impact of failures on the service interface APIs provided by OpenStack. \nThe \\emph{Workload Generator} (which impersonates a user of the cloud management system) invokes these APIs, looks for errors returned by the APIs and performs assertion checks between API calls. A fail-stop behavior occurs when an API returns an error before any failed assertion check. In such cases, the \\emph{Workload Generator} stops on the occurrence of the API error. Instead, it is possible that an API invocation terminates without returning any error, but leaving the internal resources of the infrastructure (instances, volumes, etc.) in a failed state, which is reported by assertion checks. These cases represent violations of the fail-stop hypothesis, and represent a risk for the users as they are unaware of the failure.\nTo investigate this aspect, we initially focus on the faulty round of each test, in which fault injection is enabled (\\figurename{}~\\ref{fig:workflow}). \n\n\\figurename{}~\\ref{fig:all_component_failure_types} shows the number of tests that experienced failures, divided into \\emph{API Error only}, \\emph{Assertion Failure only}, and \\emph{Assertion Failure(s), followed by an API Error}. The figure shows the data divided with respect to the subsystem where the bug was injected (respectively in Nova, Cinder, and Neutron); moreover, \\figurename{}~\\ref{fig:all_component_failure_types} shows the distribution across all fault injection tests. \nWe can see that the cases in which the system does not exhibit a fail-stop behavior (i.e., the categories \\emph{Assertion Failure only} and \\emph{Assertion Failure followed by an API Error}) represent the majority of the failures.\n\n\n\\figurename{}~\\ref{fig:all_component_failure_assertion_types} shows a detailed perspective on the failures of assertion checks. Notice that the number of assertion is greater than the number of tests classified in the Assertion failure category (i.e., \\emph{Assertion Failure only} and \\emph{Assertion Failure followed by an API Error}) since a test can generate multiple assertion failures.\nThe most common case has been one of the instances not active because the instance creation failed (i.e., it did not move into the \\textit{ACTIVE} state \\cite{openstack_instances_states}). \nIn other cases, the instance could not be reached through the network or could not be attached to a volume, even if in the \\textit{ACTIVE} state. A further common case is the failure of the volume creation, but only the faults injected in the Cinder sub-system caused this assertion failure. \n\n\n\n\n\n\n\\begin{figure}[t]\n \\begin{centering}\n \\includegraphics[width=.75\\columnwidth]{assertion_distribution.png}\n \\end{centering}\n \\vspace*{-5mm} \n \\caption{Distribution of assertion check failures.}\n \\label{fig:all_component_failure_assertion_types}\n \\vspace{-0.1cm}\n\\end{figure}\n\n\n\\begin{figure}[t]\n \\vspace{-0.4cm}\n \\begin{centering}\n \\includegraphics[width=0.6\\columnwidth]{api_distribution.png}\n \\end{centering}\n \\vspace*{-5mm} \n \\caption{Distribution of API Errors.}\n \\label{fig:all_component_api_errros_after_assertion}\n \n \\vspace{-7mm}\n\\end{figure}\n\nThese cases point out that OpenStack lacks redundant checks to assure that the state of the virtual resources after a service call is in the expected state (e.g., newly-created instances are active). Such redundant checks would assess the state of the virtual resources before and after a service invocation and would raise an error if the state does not comply with the expectation (such as a new instance could not be activated). However, these redundant checks are seldom adopted, most likely due to the performance penalty they would incur, and because of the additional engineering efforts to design and implement them. Nevertheless, the cloud management system is exposed to the risk that residual bugs can lead to non-fail-stop behaviors, where failures are notified with a delay or not notified at all. This makes not trivial to prevent data losses and to automate recovery actions.\n\n\n\\figurename{}~\\ref{fig:all_component_api_errros_after_assertion} provides another perspective on API errors. It shows the number of tests in which each API returned an error, focusing on 15 out of 40 APIs that failed at least one time. The API with the highest number of API errors is the one for adding a volume to an instance (\\textit{openstack server add volume}), provided by the Cinder sub-system. \nThis API generated errors even when faults were injected in Nova (instance management) and Neutron (virtual networking). This behavior means that the effects of fault injection propagated from other sub-systems to Cinder (e.g., if an instance is in an incorrect state, other APIs on that resource are also exposed to failures). On the one hand, this behavior is an opportunity for detecting failures, even if in a later stage. On the other hand, it also represents the possibility of a failure to spread across sub-systems, thus defeating fault containment and exacerbating the severity of the failure. We will analyze fault propagation in more detail in Section~\\ref{subsec:rq3}.\n\n\n\n\n\n\n\n\n\n\\begin{figure}[!t]\n \\vspace{-3mm}\n \\begin{centering}\n \\includegraphics[width=0.75\\columnwidth]{all_components_assertion_and_api_errors_latency.pdf}\n \\end{centering}\n \\vspace*{-5mm} \n \\caption{Cumulative distribution of API Error latency.\n \\label{fig:all_components_assertion_and_api_errors_latency}\n \n \\vspace{-0.92cm}\n\\end{figure}\n\nTo understand the extent of non-fail-stop behaviors, we also analyze the period of time (\\textbf{latency}) between the execution of the injected bug and the resulting API error. \nIt is desirable that this latency is as low as possible. Otherwise, the longer the latency, the more difficult is to relate an API error with its root cause (i.e., an API call invoked much earlier, on a different sub-system or virtual resource); and the more difficult to perform troubleshooting and recovery actions. \nTo track the execution of the injected bug, we instrumented the injected code with logging statements to record the timestamp of its execution. \nIf the injected code is executed several times before a failure (e.g., in the body of a loop), we conservatively consider the last timestamp. \nWe consider separately the cases where the API error is preceded by assertion check failures (i.e., the injected bug was triggered by an API different from the one affected by the bug) from the cases without any assertion check failure (e.g., the API error arises from the same API affected by the injected bug).\n\n\n\n\n\n\\figurename{}~\\ref{fig:all_components_assertion_and_api_errors_latency} shows the distributions of latency for API errors that occurred after assertion check failures, respectively for the injections in Nova, Cinder, and Neutron. Table~\\ref{tab:all_components_all_latencies_table_last_activation} summarizes the average, the 50$^{th}$, and the 90$^{th}$ percentiles of the latency distributions. We note that in the first category (API errors after assertion checks), all sub-systems exhibit a median API error latency longer than 100 seconds, with cases longer than several minutes. This latency should be considered too long for cloud services with high-availability SLAs (e.g., four \\emph{nines} or more \\cite{Bauer:2012:RAC:2339445}), which can only afford few minutes of monthly outage. \nThis behavior points out that the API errors are due to a ``reactive'' behavior of OpenStack, which does not actively perform any redundant check on the integrity of virtual resources, but only reacts to the inconsistent state of the resources once they are requested in a later service invocation. Thus, OpenStack experiences a long API error latency when a bug leaves a virtual resource in an inconsistent state. \nThis result suggests the need for improved error checking mechanisms inside OpenStack to prevent these failures. \n\n\nIn the case of failures that are notified by API errors without any preceding assertion check failure (the second category in Table~\\ref{tab:all_components_all_latencies_table_last_activation}), the latency of the API errors was relatively small, less than one second in the majority of cases. \nNevertheless, there were few cases with an API error latency higher than one minute. In particular, these cases happened when bugs were injected in Nova, but the API error was raised by a different sub-system (Cinder). In these cases, the high latency was caused by the propagation of the bug's effects across different API calls. These cases are further discussed in \\S{}~\\ref{subsec:rq3}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{table}\n \\begin{center}\n \\footnotesize\n \\caption{Statistics on API Error latency.}\n \\label{tab:all_components_all_latencies_table_last_activation}\n \\vspace{-3mm}\n \n \\begin{tabular}{ >{\\centering}m{0.8in} | >{\\centering}m{0.4in} | >{\\centering}m{0.45in} | >{\\centering}m{0.45in} | >{\\centering\\arraybackslash}m{0.45in} | }\n \\cline{2-5}\n & \n \\textbf{Subsys.} & \\textbf{Avg [s]} & \\textbf{50$^{th}$ \\%ile [s]} & \\textbf{90$^{th}$ \\%ile [s]} \\\\\n \\hline\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \\multicolumn{1}{ |c| }{\n \\multirow{3}{*}{\\textbf{\\shortstack{API Errors after\\\\ an Assertion\\\\ failure}}} }\n & Nova & 152.25 & 168.34 & 191.60 \\\\\n \\cline{2-5}\n \\multicolumn{1}{ |c| }{}\n & Cinder & 74.52 & 93.00 & 110.00 \\\\\n \\cline{2-5}\n \\multicolumn{1}{ |c| }{}\n & Neutron & 144.72 & 166.00 & 263.60 \\\\\n \\hline\n \n \n \n\n\n\n\n \n \n \n\n \n \n \n\n \n \n \n\n \n \\multicolumn{1}{ |c| }{\n \\multirow{3}{*}{\\textbf{\\shortstack{API Errors\\\\ only}}} }\n & Nova & 3.73 & 0.21 & 0.55 \\\\\n \\cline{2-5}\n \\multicolumn{1}{ |c| }{}\n & Cinder & 0.30 & 0.01 & 1.00 \\\\\n \\cline{2-5}\n \\multicolumn{1}{ |c| }{}\n & Neutron & 0.30 & 0.01 & 1.00 \\\\\n \\hline\n \n \n \n \\end{tabular}\n \\vspace{-0.5cm}\n \\end{center}\n\\end{table}\n\n\n\n\n\n\n\n\n\\subsection{Is OpenStack able to log failures?}\n\\label{subsec:rq2}\n\nSince failures can be notified to the end-user with a long delay, or even not at all, it becomes important for system operators to get additional information to troubleshoot these failures. In particular, we here consider log messages produced by OpenStack sub-systems.\n\n\n\n\n \n \n\n\n\n\nWe computed the percentage (\\textbf{logging coverage}) of failed tests which produced at least one high-severity log message (see also \\S{}~\\ref{subsec:failure_analysis}). Table \\ref{tab:all_components_logged_not_logged_failed_tests} provides the logging coverage for different subsets of failures, by dividing them with respect to the injected subsystem and to the type of failure. \nFrom these results, we can see that OpenStack logged at least one high-severity message (i.e., with severity level \\textit{ERROR} or \\textit{CRITICAL}) in most of the cases. The Cinder subsystem shows the best results since logging covered almost all of the failures caused by fault injection. However, in the case of Nova and Neutron, logs missed some of the failures. In particular, the failures without API errors (i.e., \\emph{Assertion Failure only}) exhibited the lowest logging coverage. This behavior can be problematic for recovery and troubleshooting since the failures without API errors lack an explicit error notification. These failures are also the ones in need of complementary sources of information, such as logs.\n\nTo identify opportunities to improve logging in OpenStack, we analyzed the failures without any high-severity log across, with respect to the bug types injected in these tests. We found that \\textit{MISSING FUNCTION CALL} and \\textit{WRONG RETURN VALUE} represent the 70.7\\% of the bug types that lead to non-logged failures (43.9\\% and 26.8 \\%, respectively). \nThe \\textit{WRONG RETURN VALUE} faults are the easiest opportunity for improving logging and failure detection since the callers of a function could perform additional checks on the returned value and record anomalies in the logs.\nFor example, one of the injected bugs introduced a \\textit{WRONG RETURN VALUE} in calls to a database API called by the Nova sub-system to update the information linked to a new instance. The bug forced the function to return a \\emph{None} instance object. The bug caused an assertion check failure, but OpenStack did not log any high-severity message. By manually analyzing the logs, we could only find one suspicious message with the only \\textit{WARNING} severity and with little information about the problem, as this message was not related to database management.\n\n\nThe non-logged failures caused by a \\textit{MISSING FUNCTION CALL} emphasize the need for redundant end-to-end checks to identify inconsistencies in the state of the virtual resources. For example, in one of these experiments, we injected a \\textit{MISSING FUNCTION CALL} in the \\textit{LibvirtDriver} class in the Nova subsystem, which allows OpenStack to interact with the \\textit{libvirt} virtualization APIs \\cite{libvirt}. Because of the injected bug, the Nova driver omits to attach a volume to an instance, but the Nova sub-system does not perform checks that the volume is indeed attached to the instance. This kind of end-to-end checks could be introduced at the service API interface of OpenStack (e.g., in \\emph{nova-api}) to test the availability of the virtual resources at the end of API service invocations (e.g., by pinging them).\n\n\n\n\n\n\n\n\n\\begin{table}\n \\begin{center}\n \\footnotesize\n \\caption{Logging coverage of high-severity log messages.}\n \\label{tab:all_components_logged_not_logged_failed_tests}\n \\vspace{-3mm}\n \n \\begin{tabular}{ >{\\centering}m{0.75in} | >{\\centering}m{0.7in} | >{\\centering}m{0.7in} | >{\\centering\\arraybackslash}m{0.6in} | }\n \n \\cline{2-4}\n & \\multicolumn{3}{c|}{\\textbf{Logging coverage}} \\\\\n \n \\hline\n \\multicolumn{1}{|c|}{\\textbf{Subsystem}} & \\textbf{API Errors\\\\only} & \\textbf{Assertion\\\\ failure only} & \\textbf{Assertion failure and API Errors} \\\\\n \\hline\n \n \\multicolumn{1}{|c|}{Nova} & 90.32\\% & 80.77\\% & 82,56\\% \\\\\n \\hline\n \\multicolumn{1}{|c|}{Cinder} & 100\\% & 95,65\\% & 100\\% \\\\\n \\hline\n \\multicolumn{1}{|c|}{Neutron} & 98.67\\% & 66.67\\% & 95\\% \\\\\n \\hline\n \\end{tabular}\n \\vspace{-0.6cm}\n \\end{center}\n\\end{table}\n\n\\subsection{Do failures propagate across OpenStack?}\n\\label{subsec:rq3}\n\n\n\n\n\nWe analyze failure propagation across sub-systems, to identify more opportunities to reduce their severity. We consider failures of both the ``faulty'' and the ``fault-free'' rounds, respectively (\\figurename{}~\\ref{fig:workflow}).\n\nIn the faulty round, we are interested in whether the injected bug impacted on sub-systems beyond the injected one. To this aim, we divide the API errors with respect to the API that raised the error (e.g., an API exposed by Nova, Neutron, or Cinder). Similarly, we divide the assertion check failures with respect to the sub-system that manages the virtual resource checked by the assertion. There is a \\textbf{spatial} fault propagation across the components if an injection on a sub-system (say, Nova) causes an assertion check failure or an API error on a different sub-system (say, Cinder or Neutron).\n\n\n\\begin{figure*}[!ht]\n \\begin{centering}\n \\begin{subfigure}{0.40\\textwidth}\n \\includegraphics[width=\\linewidth]{r1.pdf}\n \\caption{During faulty round.}\n \\label{fig:openstack_spatial_propagation_round1}\n \\end{subfigure}\n \\qquad\n \\begin{subfigure}{0.40\\textwidth}\n \\includegraphics[width=\\linewidth]{r2.pdf}\n \\caption{After removing the injected fault (fault-free round).}\n \\label{fig:openstack_spatial_propagation_round2}\n \\end{subfigure}\n \n \n \\end{centering}\n \\vspace*{-4mm} \n \\caption{Fault propagation during fault injection tests.}\n\\vspace{-5mm}\n\\end{figure*}\n\n\n\\figurename{}~\\ref{fig:openstack_spatial_propagation_round1} shows a graph with of events occurred during the faulty round of the tests with a failure. The nodes on the top of the graph represent the sub-systems where bugs were injected; the nodes on the middle represent assertion check failures; the nodes on the bottom represent API errors. The edges that originate from the nodes on the top represent the number of injections that were followed by an assertion check failure or an API error. Moreover, the edges between the middle and the bottom nodes represent the number of tests where an assertion check failure was followed by an API error. The most numerous cases are emphasized with proportionally thicker edges and annotated with the number of occurrences. We used different shades to differentiate the cases with respect to the injected sub-system.\n\n \n\n\nThe failures exhibited a propagation across OpenStack services in a significant amount of cases (37.5\\% of the failures). \nIn many cases, the propagation initiated from an injection in Nova, which caused a failure at activating a new instance; as discussed in the previous subsections, the unavailability of the instance was detected in a later stage, such as when the user attaches a volume to the instance using the Cinder API. \nEven worse, there are some cases of propagation from Neutron across Nova and Cinder.\nThese failures represent a severe issue for fault containment since an injection in Neutron not only caused a failure of their APIs but also impacted on virtual resources that were not managed by these sub-systems. Therefore, the failures are not necessarily limited to the virtual resources managed by the sub-system invoked at the time of the failure, but also to other related virtual resources. Therefore, end-to-end checks on API invocations should also include resources that are indirectly related to the API (such as, checking the availability of an instance after attaching a volume).\nFor as concerns Cinder, instead, there are no cases of error propagation from this sub-system across Nova and Neutron.\n\n\nWe further analyze the propagation of failures by considering what happens during the fault-free round of execution. The fault-free round invokes the service APIs after the buggy execution path is disabled as dead code. Moreover, the fault-free round executes on new virtual resources (i.e., instances, networks, routers, etc., are created from scratch). \nTherefore, it is reasonable to expect (and it is indeed the case) that the fault-free round executes without experiencing any failure. However, we still observe a subset of failures (7.5\\%) that propagate their effects to the fault-free round. These failures must be considered critical, since they are affecting service requests that are supposed to be independent but are still exposed to \\textbf{temporal} failure propagation through shared state and resources. \nWe remark that the failures in the fault-free round are caused by the injection in the faulty round. Indeed, we assured that previous injections do not impact on the subsequent experiments by restoring all the persistent state of OpenStack before every experiment. \n\n\n\n\n\\figurename{}~\\ref{fig:openstack_spatial_propagation_round2} shows the propagation graph for the fault-free round. The most cases, the Nova sub-system was unable to create new instances, even after the injected bug is removed from Nova. A similar persistent issue happens for a subset of failures caused by injections in Neutron. These sub-systems both manage a relational database which holds information on the virtual instances and networks, and we found that the persistent issues are solved only after that the databases are reverted to the state before fault injection. This recovery action can be very costly since it can take a significant amount of time, during which the cloud infrastructure may become unavailable. For this reason, we remark the need for detecting failures as soon as they occur, such as using end-to-end checks at the end of service API calls. Such detection would support quicker recovery actions, such as to revert the database changes performed by an individual transaction.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Discussion and lessons learned}\n\nThe experimental analysis pointed out that software bugs often cause erratic behavior of the cloud management system, hindering detection and recovery of failures. We found failures that were notified to the user only after a long delay when it is more difficult to trace back the root cause of the failure, and recovery actions are more costly (e.g., reverting the database); or, the failures were not notified at all. \nMoreover, our analysis suggests the following practical strategies to mitigate these failures. \n\n\n \n\\noindent\n$\\rhd$ \\textbf{Need for deeper run-time verification of virtual resources.} \nFault injections pointed out OpenStack APIs that leaked resources on failures, or left them in an inconsistent state, due to missing or incorrect error handlers.\nFor example, the \\emph{server-create} API failed without creating a new VM, but it did not deallocate virtual resources (e.g., instances in ``dead'' state, unused virtual NICs) created before the failure.\nThese failures can be prevented through fault injection. Moreover, residual faults should be detected and handled by means of run-time verification strategies, which perform redundant, end-to-end checks after a service API call, to assert whether the virtual resources are in the expected state. For example, these checks can be specified using temporal logic and synthesized in a run-time monitor \\cite{delgado2004taxonomy,chen2007mop,zhou2014runtime,rabiser2017comparison}, e.g., a logical predicate for a traditional OS can assert that a thread suspended on a semaphore leads to the activation of another thread \\cite{arlat2002dependability}. In the context of cloud management, the predicates should test at run-time the availability of virtual resources (e.g., volumes and connectivity), similarly to our assertion checks (\\tablename{}~\\ref{tab:assertion_and_description}).\n\n\n\n\n\n\\noindent\n$\\rhd$ \\textbf{Increasing the logging coverage.} \nThe logging mechanisms in OpenStack reported high-severity error messages for many of the failures. However, there were failures with late or no API errors that would benefit from logs to diagnose the failure, but such logs were missing. In particular, fault injection identified function call sites in OpenStack where the injected wrong return values were ignored by the caller. These cases are opportunities for developers to add logging statements and to improve the coverage of logs (e.g., by checking the outputs produced by the faulty function calls). Moreover, the logs can be complemented with the run-time verification checks. \n\n\n\n\n\n\n\n\n\\noindent\n$\\rhd$ \\textbf{Preventing corruptions of persistent data and shared state.} \nThe experiments showed that undetected failures can propagate across several virtual resources and sub-systems. Moreover, we found that these propagated failures can impact on shared state and persistent data (such as databases), causing permanent issues. Fault injection identified failures that were detected much later after their initial occurrence (i.e., with high API error latency, or no API errors at all). In these cases, it is very difficult for operators to diagnose which parts of the system have been corrupted, thus increasing the cost of recovery. Therefore, in addition to timely failure detection (using deeper run-time verification techniques, as discussed above), it becomes important to address the corruptions as soon as the failure is detected, since the scope of recovery actions can be smaller (i.e., the impact of the failure is limited specific resources involved by the failed service API call). One potential direction of research is on selectively undoing recent changes to the shared state and persistent data of the cloud management system \\cite{weber2012automatic,satyal2017rollback}.\n\n\n\n\n\\subsection{Threats to validity}\n\\label{subsec:threats_validity}\n\nThe injection of software bugs is still a challenging and open research problem. We addressed this issue by using code mutations to generate realistic run-time errors. This technique is widespread in the field of mutation testing \\cite{jia2011survey,just2014mutants,papadakis2018mutation,papadakis2019mutation} to devise test cases; moreover, it is also commonly adopted by studies on software dependability \\cite{chillarege1996:generation-error-set,voas1997:predicting,ng2001design,duraes2006emulation,giuffrida2013edfi} and on assessing bug finding tools \\cite{dolan2016lava,roy2018bug}. In our context, bug injection is meant to anticipate the potential consequences of bugs on service availability and resource integrity. \nTo strengthen the connection between the real and the experimental failures, we based our selection of code mutations on past software bugs in OpenStack. \nThe injected bug types were consistent with code mutations typically adopted for mutation testing and fault injection (e.g., the omission of statements). Moreover, the analysis of OpenStack bugs gave us insights on where to apply the injections (e.g., on method calls for controlling Nova, for performing SQL queries, etc.). \nEven if some categories of failures may have been over- or under-represented (e.g., the percentages for failures that were not detected or that propagated), our goal is to point out the existence of potential, critical classes of failures, despite possible errors in the estimates of the percentages. In our experiments, these classes were large enough to be considered a threat to cloud management platforms. \n\n\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\\input{intro.tex}\n\n\n\\section{Overview on the research problem}\n\\label{sec:research_problem}\n\\input{research_problem.tex}\n\n\n\n\\section{Methodology}\n\\label{sec:methodology}\n\\input{methodology.tex}\n\n\n\\section{Experimental results}\n\\label{sec:experiments}\n\\input{experiments.tex}\n\n\n\\section{Related work}\n\\label{sec:related}\n\\input{related.tex}\n\n\n\\section{Experimental artifacts}\n\\label{sec:artifacts}\n\\input{artifacts.tex}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\\input{conclusion.tex}\n\n\n\\section*{Acknowledgments}\nThis work has been partially supported by the PRIN 2015 project ``GAUSS'' funded by MIUR (Grant n. 2015KWREMX\\_002) and by UniNA and Compagnia di San Paolo in the frame of Programme STAR. We are grateful to Alfonso Di Martino for his help in the early stage of this work.\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n\n\\subsection{Bug analysis}\n\\label{subsec:bug_analysis}\n\n\n\n\n\n\n\n\n\n\n \n \n \n\n\n \n \n \n\n\n\n\n\n\nA key aspect to perform software fault injection experiments is to inject representative software bugs \\cite{chillarege1996:generation-error-set,duraes2006emulation}. \nSince the body of knowledge on bugs in Python software \\cite{rodriguez2018if,orru2015curated}, the programming language of OpenStack, is relatively smaller compared to other languages, we seek for more insights about bugs in the OpenStack project. Therefore, we analyzed the OpenStack issue tracker on the \\emph{Launchpad} portal \\cite{openstack_launchpad}, by looking for bug-fixes at the source code level, in order to identify \\emph{bug patterns} \\cite{duraes2006emulation,pan2009toward,martinez2013automatically,zhong2018towards,tufano2018learning} for this project. From these patterns, we defined a set of bug types to be injected.\n\nWe went through the problem reports and inspected the related source code. We looked for reports where: (i) the root cause of the problem was a software bug, excluding build, packaging and installation issues; \n(ii) the problem had been marked with the highest severity level (i.e., the problem has a strong impact on OpenStack services); \n(iii) the problem was fixed, and the bug-fix was linked to the discussion. We manually analyzed a sample of 179 problem reports from the Launchpad, focusing on entries with importance set to ``\\emph{Critical}'', and with status set to ``\\emph{Fix Committed}'' or ``\\emph{Fix Released}'' (such that the problem report also includes a final solution shipped in OpenStack). Of these problem reports, we identified 113 reports that met all of the three criteria. We shared the full set of bug reports (see Section \\ref{sec:artifacts}).\n\nThe bugs encompass several areas of OpenStack, including: bugs that affected the service APIs exposed to users (e.g., \\emph{nova-api}); bugs that affected dictionaries and arrays, such as a wrong key used in {\\fontfamily{lmtt}\\selectfont image['imageId']}; bugs that affected SQL queries (e.g., database queries for information about instances in Nova); bugs that affected RPC calls between OpenStack subsystems (e.g., \\emph{rpc.cast} was omitted, or had a wrong topic or contents); bugs that affected calls to external system software, such as \\emph{iptables} and \\emph{dsnmasq}; bugs that affected pluggable modules in OpenStack, such as network protocol plugins and agents in Neutron.\nFigure~\\ref{fig:faults_nova_neutron} shows statistics about the bug types that we identified from the problem reports and their bug-fixes. The five most frequent bug types include the following ones.\n\n\n\n\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{Wrong parameters value}: The bug was an incorrect method call inside OpenStack, where a wrong variable was passed to the method call. For example, this was the case of the Nova bug \\#1130718 (\\url{https:\/\/bugs.launchpad.net\/nova\/+bug\/1130718}, which was fixed in \\url{https:\/\/review.openstack.org\/#\/c\/22431\/} by changing the exit codes passed through the parameter {\\fontfamily{lmtt}\\selectfont check\\_exit\\_code}).\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{Missing parameters}: A method call was invoked with omitted parameters (e.g., the method used a default parameter instead of the correct one). For example, this was the case of the Nova bug \\#1061166 (\\url{https:\/\/bugs.launchpad.net\/nova\/+bug\/1061166}, which was fixed in \\url{https:\/\/review.openstack.org\/#\/c\/14240\/} by adding the parameter {\\fontfamily{lmtt}\\selectfont read\\_deleted='yes'} when calling the SQL Alchemy APIs).\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{Missing function call}: A method call was entirely omitted. For example, this was the case of the Nova bug \\#1039400 (\\url{https:\/\/bugs.launchpad.net\/nova\/+bug\/1039400}, which was fixed in \\url{https:\/\/review.openstack.org\/#\/c\/12173\/} by adding and calling the new method \n\n\\noindent\n{\\fontfamily{lmtt}\\selectfont trigger\\_security\\_group\\_members\\_refresh}).\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{Wrong return value}: A method returned an incorrect value (e.g., {\\fontfamily{lmtt}\\selectfont None} instead of a Python object). For example, this was the case of the Nova bug \\#855030 (\\url{https:\/\/bugs.launchpad.net\/nova\/+bug\/855030}, which was fixed in \\url{https:\/\/review.openstack.org\/#\/c\/1930\/} by returning an object allocated through {\\fontfamily{lmtt}\\selectfont allocate\\_fixed\\_ip}).\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{Missing exception handlers}: A method call lacks exception handling. For example, this was the case of the Nova bug \\#1096722 ({\\url{https:\/\/bugs.launchpad.net\/nova\/+bug\/1096722}}, which was fixed in \\url{https:\/\/review.openstack.org\/#\/c\/19069\/} by adding an exception handler for {\\fontfamily{lmtt}\\selectfont exception.InstanceNotFound}).\n\n\n\n\n\n\n\n\n\n\n\\subsection{Fault injection}\n\\label{subsec:fault_injection}\n\nIn this study, we perform \\emph{software fault injection} to analyze the impact of software bugs \\cite{voas1997:predicting,chillarege1996:generation-error-set,natella2016assessing}. This approach deliberately introduces programming mistakes in the source code, by replacing parts of the original source code with faulty code. \nFor example, in \\figurename{}~\\ref{fig:workflow}, the injected bug emulates a missing optional parameter (a port number) to a function call, which may cause failure under certain conditions (e.g., a VM instance may not be reachable through an intended port).\nThis approach is based on previous empirical studies, which observed that the injection of code changes can realistically emulate software faults \\cite{daran1996software,chillarege1996:generation-error-set,andrews2005mutation}, in the sense that \\emph{code changes produce run-time errors that are similar to the ones produced by real software faults}.\nThis approach is motivated by the high efforts that would be needed for experimenting with hand-crafted bugs or with real past bugs: in these cases, every bug would require to carefully craft the specific conditions that trigger it (i.e., the topology of the infrastructure, the software configuration, and the hardware devices under which the bug surfaces). \nTo achieve a match between injected and real bugs, we focus the injection on the most frequent five types found by the bug analysis. These bug types allow us to cover all of the main areas of OpenStack (API, SQL, etc.), and suffice to generate a large and diverse set of faults over the codebase of OpenStack. \n\n\n\\begin{figure*}[!ht]\n \\begin{centering}\n \n \\includegraphics[width=0.8\\textwidth]{workflow_v2.pdf}\n \\end{centering}\n \\vspace*{-5mm} \n \\caption{Overview of a fault injection experiment}\n \\label{fig:workflow}\n\\vspace{-5mm}\n\\end{figure*}\n\n\n\n\n\n\n\n\nWe emulate the bug types by mutating the existing code of OpenStack. \nThe \\figurename{}~\\ref{fig:workflow} shows the steps of a fault injection experiment. \nWe developed a tool to automate the bug injection process in Python code. The tool uses the \\emph{ast} Python module to generate an \\emph{abstract syntax tree} (AST) representation of the source code; then, it scans the AST by looking for relevant elements (function calls, expressions, etc.) where the bug types could be injected; it modifies the AST, by removing or replacing the nodes to introduce the bug; finally, it rewrites the modified AST into Python code, using the \\emph{astunparse} Python module. \nTo inject the bug types of Section~\\ref{subsec:fault_injection}, we modify or remove method calls and their parameters. We targeted method calls related to the bugs that we analyzed, by targeting calls to internal APIs for managing instances, volumes, and networks (e.g., which are denoted by specific keywords, such as \\emph{instance} and \\emph{nova} for the methods of the Nova subsystem). Wrong input and parameters are injected by wrapping the target expression into a function call, which returns at run-time a corrupted version of the expression based on its data type (e.g., a null reference in place of an object reference, or a negative value in place of an integer). Exceptions are raised on method calls according to a pre-defined list of exception types.\n\nThe tool inserts fault-injected statements into an \\emph{if} block, together with the original version of the same statements but in a different branch (as in step 2 in \\figurename{}~\\ref{fig:workflow}). \nThe execution of the fault-injected code is controlled by a \\emph{trigger} variable, which is stored in a shared memory area that is writable from an external program. This approach has been adopted for controlling the occurrence of failures during the tests. In the first phase (\\textbf{round 1}), we enable the fault-injected code, and we run a workload that exercises the service APIs of the cloud management system. During this phase, the fault-injected code could generate run-time errors inside the system, which will potentially lead to user-perceived failures. Afterward, in a second phase (\\textbf{round 2}), we disable the injected bug, and we execute the workload for a second time. This fault-free execution points out whether the scope of run-time errors (generated by the first phase) is limited to the service API invocations that triggered the buggy code (e.g., the bug only impacts on local session data). If failures still occur during the second phase, then the system has not able to handle the run-time errors of the first phase. Such failures point out the propagation of effects across the cloud management system (see \\S~\\ref{sec:research_problem}).\n\n\n\n\n\nWe implemented a workload generator to automatically exercise the service APIs of the main OpenStack sub-systems. The workload has been designed to cover several sub-systems of OpenStack and several types of virtual resources, in a similar way to integration test cases from the OpenStack project \\cite{openstack_tempest}. The workload creates VM instances, along with key pairs and a security group; attaches the instances to volumes; creates a virtual network, with virtual routers; and assigns floating IPs to connect the instances to the virtual network. Having a comprehensive workload allows us to point out propagation effects across sub-systems caused by bugs.\n\n\nThe experimental workflow is repeated several times. Every experiment injects a different fault, and only one fault is injected per experiment. Before a new experiment, we clean-up any potential residual effect from the previous experiment, in order to be able to relate failure to the specific bug that caused it. The clean-up re-deploys OpenStack removes all temporary files and processes and restores the database to its initial state. However, we do not perform these clean-up operations between the two workload rounds (i.e., no clean-up between the steps 6 and 8 of \\figurename{}~\\ref{fig:workflow}), since we want to assess the impact of residual side effects caused by the bug.\n\n\n\\begin{table}[t]\n \\begin{center}\n \\caption{Assertion check failures.}\n \\label{tab:assertion_and_description}\n \\vspace{-4mm}\n \n {\n \\scriptsize\n \n\n \n \\begin{tabulary}{\\columnwidth}{|L|L|}\n \n \\hline\n \\textbf{Name} & \\textbf{Description}\\\\\n \\hline\n FAILURE IMAGE ACTIVE & The created \\textit{image} does not transit into the \\textit{ACTIVE} state\\\\ \\hline\n FAILURE INSTANCE ACTIVE & The created \\textit{instance} does not transit into the \\textit{ACTIVE} state \\\\ \\hline\n FAILURE SSH & It is impossible to establish a \\textit{ssh} session to the created instance\\\\ \\hline\n FAILURE KEYPAIR & The creation of a \\textit{keypair} fails\\\\ \\hline\n FAILURE SECURITY GROUP & The creation of a \\textit{security group} and \\textit{rules} fails\\\\ \\hline\n FAILURE VOLUME CREATED & The creation of a \\textit{volume} fails\\\\ \\hline\n FAILURE VOLUME ATTACHED & Attaching a \\textit{volume} to an instance fails\\\\ \\hline\n FAILURE FLOATING IP CREATED & The creation of a \\textit{floating IP} fails\\\\ \\hline\n FAILURE FLOATING IP ADDED & Adding a \\textit{floating IP} to an instance fails\\\\ \\hline\n FAILURE PRIVATE NETWORK ACTIVE & The created \\textit{network} resource does not transit into the \\textit{ACTIVE} state\\\\ \\hline\n FAILURE PRIVATE SUBNET CREATED & The creation of a \\textit{subnet} fails\\\\ \\hline\n FAILURE ROUTER ACTIVE & The created \\textit{router} resource does not transit into the \\textit{ACTIVE} state\\\\ \\hline\n FAILURE ROUTER INTERFACE CREATED & The creation of a router interface fails\\\\ \\hline\n \n \n \n \\end{tabulary}\n }\n \n \\end{center}\n \n \\vspace{-0.65cm}\n \n\\end{table}\n\n\n\\subsection{Failure data collection}\n\\label{subsec:failure_data_collection}\n\nDuring the execution of the workload, we record inputs and outputs of service API calls of OpenStack. Any exception generated from the call (\\emph{API Errors}) is also recorded. In-between calls to service APIs, the workload also performs \\emph{assertion checks} on the status of the virtual resources, in order to point out failures of the cloud management system. \nIn the context of our methodology, assertion checks serve as \\emph{ground truth} about the occurrence of failures during the experiments. These checks are valuable to point out the cases in which a fault causes an error, but the system does not generate an API error (i.e., the system is unaware of the failure state). \nOur assertion checks are similar to the ones performed by the integration tests as test oracles \\cite{ju2013fault,openstack_instances_states}: they assess the connectivity of the instances through SSH and query the OpenStack API to check that the status of the instances, volumes and network is consistent with the expectation of the test cases. \nThe assertion checks are performed by our workload generator. For example, after invoking the API for creating a volume, the workload queries the volume status to check if it is available (\\emph{VOLUME CREATED assertion}). These checks are useful to find failures not notified through the API errors.\n\\tablename{}~\\ref{tab:assertion_and_description} describes the assertion checks.\n\n\nIf an API call generates an error, the workload is aborted, as no further operation is possible on the resources affected by the failure (e.g., no volume could be attached if the instance could not be created). In the case that the system fails without raising an exception (i.e., an assertion check highlights a failure, but the system does not generate an API error), the workload continues the execution (as a hypothetical end-user, being unaware of the failure, would do), regardless of failed assertion check(s). The workload generator records the outcomes of both the API calls and of the assertion checks. Moreover, we collect all the log files generated by the cloud management system. This data is later analyzed for understanding the behavior of the system under failure.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Failure analysis}\n\\label{subsec:failure_analysis}\n\nWe analyze fault injection experiments according to three perspectives discussed in Section~\\ref{sec:research_problem}. \nThe first perspective classifies the experiments \\emph{with respect to the type of failure that the system experiences}. The possible cases are the following ones.\n\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{API Error}: In these cases, the workload was not able to correctly execute, due to an exception raised by a service API call. In these cases, the cloud management system has been able to handle the failure in a fail-stop way, since the user is informed by the exception that the virtual resources could not be used, and it can perform recovery actions to address the failure. In our experiments, the workload stops on the occurrence of an exception, as discussed before.\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{Assertion failure}: In these cases, the failure was not pointed out by an exception raised by a service API. The failure was detected by the assertion checks made by the workload in-between API calls, which found an incorrect state of virtual resources. In these cases, the execution of the workload was not interrupted, as no exception was raised by the service APIs during the whole experiment, and the service API did (apparently) work from the perspective of the user. These cases point out non-fail-stop behavior.\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{Assertion failure(s), followed by an API Error}: In these cases, the failure was initially detected by assertion checks, which found an incorrect state of virtual resources in-between API calls. After the assertion check detected the failure, the workload continued the execution, by performing further service API calls, until an API error occurred in a later API call. These cases also point out issues at handling the failure, since the user is unaware of the failure state and cannot perform recovery actions.\n\n\\vspace{2pt}\n\\noindent\n$\\blacksquare$ \\textbf{No failure}: The injected bug did not cause a failure that could be perceived by the user (neither by API exceptions nor by assertion checks). It is possible that the effects of the bug were tolerated by the system (e.g., the system switched to an alternative execution path to provide the service); or, the injected source code was harmless (e.g., an uninitialized variable is later assigned before use). Since no failure occurred, these experiments are not further analyzed, as they do not allow to draw conclusions on the failure behavior of the system.\n\n\\vspace{2pt}\n\n\n\nFailed executions are further classified according to a second perspective, \\emph{with respect to the execution round in which the system experienced a failure}. The possible cases are the following ones.\n\n\n\\vspace{1pt}\n\\noindent\n$\\rhd$ \\textbf{Failure in the faulty round only}: In these cases, a failure occurred in the first (faulty) execution round (\\figurename{}~\\ref{fig:workflow}), in which a bug has been injected; and no failure is observed during the second (fault-free) execution round, in which the injected bug is disabled, and in which the workload operates on a new set of resources. This behavior is the likely outcome of an experiment since we are deliberately forcing a service failure only in the first round through the injected bug.\n\n\\vspace{1pt}\n\\noindent\n$\\rhd$ \\textbf{Failure in the fault-free round (despite the faulty round)}: \nThese cases are concerns for fault containment since the system is still experiencing failures despite the bug is disabled after the first round and the workload operates on a new set of resources. This behavior is due to residual effects of the bug that propagated through session state, persistent data, or other shared resources.\n\n\n\\vspace{1pt}\n\n\nFinally, the experiments with failures are classified from the perspective of \\emph{whether they generated logs able to indicate the failure}. In order to make more resilient a system, we are interested in whether it produces information for detecting failures and for triggering recovery actions. \nIn practice, developers are conservative at logging information for post-mortem analysis, by recording high volumes of low-quality log messages that bury the truly important information among many trivial logs of similar severity and contents, making it difficult to locate issues \\cite{zhu2015learning,li2017log,yuan2012improving}. Therefore, we cannot simply rely on the presence of logs to conclude that a failure was detected.\n\n\nTo clarify the issue, \\figurename{}~\\ref{fig:logging_stats} shows the distribution of the number of log messages in OpenStack across severity levels, \\textit{TRACE} to \\textit{CRITICAL}, during the execution of our workload generator, and \\emph{without} any failure. We can notice that all OpenStack components generate a large number of messages with severity \\textit{WARNING}, \\textit{INFO}, and \\textit{DEBUG} even when there is no failure. Instead, there are no messages of severity \\textit{ERROR} or \\textit{CRITICAL}. Therefore, even if a failure is logged with severity \\textit{WARNING} or lower, such log messages cannot be adopted for automated detection and recovery of the failure, as it is difficult to distinguish between ``informative'' messages and actual issues. Therefore, to evaluate the ability of the system to support recovery and troubleshooting through logs, we classify failures according to the presence of one or more \\emph{high-severity message} (i.e., \\emph{CRITICAL} or \\emph{ERROR}) recorded in the log files (\\textbf{logged failures}), or no such message (\\textbf{non-logged failures}). \n\n\n\n\n\\begin{figure}[t]\n \\begin{centering}\n \\includegraphics[width=0.95\\columnwidth]{severity2.pdf}\n \\end{centering}\n \\vspace*{-5mm} \n \\caption{Distribution of log messages severity during a fault-free execution of the workload.}\n \\label{fig:logging_stats}\n \\vspace{-6mm}\n\\end{figure}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn \\cite{biha4}, B. Y. Chen established sharp inequality for a submanifold in a real space form involving intrinsic invariants of the submanifolds and squared mean curvature, the main extrinsic invariant and in \\cite{biha2}, B. Y. Chen obtained the same inequality for complex space form. After that many research articles \\cite{biha5, biha6, biha1} have been published by different authors for different submanifolds and ambient spaces in complex as well as in contact version. In this article we obtain these inequalities for submanifolds in Bochner Kaehler manifold.\n\n In \\cite{bishopwrap} Bishop and O'Neil initiated the thoery of warped product submanifold as a generalization of pseudo-Riemannian product manifold. In \\cite{chen warp} Chen introduced the notion of CR-warped products. In This paper we study the CR-warped product submanifolds of Bochner Kaehler manifolds.\n\\newline\n\\section{Preliminaries}\nLet $\\mathcal{W}$ be a $n$-dimensional submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$ of dimension $2m$. Let $\\nabla$ and $\\overline{\\nabla}$ be the Levi-Civita connection on $\\mathcal{W}$ and $\\overline{\\mathcal{W}}$ respectively. Let $J$ be the complex structure on $\\overline{\\mathcal{W}}$. Then the Gauss and Weingarten formulas are given respectively by\n\\begin{eqnarray}\\label{a1}\n\\overline{\\nabla}_{X}Y = \\nabla_{X}Y + \\omega(X,Y),\n\\end{eqnarray}\n\\begin{eqnarray}\\label{a2}\n\\overline{\\nabla}_{X}V = - B_{V}X + \\nabla_{X}^{\\perp}Y ,\n\\end{eqnarray}\nfor all $X, Y$ tangent to $\\mathcal{W}$ and vector field $V$ normal to $\\mathcal{W}$. Where $\\omega$, $\\nabla_{X}^{\\perp}$, $B_{V}$ denotes the second fundamental form, normal connection and the shape operator respectively. The second fundamental form and the shape operator are related by\n\\begin{eqnarray}\\label{a3}\ng(\\omega(X,Y), V) = g(B_{V}X, Y).\n\\end{eqnarray}\nLet $R$ be the curvature tensor of $\\mathcal{W}$, Then the Gauss equation is given by \\cite{biha4}\n\\begin{eqnarray*}\\label{a4}\n\\overline{R}(X,Y,Z,W) = R(X,Y,Z,W) + g(\\omega(X,W),\\omega(Y,Z)) - g(\\omega(X,Z),\\omega(Y,W))\n\\end{eqnarray*}\nfor any vector fields $X$, $Y$, $Z$, $W$ tangent to $\\mathcal{W}$.\n\nThe curvature tensor of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$ is given by \\cite{biha7}\n\\begin{eqnarray}\\label{a5}\n\\overline{R}(X,Y,Z,W)\\nonumber &=& L(Y,Z)g(X,W) - L(X,Z)g(Y,W) + L(X,W)g(Y,Z) \\\\ \\nonumber && - L(Y,W)g(X,Z) + M(X,W)g(JX,W) - M(X,Z)g(JY,W)\\\\ \\nonumber && + M(X,W)g(JY,Z) - M(Y,W)g(JX,Z) \\\\ && - 2M(X,Y)g(JZ,W) - 2M(Z,W)g(JX,Y)\n\\end{eqnarray}\n\nwhere\n\\begin{eqnarray}\\label{a6}\nL(Y,Z) = \\frac{1}{2n+4}Ric(Y,Z) - \\frac{\\rho}{2(2n+2)(2n+4)}g(Y,Z),\n\\end{eqnarray}\n\\begin{eqnarray}\\label{a7}\nM(Y,Z) = -L(Y,JZ),\n\\end{eqnarray}\n\\begin{eqnarray}\\label{a8}\nL(Y,Z) = L(Z,Y),\\hspace{1cm} L(Y,Z) = L(JY, JZ), \\hspace{1cm} L(Y, JZ) = -L(JY,Z),\n\\end{eqnarray}\n$Ric$ and $\\rho$ are the Ricci tensor and scalar curvature of $\\mathcal{W}$.\n\nLet $x\\in \\mathcal{W}$ and $\\{e_{1}, ... , e_{n}\\}$ be an orthonormal basis of the tangent space $T_{x}\\mathcal{W}$ and $\\{e_{n+1}, ... , e_{2m}\\}$ be the orthonormal basis of $T^{\\perp}\\mathcal{W}$. We denote by $\\mathcal{H}$, the mean curvature vector at $x$, that is\n\\begin{eqnarray}\\label{a9}\n\\mathcal{H}(x) = \\frac{1}{n}\\sum_{i=1}^{n}\\omega(e_{i},e_{i}),\n\\end{eqnarray}\nAlso, we set\n\\begin{eqnarray*}\\label{a10}\n\\omega_{ij}^{r} = g(\\omega(e_{i},e_{j}),e_{r}), \\hspace{1cm} i,j \\in \\{ 1, ... , n\\},\\hspace{.3cm} r \\in \\{n+1, ... ,2m\\}\n\\end{eqnarray*}\nand\n\\begin{eqnarray}\\label{a11}\n\\|\\omega\\|^{2} = \\sum_{i,j=1}^{n}(\\omega(e_{i},e_{j}), \\omega(e_{i},e_{j})).\n\\end{eqnarray}\nFor any $x \\in \\mathcal{W}$ and $X \\in T_{x}\\mathcal{W}$, we put $JX = TX + FX$, where $TX$ and $FX$ are the tangential and normal components of $JX$, respectively.\n\n We denote by\n\\begin{eqnarray*}\\label{a12}\n\\|T\\|^{2} = \\sum_{i,j=1}^{n}g^{2}(Te_{i}, e_{j}).\n\\end{eqnarray*}\nLet $\\mathcal{W}$ be a Riemannian manifold. Denote by $\\mathcal{K}(\\pi)$ the sectional curvature of $\\mathcal{W}$ of the plane section $\\pi \\subset T_{x}\\mathcal{W}, x\\in \\mathcal{W}$. The scalar curvature $\\rho$ for an orthonormal basis$ \\{e_{1}, e_{2}, ..., e_{n}\\}$ of the tangent space $T_{x}\\mathcal{W}$ at $x$ is defined by\n\\begin{eqnarray*}\\label{a121}\n\\rho(x) = \\sum_{i2,\\\\\n\\omega_{ij}^{r} = 0, \\forall i \\neq j,\\hspace{.5cm}i,j = 3, ..., 2m, \\hspace{.5cm}r= n+1, ..., 2m,\\\\\n\\omega_{11}^{r}+\\omega_{22}^{r} = 0, \\forall r=n+2, ..., 2m,\\hspace{2cm}\\\\\n\\omega_{11}^{n+2}+\\omega_{22}^{n+1} = ... = \\omega_{11}^{m}+\\omega_{22}^{m}=0.\\\\\n\\end{cases}$\n\nNow, if we choose $e_{1},e_{2}$ such that $\\omega_{12}^{n+1}$= 0 and we denote by $\\alpha = \\omega_{11}^{r}, \\beta = \\omega_{22}^{r}$, $\\xi = \\omega_{33}^{n+1} = ... = \\omega_{33}^{r}$. Therefore by choosing the suitable orthonormal basis the shape operators take the desired forms.\n\\end{proof}\nWe conclude the following corollary from this theorem.\n\\begin{corollary}\nLet $\\mathcal{W}$ be a submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$ which is Einstein. Then, for each point $x \\in \\mathcal{W}$ and each plane section $\\pi \\subset T_{x}\\mathcal{W}$, we have\n\\begin{eqnarray*}\\label{p21}\n\\mathcal{K}(\\pi) \\geq (\\frac{5n^{2}+31n+26+3\\|T\\|^{2}}{2(2n+2)(2n+4)})\\rho - \\frac{n^{2}(n-2)}{2(n-1)}\\|\\mathcal{H}\\|^{2} - \\frac{6\\lambda}{2(2n+4)} \\|T\\|^{2}.\n\\end{eqnarray*}\nThe equality at a point $x \\in \\mathcal{W}$ holds iff there exists an orthonormal basis $\\{e_{1},e_{2}, ..., e_{n}\\}$ of $ T_{x}\\mathcal{W}$ and orthonormal basis $\\{e_{n+1},e_{n+2}, ..., e_{2m}\\}$ of $T^{\\perp}\\mathcal{W}$ such that shape operators of $\\mathcal{W}$ in $\\overline{\\mathcal{W}}$ at $x$ have the forms (\\ref{t2}) and (\\ref{t3}).\n\\end{corollary}\nSimilarly, in case if $\\mathcal{W}$ is a slant submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$. We have the following theorem\n\\begin{theorem}\nLet $\\mathcal{W}$ be a slant submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$. Then, for each point $x \\in \\mathcal{W}$ and each plane section $\\pi \\subset T_{x}\\mathcal{W}$, we have\n\\begin{eqnarray*}\\label{p22}\n\\nonumber \\mathcal{K}(\\pi) \\geq (\\frac{5n^{2}+31n+26+3cos^{2}\\theta}{2(2n+2)(2n+4)})\\rho - \\frac{n^{2}(n-2)}{2(n-1)}\\|\\mathcal{H}\\|^{2} - \\frac{6}{2(2n+4)}Ric(e_{i},Je_{j})cos\\theta.\\\\\n\\end{eqnarray*}\nEquality holds if and only if there exists an orthonormal basis $\\{e_{1} ,e_{2}, ... , e_{n}\\}$ of $T_{x}\\mathcal{W}$ and orthonormal basis $\\{e_{n+1}, e_{n+2} , ... , e_{2m}\\}$ of $T^{\\perp}\\mathcal{W}$ such that the shape operator takes the following forms\n\n\\begin{eqnarray}\\label{p23}\nB_{n+1} =\n \\begin{pmatrix}\n \\alpha & 0 & 0 & \\cdots & 0 \\\\\n 0 & \\beta & 0 & \\cdots & 0 \\\\\n 0 & 0 & \\xi &\\cdots & 0 \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 0 & 0 & 0 & \\cdots & \\xi\n \\end{pmatrix} , \\alpha+\\beta = \\xi\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{p24}\nB_{r} =\n \\begin{pmatrix}\n \\omega_{11}^{r} & \\omega_{12}^{r} & 0 & \\cdots & 0 \\\\\n \\omega_{12}^{r} & - \\omega_{11}^{r} & 0 & \\cdots & 0 \\\\\n 0 & 0 & 0 &\\cdots & 0 \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 0 & 0 & 0 & \\cdots & 0\n \\end{pmatrix} , r = n+2, ..., 2m.\n\\end{eqnarray}\n\\end{theorem}\nFrom this theorem, following corollaries can be easily deduced.\n\\begin{corollary}\nLet $\\mathcal{W}$ be a slant submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$, which is Einstein . Then, for each point $x \\in \\mathcal{W}$ and each plane section $\\pi \\subset T_{x}\\mathcal{W}$, we have\n\\begin{eqnarray*}\\label{p25}\n\\nonumber \\mathcal{K}(\\pi) \\geq (\\frac{5n^{2}+31n+26+3cos^{2}\\theta}{2(2n+2)(2n+4)})\\rho - \\frac{n^{2}(n-2)}{2(n-1)}\\|\\mathcal{H}\\|^{2} - \\frac{6\\lambda}{2(2n+4)}cos^{2}\\theta.\\\\\n\\end{eqnarray*}\nThe equality holds at a point $x \\in \\mathcal{W}$ if and only if there exists an orthonormal basis $\\{e_{1},e_{2}, ..., e_{n}\\}$ of $ T_{x}\\mathcal{W}$ and orthonormal basis $\\{e_{n+1},e_{n+2}, ..., e_{2m}\\}$ of $T^{\\perp}\\mathcal{W}$ such that shape operators of $\\mathcal{W}$ in $\\overline{\\mathcal{W}}$ at $x$ have the forms (\\ref{p23}) and (\\ref{p24}).\n\\end{corollary}\n\\begin{corollary}\nLet $\\mathcal{W}$ be a invariant submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$ . Then, for each point $x\\in \\mathcal{W}$ and each plane section $\\pi \\subset T_{x}\\mathcal{W}$, we have\n\\begin{eqnarray*}\\label{p26}\n \\mathcal{K}(\\pi) \\geq (\\frac{5n^{2}+31n+26+3}{2(2n+2)(2n+4)})\\rho - \\frac{n^{2}(n-2)}{2(n-1)}\\|\\mathcal{H}\\|^{2} - \\frac{6}{2(2n+4)}Ric(e_{i}, Je_{j}).\n\\end{eqnarray*}\nThe equality at a point $x\\in \\mathcal{W}$ holds iff there exists an orthonormal basis $\\{e_{1},e_{2}, ..., e_{n}\\}$ of $ T_{x}\\mathcal{W}$ and orthonormal basis $\\{e_{n+1},e_{n+2}, ..., e_{2m}\\}$ of $T^{\\perp}\\mathcal{W}$ such that shape operators of $\\mathcal{W}$ in $\\overline{\\mathcal{W}}$ at $x$ have the forms (\\ref{p23}) and (\\ref{p24}).\n\\end{corollary}\n\\begin{corollary}\nLet $\\mathcal{W}$ be a anti-invariant submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$ . Then, for each point $x \\in \\mathcal{W}$ and each plane section $\\pi \\subset T_{x}\\mathcal{W}$, we have\n\\begin{eqnarray*}\\label{p26}\n \\mathcal{K}(\\pi) \\geq (\\frac{5n^{2}+31n+26}{2(2n+2)(2n+4)})\\rho - \\frac{n^{2}(n-2)}{2(n-1)}\\|\\mathcal{H}\\|^{2}.\n\\end{eqnarray*}\nThe equality at a point $x \\in \\mathcal{W}$ holds iff there exists an orthonormal basis $\\{e_{1},e_{2}, ..., e_{n}\\}$ of $ T_{x}\\mathcal{W}$ and orthonormal basis $\\{e_{n+1},e_{n+2}, ..., e_{2m}\\}$ of $T^{\\perp}\\mathcal{W}$ such that shape operators of $\\mathcal{W}$ in $\\overline{\\mathcal{W}}$ at $x$ have the forms (\\ref{p23}) and (\\ref{p24}).\n\\end{corollary}\n\\section{Warped product of CR-submanifolds of Bochner Kaehler manifolds}\nLet $\\mathcal{W} = \\mathcal{W}_{T}\\times_{f}\\mathcal{W}_{\\perp}$ be the warped product CR-submanifolds of Bochner Kaehler manifold $\\overline{\\mathcal{W}}$ such that the invariant distribution is $D = T\\mathcal{W}_{T}$ and anti-invariant distribution is $D^{\\perp} = T\\mathcal{W}_{\\perp}$, where $f: \\mathcal{W}_{T} \\longrightarrow \\mathbb{R}$. Then the metric $g$ on $\\mathcal{W}$ is given by \\cite{chen warp}\n\\begin{equation*}\\label{w1}\ng(X,Y) = \\langle \\pi_{*}X, \\pi_{*}Y\\rangle + (f\\circ\\pi)^{2}\\langle \\sigma_{*}X, \\sigma_{*}Y\\rangle\n\\end{equation*}\nwhere $\\pi$ and $\\sigma$ are the projection maps from $\\mathcal{W}$ onto $\\mathcal{W}_{T}$ and $\\mathcal{W}_{\\perp}$ respectively.\n\n\nIt is easy to see that\n\\begin{eqnarray}\\label{w6}\nT\\mathcal{W} = D \\oplus D^{\\perp}\\hspace{.3cm} \\text{and} \\hspace{.3cm} T^{\\perp}\\mathcal{W} = JD^{\\perp}\\oplus \\nu,\n\\end{eqnarray}\nwhere $\\nu$ is the orthogonal distribution to $JD^{\\perp}$ in the normal bundle $T^{\\perp}\\mathcal{W}.$\n\nFrom (\\ref{w6}), we can write\n\\begin{eqnarray*}\\label{w7}\n\\omega(X,Y) = \\omega_{JD^{\\perp}}(X,Y) + \\omega_{\\nu}(X,Y)\n\\end{eqnarray*}\nAlso for warped product submanifold $\\mathcal{W}$ of $\\overline{\\mathcal{W}}$, we have \\cite{chen warp}\n\\begin{eqnarray}\\label{w8}\n\\nabla_{X}Z = X(logf)Z = \\frac{X(f)}{f}Z\n\\end{eqnarray}\nfor any vector fields $X \\in D$ and $Z \\in D^{\\perp}$.\n\nFurther, we can decompose $(\\overline{\\nabla}_{X}J)Y$ into the tangential and normal components as under\n\\begin{eqnarray}\\label{w9}\n(\\overline{\\nabla}_{X}J)Y = \\mathcal{P}_{X}Y + \\mathcal{Q}_{X}Y\n\\end{eqnarray}\nwhere $\\mathcal{P}_{X}Y$ and $\\mathcal{Q}_{X}Y$ denotes the tangential and normal components of $(\\overline{\\nabla}_{X}J)Y$\n\nFirst we prove the following lemma\n\\begin{lemma}\nLet $\\mathcal{W} = \\mathcal{W}_{T}\\times_{f}\\mathcal{W}_{\\perp}$ be a CR-warped product submanifold of a Bochner Kaehler manifold $\\overline{\\mathcal{W}}$. Then we have\n\\begin{equation*}\n\\omega_{JD^{\\perp}}(JX,Z) = J\\mathcal{P}_{Z}JX + X(log f)JZ\n\\end{equation*}\n\\begin{equation*}\ng(\\mathcal{P}_{Z}JX, W) = g(\\mathcal{Q}_{Z}X, JW)\n\\end{equation*}\n and\n\\begin{equation*}\ng(\\omega(JX,Z), J\\omega(X,Z)) -\\|\\omega_{\\nu}(X,Z)\\|^{2} = g(\\mathcal{Q}_{Z}X, J\\omega_{\\nu}(X,Z))\n\\end{equation*}\n for $X \\in D$ and $Z \\in D^{\\perp}.$\n\\end{lemma}\n\\begin{proof}\nFrom Gauss equation, we have\n\\begin{equation*}\\label{w10}\n\\nabla_{Z}JX + \\omega(JX,Z) -J(\\nabla_{Z}X) - J\\omega(X,Z) = \\mathcal{P}_{Z}X + \\mathcal{Q}_{Z}X\n\\end{equation*}\nUsing (\\ref{w8}), we infer\n\\begin{equation*}\\label{w11}\n\\omega(JX,Z)= \\mathcal{P}_{Z}X + \\mathcal{Q}_{Z}X + J[X(log f)Z] + J\\omega(X,Z) - JX(log f)Z .\n\\end{equation*}\nReplace $X$ by $JX$, we get\n\\begin{equation*}\\label{w12}\n-\\omega(X,Z)= \\mathcal{P}_{Z}JX + \\mathcal{Q}_{Z}JX + JX(log f)JZ + J\\omega(JX,Z) + X(log f)Z.\n\\end{equation*}\nWe can write the above equation as\n\\begin{equation}\\label{w13}\n-\\omega(X,Z)= \\mathcal{P}_{Z}JX + \\mathcal{Q}_{Z}JX + JX(log f)JZ + J\\omega_{JD^{\\perp}}(JX,Z) + J\\omega_{\\nu}(JX,Z) + X(log f)Z.\n\\end{equation}\nOn comparing the tangential components, we obtain\n\\begin{equation*}\\label{w14}\n\\mathcal{P}_{Z}JX + J\\omega_{JD^{\\perp}}(JX,Z) + X(log f)Z = 0,\n\\end{equation*}\nor\n\\begin{equation*}\\label{w15}\nJ\\omega_{JD^{\\perp}}(JX,Z) = -\\mathcal{P}_{Z}JX - X(log f)Z\n\\end{equation*}\nwhich shows that\n\\begin{equation}\\label{w16}\n\\omega_{JD^{\\perp}}(JX,Z) = J\\mathcal{P}_{Z}JX + X(log f)JZ,\n\\end{equation}\nfor $X \\in D$ and $Z \\in D^{\\perp}$.\n\nAgain on comparing the normal components in (\\ref{w13}), we have\n\\begin{equation*}\\label{w17}\n-\\omega(X,Z)= \\mathcal{Q}_{Z}JX + JX(log f)JZ + J\\omega_{\\nu}(JX,Z)\n\\end{equation*}\nfrom which we conclude that\n\\begin{equation*}\\label{w18}\n\\omega(JX,Z)= \\mathcal{Q}_{Z}X + X(log f)JZ + J\\omega_{\\nu}(X,Z)\n\\end{equation*}\nor\n\\begin{equation}\\label{w19}\n\\omega(JX,Z) - J\\omega_{\\nu}(X,Z)= \\mathcal{Q}_{Z}X + X(log f)JZ\n\\end{equation}\nBy taking the inner product (\\ref{w19}) with $JW$, we get\n\\begin{equation}\\label{w20}\ng(\\omega_{JD^{\\perp}}(JX,Z), JW) = g(\\mathcal{Q}_{Z}X,JW) + X(log f)g(JZ,JW)\n\\end{equation}\nFurther using (\\ref{w16}) in (\\ref{w20}), we have\n\\begin{equation*}\\label{w21}\ng(\\mathcal{P}_{Z}JX, W) + X(log f)g(Z, W) = g(\\mathcal{Q}_{Z}X,JW) + X(log f)g(Z, W)\n\\end{equation*}\nfrom which we conclude that\n\\begin{equation}\\label{w22}\ng(\\mathcal{P}_{Z}JX, W) = g(\\mathcal{Q}_{Z}X,JW)\n\\end{equation}\nAlso, by taking the inner product of (\\ref{w19}) with $J\\omega(X,Z)$, we find\n\\begin{equation*}\\label{w23}\ng(\\omega(JX,Z),J\\omega(X,Z)) - \\|\\omega_{\\nu}(X,Z)\\|^{2} = g(\\mathcal{Q}_{Z}X,J\\omega_{\\nu}(X,Z)).\n\\end{equation*}\n\\end{proof}\n\\begin{theorem}\nLet $\\mathcal{W} = \\mathcal{W}_{T}\\times_{f}\\mathcal{W}_{\\perp}$ be a warped product CR-submanifolds of Bochner Kaehler manifold $\\overline{\\mathcal{W}}$ with $\\mathcal{P}_{D^{\\perp}}D \\in D$, then the squared norm of second fundamental form of $\\mathcal{W}$ in $\\overline{\\mathcal{W}}$ satisfies the following inequality\n\\begin{equation*}\\label{w24}\n\\|\\omega\\|^{2} \\geq \\|\\mathcal{P}_{D^{\\perp}}D\\|^{2} + q\\|grad_{D}(log f)\\|^{2}.\n\\end{equation*}\n\\end{theorem}\n\n\\begin{proof}\nLet $\\{X_{1}, ..., X_{p}, X_{p+1} = JX_{1}, ..., X_{2p}= JX_{p}\\}$ be a local orthonormal frame of vector fields on $N_{T}$ and $\\{Z_{1}, ..., Z_{q}\\}$ be a local orthonormal frame of vector fields on $N_{\\perp}$, where $2p+q = n$. Then we have\n\\begin{eqnarray*}\\label{w25}\n\\nonumber \\|\\omega\\|^{2} = \\sum_{i,j=1}^{2p}g(\\omega(X_{i},X_{j}), \\omega (X_{i},X_{j})) + \\sum_{\\alpha=1}^{q}\\sum_{j=1}^{2p}g(\\omega(X_{i},Z_{\\alpha}),\\omega(X_{i},Z_{\\alpha}))\\\\ +\\sum_{\\alpha,\\beta =1}^{q}g(\\omega(Z_{\\alpha},Z_{\\beta}), \\omega(Z_{\\alpha},Z_{\\beta}))\n\\end{eqnarray*}\nfrom above equation we can say that\n\\begin{eqnarray*}\\label{w26}\n \\|\\omega\\|^{2} \\geq \\sum_{j=1}^{2p}\\sum_{\\alpha=1}^{q}g(\\omega(X_{i},Z_{\\alpha}),\\omega(X_{i},Z_{\\alpha}))\n\\end{eqnarray*}\nNow from (\\ref{w16}), we have\n\\begin{eqnarray*}\\label{w27}\n \\|\\omega\\|^{2} \\geq \\sum_{j=1}^{2p}\\sum_{\\alpha=1}^{q}g(J\\mathcal{P}_{Z_{\\alpha}}X_{i} - JX_{i}(log f)JZ_{\\alpha}, J\\mathcal{P}_{Z_{\\alpha}}JX_{i}- JX_{i}(log f)JZ_{\\alpha})\n\\end{eqnarray*}\nIn view of the assumption $\\mathcal{P}_{D^{\\perp}}D \\in D$, we have\n\\begin{eqnarray*}\\label{w28}\n\\nonumber\\|\\omega\\|^{2} &\\geq& \\sum_{j=1}^{2p}\\sum_{\\alpha=1}^{q}\\bigg[g(J\\mathcal{P}_{Z_{\\alpha}}X_{i} , J\\mathcal{P}_{Z_{\\alpha}}X_{i}) + g(JX_{i}(log f)JZ_{\\alpha}, JX_{i}(log f)JZ_{\\alpha})\\bigg]\n\\nonumber \\\\ &=& \\sum_{j=1}^{2p}\\sum_{\\alpha=1}^{q}\\bigg[g(\\mathcal{P}_{Z_{\\alpha}}X_{i}, \\mathcal{P}_{Z_{\\alpha}}X_{i}) + (JX_{i}(log f))^{2}g(Z_{\\alpha},Z_{\\alpha})\\bigg]\n\\nonumber \\\\ &=& \\|\\mathcal{P}_{D^{\\perp}}D\\|^{2} + \\sum_{j=1}^{2p}\\|JX_{i}(log f)\\|^{2}q\n \\\\ &=& \\|\\mathcal{P}_{D^{\\perp}}D\\|^{2} + q\\|grad_{D}(log f)\\|^{2}\n\\end{eqnarray*}\nwhere $grad_{D}$ denotes the gradient of some function on the distribution $D$.\n\nThus we have\n\\begin{eqnarray*}\\label{w29}\n\\|\\omega\\|^{2} \\geq \\|\\mathcal{P}_{D^{\\perp}}D\\|^{2} + q\\|grad_{D}(log f)\\|^{2}.\n\\end{eqnarray*}\n\\end{proof}\n\n\\begin{theorem}\nLet $\\mathcal{W} = \\mathcal{W}_{T}\\times_{f}\\mathcal{W}_{\\perp}$ be a compact orientable warped product CR-submanifold of Bochner Kaehler manifold $\\overline{\\mathcal{W}}$. If $\\mathcal{P}_{D^{\\perp}}D \\in D$ and $B_{\\nabla_{JX_{i}}^{\\perp}JZ}X_{i} = B_{\\nabla_{X_{i}}^{\\perp}JZ}JX_{i}$, then we have $\\rho \\leq 0$, and the equality holds iff $grad_{D}(log f) = 0.$\n\\end{theorem}\n\\begin{proof}\nLet $X \\in D$, $Z \\in D^{\\perp}$, then from (\\ref{a5}), we have\n\\begin{eqnarray}\\label{w33}\n\\overline{R} (X, JX,Z,JZ) &=& -2M(X,JX)g(Z,Z) - 2M(Z,JZ)g(X,X)\n\\end{eqnarray}\n Now Codazzi equation is\n\\begin{eqnarray*}\\label{w34}\n\\nonumber \\bigg[\\overline{R}(X,Y)Z\\bigg]^{\\perp} &=& \\bigg \\{ \\nabla_{X}^{\\perp}\\omega(Y,Z) - \\omega(\\nabla_{X}Y,Z) - \\omega(Y,\\nabla_{X}Z)\\bigg \\}\n\\\\ \\nonumber & & - \\bigg \\{ \\nabla_{Y}^{\\perp}\\omega(X,Z) - \\omega(\\nabla_{Y}X,Z) - \\omega(X,\\nabla_{Y}Z)\\bigg \\}\n\\end{eqnarray*}\nIn view of the last equation we may write\n\\begin{eqnarray}\\label{w35}\n\\nonumber \\overline{R}(X, JX, Z, JZ) &=& g( \\nabla_{X}^{\\perp}\\omega(JX,Z) - \\omega(\\nabla_{X}JX,Z) - \\omega(JX,\\nabla_{X}Z),JZ)\n\\\\ & & - g(\\nabla_{JX}^{\\perp}\\omega(X,Z) - \\omega(\\nabla_{JX}X,Z) - \\omega(X,\\nabla_{JX}Z),JZ)\n\\end{eqnarray}\nWe now compute each term of (\\ref{w35}). First we have\n\\begin{eqnarray}\\label{w36}\nXg(\\omega(JX,Z),JZ) = g(\\overline{\\nabla}_{X}\\omega(JX,Z), JZ) + g(\\omega(JX,Z),\\overline{\\nabla}_{X}JZ)\n\\end{eqnarray}\n Using Weingarten formula we have\n\\begin{eqnarray}\\label{w37}\ng(\\overline{\\nabla}_{X}^{\\perp}\\omega(JX,Z), JZ) = Xg(\\omega(JX,Z),JZ) -g(\\omega(JX,Z),\\overline{\\nabla}_{X}JZ)\n\\end{eqnarray}\nNow from (\\ref{w19})\n\\begin{eqnarray}\\label{w38}\n\\omega(JX, Z) - J\\omega_{\\nu}(X,Z) = \\mathcal{Q}_{Z}X + X(log f)JZ\n\\end{eqnarray}\nTaking the inner product of (\\ref{w38}) with $JZ$, we have\n\\begin{eqnarray}\\label{w39}\ng(\\omega(JX, Z),JZ) - g(J\\omega_{\\nu}(X,Z), JZ) = g(\\mathcal{Q}_{Z}X, JZ) + X(log f)g(JZ,JZ)\n\\end{eqnarray}\nCombining (\\ref{w22}) and (\\ref{w39}), we get\n\\begin{eqnarray}\\label{w41}\ng(\\omega(JX, Z),JZ) = g(\\mathcal{\\mathcal{P}}_{Z}JX, Z) + X(log f)\\|Z\\|^{2}\n\\end{eqnarray}\nMoreover\n\\begin{eqnarray*}\ng(\\omega(JX,Z), JZ) = X(log f)g(Z,Z)\n\\end{eqnarray*}\nHence we have\n\\begin{eqnarray}\\label{w42}\n\\nonumber Xg(\\omega(JX,Z), JZ) &=& X\\bigg\\{ X( log f) g(Z,Z) \\bigg\\}\n\\\\ \\nonumber &=& X\\big( X(log f)\\big) g(Z,Z)+ 2X(log f)g(Z,\\nabla_{X}Z)\n\\\\ \\nonumber &=& X\\big( X(log f)\\big)\\|Z\\|^{2}+ 2\\big( X(log f)\\big)^{2}\\|Z\\|^{2}\n\\\\ &=& \\bigg \\{ X (X(log f))+ 2( X(log f))^{2} \\bigg \\} \\|Z\\|^{2}\n\\end{eqnarray}\n From (\\ref{w37}) and (\\ref{w42}), we get\n\\begin{eqnarray}\\label{w43}\ng(\\nabla_{X}^{\\perp}\\omega(JX,Z), JZ) = \\bigg \\{ X (X(log f))+ 2( X(log f))^{2} \\bigg \\} \\|Z\\|^{2} - g(\\omega(JX,Z), \\overline{\\nabla}_{X}JZ)\n\\end{eqnarray}\nReplacing $X$ by $JX$ in the above equation , we find\n\\begin{eqnarray}\\label{w44}\n-g(\\nabla_{JX}^{\\perp}\\omega(X,Z), JZ) = \\bigg \\{ JX (JX(log f))+ 2( JX(log f))^{2} \\bigg \\} \\|Z\\|^{2} + g(\\omega(X,Z), \\overline{\\nabla}_{JX}JZ)\n\\end{eqnarray}\nAlso using (\\ref{w16}) and $\\mathcal{P}_{D^{\\perp}} D \\in D$, we conclude that\n\\begin{eqnarray}\\label{w45}\ng(\\omega_{JD^{\\perp}}(JX, \\nabla_{X}Z), JZ ) = g(X(log f)J\\nabla_{X}Z,JZ) = (X(log f))^{2}g(Z,Z) = (X(log f))^{2}\\|Z\\|^{2}\n\\end{eqnarray}\n Replacing $X$ by $JX$ in the above equation, we find\n\\begin{eqnarray}\\label{w46}\ng(\\omega_{JD^{\\perp}}(X, \\nabla_{JX}Z), JZ ) = -(JX(log f))^{2}\\|Z\\|^{2}\n\\end{eqnarray}\nAgain using (\\ref{w16}), we get\n\\begin{eqnarray*}\\label{w47}\n\\omega_{JD^{\\perp}}( \\nabla_{JX}X), Z ) = J\\mathcal{P}_{Z}\\nabla_{JX}X- J\\nabla_{JX}X(log f)JZ\n\\end{eqnarray*}\nor\n\\begin{eqnarray*}\\label{w48}\ng(\\omega_{JD^{\\perp}}( \\nabla_{JX}X), JZ ) = g(\\mathcal{P}_{Z}\\nabla_{JX}X, Z)- J\\nabla_{JX}X(log f)\\|Z\\|^{2}\n\\end{eqnarray*}\nThe above equation can be written as\n\\begin{eqnarray*}\\label{w49}\ng(\\omega( \\nabla_{JX}X), JZ ) = g(\\mathcal{P}_{Z}\\nabla_{JX}X, Z)- J\\nabla_{JX}X(log f))\\|Z\\|^{2}\n\\end{eqnarray*}\n But $N_{T}$ is totally geodesic in $\\overline{N}$ which implies that $\\nabla_{JX}X \\in D$. Hence $\\mathcal{P}_{Z}J\\nabla_{JX}X \\in D$. This makes the first term in the above equation zero and hence we have\n\\begin{eqnarray}\\label{w49}\ng(\\omega( \\nabla_{JX}X), JZ ) = - J\\nabla_{JX}X(log f))\\|Z\\|^{2}\n\\end{eqnarray}\nSimilarly on replacing $X$ by $JX$ in the above equation, we have\n\\begin{eqnarray*}\\label{w50}\ng(\\omega( \\nabla_{X}JX), JZ ) = - J\\nabla_{X}JX(log f))\\|Z\\|^{2}\n\\end{eqnarray*}\n Using Gauss equation, the last equation simplifies to\n\\begin{eqnarray}\\label{w51}\ng(\\omega( \\nabla_{X}JX), JZ ) = \\nabla_{X}X(log f)g(Z,Z) + \\nabla_{JX}JX(log f)g(Z,Z) - J\\nabla_{JX}X(log f)g(Z,Z)\n\\end{eqnarray}\n Putting (\\ref{w43})$\\sim$(\\ref{w51}) into (\\ref{w35}), we get\n\\begin{eqnarray}\\label{w52}\n\\nonumber \\overline{R}(X,JX,Z,JZ ) &=& \\bigg \\{ X(X(log f)) + 2(X(log f))^{2} \\bigg \\}\\|Z\\|^{2} - g(\\omega(JX,Z), \\nabla_{X}^{\\perp}JZ)\n\\nonumber \\\\ & & - \\nabla_{X}X(log f)\\|Z\\|^{2} - \\nabla_{JX}JX(log f)\\|Z\\|^{2} + J\\nabla^{JX}X(log f)\\|Z\\|^{2}\n\\nonumber \\\\ & & - (X(log f))^{2}\\|Z\\|^{2} + \\bigg \\{ JX(JX(log f)) + 2(JX(log f))^{2} \\bigg \\}\\|Z\\|^{2}\n\\nonumber \\\\ & & + g(\\omega(X,Z), \\nabla_{JX}^{\\perp}JZ) - J\\nabla_{JX}X(log f)\\|Z\\|^{2}\n\\end{eqnarray}\nFrom(\\ref{w33}) and (\\ref{w52})\n\\begin{eqnarray*}\\label{w53}\n\\noindent& &\\nonumber - 2M(X,JX)g(Z,Z) - 2M(Z,JZ)g(X,X) \\hspace{4cm}\n\\nonumber \\\\ \\text{ } = & &\\bigg \\{ X(X(log f)) + 2(X(log f))^{2} \\bigg \\}\\|Z\\|^{2}\n\\nonumber \\\\ & & - g(\\omega(JX,Z), \\nabla_{X}^{\\perp}JZ) - \\nabla_{X}X(log f)\\|Z\\|^{2} - \\nabla_{JX}JX(log f)\\|Z\\|^{2}\n\\nonumber \\\\ & & - (X(log f))^{2}\\|Z\\|^{2} + \\bigg \\{ JX(JX(log f)) + 2(JX(log f))^{2} \\bigg \\}\\|Z\\|^{2}\n\\nonumber \\\\ & & + g(\\omega(X,Z), \\nabla_{JX}^{\\perp}JZ) - (JX(log f))^{2}\\|Z\\|^{2}\n\\end{eqnarray*}\nPutting $X = X_{i}$ and taking summation from 1 to $p$, we drive\n\\begin{eqnarray*}\\label{w54}\n\\noindent & &\\nonumber -2\\|Z\\|^{2}\\sum_{i=1}{p}M (X_{i}, JX_{i}) - 2M(Z, JZ)p\n\\nonumber \\\\ \\text{ } = & & \\sum_{i=1}^{p}\\bigg \\{ X_{i}(X_{i}(log f)) + JX_{i}(JX_{i}(log f)) - \\nabla_{X_{i}}X_{i}(log f)) - \\nabla_{JX_{i}}JX_{i}(log f)\\bigg \\}\\|Z\\|^{2}\n\\nonumber \\\\ & & + \\sum_{i=1}^{p}\\bigg \\{ (X_{i}(log f))^{2} + (JX_{i}(log f))^{2}\\bigg \\}\\|Z\\|^{2}\n\\nonumber \\\\ & & + \\sum_{i=1}{p}\\bigg[ g(\\omega(X_{i},Z), \\nabla_{JX_{i}}^{\\perp}JZ) - g(\\omega(JX_{i},Z), \\nabla_{X_{i}}^{\\perp}JZ)\\bigg]\\|Z\\|^{2}\n\\end{eqnarray*}\n from which we have\n\\begin{eqnarray*}\\label{w55}\n\\noindent \\nonumber & &-2\\|Z\\|^{2}\\sum_{i=1}{p} M (X_{i}, JX_{i}) - 2M(Z, JZ)p\n\\nonumber \\\\ & =& \\Delta_{D}(log f)\\|Z\\|^{2} + \\|grad_{D}(log f)\\|^{2}\\|Z\\|^{2}\n\\nonumber \\\\ & &+ \\sum_{i=1}{p}\\bigg[ g(\\omega(X_{i},Z), \\nabla_{JX_{i}}^{\\perp}JZ) - g(\\omega(JX_{i},Z), \\nabla_{X_{i}}^{\\perp}JZ)\\bigg]\\|Z\\|^{2}\n\\end{eqnarray*}\nUsing (\\ref{a6}) and (\\ref{a7}) in the last equation we arrive at\n\\begin{eqnarray*}\\label{w56}\n\\noindent \\nonumber \\frac{-1}{n+2}\\sum_{i=1}^{p}\\bigg[\\|Z\\|^{2}Ric(X_{i}, X_{i}) + \\|X_{i}\\|^{2} Ric(Z,Z)\\bigg] + \\frac{\\rho \\|X_{i}\\|^{2}\\|Z\\|^{2}}{2(n+1)(n+2)}\n\\nonumber \\\\ = \\Delta_{D}(log f)\\|Z\\|^{2} + \\|grad_{D}(log f)\\|^{2}\\|Z\\|^{2}\n\\nonumber \\\\+ \\sum_{i=1}^{p}\\bigg[ g(\\omega(X_{i},Z), \\nabla_{JX_{i}}^{\\perp}JZ) - g(\\omega(JX_{i},Z), \\nabla_{X_{i}}^{\\perp}JZ)\\bigg]\\|Z\\|^{2}\n\\end{eqnarray*}\nfrom which we have\n\\begin{eqnarray*}\\label{w58}\n\\noindent \\nonumber \\frac{-1}{n+2}\\bigg[\\|Z\\|^{2}\\sum_{i=1}^{p}Ric(X_{i}, X_{i}) + p Ric(Z,Z)\\bigg] + \\frac{\\rho p\\|Z\\|^{2}}{2(n+1)(n+2)}\n\\nonumber \\\\ = \\Delta_{D}(log f)\\|Z\\|^{2} + \\|grad_{D}(log f)\\|^{2}\\|Z\\|^{2}\n\\nonumber \\\\+ \\sum_{i=1}^{p}\\bigg[ g(B_{\\nabla_{JX_{i}}^{\\perp} JZ}X_{i}, Z) - g(B_{\\nabla_{X_{i}}^{\\perp} JZ}JX_{i}, Z)\\bigg]\\|Z\\|^{2}\n\\end{eqnarray*}\nSince by assumption, we have $B_{\\nabla_{JX_{i}}^{\\perp} JZ}X_{i} = B_{\\nabla_{X_{i}}^{\\perp} JZ}JX_{i} $, then (\\ref{w58}) becomes\n\\begin{eqnarray*}\\label{w59}\n\\noindent \\nonumber \\frac{-1}{n+2}\\bigg[\\sum_{i=1}^{p}Ric(X_{i}, X_{i}) +\\frac{p}{\\|Z\\|^{2}} Ric(Z,Z)\\bigg] + \\frac{p \\rho \\|Z\\|^{2}}{2(n+1)(n+2)}\n\\nonumber \\\\ = \\Delta_{D}(log f) + \\|grad_{D}(log f)\\|^{2}\n\\end{eqnarray*}\nIntegrating both sides and using Green's equation, the last equation simplifies to\n\\begin{eqnarray}\\label{w60}\n\\nonumber\\frac{-1}{n+2}\\int\\bigg[\\sum_{i=1}^{p}Ric(X_{i}, X_{i}) +\\frac{p}{\\|Z\\|^{2}} Ric(Z,Z)\\bigg]dv + \\int\\frac{p \\rho \\|Z\\|^{2}}{2(n+1)(n+2)}dv\n\\\\ = \\int \\|grad_{D}(log f)\\|^{2}dv\n\\end{eqnarray}\nSimilarly we have\n\\begin{eqnarray}\\label{w61}\n\\frac{-1}{n+2}\\int\\bigg[\\sum_{i=1}^{p}Ric(JX_{i}, JX_{i}) +\\frac{p}{\\|Z\\|^{2}} Ric(Z,Z)\\bigg]dv + \\int\\frac{p \\rho \\|Z\\|^{2}}{2(n+1)(n+2)}dv \\nonumber \\\\ = \\int \\|grad_{D}(log f)\\|^{2}dv\n\\end{eqnarray}\nAdding (\\ref{w60}) and (\\ref{w61}), we find\n\\begin{eqnarray*}\\label{w62}\n\\frac{-1}{n+2}\\int\\bigg[\\sum_{i=1}^{p}Ric(X_{i}, X_{i})+ \\sum_{i=1}^{p}Ric(JX_{i}, JX_{i}) +\\frac{2p}{\\|Z\\|^{2}} Ric(Z,Z)\\bigg]dv \\nonumber \\\\ + \\int\\frac{2p \\rho \\|Z\\|^{2}}{2(n+1)(n+2)} dv = 2\\int \\|grad_{D}(log f)\\|^{2}dv\n\\end{eqnarray*}\n from which we have\n\\begin{eqnarray*}\\label{w63}\n\\frac{-1}{n+2}\\int\\bigg[\\rho_{D} +\\frac{2p}{\\|Z\\|^{2}} Ric(Z,Z)\\bigg]dv + \\int\\frac{p \\rho \\|Z\\|^{2}}{(n+1)(n+2)} dv = 2\\int \\|grad_{D}(log f)\\|^{2} dv\n\\end{eqnarray*}\n where $\\rho _{D}$ is the scalar curvature of distribution $D$.\n Further replacing $Z$ by $Z_{\\alpha}$ and taking summation from $1$ to $q$ on both sides.\n As\n\\begin{eqnarray*}\\label{w64}\nq\\int \\|grad_{D}(log f)\\|^{2}dv \\geq 0\n\\end{eqnarray*}\nwe conclude that\n\\begin{eqnarray*}\\label{w65}\n\\frac{-1}{n+2}\\int\\bigg[q\\rho_{D} +2p\\rho_{D^{\\perp}}\\bigg]dv + \\int\\frac{pq^{2} \\rho}{(n+1)(n+2)} dv\\geq 0\n\\end{eqnarray*}\nThis shows that\n\\begin{eqnarray*}\\label{w66}\n\\frac{pq^{2}}{(n+1)(n+2)}\\int \\rho dv\\geq \\frac{1}{n+2}\\int\\bigg[q\\rho_{D} +2p\\rho_{D^{\\perp}}\\bigg]dv\n\\end{eqnarray*}\nor\n\\begin{eqnarray}\\label{w67}\n\\int \\rho dv \\geq (n+1)\\int\\bigg[\\frac{\\rho_{D}}{pq} +\\frac{2(n+1)}{q^{2}}\\rho_{D^{\\perp}}\\bigg]dv\n\\end{eqnarray}\nThus we have\n\\begin{eqnarray*}\\label{w68}\n\\int\\bigg[\\rho_{D} + \\rho_{D^{\\perp}}\\bigg]dv \\geq \\int \\bigg[\\frac{(n+1)}{pq}\\rho_{D} +\\frac{2(n+1)}{q^{2}} \\rho_{D^{\\perp}}\\bigg]dv\n\\end{eqnarray*}\nFrom we have the following observations.\nEither $(n+1) \\leq pq$ and $2(n+1) \\leq q^{2}$ or $\\rho_{D} \\leq 0$ and $\\rho_{D^{\\perp}} \\leq 0$ that id $\\rho = \\rho_{D} + \\rho_{D^{\\perp}} \\leq 0$. Equality holds if and only if either $(n+1) = pq$ and $2(n+1) = q^{2}$ or $grad_{D}(log f) = 0.$\n\\end{proof}\n\\hspace{6cm}{\\bf{References}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nSelective control with resonant light pulses is an emerging route for manipulating the properties of transition metal oxides. Selectivity has been achieved through resonant excitation of magnetic dipole modes in antiferromagnets~\\cite{Kampfrath2011} and multiferroics~\\cite{Kubacka2014}, or through IR-active phonon modes, in order to drive insulator-metal transitions~\\cite{Rini2007} and superconductivity~\\cite{Fausti2011, Mitrano2016}. \n\nUltrafast control of the magnetic state, in particular, could have a strong impact on magnetic recording technology. Ultrafast demagnetization is a complex process involving strong coupling between electronic, spin, and structural degrees of freedom which is dependent on the type of magnetic order and band structure~\\cite{Koopmans2010,Kirilyuk2010}. Controlling these interactions is key for developing magnetic devices that can fully exploit femto-magnetism. Antiferromagnetic (AFM) materials are insensitive to external magnetic fields~\\cite{Marti2014a} and are more stable when miniaturized ~\\cite{Loth2012} than ferromagnets. Moreover the absence of a net magnetic moment can enable much faster control of spin dynamics ~\\cite{Kimel2004}. \n\nDespite these possibilities, spin dynamics in AFM materials are still not well understood. Conventional techniques used to study the femtosecond dynamics of magnetic materials cannot be applied to AFM materials as they lack a net magnetic moment and alternative techniques are required. The advent of free electron lasers has enabled the study of spin dynamics through resonant elastic ~\\cite{Ehrke2011a, Forst2011, Tobey2012, Kubacka2014} or inelastic magnetic scattering~\\cite{Dean2016}. However, optical techniques such as linear magnetic birefringence~\\cite{Bossini2014, Bossini2015} or second harmonic generation (SHG)~\\cite{Fiebig2005} can also be used. \n\nDue to the small band gaps in many transition metal oxides, selective control has focused on the THz or mid-IR spectral region, which may reduce the speed at which the material can be driven. This is observed in the manganites, where it has been shown that the melting of antiferromagnetism via IR-active phonons is an order of magnitude slower and less efficient than when exciting above the band gap at 1.5\\,eV~\\cite{Forst2011}.\n\nCr$_2$O$_3$ is an ideal material to demonstrate ultrafast control of demagnetization at higher speeds. The band-gap is in the near UV, enabling optical control over a broader range of wavelengths. The magnetic state has been well characterized by SHG spectroscopy~\\cite{Fiebig1994, Fiebig2005}, thus experiments can be performed with a regular laboratory setup and the static optical and electronic properties of the material are significantly better understood than those of other correlated materials, enabling an greater chance for understanding the dynamics induced by the laser pulse.\n\nIn this paper, we show that electronic visible photoexcitation can be used to selectively control the structural and magnetic properties of Cr$_2$O$_3$. By measuring the time- and polarization-resolved second harmonic signal, we show that different electronic in-gap states couple to different phonon modes. As these modes are responsible for the spin scattering process, we can control the demagnetization rate by as much as 25\\% by changing the photon energy used to excite the system.\n\n\\section{Sample Characterization and Experimental Details}\n\nA single crystal of Cr$_2$O$_3$, cut perpendicular to the trigonal $z$ axis, was grown using the flux method and was polished to a thickness of approximately 10\\,$\\mu$m and, unless otherwise stated, cooled to 77\\,K. The measured sample absorbance is shown in Fig~\\ref{fig:Static}a. Due to the small sample size and thickness it was not possible to measure attenuations greater than 2 orders of magnitude due to the dominance of scattered light around the sample edges. Optially, Cr$_2$O$_3$ can be considered a 100\\% doped ruby crystal. The absorbance shows two broad peaks ($^4T_1$ and $^4T_2$), corresponding to crystal field excitations within the 3$d$ levels of the Cr ion which preserve the spin orientation of the photo-excited electron. The resonances are broadened due to a strong electron phonon coupling. In addition, a low energy narrow resonance ($^2E$) is observed which corresponds to a dipole forbidden transition in which the spin is flipped but the spatial part of the wavefunction is the same as the ground state~\\cite{Muto1998}. The high contrast between the transmission of light on and between the resonances and the narrow Raman spectra (Fig~\\ref{fig:Static}d) indicate that the sample is of high quality. In the ruby laser, the $^4T_1$ and $^4T_2$ levels are optically pumped and the excited electrons decay non-radiatively to the $^2E$ state on sub picosecond timescales~\\cite{Fonger1975}. As this process results in a spin flip, the lifetime of the excited state is very long and is used for lasing. Cr$_2$O$_3$, unlike Ruby, is anti-ferromagnetic, so the spin flip process can lead to rapid demagnetization. In this paper we track this process using time-resolved SHG. \n\n\\begin{figure}\n\\centering\n\\includegraphics{figure1.pdf}\n\\caption{(a) The absorbance of Cr$_2$O$_3$. Black markers correspond to measured values of our 10 $\\mu$m thick sample at 77\\,K. Blue line is a guide to the eye based on the measurements found in Ref~\\cite{McClure1963}. The $^4T_1$ and $^4T_2$ states correspond to transitions within crystal field split states of the Cr 3$d$ levels which preserve the spin of the electron. The $^2E$ state corresponds to a spin-forbidden transition. The colored arrows indicate the pump wavelengths used in the time resolved experiments and the black dashed arrow corresponds to the second harmonic of the probe. (b) Polarization and temperature dependence of the SHG signal from Cr$_2$O$_3$ measured with a fundamental photon energy of 1.08\\,eV. Solid lines are fits to the data using Eq.~\\ref{eq:angle}. (c) Temperature dependence of the antiferromagnetic order parameter $l(T)$ extracted from the fits from (b) giving $T_\\mathrm{N} = 314\\pm5$\\,K. (d) Raman spectrum for two different excitation wavelengths, demonstrating coupling to $E_g$ modes is stronger for longer wavelengths at room temperature. \n\\label{fig:Static}}\n\\end{figure}\n\nCr$_2$O$_3$ is centrosymmetric with the crystallographic point group $\\bar{3}m$. The presence of inversion symmetry forbids conventional SHG of the electric field, $E$; however an axial source term $\\bm M \\propto \\bm{\\chi_m^{(2)}}\\bm{EE}$ is allowed. Below the N\\'eel temperature, $T_\\mathrm{N}= 308$\\,K, the Cr ions order antiferromagnetically along the trigonal z-axis, enabling an additional polar source term $\\bm P \\propto\\bm{\\chi_e^{(2)}}\\bm{EE}$. Both $\\bm{\\chi_e^{(2)}}$ and $\\bm{\\chi_m^{(2)}}$ satisfy the symmetry relations of the point group 32: $\\chi_{xxx}^{(2)}=-\\chi_{xyy}^{(2)}=-\\chi_{yxy}^{(2)}=-\\chi_{e\/m}^{(2)}$~\\cite{Boyd2003}. For light traveling along the trigonal $z$ axis, this gives rise to a SHG scattering pattern that depends on the linear polarization in the $(x, y)$ plane as\n\\begin{equation}\nI_{\\mathrm{SHG}}(\\theta) \\propto |\\chi_e^{(2)}\\sin(3\\theta)-\\chi_m^{(2)}\\cos(3\\theta) |^2 I_\\mathrm{f}^2,\n\\label{eq:angle}\n\\end{equation}\nwhere $I_\\mathrm{f}$ is the intensity of the fundamental beam and $\\theta$ is the polarization angle relative to the crystallographic $y$-axis. Furthermore, the non-linear susceptibilities can be expanded as a function of the antiferromagnetic order parameter, $l$, as $\\chi_e^{(2)} = c_1 l(T)$ and $\\chi_m^{(2)} = c_2 + c_3 l^2(T)$~\\cite{Muto1998, Tanabe1998, Muthukumar1996}. \n\nThe $c_i$ coefficients are temperature independent, but depend on the photon energy of the light used to probe the sample; resonant enhancement occurs for photon energies around 1.08\\,eV due to the crystal field effects in the final state~\\cite{Tanabe1998, Fiebig1994}. Thus resonant SHG is both sensitive to the magnetic order and the crystallographic structure surrounding the Cr$^{3+}$ ions. \n\nThe temperature dependence of the static SHG signal was measured with the output of an optical parametric amplifier (OPA), pumped by the output of a Ti:sapphire laser at a repetition rate of 5 kHz and tuned to the 1.08 eV (1150\\,nm) resonance. The pulse duration was approximately 60 fs. The SHG signal was measured in transmission with a high and low pass filter to separate out the 2.16\\, eV (575\\,nm) SHG signal from the fundamental and third harmonic light. A polarizer and half-waveplate set the linear polarization of the probe beam before the sample and a second polarizer after the sample selects the SHG signal at a polarization parallel to the incident beam. These were rotated to measure the polarization dependence of the SHG for the different temperatures shown in Fig.~\\ref{fig:Static}b. From this data, the complex $c_i$ coefficients and the temperature-dependent order parameter, $l$, (Figs.~\\ref{fig:Static}c) were determined, in good agreement with the literature~\\cite{Fiebig1994, Fiebig2005}. \n\nFor the time resolved experiments, a second OPA generated the pump pulse with central wavelengths at 3\\,eV (400\\,nm), 2.5\\,eV (500\\,nm), and 1.8\\,eV (700\\,nm) as indicated in Fig.~\\ref{fig:Static}a, with pulse durations of order 50\\,fs. These pump wavelengths are chosen in order to populate different excited states. 2.5\\,eV and 1.8\\,eV photons predominately populate the $d$ levels of the Cr ion, while the higher energy 3\\,eV excitation also causes charge transfer excitation between O and Cr ions. Furthermore, the photon energies are detuned from the peak absorption to ensure a greater penetration depth of the pump light. \n\nThe pump is set in a counter-propagating configuration, relative to the probe, exciting the sample from the backside and at a small angle as shown in the insert of Fig~\\ref{fig:dynamic}a. This interaction geometry is chosen because the SHG of the probe is absorbed in the crystal and thus only the SHG generated at the back surface leaves the crystal. Due to the large attenuation at 2.1\\,eV compared to the pump wavelengths chosen, the probed volume is uniformly excited. \n\n\\section{Results}\n\\begin{figure}\n\\centering\n\\includegraphics{figure2.pdf}\n\\caption{(a) Visible-pump SHG-probe signal at the three different pump photon energies, 3.0\\,eV (blue circles), 2.5\\,eV (green triangles) and 1.8\\,eV (red squares). Dashed lines correspond to the fits to Eq.~\\ref{eq:fit}. Arrows indicate the times at which the polarization dependence shown in Fig.~\\ref{fig:pol} were performed. The insert shows a schematic of the counter-propagating pump-probe setup (P - Polarizer, WP - Wave Plate, IF - Interference filters). (b) Fluence dependence of the SHG transient signal at 3\\,eV from 0.25-2 mJ\\,cm$^{-2}$ showing linear behavior. Dashed lines correspond to fits using Eq.~\\ref{eq:fit}. (c) Fit parameters of the data in (b) showing that both the amplitude of the peak and plateau signal vary linearly with fluence. The $\\tau_\\mathrm{pm}$ time-constant is independent of fluence. \\label{fig:dynamic}}\n\\end{figure}\n\nFigure~\\ref{fig:dynamic} shows the time-resolved change in the SHG signal ${\\Delta I}\/{I_0}$ for the three pump photon energies. In the traces shown, the pump, probe and SHG polarizations were all parallel to the crystallographic $x$-axis, however, other combinations were also measured. For all pump photon energies, a decrease in the SHG signal is measured. At delays $> 1$\\,ps, all three signals settle to a plateau that lasts for more than 400\\,ps. However, at shorter delays ($< 1$\\,ps) additional fast dynamics are observed, which strongly depend on the pump photon energy. Due to the differences in the absorption coefficient for each pump pulse, the pump fluences were controlled in order to give roughly the same decrease in the SHG signal at 2\\,ps. However, in all cases the transient signal was found to vary linearly with the pump fluence for all time delays as demonstrated in Figure~\\ref{fig:dynamic}b for the case of 3\\,eV excitation. Thus the different dynamics observed at early delays are not due to differences in the absorption and the response measured at 1.8\\,eV excitation cannot be recreated by changing the intensity of a 3\\,eV pump. \n\nThe dynamics can be fit well by \n\\begin{equation}\n\t\\Delta I(t)\/I_0 = \\Theta(t)[A_\\mathrm{p} e^{-t\/\\tau_\\mathrm{pm}} + A_m(1-e^{-t\/\\tau_\\mathrm{pm}})],\n\\label{eq:fit}\n\\end{equation}\nwhere $\\Theta(t)$ is the error function with a 75\\,fs rise, fixed for all wavelengths, $A_\\mathrm{p}$ and $A_\\mathrm{m}$ are the magnitudes of changes at the spike at time zero and plateau respectively, and $\\tau_\\mathrm{pm}$ is the rapid recovery time constant, which was found to vary from $300\\pm 20$\\,fs at 3.0\\,eV pumping to $400\\pm 50$\\,fs at 1.8\\,eV pumping. Figure~\\ref{fig:dynamic}c shows that the time constant obtained and the ratio between the peak and plateau changes is independent of fluence, again demonstrating the linearity of the signal in this regime. We note that similar transients were observed when measuring the probe transmission at the fundamental photon energy (data not shown). However, we can exclude pump-induced changes in the fundamental intensity as the origin of the second harmonic dynamics because this would give rise to a signal that scales with the square of the pump fluence. As a result, transient changes in the linear optical properties have a negligible effect on the SHG transients.\n\n\\begin{figure}\n\\centering\n\\includegraphics{figure3.pdf}\n\\caption{(a) Change in the polarization dependence of the SHG probe at 0\\,ps for two perpendicular pump polarizations ($x$, $y$). Lighter points are not measured and reconstructed from the symmetry of the signal. Dashed lines in the 3.0\\,eV ($A_{1g}$) and 1.8\\,eV ($E_g$) plots are fits for the $c_i$ coefficients at constant magnetization. (b) Change in the probe polarization dependence at 2 ps and corresponding fit to the scattering pattern when the AFM order parameter, $l$, is reduced by 0.75\\%. \\label{fig:pol}}\n\\end{figure}\n\nIn equilibrium, the intensity of the SHG signal can be directly related to the magnetic order. However, out of equilibrium this does not have to be the case~\\cite{Huber2015}. In order to assign an origin to the observed dynamics we measured the polarization dependence of the transient SHG signal at the peak of the change and at long times after pumping, as shown in Fig.~\\ref{fig:pol}.\n\nWhen the system is in the long lived state, the SHG signal shows the same angular dependence irrespective of the pump photon energy or polarization. In this case the data can be accurately fitted by {\\em only} reducing the antiferromagnetic order parameter, $l$, in Eq.~\\ref{eq:angle}. Fig. ~\\ref{fig:pol}b demonstrates the quality of the fit for $\\Delta l = 0.75$\\,\\%. Thus the long-lived quenching of the SHG signal results from demagnetization. However, the angular dependence at short delays (Fig. ~\\ref{fig:pol}a) shows a different behavior that strongly depends on the pump photon energy and polarization. This pattern cannot be fitted by changing $\\Delta l$. Excitation at 3\\,eV produces an isotropic change, while lowering the photon energy to 1.8\\,eV breaks the symmetry of the SHG signal. Furthermore, the direction of the asymmetry can be controlled by the polarization of the pump. In this case it is clear that the SHG dynamics near time zero are not due to the spin system alone. \n\n\\section{Discussion}\n\\begin{figure}\n\\centering\n\\includegraphics{figure4.pdf}\n\\caption{(a) Sketch of the Cr$_2$O$_3$ unit cell in the $xy$-plane showing the $A_{1g}$ and (b) $E_g$ displacements of the oxygen ions around a Cr ion (darker colors correspond to atoms at different $z$ positions). (c) DFT calculations for changes in the magnitude of the nonlinear susceptibility for $A_{1g}$ and $E_g$ displacements of {0.05\\,\\AA}. The shaded area corresponds to the measured region. The dashed line shows the calculated equilibrium spectrum.\\label{fig:theory}}\n\\end{figure}\n\nAs the excited states that are pumped are broadened due to electron-phonon coupling, we consider how crystal distortions can change the SHG signal, both in terms of amplitude and symmetry. Cr$_2$O$_3$ possesses $A_{1g}$ and $E_g$ Raman active modes, the motions of which are shown in Figs.~\\ref{fig:theory}a and~\\ref{fig:theory}b. $A_{1g}$ modes preserve the symmetry of the crystal and thus can only modify the efficiency of the SHG process through a crystal-field-induced shift in the resonance condition and should not change the polarization dependence. For small displacements the $c_i$ coefficients change as as $c_i\\rightarrow c_i (1+\\delta Q_{A_{1g}})$, where $Q_{A_{1g}}$ is the phonon amplitude and $\\delta$ is the coupling constant. Such a change can fit the polarization dependence measured for excitation at 3\\,eV as shown in Fig.~\\ref{fig:pol}a when the magnetization is kept constant and $\\delta Q_{A_{1g}} = -0.05$.\n\n$E_g$ displacements lower the crystallographic point group symmetry from $\\bar{3}m$ to $2\/m$, thus they can also change the polarization dependence as well as efficiency. As the $E_g$ mode is doubly degenerate, the direction along which the distortion occurs can be varied in the plane. Figure~\\ref{fig:theory}b depicts the case for oxygen displacements along the $y$-direction. In this case the three non-zero components of the susceptibility become independent, which can be captured by $c_i^{jkl}\\rightarrow c_i (1+\\delta^{jkl}Q_{E_g})$, where $jkl= xxx,xyy,yxy$ are the indexes of the tensor elements. Thus a minimum of three parameters are needed to describe the effect. The resulting change can fit the 1.8\\,eV experimental data, with $c_i^{xxx}=0.9881c_i ~(\\delta^{xxx}Q_{E_g}=-0.0119), c_i^{xyy}=1.0005c_i ~(\\delta^{xyy}Q_{E_g}=0.0005)$ and $c_i^{xyx}=0.9915\\,c_i ~ (\\delta^{xyx}Q_{E_g}=-0.0085)$ as shown in Fig.~\\ref{fig:pol}a. The perpendicular scattering pattern can be achieved when the symmetry breaking axis is rotated by 90$^\\circ$. In this case the original non-zero components of the susceptibility remain unchanged and the corresponding components along the $y$ axis become non-zero. The data for pumping at 2.5\\,eV can be described by a combination of the $A_{1g}$ and $E_g$ responses. \n\nThese symmetry considerations are confirmed by calculations of the SHG signal under the distorted crystal structures obtained from density functional theory (DFT). We obtained the electronic structure for the ground state of Cr$_2$O$_3$ using the Abinit code, based on plane waves and pseudopotentials~\\cite{Gonze2005,Gonze2009}. The antiferromagnetic state was taken into account in the calculations. Then the second order susceptibility was computed in Time-Dependent DFT using the 2light code based on the formalism developed in Ref.~\\cite{Luppi2010}. The second order susceptibility is split into two parts, associated with the spin-up and spin-down components. Spin-orbit coupling is not included, as it is far beyond the reach of nonlinear ab initio calculations. Therefore, as expected, see for instance Ref.~\\cite{Muto1998}, both components cancel. However, trends can be obtained for the effects of the crystal distortion by looking at one component only. The second order spin-up susceptibility for the distorted structures was obtained by moving the oxygen atoms along the $A_{1g}$ and $E_g$ modes at the level of the DFT calculations. The spectra were calculated in the independent particle approximation. Convergence was achieved with 432 off-symmetry shifted $k$ points in the full Brillouin zone and 90 unoccupied states. \n\nThe results are shown in Fig.~\\ref{fig:theory}c. The $A_{1g}$ displacement induces an equal decrease of all elements, whereas the $E_g$ distortion breaks the degeneracy of the elements, as expected from the above symmetry analysis. Furthermore, these calculations enable us to estimate the size of the induced displacements from the magnitude of our signal. The calculations were performed with a displacement of {0.05\\,\\AA} and induce changes that are larger than those measured experimentally, thus resonant SHG is extremely sensitive to highly symmetric atomic motion.\n\n\\begin{figure}\n\\centering\n\\includegraphics{figure5.pdf}\n\\caption{Part of the unit cell centered on the Cr ion and nearest oxygen neighbors. Bold arrows correspond to the spin state. Photoexcitation at 1.8\\,eV excites electrons from the $t_{2g}$ levels into the $e_g$ levels while preserving the spin state. The resulting charge redistribution is anisotropic (blue corresponds to more positively charged regions, red more negative) and couples strongly to the anisotropic $E_g$ phonon mode. The electrons are then scattered to a spin-flipped $t_{2g}$ state in 400\\,fs. Thus the distortion is relaxed and the system is demagnetized. Excitation at 3\\,eV causes charge transfer from O to Cr. This causes a symmetric charge redistribution mapping onto the $A_{1g}$ phonon mode and scattering occurring within 300\\,fs. \\label{fig:story}}\n\\end{figure}\n\nThese results enable us to build a picture for the demagnetization process, which is sketched in Fig.~\\ref{fig:story}. The pump pulse excites electrons from the occupied $^4A_2$ ground state which is composed of the $t_{2g}$ levels of the Cr ion, into final states that depend on the photon energy. Lower energy photons transfer electrons into the unoccupied T states, which mix the $3d-e_g$ and $t_{2g}$ levels of the Cr ion~\\cite{Muto1998}, changing the charge distribution in the vicinity of the Cr ion. As these orbitals are anisotropic, neighboring ions experience a different force, causing oxygen atomic motion that is along the $E_g$ phonon coordinate. Higher energy photons trigger transitions that are more of a charge transfer character, i.e. charge is redistributed between chrome and oxygen ions. In the limit that the charge transfer is complete, this produces a symmetric force on the oxygen ions which couples to $A_{1g}$ motion. The preferential coupling to the $A_{1g}$ mode for high photon energies is also confirmed by resonant Raman scattering measurements shown in Fig~\\ref{fig:Static}d. Furthermore, the timescale for this processes is in good agreement for the equivalent process that occurs in ruby~\\cite{Fonger1975}.\n\nAs both Raman modes have frequencies in the region 9 - 18\\,THz, they respond adiabatically during the excitation pulse and coherent motion of the lattice is not triggered. In the distorted state, the excited electrons in both cases can be efficiently scattered into the spin-flipped $^2E$ states of Fig.~\\ref{fig:Static}a. These states have the same spatial distribution of charge as the ground state, but the electron's spin is flipped~\\cite{Muto1998}. Therefore, the force causing the displacements is lost and the crystal relaxes, while at the same time the magnetization is reduced. Once relaxed, the equilibrium spin-lattice relaxation process dominates, which is much slower. These results differ from previous measurements of spin dynamics in Cr$_2$O$_3$~\\cite{Satoh2007, Dodge1999}, since we operate at lower temperatures and lower excitation fluences. Higher fluences and temperatures may enable different scattering processes. However, the 300-400\\,fs timescales obtained for Cr$_2$O$_3$ are very similar to the timescale observed in antiferromagnetic order melting in insulating Sr$_2$IrO$_4$ measured using time-resolved resonant diffraction~\\cite{Dean2016}, thus demonstrating that rapid control of the magnetic state is possible in antiferromagnetic insulators when exciting above the band gap. \n\n\\section{conclusion}\nIn summary, we have shown how specific lattice modes dictate the demagnetization rate in Cr$_2$O$_3$ and that the lattice modes can be controlled by resonantly tuning the light pulse in the visible spectral region, enabling a modulation of the demagnetization rate by 25\\%. This control is achieved through the presence of in-gap states that occur within the band gap of insulating materials. Through the control of the crystallographic structure, presented here, or through direct excitation of the magnetic subsystem with below gap excitation~\\cite{Bossini2014}, insulating antiferromagnets present new opportunities for light control that are not available in ferromagnetic metals. As the coupling between charge excitations and Raman active modes is direct and occurs in the optical regime, it also has the potential to be more efficient and easier to implement that selective methods that require non-linear coupling between IR and Raman active modes~\\cite{Forst2011a}.\n\nFurthermore, we have shown that, out of equilibrium, time resolved SHG is a powerful tool for monitoring {\\em both} magnetic and structural degrees of freedom and is capable of measuring small, symmetric atomic displacements. When combined with theory, the ability to observe small changes in the lattice simultaneously with the evolution of the magnetic degree of freedom, enables measurements in a standard laboratory which would otherwise require large scale user facilities. \n\n\n\\begin{acknowledgments}\n The research leading to these results has received funding from LASERLAB-EUROPE (grant agreement no. 284464, EC's Seventh Framework Program). We acknowledge GENCI (project 544) for the computational support provided. SW received financial support from Ramon y Cajal program RYC-2013-14838 and Marie Curie Career Integration Grant PCIG12-GA-2013-618487. VGS, TAM and SW acknowledge support from Severo Ochoa Excellence Grant and Fundaci\\`{o} Privada Cellex. GC acknowledges support by the European Union's Seventh Framework Program (FP7\/2007-2013) Grant No. CNECT-ICT-604391 (Graphene Flagship).\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn models of supergravity (SUGRA)~\\cite{Ref:SUGRA,WessBagger}, superweakly interacting massive fields are generally involved, such as the gravitino and modulus fields. They often cause cosmological problems, such as the moduli or Polonyi problem~\\cite{Coughlan:1983ci,Ref:Moduli}, and gravitino problems~\\cite{Ref:Gravitino,Kawasaki:2008qe}. If they are generated in an early stage of the universe, they easily spoil big bang nucleosynthesis (BBN) when they decay, or tend to overclose the universe if stable. Hence, their abundances are severely restricted. Among the SUGRA particles, gravitinos are thermally produced in hot plasma. Moreover, it has been pointed out that a large amount of gravitinos are generated via decays of \ncoherent oscillations of\nheavy scalars, such as moduli and inflatons~\\cite{MIGP,Kawasaki:2006gs,Asaka:2006bv,Endo:2007sz}. Therefore, the cosmology constrains the reheating temperature and models of the scalars.\n\nThe partial decay rate of a scalar into a pair of gravitinos depends on the structures of the scalar and supersymmetry (SUSY)-breaking sectors. The production rate has been studied in detail by Ref.~\\cite{Endo:2006tf} when the SUSY breaking is caused by a vacuum expectation value (VEV) of the F term of the SUSY breaking field. In this case, the goldstino, which is the longitudinal component of the gravitino, is composed of the spinor component of the SUSY-breaking chiral supermultiplet. \nIn the literature which studies the gravitino production, D-term contributions to the scalar potential have been discarded. If the SUSY breaking is dominated by a D-term VEV, the goldstino mainly consists of a gaugino, and the above result can significantly change. \nIn this paper, direct gravitino production by scalar decay will be studied when the D-term VEV is sizable. \n\n\nThis paper is organized as follows. In Sec.~\\ref{sec:general}, the production rate of a pair of gravitinos will be provided, taking the D-term potential into account. Scalar mixing with the fields in the SUSY-breaking sector will be investigated in the SUGRA framework. In Sec.~\\ref{sec:example}, the result will be applied to a D-term SUSY-breaking model, and its cosmological implications will be discussed. Sec.~\\ref{sec:conclusion} is devoted to a summary and discussions.\n\n\\section{Direct Gravitino Production Rate}\\label{sec:general}\n\nIn this section, we evaluate the partial decay rate of a scalar field such as the modulus or inflaton field into a pair of gravitinos. It is derived at the leading order of Planck-suppressed interactions in the framework of SUGRA.\nBoth the D-term and F-term SUSY-breaking contributions are taken into account,\nand sources of the D-term contributions are clarified.\n\n\n\n\\subsection{Gravitino Production Rate in the Mass-Eigenstate Basis}\n\nThe SUSY-breaking sector is supposed to have nonzero VEVs of the D-term potentials for gauge symmetries which are not included in the Standard Model (SM) gauge groups. The gauge symmetries can be $U(1)$ and\/or non-Abelian symmetries. \nIf the symmetry is $U(1)$, we assume that there is no genuine Fayet-Iliopoulos term, since it is difficult to embed the Fayet-Iliopoulos term in SUGRA~\\cite{FI-SUGRA}.\nThe VEVs of the D-term potentials are considered to be dynamically generated by the fields in the SUSY-breaking sector, $z^i$, which are charged under the gauge symmetries. See Sec.~\\ref{sec:example} for an example of such a model.\n\nIn gravitino production, the relevant fields are the scalar field, $\\phi$, the SUSY-breaking fields, $z^i$, and the gravitino. \nIt is assumed that the scalar $\\phi$ is much heavier than the gravitino due to the SUSY-invariant mass.\\footnote{This is the case for the inflaton in many inflation models. In the case of modulus, if the modulus mass is $m_\\phi < {\\cal O}(10 \\rm{TeV})$, its decay occurs after BBN starts and spoils the success of the standard cosmology. Thus, we assume $m_\\phi \\gg 10 \\rm{TeV} \\gtrsim m_{3\/2}$.}\nThe scalar is also assumed to be a singlet under the extra gauge symmetry, \nand couples with $z^i$ only through the terms suppressed by powers of the (reduced) Planck scale, $M_P = 2.4 \\times 10^{18} \\rm{GeV}$, which generally exist in the SUGRA Lagrangian.\\footnote{Otherwise, $\\phi$ decays into the SUSY-breaking sector much faster, which worsens the cosmological gravitino problem.}\n\nThe superpotential, K\\\"ahler potential and gauge kinetic function are represented as\\footnote{In this paper, we follow the conventions of Ref.~\\cite{WessBagger}. \nDerivatives with respect to fields are denoted by subscripts, e.g., $G_{\\phi}=\\partial G\/\\partial \\phi$.\nAlso, we omit symbols of VEV, $\\vev{\\cdots}$, if not otherwise specified.\n}\n\\begin{align}\nK &= K(\\phi ,\\bar{\\phi})+K(z^{i} ,\\bar{z}^{i}) \n+ \\sum_{n\\ge 1}\\frac{1}{M_P^n}K_{\\rm mix}^{(n)}(\\phi, \\bar{\\phi}, z^i, \\bar{z}^i), \\nonumber\\\\ \nW &= W(\\phi) + W(z^i) + \\sum_{n\\ge 1}\\frac{1}{M_P^n}W_{\\rm mix}^{(n)}(\\phi, z^i), \\nonumber\\\\\nh_{AB}&=\\left( 1+\\frac{\\theta g^{2}}{8 \\pi i} \\right) \\delta_{AB} \n+\\sum_{n\\ge 1}\\frac{1}{M_{P}^{n}}h_{AB}^{(n)}(\\phi, z^{i}),\n\\label{eq:model-base}\n\\end{align}\nwhere $K_{\\rm mix}^{(n)}$ and $W_{\\rm mix}^{(n)}$ are the interaction terms between $\\phi$ and $z^i$ whose Planck suppressions are displayed explicitly. The first term in the right-hand side of $h_{AB}$ corresponds to the gauge kinetic and $\\theta$ terms, and $g$ is the gauge coupling constant of the extra gauge symmetry.\nLet us call the basis in Eq.~\\eqref{eq:model-base} the ``model basis'' in this paper. \nNote that the VEVs of $\\phi$ and $z^i$ can induce kinetic mixings through higher-dimensional terms in the K\\\"ahler potential. In the following, we assume that those VEVs are much smaller than the Planck scale. Otherwise, the following discussion applies after the fields are shifted to absorb the large VEVs.\n\nThe decay rate is evaluated in the mass-eigenstate basis. The fields in the mass-eigenstate basis, $X^{a}$, are related to those in the model basis, $x^{\\alpha}$, as\n\\begin{eqnarray}\nX^{a}=A^{X^{a}}{}_{x^{\\alpha}} \\delta x^{\\alpha},\n\\end{eqnarray}\nwhere $\\delta x^{\\alpha}= x^{\\alpha}-\\vev{x^{\\alpha}}$ is the fluctuation around the VEV, and $A$ is a matrix to diagonalize the mass matrix and to canonicalize the kinetic terms at the potential minimum. The model basis fields, $x^{\\alpha}$, consist of $x^{\\alpha}=\\varphi^{\\alpha}$ and its Hermitian conjugate $\\bar{\\varphi}^{\\alpha}$, with $\\varphi^{\\alpha}=\\phi$ and $z^{i}$. \nOn the other hand, the mass eigenstates are generally represented by\nreal fields, $X^{a}=\\Phi_{R}, \\Phi_{I}, Z_{R}^{i}$, and $Z_{I}^{i}$, which primarily consist of the \nreal and imaginary parts of $\\phi$ and $z^i$, respectively.\nHence, $(A^{X^{a}}{}_{\\varphi^{\\alpha}})^* = A^{X^{a}}{}_{\\bar{\\varphi}^{\\alpha}}$ is satisfied.\nSince the interactions between $\\phi$ and $z^i$ are given by higher-dimensional operators,\n$A^{\\Phi_{R,I}}{}_{z^{i}}, A^{\\Phi_{R,I}}{}_{\\bar{z}^{i}}, A^{Z^{i}_{R,I}}{}_{\\phi}$, and $A^{Z^{i}_{R,I}}{}_{\\bar{\\phi}}$ are suppressed by the Planck scale.\nThe matrix $A$ will be evaluated later.\n\n\n The Lagrangian terms which are relevant for the tree-level decay of the scalar into a pair of gravitinos are found to be~\\cite{WessBagger}\n \\begin{align}\n\\mathcal{L}=& \\frac{1}{8i}\\epsilon ^{\\mu \\nu \\rho \\sigma}( G_{\\varphi^{\\alpha}} \\partial_{\\rho} \\delta \\varphi^{\\alpha} - G_{\\bar{\\varphi}^{\\alpha}}\\partial_{\\rho}\\delta \\bar{\\varphi}^{\\alpha} )\\bar{\\psi}_{\\mu}\\gamma_{\\nu}\\psi_{\\sigma} \\nonumber \\\\\n& +\\frac{1}{4}m_{3\/2} ( G_{\\varphi^{\\alpha}} \\delta \\varphi^{\\alpha} + G_{\\bar{\\varphi}^{\\alpha}}\\delta \\bar{\\varphi}^{\\alpha} ) \\bar{\\psi}_{\\mu} \\gamma ^{\\mu \\nu} \\psi_{\\nu}, \\label{Lm22g}\n\\end{align}\nwhere $\\varphi^{\\alpha}$ stands for $\\phi$ and $z^{i}$ in the model basis, while $\\psi_{\\mu}$ is the gravitino, $m_{3\/2}$ the gravitino mass, and $G=K+\\ln |W|^{2} $ the total K\\\"{a}hler potential.\nHere and hereafter, the Planck unit, $M_{P} = 1$, is used if not otherwise specified.\nThis Lagrangian is the same as that in the F-term SUSY-breaking case~\\cite{MIGP}.\nIn terms of the mass-eigenstate basis, the above Lagrangian becomes \n\\begin{align}\n\\mathcal{L}=& \\frac{1}{8i}\\epsilon ^{\\mu \\nu \\rho \\sigma} \\mathcal{G}^{\\text{(1)}}_{\\Phi_{R}} \\partial_{\\rho} \\Phi_{R} \\bar{\\psi}_{\\mu}\\gamma_{\\nu}\\psi_{\\sigma} +\\frac{1}{4}m_{3\/2} \\mathcal{G}^{\\text{(2)}}_{\\Phi_{R}} \\Phi_{R} \\bar{\\psi}_{\\mu} \\gamma ^{\\mu \\nu} \\psi_{\\nu} + (R \\rightarrow I) + \\dots,\n\\label{eq:g12RI}\n\\end{align}\nwhere the coefficients are\n\\begin{align}\n\\mathcal{G}^{\\text{(1)}}_{\\Phi_{R,I}} =2i\\, \\Im ( G_{\\varphi^{\\alpha}}(A^{-1})^{\\varphi^{\\alpha}}{}_{\\Phi_{R,I}} ),~~~\n\\mathcal{G}^{\\text{(2)}}_{\\Phi_{R,I}} =2\\, \\Re ( G_{\\varphi^{\\alpha}}(A^{-1})^{\\varphi^{\\alpha}}{}_{\\Phi_{R,I}} ),\n\\end{align}\nwhere $A^{-1}$ is the inverse matrix of $A$, and \n $\\Re(\\cdots)$ and $\\Im(\\cdots)$ represent the real and imaginary parts, respectively.\nThe omitted terms in Eq.~(\\ref{eq:g12RI}) include $Z_{R,I}^{i}$, which are irrelevant for the gravitino production by the $\\Phi_{R,I}$ decay. Note that the gravitino production rate will be derived at the tree level in this section. The rate can receive radiative corrections, which will be mentioned later.\n\nFrom the above interactions, the partial decay rate of the scalar into a pair of gravitinos is evaluated as~\\cite{MIGP}\n\\begin{align}\n\\Gamma(\\Phi_{R,I}\\to \\psi_{3\/2}\\psi_{3\/2}) =\n\\frac{|\\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R,I}}|^{2}m_{\\Phi_{R,I}}^{5}}{288 \\pi m^{2}_{3\/2 }}, \n\\label{gravitinorate}\n\\end{align}\nwhere we have used $m_{\\Phi_{R,I}}\\gg m_{3\/2}$.\nThe effective coupling constants are defined as\n\\begin{eqnarray}\n\\left | \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R,I}} \\right |^{2}\n=\\frac{1}{2}\\left( \\left| \\mathcal{G}^{\\text{(1)}}_{\\Phi_{R,I}} \\right|^{2} + \\left| \\mathcal{G}^{\\text{(2)}}_{\\Phi_{R,I}} \\right|^{2} \\right) \n=2\\left| \\sum_{\\varphi^{\\alpha}=\\phi,z^i} G_{\\varphi^{\\alpha}} (A^{-1}) ^{\\varphi^{\\alpha}}{}_{ \\Phi_{R,I}} \\right|^{2}.\\label{Geffective}\n\\end{eqnarray}\nThe rate is apparently the same as that in the F-term SUSY-breaking models~\\cite{MIGP} because the relevant Lagrangian [Eq.~\\eqref{Lm22g}] is the same. The effective coupling constants [Eq.~\\eqref{Geffective}] are governed by $G_{\\varphi^{\\alpha}}$, which is related to the F term as \n\\begin{align}\nF^i = -e^{G\/2} g^{i \\bar j} G_{\\bar j},\n\\end{align}\nwhere $g^{\\bar{i}j}$ is the inverse of the K\\\"{a}hler metric, $g_{i\\bar{j}}=\\vev{G_{i\\bar{j}}}$.\nHowever, this does not mean that the D term is unimportant.\nAs we will see, the magnitudes of F terms are controlled by vacuum conditions,\nwhich are affected by the D-term potential.\nMoreover, the D term contributes to the scalar mass matrix, \nwhich determines the mixing of scalar fields with the SUSY-breaking fields.\nThese D-term contributions to the effective coupling constants are studied in the following subsection. \n\n\n\n\\subsection{Effective Coupling Constants}\n\\label{sec:EffCoupling}\n\nIn this subsection, the effective coupling constants of gravitino pair production, $\\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R,I}}$, are evaluated in the model basis within the SUGRA framework. Since we are interested in the cosmological applications of gravitino production, it is sufficient to evaluate them at the leading order with respect to the Planck-suppressed couplings. Higher-order corrections are safely neglected. \nThe interactions are classified by powers of the inverse of the Planck scale. \nAt the zeroth order, i.e., in the global SUSY limit, $\\phi$ is secluded from the SUSY-breaking sector by the assumptions. On the other hand, SUGRA corrections which include Planck-suppressed interactions belong to higher orders of the perturbation. Turning on the corrections, the two sectors communicate with each other, and gravitino production occurs.\n\nIn Eq.~\\eqref{Geffective}, it is sufficient to evaluate $G_{z^{i}}$ and $(A^{-1})^{\\phi}{}_{ \\Phi_{R,I}}$ at the zeroth order, because $G_{\\phi}$ and $(A^{-1})^{z^{i}}{}_{ \\Phi_{R,I}}$ start from the first order of the perturbation. Let us start from $(A^{-1})^{\\phi}{}_{ \\Phi_{R,I}}$.\nIn the global SUSY limit, the mass eigenstates in the $\\phi$ sector are simply given by $\\Phi_{R}=\\sqrt{g_{\\phi\\bar{\\phi}}\/2}(\\delta\\phi+\\delta\\bar{\\phi})$ and $\\Phi_{I}=-i\\sqrt{g_{\\phi\\bar{\\phi}}\/2}(\\delta\\phi-\\delta\\bar{\\phi})$, with their masses\n\\begin{align}\nm^{2}_{\\Phi_R}=\\frac{1}{g_{\\phi \\bar{\\phi}}} \\left( V^{\\rm (g)}_{\\phi \\bar{\\phi}} + V^{\\rm (g)}_{\\phi \\phi} \\right) ,~~~~~\nm^{2}_{\\Phi_I}=\\frac{1}{g_{\\phi \\bar{\\phi}}} \\left( V^{\\rm (g)}_{\\phi \\bar{\\phi}} - V^{\\rm (g)}_{\\phi \\phi} \\right) ,\n\\end{align}\nwhere $V^{\\rm (g)}$ denotes the scalar potential in the global SUSY limit, and \n\\begin{eqnarray}\nV_{x^\\alpha x^\\beta} = \\frac{\\partial^2 V}{\\partial x^\\alpha \\partial x^\\beta}\n\\end{eqnarray}\nis the second derivative of the potential in the model basis. Here, we assume $V^{\\rm (g)}_{\\phi \\phi} = V^{\\rm (g)}_{\\bar{\\phi}\\bar{\\phi}}$ for simplicity. It is straightforward to include the phase.\nThus, the mixings of the scalar $\\phi$ itself become $(A^{-1})^{\\phi}{}_{ \\Phi_{R}}=1\/\\sqrt{2g_{\\phi\\bar{\\phi}}}$ and $(A^{-1})^{\\phi}{}_{ \\Phi_{I}}=i\/\\sqrt{2g_{\\phi\\bar{\\phi}}}$ at the zeroth order. \nThe effective coupling constants [Eq.~\\eqref{Geffective}] are approximated to be\n\\begin{align}\n\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R}}\\right|^{2}=2\\left|\\frac{1}{\\sqrt{2g_{\\phi \\bar{\\phi}}}} G_{\\phi} + G_{z^{i}} (A^{-1})^{z^{i}}{}_{\\Phi_{R}} \\right|^{2}, \\label{GeffR0} \\\\\n\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{I}}\\right|^{2}=2\\left|\\frac{i}{\\sqrt{2g_{\\phi \\bar{\\phi}}}} G_{\\phi} + G_{z^{i}} (A^{-1})^{z^{i}}{}_{\\Phi_{I}} \\right|^{2}. \\label{GeffI0}\n\\end{align}\nNext, the F terms of the SUSY-breaking fields, $G_{z^{i}}\\simeq W_{z^i}\/W$, are evaluated at the zeroth order, namely by using the field VEVs in the global SUSY limit, as discussed above. This depends on the model, and we will demonstrate it in a D-term SUSY-breaking model in the next section. Although the SUSY-breaking sector may be involved, this procedure is straightforward. \n\nThe remaining task is to evaluate $G_{\\phi}$ and $(A^{-1})^{z^{i}}{}_{\\Phi_{R,I}}$ at the first order. \nThey include contributions from the D-term potential, which is represented by the Killing potential, $D_{A}$. \nBy the gauge invariance of the action, the Killing potential satisfies\n\\begin{align}\nD_A = i X_A^i G_i = i X_A^i K_i,\n\\label{eq:D-term}\n\\end{align}\nwhere $X_A^i =-ig^{i\\bar{j}}D_{A \\bar{j}}$ is the holomorphic Killing vector, by which the gauge transformation of a chiral superfield, $\\Phi^i$, is defined as $\\delta \\Phi^i = \\Lambda^A X_A^i (\\Phi^j)$ with $\\Lambda^A$ being the gauge transformation parameters. The second equality in Eq.~\\eqref{eq:D-term} is derived by the gauge invariance of the superpotential, $\\delta W = W_i \\delta \\Phi^i = 0$. \n\nThe F term of $\\phi$ is evaluated at the minimum of the scalar potential in SUGRA. The conditions of the vanishing cosmological constant and the potential minimization are\\begin{align}\n&V = \\frac{1}{2}g^{2} D^{A}D_{A} + e^{G}(G^{i}G_{i}-3) = 0, \\label{V} \\\\\n&V_{i}= g^{2} \\left(-\\frac{1}{2}h^{R}_{ABi}D^{A}D^{B}+D^{A}D_{Ai} \\right)+e^{G}\\left( G_{i}G^{j}G_{j}-2G_{i}+G^{j}\\nabla _{i} G_{j} \\right)=0, \\label{Vi} \n\\end{align}\nwhere $V$ is the scalar potential in SUGRA, $h^{R}_{AB}\\, (h_{R}^{AB})$ is (the inverse of) the real part of the gauge kinetic function,\nand $\\nabla_{i}G_{j}=G_{ij}-\\Gamma^{k}_{ij}G_{k}$ is the covariant derivative with $\\Gamma^{k}_{ij}=g^{\\bar{l}k}G_{ij\\bar{l}}$.\nFrom Eq.~(\\ref{Vi}), $G_{\\phi}$ is obtained as\n\\begin{align}\ng^{\\phi\\bar{\\phi}} G_{\\phi} \\simeq \n\\frac{1}{\\nabla_{\\bar{\\phi}}G_{\\bar{\\phi}}} \\left[ -g^{\\bar{z}^{i}z^{j}}G_{z^{j}} \\nabla_{\\bar{\\phi}}G_{\\bar{z}^{i}} +\\frac{g^{2}}{m_{3\/2}^{2}} \\left( \\frac{1}{2}h^{R}_{AB\\bar{\\phi}}D^{A}D^{B} - D^{A}D_{A\\bar{\\phi}} \\right) \\right]\n- g^{z^{i}\\bar{\\phi}} G_{z^{i}}, \n\\label{Gphi}\n\\end{align}\nwhere the gravitino mass is from $m_{3\/2} = \\vev{e^{G\/2}}$. \nIn this expression, we have used $|\\nabla_{\\phi}G_{\\phi}| = W_{\\phi\\phi}\/W + \\cdots \\sim m_\\phi\/m_{3\/2} \\gg 1$ and $G_i \\lesssim {\\mathcal O}(1)$ from Eq.~\\eqref{V}.\nIt is found that $G_{\\phi}$ vanishes in the global SUSY limit, when $\\phi$ is secluded from the SUSY-breaking sector in the limit and is a singlet under the extra gauge symmetry.\n\nNext, let us evaluate the mixing matrix $(A^{-1})^{z^{i}}{}_{\\Phi_{R,I}}$. There are two sources of mixing between $\\phi$ and $z^{i}$. The first one is from kinetic mixing, $g_{z^{i}\\bar{\\phi}}$, which is induced by higher-dimensional operators with the field VEVs. The kinetic terms are canonicalized by redefining the fields as\n\\begin{align}\n\\phi'=\\sqrt{g_{\\phi\\bar{\\phi}}}\\;\\delta\\phi+\\frac{g_{z^{i}\\bar{\\phi}} }{\\sqrt{g_{\\phi\\bar{\\phi}}} }\\delta z^{i},~~~\nz'^{i}= (C^{-1})^{i}{}_{j}\\delta z^{j},\n\\label{non-unitary}\n\\end{align}\nat the leading order of $g_{z^{i}\\bar{\\phi}}$,\nwhere $C$ is a matrix that canonicalizes the kinetic term of the SUSY-breaking sector,\n\\begin{align}\nC^{\\dag}{}_{i}{}^{j} g_{\\bar{z}^{j}z^{k}} C^{k}{}_{l}=\\delta_{il}.\n\\end{align}\nHere, it is sufficient to evaluate $g_{z^{i}\\bar{\\phi}}$ and $g_{\\phi\\bar{\\phi}}$ \nby using the field VEVs in the global SUSY, when the first-order perturbation is considered. \nThe second source of the mixing comes from the mass term. By canonicalizing the kinetic terms, the mixings in the mass matrix become\n\\begin{align}\nV_{\\phi' \\bar{z}^{\\prime i}} &= \n\\frac{1}{\\sqrt{g_{\\phi\\bar{\\phi}}}} (C^\\dagger)_i{}^j\n\\left[ V_{\\phi \\bar{z}^{j}} - \\frac{g_{\\phi \\bar{z}^{j}}}{g_{\\phi\\bar{\\phi}}} V^{\\rm (g)}_{\\phi\\bar{\\phi}} \\right] \n\\equiv \\frac{1}{\\sqrt{g_{\\phi\\bar{\\phi}}}} (C^\\dagger)_i{}^j\\, \\tilde{V}_{\\phi \\bar{z}^{j}}, \n\\label{eq:mass-matrix-1} \\\\\nV_{\\phi' z^{\\prime i}} &= \n\\frac{1}{\\sqrt{g_{\\phi\\bar{\\phi}}}} C^j{}_i\n\\left[ V_{\\phi z^{j}} - \\frac{g_{z^{j}\\bar{\\phi}}}{g_{\\phi\\bar{\\phi}}} V^{\\rm (g)}_{\\phi \\phi} \\right]\n\\equiv \\frac{1}{\\sqrt{g_{\\phi\\bar{\\phi}}}} C^j{}_i\\, \\tilde{V}_{\\phi {z}^{j}},\n\\label{eq:mass-matrix-2}\n\\end{align}\nat the leading order of $g_{z^{i}\\bar{\\phi}}$. \nExplicit forms of $V_{\\phi z^{i}}$ and $V_{\\phi \\bar{z}^{j}}$ will be given later.\nSince the mixing terms are small, the mass matrix is diagonalized by means of the perturbation theory. At the leading order of the perturbation, $(A^{-1})^{z^{i}}{}_{\\Phi_{R,I}}$ is obtained by combining the canonicalization and the diagonalization as\n\\begin{align}\n(A^{-1})^{z^{i}}{}_{ \\Phi_{R}}&= \\frac{1}{\\sqrt{2g_{\\phi\\bar{\\phi}}}}\n\\left[ \\left( m^{2}_{\\Phi_{R}} g_{z} -m_z^{2} \\right)^{-1} \\right]^{z^{i}\\tilde{z}^{j}} \\left( \\tilde{V}_{\\tilde{z}^{j} \\phi}+ \\tilde{V}_{\\tilde{z}^{j} \\bar{\\phi}} \\right), \\\\\n(A^{-1})^{z^{i}}{}_{ \\Phi_{I}}&= \\frac{i}{\\sqrt{2g_{\\phi\\bar{\\phi}}}}\n\\left[ \\left( m^{2}_{\\Phi_{I}} g_{z} -m_z^{2} \\right)^{-1}\\right]^{z^{i}\\tilde{z}^{j}} \\left( \\tilde{V}_{\\tilde{z}^{j} \\phi}- \\tilde{V}_{\\tilde{z}^{j} \\bar{\\phi}} \\right),\n\\end{align}\nwhere the indices $\\phi$ and $z^i$ are not in the primed basis of Eq.~\\eqref{non-unitary}, but in the model one. \nHere, $\\left( m^{2}_{\\Phi} g_{z} - m_z^{2} \\right)^{-1}$ is an inverse matrix, and the index $\\tilde{z}^{j}$ runs over both $z^{j}$ and $\\bar{z}^{j}$. The matrices, $m_z^{2}$ and $g_z$, are defined in the model basis as \n\\begin{align}\nm_z^2 = &\\begin{pmatrix}\nV^{\\rm (g)}_{z^i\\bar z^j} & V^{\\rm (g)}_{z^i z^j} \\\\\nV^{\\rm (g)}_{\\bar z^i \\bar z^j} & V^{\\rm (g)}_{\\bar z^i z^j} \n\\end{pmatrix},\n~~~~~\ng_{z}=\n\\begin{pmatrix}\ng_{z^i\\bar z^j} & 0 \\\\\n0 & g_{\\bar z^i z^j}\n\\end{pmatrix}.\\label{zmatrices}\n\\end{align}\nSince the mixing \n$\\tilde{V}_{\\tilde{z}^{j} \\phi}\\pm \\tilde{V}_{\\tilde{z}^{j} \\bar{\\phi}}$\nbelongs to at least the first order of the perturbation, \nthe inverse matrix is evaluated in the global SUSY.\n\nThe mixing terms $V_{\\phi \\bar{z}^{i}}$ and $V_{\\phi z^{i}}$ are obtained from the scalar potential of SUGRA. \nBy using Eqs.~\\eqref{V} and \\eqref{Vi},\nthe mass matrices are derived as\n\\begin{align}\n\\label{hmass} \nV_{i \\bar{j}} &= \n\\mathrm{e}^{G} \n\\left( \\nabla _{i} G_{k} \\nabla _{\\bar{j}} G^{k} - R_{i \\bar{j} k \\bar{l} } G^{k}G^{\\bar{l}} + g_{i \\bar{j}} \\right) \\\\\n& + g^{2} \\left( \\frac{1}{2} \\left( G_{i}G_{\\bar{j}} - g_{i \\bar{j}} \\right) D^{A}D_{A} -G_{i}D^{A}D_{A \\bar{j}} - G_{\\bar{j}}D^{A}D_{Ai} +\\frac{1}{2} \\left( h^{R}_{ABi}G_{\\bar{j}} + h^{R}_{AB \\bar{j}}G_{i} \\right) D^{A}D^{B} \\right) \\nonumber \\\\\n& + g^{2} \\left( h_{R}^{AB}D_{Ai}D_{B\\bar{j}} + h_{R}^{AB}{}_{i}D_{A}D_{B\\bar{j}}+ h_{R}^{AB}{}_{\\bar{j}}D_{A}D_{Bi} +D^{A}D_{Ai\\bar{j}} + h^{R}_{ACi}h_{R}^{CD}h^{R}_{DB\\bar{j}} D^{A}D^{B} \\right), \\nonumber \\\\\n\\label{ahmass}\nV_{ij} &= \n\\mathrm{e}^{G}\n\\left( 2\\nabla _{i} G_{j} + G^{k}\\nabla _{i} \\nabla _{j} G_{k} \\right) \\nonumber \\\\\n& + g^{2} \\left( \\frac{1}{2} \\left( G_{i}G_{j} -\\nabla _{i}G_{j} \\right)D^{A}D_{A} -G_{i}D^{A}D_{Aj} - G_{j}D^{A}D_{Ai} \\right. \\\\\n& ~~~~~~~~\n\\left. +\\frac{1}{2} \\left( h^{R}_{ABi}G_{j} + h^{R}_{ABj}G_{i} \\right)D^{A}D^{B} +\\Gamma ^{k}_{ij} \\left( -D^{A}D_{Ak}+\\frac{1}{2}h^{R}_{ABk}D^{A}D^{B} \\right) \\right) \\nonumber \\\\\n & +g^{2} \\left( h_{R}^{AB}D_{A}D_{Bij}+ h_{R}^{AB}D_{Ai}D_{Bj} \n + h_{R}^{AB}{}_iD_{A}D_{Bj} \n + h_{R}^{AB}{}_jD_{A}D_{Bi} \n + \\frac{1}{2}h_{R}^{AB}{}_{ij}D_{A}D_{B} \\right), \\nonumber\n\\end{align}\nwith\n\\begin{align}\n\\nabla_{i}\\nabla_{j}G_{k}\n&= (\\nabla_{j}G_{k})_{i} -\\Gamma^{l}_{ij}\\nabla_{l}G_{k}-\\Gamma^{l}_{ik}\\nabla_{j}G_{l} \\nonumber \\\\\n&= G_{ijk}-G_{ijk\\bar{l}}G^{\\bar{l}}-3\\Gamma^{l}_{(ij}G_{k)l}+3\\Gamma^{l}_{(ij}\\Gamma^{m}_{k)l}G_{m},\n\\end{align}\nwhere the indices in a parenthesis are totally symmetrized.\nIn each of Eqs. \\eqref{hmass} and (\\ref{ahmass}), the first parenthesis is induced by the F-term VEVs, while the others are finite when the D term contributes. \nAt the first order, the mixing terms of the mass matrices which are (potentially) relevant for the cosmology are obtained as\n\\begin{align}\nV_{\\phi \\bar{z}^{i}} &\\simeq \ne^{G} \\left( g^{\\phi\\bar{\\phi}}\\nabla_{\\phi}G_{\\phi}\\nabla_{\\bar{z}^{i}}G_{\\bar{\\phi}} \n+ g^{z^{j}\\bar{z}^{k}}\\nabla_{\\phi}G_{z^{j}}\\nabla_{\\bar{z}^{k}}G_{\\bar{z}^{i}}\n\\right. \\nonumber\\\\ &~~~~~~~~~~\\left.\n+ g^{\\phi \\bar{z}^{j}}\\nabla_{\\phi}G_{\\phi}\\nabla_{\\bar{z}^{i}}G_{\\bar{z}^{j}} \n- R_{\\phi \\bar{z}^i z^j \\bar{z}^k} g^{z^{j}\\bar{z}^{l}}g^{\\bar{z}^{k}z^{m}}G_{\\bar{z}^l} G_{z^m} \\right) \n\\nonumber \\\\ &~~~\n+ g^{2} \\left( h_{R}^{AB}D_{A\\phi}D_{B\\bar{z}^{i}}+h_{R}^{AB}{}_{\\phi}D_{A}D_{B\\bar{z}^{i}}+D^{A}D_{A\\phi\\bar{z}^{i}}\\right), \\label{Mphizbar} \\\\\nV_{\\phi z^{i}} &\\simeq\n- e^{G} \\left( G_{\\bar{\\phi}z^{i}z^{j}}G_{\\phi\\phi}+2g^{z^{k}\\bar{z}^{l}}G_{\\bar{z}^{l}\\phi (z^{i}}G_{z^{j})z^{k}} \\right) g^{z^{j}\\bar{z}^{m}} G_{\\bar{z}^{m}} \n\\nonumber \\\\ &~~~\n+ g^{2} \\left( -g^{z^{j}\\bar{z}^{k}}G_{\\phi z^{i}\\bar{z}^{k}}D^{A}D_{A z^{j}} \n+ h_{R}^{AB}D_{A}D_{B\\phi z^{i}} \n+ h_{R}^{AB}D_{A\\phi}D_{B z^{i}} \n+ h_{R}^{AB}{}_{\\phi}D_{A}D_{B z^{i}} \\right). \\label{Mphiz} \n\\end{align}\nHere, we have used $|G_\\phi| \\ll 1$ and kept only the terms which can be enhanced by\nthe following means: (i) the modulus\/inflaton is much heavier than the gravitino due to its SUSY mass, $|\\nabla_{\\phi}G_{\\phi}| \\gg 1$, (ii) in the D-term SUSY-breaking models, the fields in the SUSY-breaking sector, $z^i$, can have a large SUSY-invariant mass, $|\\nabla_{z^i}G_{z^j}| \\gg 1$, and (iii) derivatives of $D_A$ with respect to $z^i$ can be enhanced, since the VEVs of $z^i$ are much smaller than $M_P$. \n\nIn summary, the effective coupling constants are represented in the model basis as\n\\begin{align}\n\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R}}\\right|^{2}=\n\\frac{1}{g_{\\phi\\bar{\\phi}}} \\left| G_{\\phi}+ G_{z^{i}} \\left[ \\left( m^{2}_{\\phi_{R}} g_{z} - m_z^{2} \\right)^{-1} \\right]^{z^{i}\\tilde{z}^{j}} \\left( \\tilde{V}_{\\tilde{z}^{j} \\phi}+ \\tilde{V}_{\\tilde{z}^{j} \\bar{\\phi}} \\right) \\right|^{2}, \\label{GeffR} \\\\\n\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{I}}\\right|^{2}=\n\\frac{1}{g_{\\phi\\bar{\\phi}}}\\left| G_{\\phi}+ G_{z^{i}} \\left[\\left( m^{2}_{\\phi_{I}} g_{z} - m_z^{2} \\right)^{-1}\\right]^{z^{i}\\tilde{z}^{j}} \\left( \\tilde{V}_{\\tilde{z}^{j} \\phi}- \\tilde{V}_{\\tilde{z}^{j} \\bar{\\phi}} \\right) \\right|^{2}, \\label{GeffI}\n\\end{align}\nwhere $G_\\phi$ is found in Eq.~\\eqref{Gphi}, $m^{2}_{z}$ and $g_{z}$ are in Eq.~\\eqref{zmatrices}, and the mixing terms are in Eqs.~\\eqref{eq:mass-matrix-1} and \\eqref{eq:mass-matrix-2} with Eqs.~\\eqref{Mphizbar} and \\eqref{Mphiz}. They are independent of the matrix $C$ in Eq.~\\eqref{non-unitary}. The direct gravitino production rate is evaluated by substituting the above coupling constants into Eq.~\\eqref{gravitinorate}. \n\nThe D-term contributions are found in the terms proportional to $g^2$ in Eqs.~\\eqref{Gphi}, \\eqref{Mphizbar}, and \\eqref{Mphiz}. They are classified into two groups: i) contributions of $\\phi$ to the gauge kinetic term such as $h_{AB} \\sim \\phi\/M_P$, and ii) contributions to the D-term potential such as $G_{\\phi z^{i}\\bar{z}^{j}}g^{\\bar{z}^{j} z^{k}}D_{Az^{k}}$ and $D_{A\\phi}=ig_{\\phi \\bar{z}^{i}}X^{\\bar{z}^{i}}_{A}=g_{\\phi \\bar{z}^{i}}g^{\\bar{z}^{i} z^{j}}D_{A z^{j}}$, due to an effective charge which is induced by mixings with the SUSY-breaking fields from higher-dimensional operators.\nSee Eq.~\\eqref{eq:D-term} for the expression of the D term.\n Thus, all the D-term contributions vanish if the following two conditions are satisfied:\n\\begin{enumerate}\n\\item[(I)] $\\phi$ does not appear in the gauge kinetic function of the extra gauge symmetry, $h_{AB\\phi}=0$.\n\\item[(II)] $\\phi$ does not have the specific K\\\"{a}hler mixings with the fields in the SUSY-breaking sector which are charged under the extra gauge symmetry, $g_{\\phi\\bar{z}^i}=G_{\\phi z^{i}\\bar{z}^{j}}=0$.\n\\end{enumerate}\nIf these conditions are satisfied, the gravitino production is only from the F-term contributions. (See Appendix~\\ref{app:F-term} for a revision of the F-term contribution.) It is found that the rate decreases as the D-term VEV increases. This is because \nthe VEV of the F-term potential satisfies the vanishing cosmological constant condition [Eq.~\\eqref{V}]. If the D-term potential dominates the SUSY breaking, the F-term VEVs become suppressed. Then the cosmological problem of the direct gravitino production can be relaxed. \n\nFinally, let us comment on radiative corrections to pair gravitino production. The scalar field couples to a pair of gauginos via matter loops analogously to the anomaly, where the scalar interactions with the matter come from gravitational effects or higher-dimensional operators~\\cite{Endo:2007ih,Endo:2007sz}. Since the gaugino in the SUSY-breaking sector is a main component of the goldstino, the anomaly-induced decay is considered to contribute to the pair gravitino production. This decay induced by the gravitational anomalies works when the matter fields in the loop are lighter than the scalar. \n\nWhen the scalar $\\phi$ is heavier than the fields in the SUSY-breaking sector, the latter fields other than the goldstino are also produced by the scalar decay through SUGRA interactions~\\cite{Endo:2006qk,Endo:2007sz}. The decay products in the SUSY-breaking sector are considered to decay into lighter fields including the gravitino. This increases the abundance of the gravitino, and thus worsens the cosmology. \n\n\\section{Example}\\label{sec:example}\n\nIn this section, we apply the formulas in the previous section to a model of the D-term SUSY breaking.\nA fraction of the D-term SUSY breaking is defined as\n\\begin{align}\n\\delta=\\frac{V_{D}}{V_{F}+V_{D}}, \\label{delta}\n\\end{align}\nwhere the F-term SUSY breaking is represented by the F-term potential, $V_{F}=e^{G}G^{i}G_{i}$, and the D-term SUSY breaking is represented by the D-term potential, $V_{D}=(g^{2}\/2)D^{A}D_{A}$.\nIn the global SUSY, the F-term potential leads to $V_{F}=g^{i\\bar{j}}\\overline{W}_{\\bar{j}}W_{i}$.\nThe denominator of the right-hand side of Eq.~\\eqref{delta} is related to the gravitino mass through the cosmological constant condition [Eq.~\\eqref{V}], i.e.,\n\\begin{align}\n V_F + V_D = g^{i\\bar{j}}\\overline{W}_{\\bar{j}}W_{i} + \\frac{1}{2}g^{2} D^{A}D_{A}=3 m_{3\/2}^2 . \\label{V2}\n\\end{align}\nThe fraction $\\delta$ takes a value of $0\\leq \\delta <1$, where a pure D-term SUSY breaking, $\\delta=1$, is excluded because of the gauge invariance condition, $D_{A}=D_{A}^{i}G_{i}$~\\cite{WessBagger}. \n\nIn Sec.~\\ref{sec:SUSYbreaking}, a model of SUSY breaking is introduced. This model is extended to include a scalar field (such as modulus or inflaton), and the direct gravitino production rate is studied in Sec.~\\ref{sec:combined}. In particular, the conditions (I) and (II) obtained in the previous section are assumed to be satisfied. It will be explicitly shown that the rate is reduced compared to the F-term SUSY-breaking models, as the D-term SUSY-breaking effect increases. The cosmological implications are discussed in Sec.~\\ref{sec:cosmology}.\n\n\n\\subsection{D-term SUSY-Breaking Model}\n\\label{sec:SUSYbreaking}\n\nLet us consider a model of the D-term SUSY breaking explored in Ref.~\\cite{Gregoire:2005jr}.\nThe model has a $U(1)$ gauge superfield and six chiral superfields, $z_{-1}, z_{1}, z_{-1\/N}, z_{1\/N}, z_{0}$ and $z'_{0}$, with their $U(1)$ charges denoted by the subscripts.\nThe K\\\"{a}hler potential and the gauge kinetic function are minimal, while the superpotential is\n\\begin{eqnarray}\nW(z^i) = \\lambda_1 z_0 \\left( m^{1-N} z_1 z_{-1\/N}^N-m^{2} \\right) + \\lambda_2 m z_1 z_{-1}\n+ \\lambda_3 z'_0 z_{1\/N} z_{-1\/N},\n\\label{eq:modelW}\n\\end{eqnarray}\nwhere $m$ determines the SUSY-breaking scale. \nFor $\\lambda_{3}\\gg \\lambda_{1} \\gg \\lambda_{2} \\gg g$, \nthe field VEVs in the global SUSY limit are given by~\\cite{Gregoire:2005jr}\n\\begin{eqnarray}\n&& \n\\vev{z_{-1}}= \\vev{z_{1\/N}}= \\vev{z_{0}}=\\vev{z'_{0}}=0,\n\\label{eq:vev}\n\\\\\n&&\n|\\vev{z_{-1\/N}}|\\simeq m\\left( N^{3}\\hat{\\lambda}_{2}^{2} \\right) ^{\\frac{1}{2(N+2)}}, \\label{eq:vev2}\n \\\\\n&& \\vev{z_{1}}\\simeq m^{N+1}\\vev{z_{-1\/N}}^{-N}, \\label{eq:vev3}\n\\end{eqnarray}\nwhere $\\hat{\\lambda}_{2} \\equiv \\lambda_2\/g$. \nAt the vacuum, the scalar potential in the global SUSY is~\\cite{Gregoire:2005jr}\n\\begin{align}\nV \\simeq g^{2} m^{4} \\left( N^{3}\\hat{\\lambda}_{2}^{2} \\right)^{\\frac{2}{N+2}} \\left( \\frac{1}{N^{3}}+\\frac{1}{2 N^{2}} \\right).\\label{Vapprox}\n\\end{align}\nHere, $1\/N^{3}$ and $1\/2N^{2}$ in the parenthesis correspond to the F-term and D-term potentials, respectively. \nThus, the model involves the D-term SUSY-breaking effect as well as that of the F term, and the fraction of the D-term SUSY breaking is approximately given by $\\delta\\simeq N\/(N+2)$. The D-term contributions increase as $N$ becomes larger.\n\nIn the following numerical calculations, we use more precise values of the field VEVs, which are obtained by following the analysis in Appendix~\\ref{sec:potential}, and the mass spectrum, which is obtained using the mass matrices listed in Appendix~\\ref{sec:mass}.\nThe nonzero VEVs and mass spectra are shown in Table~\\ref{tab:mass_spectrum}\nfor some model points, as well as the SUSY-breaking fraction $\\delta$. The numerical results in Table~\\ref{tab:mass_spectrum} agree with the approximate ones within $\\sim 20\\%$.\n\n\nAs shown in the table, all the fields in the SUSY-breaking sector have masses scaled by $m$, which is related to the gravitino mass $m_{3\/2}$ by Eq.~\\eqref{eq:m_and_m32}, or approximately \n\\begin{align}\nm \\simeq 6^{\\frac{1}{4}}N^{\\frac{1}{2}}g^{-\\frac{1}{2}}\\sqrt{m_{3\/2}M_{P}},\n\\end{align}\nfor large $N$, where we have used Eqs.~\\eqref{V2}, \\eqref{eq:vev2}, and \\eqref{eq:vev3}.\nNote that there is no light scalar field (the so-called Polonyi field) in the SUSY-breaking sector with a mass on the order of the gravitino mass. \n\n\n\\begin{table}[t!]\n\\begin{center}\n\\caption{\nThe fractions of the D-term SUSY breaking $(\\delta )$, the field VEVs, and the mass spectra for $N=1, 10$, and 100. Here, two sets of the parameters, P1 and P2, are chosen. \nSee Appendixes~\\ref{sec:potential} and \\ref{sec:mass} for details of the analysis.\nThe field VEVs and masses are written in units of $m$, except for the gravitino mass $m_{3\/2}$.\n}\n\\begin{tabular}{|c|l|c|c|cc|l|c|}\n\\hline\n& $\\lambda_{3}, \\lambda_{1},$\n& \n& \n& &\n& \\multicolumn{2}{|c|}{mass spectrum}\n\\\\ \\cline{7-8}\n& $\\lambda_{2}, g$ \n& $N$\n& $\\delta$\n& $\\vev{z_1}$ & $\\vev{z_{-1\/N}}$\n& \\multicolumn{1}{|c|}{\\cancel{SUSY} sector}\n& $m_{3\/2}$\n\\\\ \\hline\\hline\nP1 \n& $1, 10^{-1},$ \n& $1$\n& $0.31$\n& $0.46$ & $2.2$\n& $3.7\\times 10^{-3} - 2.2$ \n& $3.2\\times 10^{-3}\\, m^2\/M_P $\n\\\\\n& $10^{-2}, 10^{-3}$\n& $10$\n& $0.83$\n& $8.3\\times 10^{-3}$ & $1.6$\n& $5.6\\times 10^{-4} - 12$ \n& $1.2\\times 10^{-4}\\, m^2 \/M_P$\n\\\\\n&\n& $100$\n& $0.97$\n& $1.4 \\times 10^{-4}$ & $1.1$\n& $1.3\\times 10^{-4} - 7.0\\times 10^{2}$ \n& $5.0 \\times 10^{-6}\\, m^2\/M_P$\n\\\\ \\hline\\hline\nP2\n& $1, 4^{-1}$, \n& $1$\n& $0.27$\n& $0.61$ & $1.6$\n& $4.0\\times 10^{-2} - 1.6$ \n& $2.6\\times 10^{-2} m^2 \/ M_P$\n\\\\\n& $4^{-2}, 4^{-3}$\n& $10$\n& $0.83$\n& $1.8 \\times 10^{-2}$ & $1.5$\n& $8.1 \\times 10^{-3} - 14$ \n& $1.6 \\times 10^{-3} m^2 \/ M_P$\n\\\\\n&\n& $100$\n& $0.98$\n& $2.9\\times 10^{-4}$ & $1.1$\n& $1.7\\times 10^{-3} - 8.5 \\times 10^{2}$ \n& $7.6 \\times 10^{-5} m^2 \/ M_P$\n\\\\ \\hline\n\\end{tabular}\n\n\\label{tab:mass_spectrum}\n\\end{center}\n\\end{table}\n\n\n\n\\subsection{Gravitino Production Rate}\n\\label{sec:combined}\n\nThe model in the previous subsection is extended to include the scalar $\\phi$. For simplicity, it is assumed that the K\\\"{a}hler potential and gauge kinetic function are minimal,\n\\begin{align}\nK=\n|\\phi |^{2}+\\left|z_{ 0} \\right|^{2}+\\left|z'_{0 } \\right|^{2}+\n\\left|z_{1 } \\right|^{2}+\\left|z_{-1 } \\right|^{2}+\n\\left|z_{1\/N } \\right|^{2}+\\left|z_{-1\/N } \\right|^{2},~~~\nh_{U(1)}=1.\n\\label{eq:modelKh}\n\\end{align}\nIt is also assumed that the mixings between $\\phi$ and $z^i$ are absent in the superpotential, $W^{(n)}_{\\rm mix}(\\phi, z^i)=0$, and the total superpotential is given by\n\\begin{align}\nW=W(z^i) +W(\\phi),\n\\end{align}\nwith $W(z^i)$ given by Eq.~\\eqref{eq:modelW}.\nIt is found that conditions (I) and (II) discussed in the previous section are satisfied. \n\nLet us now calculate the partial decay rate of the scalar $\\phi$ into gravitinos, or the effective coupling constants in Eqs.~\\eqref{GeffR} and \\eqref{GeffI}, following the discussion in the previous section. They are determined by $G_\\phi$ and the mixings between the scalar and the SUSY-breaking fields.\nIn the present model,\nfrom Eq.~\\eqref{Gphi}, $G_\\phi$ becomes\n\\begin{align}\nG_{\\phi} \\simeq - \\frac{G_{z^{i}} \\nabla_{\\bar{\\phi}}G_{\\bar{z}^{i}}}{\\nabla_{\\bar{\\phi}}G_{\\bar{\\phi}}},\n\\label{eq:Gphi_approx}\n\\end{align}\nand the mixings in Eqs.~\\eqref{eq:mass-matrix-1}, \\eqref{eq:mass-matrix-2}, \\eqref{Mphizbar}, and \\eqref{Mphiz} are\n\\begin{align}\n&\\tilde{V}_{\\phi \\bar{z}^{i}} = V_{\\phi \\bar{z}^{i}} \\simeq \ne^{G} \\left( \\nabla_{\\phi}G_{\\phi}\\nabla_{\\bar{z}^{i}}G_{\\bar{\\phi}} \n+ \\nabla_{\\phi}G_{z^{j}}\\nabla_{\\bar{z}^{j}}G_{\\bar{z}^{i}} \\right),\n\\label{eq:mix_approx}\\\\\n&\\tilde{V}_{\\phi z^{i}} = V_{\\phi z^{i}} \\simeq 0.\n\\end{align}\nIn Eqs.~\\eqref{eq:Gphi_approx} and \\eqref{eq:mix_approx}, the factors $e^G$, $\\nabla_\\phi G_\\phi$, and $\\nabla_{z^i}G_{z^j}$ are of zeroth order in the perturbation, and are given by\n\\begin{align}\ne^{G\/2} = m_{3\/2} \\simeq \\vev{W},~~~\n\\nabla_\\phi G_\\phi \\simeq \\frac{m_\\phi}{m_{3\/2}},~~~\n\\nabla_{z^i}G_{z^j} = \\frac{W_{z^i z^j}}{W} - \\frac{W_{z^i}}{W} \\frac{W_{z^j}}{W}.\n\\label{eq:nablaGzeroth}\n\\end{align}\nOn the other hand, the factor $\\nabla_{z^i} G_{\\phi} = \\nabla_{\\phi} G_{z^i}$ is suppressed by the Planck scale. Due to the absence of the $\\phi - z^i$ mixings in $K$ and $W$\nin the present model, it is given by\n\\begin{align}\n\\nabla_{\\phi} G_{z^i} =\n-\n\\frac{W_\\phi}{W}\n\\frac{W_{z^i}}{W}\n=\n(K_\\phi-G_\\phi)\\frac{W_{z^i}}{W}\n\\simeq K_\\phi\\frac{W_{z^i}}{W}.\n\\label{eq:Gzi_approx}\n\\end{align}\nHere, in the last equality, we have used $G_\\phi \\simeq K_\\phi \\times {\\cal O}(m_{3\/2} \/ m_\\phi) \\ll K_\\phi$, which can be shown by using Eq.~\\eqref{eq:Gphi_approx}, $\\nabla_\\phi G_\\phi \\simeq m_\\phi\/m_{3\/2}$, and $G_{z^i}\\simeq W_{z^i}\/W \\lesssim 1$.\nSee also Appendix~\\ref{app:F-term}.\nThe remaining task is to evaluate the F-term VEVs of the SUSY-breaking fields, $W_{z^i}$, in the global SUSY limit. From Eqs.~\\eqref{eq:modelW} -- \\eqref{eq:vev3}, the finite F terms are\n\\begin{align}\nW_{z_0} = \\lambda_{1}\\left(m^{1-N}\\vev{z_{1}}\\vev{z_{-1\/N}}^{N}-m^{2}\\right),~~~\nW_{z_{-1}} = \\lambda_2 m \\vev{z_1}.\n\\label{eq:Wderiv}\n\\end{align}\nFrom Eqs.~\\eqref{eq:vev} -- \\eqref{eq:vev3}, it is found that $W_{z_{-1}}$ dominates the F-term potential.\\footnote{Although $W_{z_0}$ seems to vanish when the VEVs in Eqs.~\\eqref{eq:vev} -- \\eqref{eq:vev3} are naively applied, it is finite as long as the D term is nonzero, because the right-hand side of $W_{z_0}$ is analytically related to the D-term VEVs via the stationary condition of the scalar potential [Eq.~\\eqref{eq:appV}]. \nThe gaugino masses of the SUSY SM can be given by $W_{z_0}$, or by the anomaly mediation.\n}\nThe other F terms are zero in the global SUSY. \n From Eqs.~\\eqref{eq:Gzi_approx} and \\eqref{eq:Wderiv}, it is found that $\\nabla_{\\phi} G_{z^i}$ \nis dominated by $\\nabla_{\\phi}G_{z_{-1}}$ as\n\\begin{align}\n\\nabla_{\\phi}G_{z_{-1}} \\simeq \\frac{\\lambda_{2} m}{m_{3\/2}} \\vev{z_{1}}\\vev{\\bar{\\phi}},\n\\end{align}\nwhere $\\vev{W} \\simeq m_{3\/2}$ is used. \nThus, the couplings are dominated by $z^i = z^j = z_{-1}$.\nPutting the above equations altogether, one obtains\n\\begin{align}\n\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R,I}} \\right|^{2}\n\\simeq \\left| \n\\left( 1- m^{2}_{\\phi} \\left [ \\left( m^{2}_{\\phi}-m^{2}_{z} \\right)^{-1} \\right ]^{z_{-1}\\bar{z}_{-1}} \\right)\n\\frac{m_{3\/2}}{m_\\phi} G_{z_{-1}} \\nabla_{\\bar{\\phi}} G_{\\bar{z}_{-1}}\n\\right|^2.\n\\end{align}\nFor the purpose of the cosmological application in the next subsection, we concentrate on the case in which the scalar $\\phi$ is lighter than the fields of the SUSY-breaking sector $z^{i}$. Then, the effective coupling constants are reduced to\n\\begin{align}\n\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R,I}} \\right|^{2}\n\\simeq\n\\left| \n\\frac{m_{3\/2}}{m_\\phi} G_{\\bar{z}_{-1}} \\nabla_{\\phi} G_{z_{-1}}\n\\right|^2\n=\n\\left| \n\\frac{\\lambda_{2}^2 m^2}{m_{3\/2} m_\\phi} \\vev{z_{1}}^2\\vev{\\bar{\\phi}}\n\\right|^2\n\\simeq\n\\frac{36}{N^2} \\frac{m_{3\/2}^2}{m_\\phi^2}\\vev{\\phi}^2.\n\\label{eq:modelGeff}\n\\end{align}\nIn the last equality we have used Eqs.~\\eqref{eq:vev2} and \\eqref{eq:vev3}, and $N\\gg 1$. Here, the effective couplings become insensitive to the model parameters of the SUSY-breaking sector except for $N$.\n\nIn Fig.~\\ref{fig:Geff}, we show the effective coupling constants as a function of the fraction of the D-term SUSY breaking $\\delta$, which is varied by changing the $N$ of the model. Here, we have used the full formulas in Eqs.~\\eqref{GeffR} and \\eqref{GeffI} with the numerically obtained field VEVs.\\footnote{We have also used Eqs.~\\eqref{eq:nablaGzeroth} and \\eqref{eq:Gzi_approx} to evaluate the derivatives of $W(\\phi)$.} The mass of the scalar $\\phi$\nis set to be smaller than the masses of the fields in the SUSY-breaking sector.\nFor the sake of comparison, we have also shown, by the star sign in the figure, the case where the SUSY is broken only by a single F-term VEV, assuming the minimal K\\\"{a}hler potential:\n\\begin{align}\n\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R,I}} \\right|^{2}\n\\simeq\n\\left|\n\\frac{G_z \\nabla_{\\bar{\\phi}} G_{\\bar{z}}}{\\nabla_{\\bar{\\phi}} G_{\\bar{\\phi}}} \n\\right|^2\n\\simeq\n\\left| \\frac{3m_{3\/2}}{m_\\phi} \\vev{\\phi} \\right|^2\n\\quad (\\text{pure F term}),\n\\end{align}\nwhere we have used $|G_z|\\simeq \\sqrt{3}$.\nIt is found that the effective coupling constants diminish as the D-term SUSY-breaking contribution increases. \n\n\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=85mm]{Geff.eps}\n \\end{center}\n \\caption{Dependence of the effective coupling squared $\\left | \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R}} \\right |^{2} \\simeq \\left | \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{I}} \\right |^{2} \\equiv \\left | \\mathcal{G}^{\\text{(eff)}}_{\\Phi} \\right |^{2}$ on the fraction of D-term SUSY breaking $\\delta$. $m_{3\/2}=10^3\\rm{GeV}$, $m_{\\phi}=10^{6}\\rm{GeV}$, and $\\vev{\\phi}=10^{12}\\rm{GeV}$ are used. The lines for parameter sets P1 and P2 in Table~\\ref{tab:mass_spectrum} coincide.\nThe fraction $\\delta$ has been extrapolated to $N<1$, which is drawn by the dashed line. The star mark represents the case of pure F-term SUSY breaking.}\n \\label{fig:Geff}\n\\end{figure}\n\n\n\\subsection{Cosmological Implications}\n\\label{sec:cosmology}\n\nFinally, we estimate the total gravitino abundance.\nThe coherent oscillation of the scalar $\\phi$ is assumed to dominate the energy density of the universe and then decay to create the thermal bath.\nThere are two types of gravitino production:\none is thermal production, where the gravitinos are produced through scattering processes in the thermal bath, and the other is direct production, which is the main subject of this paper.\nIn the following, we focus on the parameter region where the scalar is lighter than the fields in the SUSY-breaking sector except for the gravitino. Otherwise, the anomaly-induced gravitino production and the direct production of the SUSY-breaking fields which are mentioned in the last section can spoil the success of the standard cosmology.\n\nFirst of all, let us consider thermal production of the gravitino.\nThe gravitino yield is defined as $Y_{3\/2}=n_{3\/2}\/s$, where $n_{3\/2}$ is the number density of the gravitino, and $s$ is the entropy density. The thermal gravitino yield is approximately given by~\\cite{Kawasaki:2008qe}\n\\begin{align}\nY_{3\/2}^{\\text{thermal}}\\simeq \n\\left( 1.3\\times 10^{-14}+8.8\\times 10^{-15} \\left( \\frac{m_{1\/2}}{m_{3\/2}}\\right)^{2} \\right)\n\\left( \\frac{T_{R}}{10^{8}\\rm{GeV}} \\right),\n\\label{Ythermal}\n\\end{align}\nwhere $T_{R}=(\\pi^{2}g_{*}(T_{R})\/90)^{-1\/4}\\sqrt{\\Gamma_\\phi}$ is the reheating temperature after the $\\phi$ decay, with the total decay rate $\\Gamma_\\phi$, and $m_{1\/2}$ is the unified gaugino mass at the grand unified theory (GUT) scale.\\footnote{The definition of $T_R$ here is different from the one in Ref.~\\cite{Kawasaki:2008qe}.\nLogarithmic corrections are omitted in Eq.~\\eqref{Ythermal}.}\n\n\nNext, let us consider the direct production of gravitinos.\nThe direct production of a pair of gravitinos provides\n\\begin{align}\nY_{3\/2}^{\\text{decay}} \n&= \\frac{3T_{R}}{4m_{\\phi}} \\times 2B_{3\/2} \n= \\frac{1}{192\\pi} \\left( \\frac{90}{\\pi^{2} g_{*}(T_{R})} \\right)^{\\frac{1}{2}} \\frac{m_{\\phi}^{4} \\left| \\mathcal{G}_\\Phi^{\\text{(eff)}} \\right|^{2}}{m_{3\/2}^{2}T_{R}}.\n\\label{eq:yield}\n\\end{align}\nHere, $m_{\\Phi_{R}} \\simeq m_{\\Phi_{I}} \\equiv m_{\\phi}$ and $\\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{R}} \\right|^{2} \\simeq \\left| \\mathcal{G}^{\\text{(eff)}}_{\\Phi_{I}} \\right|^{2} \\equiv \\left| \\mathcal{G}_\\Phi^{\\text{(eff)}} \\right|^{2}$ are used. \n Substituting Eq.~\\eqref{eq:modelGeff} into Eq.~\\eqref{eq:yield}, the yield is reduced to be\n\\begin{align}\nY_{3\/2}^{\\text{decay}} \n&\\simeq\n\\frac{1}{192\\pi} \\left( \\frac{90}{\\pi^{2} g_{*}(T_{R})} \\right)^{\\frac{1}{2}} \n\\frac{1}{T_R} m_\\phi^2 \\vev{\\phi}^2\n\\left(\\lambda_2^4 \\frac{\\vev{z_1}^4 m^4}{m_{3\/2}^4} \\right) \\nonumber\\\\\n&\\simeq\n8.2 \\times 10^{-15} \\times\n\\frac{1}{N^2} \\left( \\frac{\\vev{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \\left( \\frac{m_{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \\left( \\frac{10^{5}\\rm{GeV}}{T_{R}} \\right),\n\\label{Ydecay}\n\\end{align}\nfor large $N$, where $g_{*}(T_{R})=228.75$ is used. It is found that the yield is proportional to $m_\\phi^2, \\vev{\\phi}^2$, and $T_R^{-1}$. In particular, \nit becomes independent of $m_{3\/2}$ because $\\vev{z_1}\\propto m \\propto \\sqrt{m_{3\/2}}$. It is stressed that the abundance decreases as $N$ increases; namely, the D-term contributions dominate (See also Fig. \\ref{fig:Geff}). \n\n\\begin{figure}[thb]\n \\begin{center}\n \\includegraphics[width=85mm]{APR_typ.eps}\n \\end{center}\n \\caption{Contours of the cosmological constraints on the $m_{\\phi}$ -- $\\vev{\\phi}$ plane from the thermal and direct production of gravitinos. The upper-right regions are excluded.\nSeveral values of the D-term SUSY-breaking contributions are taken: $\\delta=0$ (the pure F-term; see Fig.~\\ref{fig:Geff} and the text), $0.27\\ (N=1)$, and $0.98\\ (N=100)$. \nThe dashed lines correspond to $m_{3\/2}=10$GeV, and the solid ones correspond to $m_{3\/2}=1$TeV.\nThe former lines end in the figure because $m_{\\phi}$ is restricted to be smaller than the smallest mass of the fields in the SUSY-breaking sector.\nThe reheating temperature is provided by $\\Gamma_\\phi=m_{\\phi}^{3}\/8\\pi$. \nThe thermal abundance is subdominant in this figure.\n }\n \\label{fig:APR_typ}\n\\end{figure}\n\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=85mm]{APR_opt.eps}\n \\end{center}\n \\caption{Same as Fig.~\\ref{fig:APR_typ}, except that the reheating temperature is optimized such that the total gravitino abundance is minimized. The bound on $m_\\phi$ and $\\vev{\\phi}$ is considered to be most conservative. The gravitino mass is set to be $m_{3\/2}=1\\rm{TeV}$. There are no bounds in the case of $m_{3\/2}=10$GeV in this parameter space.\n }\n \\label{fig:APR_opt}\n\\end{figure}\n\nThe cosmological constraint on the gravitino abundance depends on the gravitino mass. \nWhen the gravitino is unstable, the most stringent bound comes from BBN.\nThe precise bound depends on details of the mass spectrum including the SUSY SM.\nHere, we adopt the constraint for the model point in Case 2 of Ref.~\\cite{Kawasaki:2008qe},\\footnote{This model point is excluded by the recent SUSY search at the LHC, but we adopt it just for illustration. The main conclusion does not depend much on the details of the mass spectrum.}\n where the bounds are given by $Y_{3\/2}< 2\\times 10^{-16}$ and $Y_{3\/2}< 2\\times 10^{-12}$\nfor the gravitino masses of $m_{3\/2}=1 \\rm{TeV}$ and $30 \\rm{TeV}$, respectively.\nThus, from Eqs.~\\eqref{Ythermal} and \\eqref{Ydecay}, the allowed region of the reheating temperature is \n\\begin{align}\n3 \\times 10^6\\rm{GeV} \\times \\frac{1}{N^2}\n\\left( \\frac{\\vev{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \n\\left( \\frac{m_{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \n< &\\ T_{R} < 2\\times 10^{6}\\rm{GeV}\n& (m_{3\/2}=1\\rm{TeV}),\\\\\n5 \\times 10^2\\rm{GeV} \\times \\frac{1}{N^2}\n\\left( \\frac{\\vev{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \n\\left( \\frac{m_{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \n< &\\ T_{R} < 1\\times 10^{10}\\rm{GeV}\n& (m_{3\/2}=30\\rm{TeV}),\n\\end{align}\nwhere the upper bound comes from the thermal gravitino abundance, while the lower bound is from direct production. This is because the thermal abundance is proportional to $T_R$, while the abundance from the latter is to $1\/T_R$. \nIt is noticed that the window tends to close as the energy scale of the scalar model is higher. In order to avoid the cosmological bound, the D-term contributions must be enhanced by increasing $N$. For instance, the window opens only when $N \\gg 1$ for $m_{3\/2}=1\\rm{TeV}$, $\\vev{\\phi} = 10^{16}\\rm{GeV}$, and $m_\\phi = 10^9\\rm{GeV}$. \n\nWhen the gravitino is the lightest among the SUSY particles, the gravitino becomes stable. \nSince the massive gravitino contributes to the dark matter abundance, the gravitino abundance cannot exceed the cold dark matter abundance as\n\\begin{align}\nm_{3\/2} Y_{3\/2} < \\frac{\\rho_c}{s} \\Omega_{\\rm DM} < 4.4 \\times 10^{-10}\\rm{GeV},\n\\end{align}\nwhere $\\rho_c$ is the critical density and $\\Omega_{\\rm DM} h^2 < 0.12$ at $2\\sigma$~\\cite{Beringer:1900zz} is used. \nFor the light stable gravitino, the bound becomes\n\\begin{align}\n1.9 \\rm{GeV} \\times\\frac{1}{N^2}\\lrf{m_{3\/2}}{1 \\rm{GeV}}\n\\left( \\frac{\\vev{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \n\\left( \\frac{m_{\\phi}}{10^{12}\\rm{GeV}} \\right)^{2} \n< &\\ T_{R} <\n1.4 \\times 10^{7}\\rm{GeV} \\times \\lrf{m_{3\/2}}{1\\rm{GeV}},\n\\end{align}\nwhere the gaugino mass is taken to be 600 GeV at the GUT scale, which affects the upper bound via the thermal gravitino abundance.\nAgain, the lower bound becomes milder as the D-term contributions dominate.\n\n\n\nFinally, let us show how D-term SUSY-breaking contributions enlarge the allowed parameter space spanned by $m_{\\phi}$ and $\\vev{\\phi}$. \nIn Figs.~\\ref{fig:APR_typ} and \\ref{fig:APR_opt}, contours of the cosmological bounds are drawn for various $\\delta$ in the $m_{\\phi}$ -- $\\vev{\\phi}$ plane. The region of lower $m_{\\phi}$ and $\\vev{\\phi}$ is allowed. \nIn the figures, $m_\\phi$ is restricted to be smaller than the smallest mass of the fields in the SUSY-breaking sector, for simplicity. \nIt can be seen that as the D-term SUSY-breaking contributions increase---i.e., as $\\delta$ becomes close to unity---the allowed region becomes wider. \n\n\nThe bound depends on the reheating temperature as seen in Eqs.~\\eqref{Ythermal} and \\eqref{Ydecay}. \nIn Fig.~\\ref{fig:APR_typ}, the temperature is estimated by assuming that the total decay rate is governed by a Planck-suppressed operator, $\\Gamma (\\phi \\rightarrow \\text{all}) = m_{\\phi}^{3}\/8 \\pi$.\nThe thermal production is subdominant in the figure.\nOn the other hand, the reheating temperature is taken to be most conservative in Fig.~\\ref{fig:APR_opt}. Since the thermal gravitino abundance is proportional to $T_R$, and that from the direct production is to its inverse, the total gravitino abundance becomes the minimum value by tuning $T_R$. Therefore, the bound in Fig.~\\ref{fig:APR_opt} is the most conservative one, and the model is refuted if it is excluded in Fig.~\\ref{fig:APR_opt}, unless there is an entropy production. \n\n\n\n\\section{Summary}\\label{sec:conclusion}\n\nIn this paper, we studied scalar decay into gravitinos in the presence of D-term SUSY-breaking and multiple SUSY-breaking fields, taking into account the mixing between the scalar and the fields in the SUSY-breaking sector.\nWe obtained the general formulas of the effective coupling constants for gravitino pair production, Eqs.~\\eqref{GeffR} and \\eqref{GeffI}.\nIt is found that the gravitino production is suppressed when the D-term VEV is sizable if conditions (I) and (II) in Sec.~\\ref{sec:EffCoupling} are satisfied.\n\nAs an example, we applied the formulas to the D-term SUSY-breaking model in Ref.~\\cite{Gregoire:2005jr} and showed that the D-term SUSY breaking suppresses the gravitino production.\nMoreover, an interesting feature of the model is that all the fields in the SUSY-breaking sector have masses of ${\\cal O}(\\sqrt{M_p m_{3\/2}})$.\nThis is very different from F-term SUSY-breaking models, \nin which the SUSY-breaking field tends to have a mass of $\\mathcal{O}(m_{3\/2})$, \ncausing the Polonyi problem~\\cite{Coughlan:1983ci}. \nThis may be a generic feature of D-term SUSY-breaking models. (See also\na recent work, Ref.~\\cite{Azeyanagi:2012pc}.)\nThese observations can open a new window for model building respecting cosmology.\n\n\\section*{Acknowledgments}\nThis work was supported by Grants-in-Aid for Scientific Research from\nthe Ministry of Education, Science, Sports, and Culture (MEXT), Japan,\nGrants No. 23740172 (M.E.), No. 21740164 (K.H.), and No. 22244021 (K.H.).\nThe work of T.T. was supported by an Advanced Leading Graduate Course for Photon Science grant.\nThis work was supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn this paper we consider a model for two-component Bose-Einstein condensates irradiated by an external electromagnetic field. The dynamics are described by two coupled nonlinear Schr\\\"odinger equations\n\\begin{equation}\\label{eq:nlss1}\ni\\d_t\\psi_j=-\\frac12\\Delta\\psi_j+V(x)\\psi_j+\\beta_{jj}|\\psi_j|^2\\psi_j+\\beta_{jk}|\\psi_k|^2\\psi_j+\\lambda\\psi_k,\n\\end{equation}\n$j, k = 1, 2$, in $(t, x)\\in\\R\\times\\R^N$, $N = 1, 2, 3$. Here $V(x)=\\frac{\\gamma^2}{2}|x|^2$ is the trapping potential, $\\beta_{jj}$ and $\\beta_{jk}$ with $\\beta_{12}=\\beta_{21}$ are the (scaled) intra- and inter-specific scattering lengths, respectively, $\\lambda$ is the Rabi frequency related to the external electric field.\n\\newline\nOur main goal in this paper is to investigate the asymptotic behavior of solutions to equation \\eqref{eq:nlss1} when the Rabi frequency $\\lambda$ becomes very large. In particular, we shall show that, after a suitable transformation of the system, the asymptotic behavior is described by two effective coupled nonlinear Schr\\\"odinger equations with the same inter- and intra-specific scattering lengths.\n\\newline\nThe equations we consider here arise in the modeling of a Bose-Einstein condensate formed of atoms in two different hyperfine states in the same harmonic trap \\cite{Ballagh}. A binary BEC of $^{87}$Rb atoms in different spin states was produced for the first time at JILA \\cite{Myatt97}. The irradiation of the condensate with an electric field induces a linear coupling in the overlap region, which causes a Josephson-type oscillation between the two species \\cite{Will99}. Such condensates are very interesting in physics, since it is possible to measure the relative phase of one component with respect to the other one \n\\cite{Hall:1998fk}. Furthermore, by controlling locally the relative phase, it is also possible to produce vortices \n\\cite{williams1999preparing}, \\cite{matthews1999vortices}. Another interesting application of this model is the creation of a stable \\emph{BEC droplet} \\cite{Saito} without using highly oscillating magnetic fields through Feshbach resonance methods: indeed the external electric field is constant but it induces an effective oscillation of the scattering lengths.\n\\newline\nThere is an extensive literature on systems of nonlinear Schr\\\"odinger equations, and giving here a comprehensive picture of all the results in different cases would be a quite hard task.\nIn particular, the case without linear coupling, i.e. $\\lambda=0$ in \\eqref{eq:nlss1}, is considered in most of the cases. \n\\cite{FanMont, LiWuLai, MaZhao} deal with the system without trapping potential and a more general class of power-type nonlinearities, giving sufficient conditions for global existence of solutions, \\cite{MR2283958} analyses the existence of ground state solutions when the nonlinearities are cubic, and \\cite{MR2449345} studies vortices in the same framework. In the presence of a trapping potential we mention \\cite{LinWei, ChenWei, Prytula}.\nOn the other hand, the case with linear coupling, i.e. $\\lambda\\neq0$ in \\eqref{eq:nlss1}, is less treated, in \n\\cite{Bao} existence and uniqueness of ground states is proved, performing also asymptotics for such solutions when \n$|\\lambda|\\to\\infty$, whereas in \\cite{Juengel11} the time-dependent system with a more general class of power-type nonlinearities is considered. Here the authors give sufficient conditions for global existence or blow up, and a semi-implicit formula shows the mass transport between the two components.\n\\newline\nMoreover, as it will be clear in the next Sections, the analysis of system \\eqref{eq:nlss1} is strictly related to the study of nonlinear Schr\\\"odinger equations with time-oscillating nonlinearities. This link is well known in physics, see \\cite{Saito}. Cazenave, Scialom \\cite{Caz10} were the first to rigorously study such problems, providing a convergence result in the highly oscillating regime, see also \\cite{Abdullaev} for numerical results, \\cite{Fang} for the case with an energy-critical nonlinearity, \n \\cite{CarGamPan} for a system of coupled nonlinear Schr\\\"odinger equations. Similar problems are also studied in some related frameworks, for example in \\cite{AS10} the authors carry out an asymptotic analysis for a \\emph{dispersion managed} NLS, i.e. when there is a highly oscillating coefficient in front of the Laplacian, or in \\cite{MR2809621} the authors consider the KdV equation with an time-dependent nonlinearity.\n \n\n\n\nIn Section \\ref{sect:prel} we introduce the notations and state the main result of our paper. We then dedicate Section \n\\ref{sect:LGWP} to the analysis of system \\eqref{eq:nlss1}, establishing the cases of global existence of solutions or possible occurrence of blow-up, depending on the different choices of the coefficients $\\beta_{jk}$'s.\nThen in Section \\ref{sect:trans} we transform \\eqref{eq:nlss1} in a way which is more suitable to study the asymptotics when \n$|\\lambda|$ becomes large. We then recover the limiting system, and in Section \\ref{sect:as} we prove rigorous convergence. \n\\section{Preliminaries and Main Result}\\label{sect:prel}\nIn what follows $C$ will denote a generic constant greater than 1, which may possibly change from line to line.\nWith $\\Re$ and $\\Im$ we denote the real and imaginary part of a complex number, respectively. By $z^*$ we denote the complex conjugate of $z$. The scalar product between two vectors $v, v'$ will be denoted by $\\langle v, v'\\rangle$.\n\\newline\nSince we are dealing with two-component Schr\\\"odinger systems, we shall indicate with capital letters the two-dimensional vector fields describing the wave-function of a two-component quantum system. For example we shall write \n$\\Psi^t=(\\psi_1, \\psi_2)$, or $\\Psi_0^t=(\\psi_{1, 0}, \\psi_{2, 0})$, $\\Psi^*=(\\psi_1^*, \\psi_2^*)$ and so on. Consequently, we may also denote\n\\begin{equation*}\n|\\Psi|^2=|\\psi_1|^2+|\\psi_2|^2.\n\\end{equation*}\nWe shall denote by $L^p(\\R^N), W^{1, p}(\\R^N)$, the usual Lebesgue and Sobolev spaces, respectively. We shall also make use of the mixed space-time Lebesgue (or Sobolev) spaces, so that for example $L^q(I;L^r(\\R^N))$ denote the space of those functions $\\Psi$ having the following norm finite,\n\\begin{equation*}\n\\|\\Psi\\|_{L^q(I;L^r(\\R^N))}:=\\pare{\\int_I\\pare{\\int_{\\R^N}|\\Psi(t, x)|^rdx}^{q\/r}dt}^{\\frac{1}{r}}.\n\\end{equation*}\nWe often shorten notation $L^q_tL^r_x(I\\times\\R^N)=L^q(I;L^r(\\R^N))$ or $L^q_tL^r_x$ if there is no source of ambiguity.\n\\newline\nWe are interested in studying system \\eqref{eq:nlss1} in the \\emph{energy space}, which is defined by\n\\begin{equation*}\n\\Sigma(\\R^N):=\\{\\Psi\\in H^1(\\R^N);\\;|\\cdot|\\Psi\\in L^2(\\R^N)\\}.\n\\end{equation*}\nLet us consider the Hamiltonian with confining potential, $H=-\\frac12\\Delta+\\frac{\\gamma^2}{2}|x|^2$. The associated propagator is $S_0(t):=e^{-itH}$, is unitary on $L^2(\\R^N)$ and from Melher's formula (see for example \\cite{Car05,Oh}) we may show it satisfies a dispersive estimate for short times, namely\n\\begin{equation}\\label{eq:disp_est}\n\\|S_0(t)f\\|_{L^\\infty(\\R^N)}\\lesssim|t|^{-\\frac{N}{2}}\\|f\\|_{L^1(\\R^N)},\\quad|t|\\leq\\delta,\n\\end{equation}\nfor some small $\\delta>0$ depending on $\\gamma$. \n\\newline\nFurthermore, in what follows we also need the following commutation formulas for $H$:\n\\begin{equation}\\label{eq:comm}\n[\\nabla, H]=\\gamma^2x,\\quad[x, H]=\\nabla.\n\\end{equation}\nAlthough \\eqref{eq:disp_est} holds only for short times, we may infer the same Strichartz estimates for $S_0(t)$ as for the free Schr\\\"odinger propagator without confining potential, but only locally in time, i.e. the constants appearing in the inequalities below depend on the length of the time interval. Here we state the results we need, for further details we address the reader to \\cite{Car05}, Section 2.\n\\begin{definition}\nWe say $(q, r)$ is an \\emph{admissible pair} if $2\\leq r\\leq\\frac{2N}{N-2}$ ($2\\leq r\\leq\\infty$ for $N=1$, $2\\leq r<\\infty$ for $N=2$), and\n\\begin{equation}\\label{eq:adm_pair}\n\\frac{1}{q}=\\frac{N}{2}\\pare{\\frac{1}{2}-\\frac{1}{r}}.\n\\end{equation}\n\\end{definition}\n\\begin{proposition}\\label{prop:strich}\nLet $(q, r), (\\tilde q, \\tilde r)$ be two arbitrary admissible pairs. Then, for any compact time interval, we have\n\\begin{equation*}\n\\begin{aligned}\n\\|S_0(t)f\\|_{L^q_tL^r_x(I\\times\\R^N)}\\leq &C(|I|, r)\\|f\\|_{L^2(\\R^N)},\\\\\n\\|\\int_0^tS_0(t-s)F(s)ds\\|_{L^q_tL^r_x(I\\times\\R^N)}\\leq &C(|I|, r, \\tilde r)\\|F\\|_{L^{\\tilde q'}_tL^{\\tilde r'}_x(I\\times\\R^N)}.\n\\end{aligned}\n\\end{equation*}\n\\end{proposition}\n We may rewrite the system of equations \\eqref{eq:nlss1} in the following compact way\n\\begin{equation}\\label{eq:nlss}\n\\left\\{\\begin{aligned}\ni\\d_t\\Psi=&-\\frac12\\Delta\\Psi+\\frac{\\gamma^2}{2}|x|^2\\Psi+\\tilde B[\\Psi]\\Psi+A\\Psi,\\\\\n\\Psi(0)=&\\Psi_0,\n\\end{aligned}\\right.\n\\end{equation}\nwhere the nonlinearity is given by the matrix\n\\begin{equation}\\label{eq:B_tilde}\n\\tilde B[\\Psi]=\\left(\\begin{array}{cc}\\beta_{11}|\\psi_1|^2&\\beta_{12}\\psi_1\\psi_2^*\\\\\n\\beta_{12}\\psi_1^*\\psi_2&\\beta_{22}|\\psi_2|^2\\end{array}\\right),\n\\end{equation}\nand $A$ determines the linear coupling,\n\\begin{equation}\\label{eq:A}\nA=\\left(\\begin{array}{cc}&\\lambda\\\\\\lambda&\\end{array}\\right).\n\\end{equation}\nThe energy and the mass associated to system \\eqref{eq:nlss} are the following\n\\begin{equation}\\label{eq:nlss_en2}\nE(t)=\\int_{\\R^N}\\Big(\\frac12|\\nabla\\Psi|^2+\\frac{\\gamma^2}{2}|x|^2|\\Psi|^2+\\frac12\\Psi^*\\tilde B[\\Psi]\\Psi\n+2\\lambda\\Re(\\psi_1^*\\psi_2) \\Big)(x,t)dx.\n\\end{equation}\n\\begin{equation}\\label{eq:nlss_mass2}\nM(t)=\\int_{\\R^N}|\\Psi(x,t)|^2dx,\n\\end{equation}\nand they are conserved quantities along the flow of solutions to \\eqref{eq:nlss}.\nIt is straightforward to see that, for any $\\Psi\\in\\Sigma(\\R^N)$, the energy in \\eqref{eq:nlss_en2} is finite.\n\\newline\nAs we already anticipated in the Introduction, in this paper we shall prove that, after a suitable transformation of the system, the asymptotic behavior is described by two effective coupled NLS equations which have the same inter- and intra-specific scattering lengths. Indeed, the two effective NLS equations are given by\n\\begin{equation}\\label{eq:u_sys_main}\ni\\d_tu_j=-\\frac12\\Delta u_j+\\frac{\\gamma^2}{2}u_j+\\chi|u_j|^2u_j+\\tilde{\\chi} |u_k|^2u_j,\\quad j, k = 1, 2,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:chi}\n\\tilde\\chi:=\\frac{\\beta_{11}+\\beta_{22}}{2} \\quad \\chi:=\\frac{\\beta_{11}+2\\beta_{12}+\\beta_{22}}{4}.\n\\end{equation}\nThe main Theorem we prove in this paper is\n\\begin{theorem}\\label{thm:main1}\nLet $\\Psi_0\\in\\Sigma(\\R^N)$. For any $\\lambda\\in\\R$, let $\\Psi^\\lambda$ be the unique maximal solution to \\eqref{eq:nlss}. Let $U^t=(u_1,u_2)$ be the solution of the system \\eqref{eq:u_sys_main} in $[0, S_{max})\\times\\R^N$, with initial datum\n\\begin{equation*}\n\\left(\\begin{array}{c}u_1(0)\\\\u_2(0)\\end{array}\\right)=\\left(\\begin{array}{c}\\frac{1}{\\sqrt2}(\\psi_{1, 0}+\\psi_{2, 0})\\\\\n\\frac{1}{\\sqrt2}(\\psi_{1, 0}-\\psi_{2, 0})\\end{array}\\right),\n\\end{equation*}\nwhere $S_{max}$ is the maximal existence time for $U$. Then for any time $00$ and a unique solution\n$\\Psi\\in\\mathcal C([0, \\delta];\\Sigma(\\R^N))$ to \\eqref{eq:nlss2}. Moreover we have\n\\begin{equation}\\label{eq:lin_pert}\n\\|\\Psi\\|_{L^\\infty([0, \\delta];\\Sigma(\\R^N))}\\leq2C\\|\\Psi_0\\|_{\\Sigma(\\R^N)}.\n\\end{equation}\nFurthermore, the solution $\\Psi$ can be extended to a maximal interval $[0, T_{max})$, and the blow-up alternative holds true, namely if $T_{max}<\\infty$, then\n$$\\lim_{t\\to T_{max}}\\|\\nabla\\Psi(t)\\|_{L^2(\\R^N)}=\\infty;$$\nFinally, for any $00$ will be chosen later, endowed with the distance\n\\begin{equation*}\nd(\\Psi, \\tilde\\Psi):=\\|\\Psi-\\tilde\\Psi\\|_{L^\\infty_tL^2_x([0, \\delta]\\times\\R^N)}\n+\\|\\Psi-\\tilde\\Psi\\|_{L^{8\/N}_tL^4_x([0, \\delta]\\times\\R^N)}.\n\\end{equation*}\nBy the hypotheses on $\\mathcal N(\\Psi)$ we have\n\\begin{equation*}\n\\begin{aligned}\n\\|\\mathcal N_1(\\Psi)-\\mathcal N_1(\\tilde\\Psi)\\|_{L^2(\\R^N)}\\leq &C\\|\\Psi-\\tilde\\Psi\\|_{L^2(\\R^N)}\\\\\n\\|\\mathcal N_2(\\Psi)-\\mathcal N_2(\\tilde\\Psi)\\|_{L^{4\/3}(\\R^N)}\\leq &C(\\|\\Psi\\|_{L^4(\\R^N)}^2+\\|\\tilde\\Psi\\|_{L^4(\\R^N)}^2)\\|\\Psi-\\tilde\\Psi\\|_{L^4(\\R^N)}\\\\\n\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\mathcal N_1(\\Psi)\\|_{L^2(\\R^N)}\\leq &C\n\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\Psi\\|_{L^2(\\R^N)}\\\\\n\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\mathcal N_2(\\Psi)\\|_{L^{4\/3}(\\R^N)}\n\\leq &C\\|\\Psi\\|_{L^4(\\R^N)}^2\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\Psi\\|_{L^4(\\R^N)}.\n\\end{aligned}\n\\end{equation*}\nLet us now consider the operator\n\\begin{equation*}\nG[\\Psi]:=S_0(t)\\Psi_0-i\\int_0^tS_0(t-s)\\mathcal N(\\Psi)(s)ds.\n\\end{equation*}\nBy using the commutation rules for the Hamiltonian $H$ and Strichartz estimates, we have on the space-time slab $[0, \\delta]\\times\\R^N$\n\\begin{equation*}\n\\begin{aligned}\n\\|G[\\Psi]\\|_{L^\\infty_t\\Sigma_x}+&\n\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)G[\\Psi]\\|_{L^{8\/N}_tL^4_x}\\\\\n&\\leq C\\|\\Psi_0\\|_{\\Sigma(\\R^N)}\n+C\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\mathcal N_1(\\Psi)\\|_{L^1_tL^2_x}\n+C\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\mathcal N_2(\\Psi)\\|_{L^{\\frac{8}{8-N}}_tL^{4\/3}_x}\\\\\n&\\leq C\\|\\Psi_0\\|_{\\Sigma(\\R^N)}\n+C\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\Psi\\|_{L^1_tL^2_x}\n+C\\|\\Psi\\|_{L^\\infty_tL^4_x}^2\n\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\Psi\\|_{L^{\\frac{8}{8-N}}_tL^4_x}.\n\\end{aligned}\n\\end{equation*}\nAnalogously we obtain\n\\begin{equation*}\n\\begin{aligned}\n\\|G[\\Psi]-G[\\tilde\\Psi]\\|_{L^\\infty_tL^2_x}+&\\|G[\\Psi]-G[\\tilde\\Psi]\\|_{L^{8\/N}_tL^4_x}\n\\leq C\\|\\mathcal N_1(\\Psi)-\\mathcal N_1(\\tilde\\Psi)\\|_{L^1_tL^2_x}+C\\|\\mathcal N_2(\\Psi)-\\mathcal N_2(\\tilde\\Psi)\\|_{L^{\\frac{8}{8-N}}_tL^{4\/3}_x}\\\\\n&\\leq C\\|\\Psi-\\tilde\\Psi\\|_{L^1_tL^2_x}+C(\\|\\Psi\\|_{L^\\infty_tL^4_x}^2+\\|\\tilde\\Psi\\|_{L^\\infty_tL^4_x}^2)\n\\|\\Psi-\\tilde\\Psi\\|_{L^{\\frac{8}{8-N}}_tL^4_x}.\n\\end{aligned}\n\\end{equation*}\nBy the Sobolev embedding $H^1\\hookrightarrow L^4$ and by using H\\\"older's inequality in time in the previous expressions, we have\n\\begin{equation*}\n\\begin{aligned}\n\\|G[\\Psi]\\|_{L^\\infty_t\\Sigma_x}\n+\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)G[\\Psi]\\|_{L^{8\/N}_tL^4_x}\n\\leq &C\\|\\Psi_0\\|_{\\Sigma}\n+C(\\delta+\\delta^{\\frac{8-2N}{8}}M^2)M\\\\\n\\|G[\\Psi_1]-G[\\Psi_2]\\|_{L^{8\/N}_tL^4_x}\\leq &\nC(\\delta+\\delta^{\\frac{8-2N}{8}}M^2)d(\\Psi, \\tilde\\Psi).\n\\end{aligned}\n\\end{equation*}\nNow, we choose $M, \\delta$ such that\n\\begin{equation*}\n\\begin{aligned}\nC\\|\\Psi_0\\|_{\\Sigma(\\R^N)}=&\\frac{M}{2}\\\\\nC(\\delta+\\delta^{\\frac{8-2N}{8}}M^2)\\leq&\\frac12,\n\\end{aligned}\n\\end{equation*}\nso that we have\n\\begin{equation}\\label{eq:21}\n\\begin{aligned}\n&\\|G[\\Psi]\\|_{L^\\infty_t\\Sigma_x}\n+\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)G[\\Psi]\\|_{L^{8\/N}_tL^4_x}\\leq M\\\\\n&d(G[\\Psi], G[\\tilde\\Psi])\\leq\\frac12d(\\Psi, \\tilde\\Psi).\n\\end{aligned}\n\\end{equation}\nThis implies $G$ is a contraction in $K$, therefore there exists a unique $\\Psi\\in K$ such that\n\\begin{equation*}\n\\Psi(t)=G[\\Psi](t)=S_0(t)\\Psi_0-i\\int_0^tS_0(t-s)\\mathcal N(\\Psi)(s)ds.\n\\end{equation*}\nHence $\\Psi\\in\\mathcal C([0, \\delta];\\Sigma(\\R^N))\\cap L^{8\/N}([0, \\delta];L^4(\\R^N))$ is a solution to \\eqref{eq:nlss2} in $[0, \\delta]$.\nFrom \\eqref{eq:21} we also see\n\\begin{equation*}\n\\|\\Psi\\|_{L^\\infty([0, \\delta];\\Sigma(\\R^N))}\\leq M=2C\\|\\Psi_0\\|_{\\Sigma(\\R^N)},\n\\end{equation*}\nwhich proves \\eqref{eq:lin_pert}. Analogously, by Strichartz estimates we also have\n\\begin{equation*}\n\\|\\left(\\begin{array}{c}1\\\\\\nabla\\\\|\\cdot|\\end{array}\\right)\\Psi\\|_{L^q_tL^r_x([0, \\delta]\\times\\R^N)}\\leq \nC\\|\\Psi_0\\|_{\\Sigma(\\R^N)}+C\\delta^{\\frac{8-2N}{N}}(1+M^2)M\\leq M.\n\\end{equation*}\nFurthermore, from the proof in the fixed point argument above we also infer that we may extend the solution as long as the \n$L^2-$norm of the gradient of the solution remains bounded, hence the blow-up alternative holds true. This implies we can extend the solution to a maximal interval $[0, T_{max})$, and moreover for any $00$. In this case indeed the matrix\n\\begin{equation*}\nB=\\left(\\begin{array}{cc}\\beta_{11}&\\beta_{12}\\\\\\beta_{12}&\\beta_{22}\\end{array}\\right)\n\\end{equation*}\nis positive definite, it follows that \n \\begin{equation*}\n \\int_{\\R^N}\\big(\\frac{\\beta_{11}}{2}|\\psi_2|^4+\\beta_{12}|\\psi_1|^2|\\psi_2|^2+\\frac{\\beta_{22}}{2}|\\psi_2|^2\\big)(x,t)dx \\geq 0,\n \\end{equation*}\nthus using the energy conservation we get uniform bounds on the $\\Sigma$-norm of the solution $\\Psi$.\n \\begin{equation*}\n \\frac{1}{2}\\|\\nabla\\Psi(t)\\|_{L^2}^2+\\frac{\\gamma^2}{2}\\||\\cdot|\\Psi\\|_{L^2}^2\n \\leq E(t)-2\\lambda\\int_{\\R^N}\\Re(\\psi_1^*\\psi_2)(x,t)dx\n \\leq E(t)+|\\lambda| M(t)=E(0)+|\\lambda| M(0).\n \\end{equation*}\n For all other cases see \\cite{Juengel11}.\n\\end{proof}\nIn the next Theorem we show the cases in which there is possible occurrence of blow-up in finite time. We describe two situations, the first one in which the nonlinearity is negative definite. In this case we apply a method introduced by Carles \\cite{Car} for focusing NLS equations with a confining potential, by using a modified energy functional (for more discussions about this modified energy and its interpretation see \\cite{Car}). In the second case we assume that at least one of the coefficients $\\beta_{ij}$'s is focusing and that the energy is negative enough so that conditions \n(ii) or (iii) below are fulfilled. In this case we apply the method by Glassey \\cite{G} using virial identities.\n\\begin{remark}\nNote that the following sufficient condition for blow-up is valid also in the supercritical case $N=3$, whereas in \\cite{Juengel11} the authors consider only the mass-critical case $N=2$.\n\\end{remark}\n\\begin{theorem}\\label{thm:blow-up}\nLet $\\Psi\\in\\mathcal C([0, T_{max});\\Sigma(\\R^N))$ be the solution to \\eqref{eq:nlss} as in Corollary \\ref{cor:lwp1}, and let us define the virial potential \n\\begin{equation*}\nI(t)=\\int_{\\R^N}|x|^2|\\Psi(x,t)|^2dx.\n\\end{equation*}\nLet us assume $N\\geq2$ and one of the following conditions is satisfied\n\\begin{itemize}\n\\item[(i)] the nonlinearity is \\emph{negative definite}, i.e. we either have $\\beta_{12}^2-\\beta_{11}\\beta_{22}<0$ with \n$\\beta_{11}<0$ and $\\beta_{12}\\geq 0$ or $\\beta_{11}, \\beta_{12}, \\beta_{22}<0$, and we also assume that\n\\begin{equation*}\nE(0)+\\frac{|\\lambda|}{2}M(0)<\\frac{\\gamma^2}{2}I(0);\n\\end{equation*}\n\\item[(ii)] $\\min\\{\\beta_{11},\\beta_{22},\\beta_{12}\\}<0$\n$$\\frac{2N}{N+2}\\pare{E(0)+|\\lambda|M(0)}<\\frac{\\gamma^2}{2}I(0);$$\n\\item[(iii)] $\\min\\{\\beta_{11},\\beta_{22},\\beta_{12}\\}<0$; \n$I'(0)<0$ and $$\\frac{2N}{N+2}\\pare{E(0)+|\\lambda|M(0)}<-\\frac{\\gamma}{\\sqrt{2+N}}I'(0).$$\n\\end{itemize}\nThen the solution blows-up at a finite time, i.e. $\\exists\\;00$ sufficiently large. \\newline\nWe can thus repeat the same argument, starting at time $t=\\ell$. Consequently we have the solution $\\Phi^\\lambda$ exists in \n$[0, \\delta+\\ell]$ and we have a uniform bound for the $\\Sigma-$norm of $\\Phi^\\lambda$,\n\\begin{equation*}\n\\sup_{|\\lambda|\\geq\\Lambda}\\|\\Phi^\\lambda\\|_{L^\\infty((0, \\delta+\\ell); \\Sigma(\\R^N))}\\leq 2CM.\n\\end{equation*}\nThus we can apply again Lemma \\ref{lemma:14} in the time interval $[\\ell, \\delta+\\ell]$ and obtain the convergence of \n$\\Phi^\\lambda$ to $U$. Again, because of the convergence we have\n\\begin{equation*}\n\\lim_{|\\lambda|\\to\\infty}\\|\\Phi^\\lambda(\\delta+\\ell)-U(\\delta+\\ell)\\|_{\\Sigma(\\R^N)}=0,\n\\end{equation*}\nand in particular\n\\begin{equation*}\n\\sup_{|\\lambda|\\geq\\Lambda'}\\|\\Phi^\\lambda(\\delta+\\ell)\\|_{\\Sigma(\\R^N)}\\leq M.\n\\end{equation*}\nThus we repeat this argument to prove the result in the whole time interval $[0, T]$ and Theorem \\ref{thm:main} is proved.\n\\qed\\newline\n\\section*{Acknowledgements}\n The work of the second author has been supported by the Hertha-Firnberg Program \n of the Austrian Science Fund (FWF), grant T402-N13.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}