∎

11institutetext: Alona Zharova 22institutetext: 22email: alona.zharova@hu-
berlin.de 33institutetext:  1 School of Business and Economics, Humboldt-
Universität zu Berlin, Germany
2 domino e.V., Germany
3 Singapore Management University, Singapore

# How to Measure the Performance of a Collaborative Research Center 111
Financial support from the German Research Foundation (DFG) via Collaborative
Research Center 649 ”Economic Risk” and International Research Training Group
1792 ”High Dimensional Nonstationary Time Series”, Humboldt-Universität zu
Berlin, is gratefully acknowledged. We are thankful for the assistance
provided by Nicole Hermann und Dominik Prugger.

Alona Zharova1 Janine Tellinger-Rice1,2 Wolfgang Karl Härdle1,3

(Received: 3 March 2018 / Accepted: 22 September 2018)

###### Abstract

New Public Management helps universities and research institutions to perform
in a highly competitive research environment. Evaluating publicly financed
research improves transparency, helps in reflection and self-assessment, and
provides information for strategic decision making. In this paper we provide
empirical evidence using data from a Collaborative Research Center (CRC) on
financial inputs and research output from 2005 to 2016. After selecting
performance indicators suitable for a CRC, we describe main properties of the
data using visualization techniques. To study the relationship between the
dimensions of research performance, we use a time fixed effects panel data
model and fixed effects Poisson model. With the help of year dummy variables,
we show how the pattern of research productivity changes over time after
controlling for staff and travel costs. The joint depiction of the time fixed
effects and the research project’s life cycle allows a better understanding of
the development of the number of discussion papers over time.

JEL C23 $\cdot$ C13 $\cdot$ M19

###### Keywords:

Research performance Fixed effects panel data model Network Collaborative
research center

###### MSC:

62-07 62-09 62P20

##  1 Introduction

New Public Management (NPM) emerged in the 1980s (Hood 1991) with the goal of
improving efficiency and overall performance of public sector institutions by
using business management approaches and models. NPM places a strong focus on
permanent monitoring and evaluation of performance. Measuring research
performance allows an analysis of the structural issues in science. It can
thus facilitate the development of a scientific system and strengthen
excellence in research.

This paper discusses Collaborative Research Centers (CRC) – long-term
university-based research institutions funded by the German Research
Foundation (DFG 2018). Evaluating publicly financed research results improves
transparency, helps in reflection and self-assessment, and provides
information for strategic decision making. Periodic monitoring of resource use
and interim results allows CRC management to keep the finger on the pulse and
to react to unfavourable phenomena promptly or to develop options for
improvement; thereby, supporting success of the CRC.

There are numerous studies that concentrate on the evaluation of university
research or research institutions in general (Pastor et al. 2015, Van den
Berghe et al. 1998). Lee (2010) and Bolli and Somogyi (2011) discuss
performance measurements for departments and research units. Jansen et al.
(2007) and Carayol and Matt (2004) further investigate performance indicators
for research groups. However, a CRC differs from common research units or
institutions, because of its interdisciplinary background. The performance
indicators used for the evaluation of a CRC should be designed specifically
for its needs and purposes in order to reflect the behaviour of involved
research fields and other underlying characteristics.

In this paper we focus on a selection of performance indicators for
intermediate and final evaluations suitable for broad applicability within
CRCs and identifying a relationship between productivity and resource use of
CRCs that may have implications for funding policy. The goals of this paper
include: (i) selecting performance indicators suitable for a CRC; (ii)
visualizing goals vs. results, societal impact and the interdisciplinarity
structure of research results of a CRC; (iii) analysis of a dependence
structure between financial inputs and research output of a CRC and
development of research productivity over time.

To achieve these objectives, we use twelve years (2005 – 2016) of
Collaborative Research Center 649 "Economic Risk" (CRC 649) data on 35 sub-
projects. For each sub-project we observe yearly staff costs, travel costs and
number of discussion papers (DPs). The life span of each sub-project varies,
which results in an unbalanced panel.

Schröder et al. (2014) indicate that the proposal for funding determines
objectives for the research activity. To examine the correspondence between
objectives and research results of the CRC, we carry out a semantic analysis
of proposals and abstracts from published DPs. As a result, we find that both
use 50% of the same words.

Apart from research activity, a CRC has an impact on society through public
events, transfer of knowledge or promotion of young researchers. For instance,
young researchers usually perform specific theoretical or practical research
that is also used for their Ph.D. thesis. Collecting data on their further
career helps to better understand this impact. With the help of a mosaic plot,
we visualize three important dimensions of young researchers careers after
receiving their Ph.D. within the CRC: gender, location and area of work. For
example, we show that almost 70% of young researchers who received their Ph.D.
during CRC membership found later a job in academia.

Through a network analysis, we illustrate the interdisciplinarity structure of
the research results and find out that most DPs were published in the fields
of mathematical and quantitative methods, followed by financial economics,
macroeconomics and monetary economics.

To study the relationship between research outcomes and funding for the CRC,
we regress the number of DPs on staff and travel costs using sub-project-level
data. With the help of year dummy variables added to the model, we show how
the pattern of the sub-projects’ productivity changed from 2005 to 2016 after
controlling for staff and travel costs. Since the level of spending from the
previous year and the preceding number of DPs may influence the current number
of DPs, we additionally control for the lagged variables. The productivity of
each sub-project may differ due to some heterogeneity or individual effects,
such as the skills of a principal investigator (PI), average abilities or
skills of researchers employed at the sub-project, or the specific behavior of
a research field. For instance, working on a publication with one vs. more co-
authors, writing in English vs. other languages, or publishing in books vs.
articles may affect the research outcomes (Zharova et al. 2017). Therefore, we
allow for the possibility of individual sub-project’s effects. Considering the
data structure, we apply a time fixed effects panel data (FE) model. Since the
number of DPs is a count variable, we also apply a fixed effects Poisson (FEP)
model.

We show that an increase of staff costs by 100% leads to an expected increase
in the number of DPs by roughly 43% (FE) or 1.62 DPs (FEP). Travel costs have
a diminishing effect on the number of DPs according to estimation results of
the considered models. The previous level of both staff and travel costs
negatively influence the number of DPs. We depict the estimates of
coefficients of the dummy variables for years and find that the development
trend corresponds with the stages of a project’s life cycle. For instance, the
most significant declines in the number of DPs take place during the stage of
theoretical and empirical research, whereas the finalization stage corresponds
with the growth in the number of published DPs.

The programmed R codes are available on the web-based repository hosting
service and collaboration platform GitHub.

The remainder of the paper is structured as follows. Literature review on
performance indicators is presented in Section 2. Section 3 describes the data
and provides some preliminary descriptive analyses. Section 4 introduces the
methodology and shows empirical results. Finally, Section 5 summarizes the
results.

##  2 Literature Review

The combination of a peer-reviewed process and quantitative indicators is
common practice in research performance assessment. The German Council of
Science and Humanities (WR, germ. - Wissenschaftsrat) suggests evaluating the
research institutions within three dimensions (research, promoting young
researchers and knowledge transfer), which contain nine research performance
criteria (WR 2004). We select five criteria relevant to a CRC and provide a
literature review on suitable indicators that may reflect the performance of
the CRC.

1\. Research quality shows originality and novelty of research outputs,
trustworthiness of methodology, impact and relevance for further research.

Table 1: Research quality. Indicator |  Definition |  Literature
---|---|---
Relative reception success |  |
CPub |  Relation of total number of citations (NCPub) to the total number of publications (NPub) |  WR (2012), Diem and Wolter (2013), Donner and Aman (2015)
CPub/FCm |  Number of citations per publication in relation to the citation’s average of the field |  WR (2012), Abramo and D’Angelo (2011), Moed et al. (2011), Van den Berghe et al. (1998)
CPub/JCm |  Number of citations per publication in relation to the citation’s average of the journal |  Moed (2010), WR (2012)

2\. Effectiveness reflects the contribution of all sub-projects to the
development of expertise in the research field within the CRC and beyond.

Table 2: Effectiveness. Indicator |  Definition |  Literature
---|---|---
Research activity |  |
NCosts |  Total amount of the third party expenses (TPE) |  WR (2012), Schmoch and Schubert (2009)
NStaff |  Total number of staff financed from third party funds (TPF) |  Carayol and Matt (2004), WR (2012)
RAunit |  Research activity of unit (sub-project, SP) – multiplication of the total number of publications and the total number of citations of a unit with regard to the institutions-wide number of citations for the analyzed period (RASP=NPubSP*CPubSP/CPubCRC) |  Pastor et al. (2015)
Research productivity |  |
NPub |  Total number of publications |  WR (2012), Abramo and D’Angelo (2011), Diem and Wolter (2013), Moed et al. (2011), Hornbostel (1991)
NCPub |  Total number of citations |  WR (2012)
FNPub |  Fractional productivity – total number of contributions to publications, where each contribution is a publication divided by the number of co-authors |  Abramo et al. (2009), Abramo and D’Angelo (2011)
ScSPub |  Scientific strength – weighted sum of publications authored by each person, where the weights for each publication is the number of citations per publication in relation to the citation’s average of the field (CPub/FCm) |  Abramo and D’Angelo (2011), Abramo et al. (2009)
h |  h-index |  Hirsch (2005), Bornmann (2013)
Visibility of the CRC |  |
AbsCPub |  Absolute citation count in the light of maximum citation count of a single publication (CPubmax) and the number of non-cited publications (NncPub) |  WR (2012)
Reputation |  |
|  List of scientific prizes and awards |  Zheng and Liu (2015), WR (2012)
Professional activity |  |  WR (2012)
|  Editorships |
|  Review activities |
|  Editorial board memberships |
|  Academic functions |
|  Academic memberships |
|  Organized conferences and workshops |

3\. The efficiency criterion describes a quantity of research outputs in
relation to a specific input, i.e. total costs, staff expenditures, number of
staff, etc.

Table 3: Efficiency. Indicator |  Definition |  Literature
---|---|---
NPub/NStaff |  Relation of the number of publications (NPub) to the number of research staff (NStaff) |  Pastor and Serrano (2016), WR (2012), Abramo and D’Angelo (2011)
NCPub/NStaff |  Relation of the number of citations of publications (NPub) to the number of research staff (NStaff) |  WR (2012), Lee (2010)
NCosts/NStaff |  Relation of the TPE to the total number of research staff (NStaff) |  WR (2012), Pastor and Serrano (2016), Barra and Zotti (2016)

4\. Research enabling relates to scientific activities that facilitate and
support the research of young researchers.

Table 4: Research Enabling / Promotion of young researchers. Indicator |  Definition |  Literature
---|---|---
Promotion of young |  |
researchers |  |
NYR |  Total number of positions for young researchers |  WR (2012)
NPh.D. |  Total number of defended Ph.D. |  WR (2012), Diem and Wolter (2013), Grözinger and Leusing (2006), Schmoch and Schubert (2009)
DPh.D. |  Average duration of Ph.D. study |  WR (2004)
NPubPh.D. |  Total number of publications by young researchers |  WR (2004)
|  List of awards and prizes of young researchers |  WR (2012)
|  List of calls and appointments for young researchers |  WR (2012)

5\. Knowledge transfer defines the transfer of research results and products
or distribution of knowledge.

Table 5: Knowledge Transfer. Indicator |  Definition |  Literature
---|---|---
NPat |  Number of patents |  WR (2011), Carayol and Matt (2004)
|  List of Transfer projects |
|  List of activities in public relations |  WR (2012)
|  List of research products and teaching materials |  WR (2012)

##  3 Data

Collaborative Research Centers (CRC) are interdisciplinary research
institutions financed through the German Research Foundation (germ. - Deutsche
Forschungsgemeinschaft, DFG). The goal of a CRC is to pursue interdisciplinary
innovative research by bringing together scholars from different research
fields within multiple research projects, also called sub-projects. The
classical CRC consolidates cooperation between several universities or non-
university research institutions with at least 60% of all sub-projects based
in the coordinating university (DFG 2018).

CRCs are granted for four years and depending on the results of the interim
evaluations can be prolonged twice for a maximum period of twelve years.
During the assessment each sub-project undergoes a critical appraisal.
Depending on a change in research program or staff turnover (professors), a
CRC can also submit proposals for new sub-projects. As a result, the number of
research projects may vary between phases.

In this paper we provide empirical evidence using data from a Collaborative
Research Center 649 "Economic Risk" (hereinafter referred to as the CRC). The
CRC was launched in 2005 for a four-year term and extended twice, for a total
life span of twelve years. As an interdisciplinary research center, it
combined economics, mathematics and statistics and pursued research within
three primary areas: i) microeconomics, in particular individual and
contractual answers to risk; ii) quantitative projects, in particular
financial markets and risk assessment; iii) macroeconomic risks. For more
information, we refer to the website of the CRC (CRC 649 2016).

The total number of the CRC sub-projects within three four-year phases is 35,
but the number of sub-projects per phase varies from 16 to 21. Since the sub-
projects of the CRC have different life periods, the data set does not have
the observations for all years that indicates an unbalanced panel, see Figure
1. The main reason for the panel being unbalanced is the attrition of sub-
projects, as a result of research project’s termination or the leave of
principal investigators to other universities, and the establishment of new
research projects during the prolongation phases. For instance, twelve sub-
projects had a life cycle of four years, eleven sub-projects lasted for eight
years and five sub-projects existed twelve years (see Figure 1).

Figure 1: Distribution of sub-projects (SP) over life span in years.

Principal investigators (PIs) lead sub-projects. From 35 sub-projects 83% have
one PI and 17% have two PIs. Since three PIs participate in two sub-projects,
the CRC counts 38 PIs in total over twelve years. PIs of all three academic
ranks participate in the CRC: full professors (76%), junior professors (19%)
and postdoctoral researchers (5%).

The CRC uses 62% of resources on average to finance the research staff working
within sub-projects, in particular doctoral (Docs) and postdoctoral (PostDocs)
researchers. In addition, all members of the CRC may use its central funds for
travel costs, organizing conferences and workshops, inviting guest lecturers
and researchers, gender equality etc.

The amount of research staff working within sub-projects differs, depending on
the scope and complexity of the research program. Each sub-project counts from
0.5 to 2.5 full-time equivalents (FTEs) of researcher positions per year. The
FTEs are often split and used to hire more research staff, i.e. 2 researchers
with 50% financing, or to top up researchers that are already employed and who
are financed by other sources. Figure 2 shows the distribution of sub-projects
according to the number of FTEs per year. For instance, 21 sub-projects have
one FTE per year on average, eight sub-projects hire staff on 0.5 FTEs, four
sub-projects use 1.5 FTEs and two sub-projects have each 2 and 2.5 FTEs.

Figure 2: Distribution of sub-projects according to the number of research
staff (in FTE per year).

In this paper we use data from annual financial reports, internal
publications’ and discussion papers’ (DPs) databases and CRC’s newsletter.
Additional insight is gathered from the texts of one proposal for a launch and
two proposals for a prolongation of the CRC 649 (2005–2008, 2009–2012,
2013–2016) which were submitted to the DFG. On the one hand, one can see such
proposals as goals that the CRC sets for each period. On the other hand, the
published DPs encompass the achieved results of the research activity. We
undertake a semantic analysis on both informational sources, i.e. 61 summaries
of sub-projects from three proposals and abstracts of 771 DPs. The two word
clouds of the top 75 keywords are illustrated in Figure 3. We find that both
use 50% of the same words. The different size of the same words, for instance
the word "risk", indicates that the number of times the word is mentioned in
the proposals and abstracts differs.

Figure 3: Semantic analysis of goals (left; 61 summaries from sub-projects of
three proposals for the CRC) vs. results (right; 771 abstracts from DP).
Figure 4: Mosaic plot of job type, location and gender of 65 CRC members who
received their Ph.D. between 2005 and 2016 (as of Dec 2016)

One of the primary goals of a CRC is the high-quality instruction, supervision
and support of young researchers. The common result of this process is a Ph.D.
defence. Collecting data on the further career of the young researchers helps
to better understand the impact on society. For instance, one may wonder how
many females that worked and defended their Ph.D. thesis at the CRC are
afterward working in academia in Germany? To visualize such data we use a
mosaic plot in Figure 4.

The vertical axis splits the individuals according to their gender. The data
are further divided into two groups on the upper horizontal axis according to
the location of the job. The lower horizontal axis shows how many people
received a contract in academia or other fields. The width and height of each
segment represent the number of observations within each group. Consider the
65 members of the CRC that received their Ph.D. from 2005 to 2016. There are
11 female researchers that received jobs in academia in Germany and 6 in other
countries. For males that stayed in academia, the number is 21 for Germany and
7 for other countries. This means that almost 70% of young researchers who
received their Ph.D. during CRC membership found later a job in an academic
institution.

Figure 5: Network of 760 discussion papers (yellow) and 20 JEL codes (blue)
published from 2005 to 2016.

The proportion of 36.9% of female researchers is quite low in comparison to
50.4% for female doctoral students within CRCs in social sciences and
humanities, but higher than 25.7% within CRC in mathematical and natural
sciences (DFG 2017). However, since the CRC pursued interdisciplinary research
in both social and mathematical sciences, the CRC proportion corresponds to
the value in-between. As a part of the communication processes with alumni and
mentoring of CRC young researchers, the CRC invited its former members who got
promoted in academia as guest lecturers for CRC seminars or as guest
researchers to work on papers jointly with PIs and/or younger CRC generations.

In order to understand if the intended interdisciplinarity occurred, we
analyze DPs that serve as an outcome of the CRC research activity. Almost each
DP has codes indicating subject fields according to the Journal of Economic
Literature (JEL) classification in the economic sciences (see JEL 2018).

We show the network of collaborating disciplines in Figure 5. The small gold
circles introduce the DPs, whereas the nodes leading to the bigger blue
circles indicate the JEL code of the corresponding research area. The size of
each blue circle reflects the relative number of references to DPs. The
explanation of JEL codes is given in Table LABEL:JELcodes. For instance, most
of the DPs were published in the C area, i.e. mathematical and quantitative
methods. They are followed by G (financial economics), E (macroeconomics and
monetary economics) and D (microeconomics). These four fields with higher
research output correspond to the three primary areas of the CRC. Note that
the DPs that involve research in more than one field are connected to two or
more JEL codes simultaneously. This confirms the interdisciplinary character
of the CRC research output.

Table 6: JEL Classification System Code |  Research field
---|---
A |  General Economics and Teaching
B |  History of Economic Thought, Methodology, and Heterodox Approaches
C |  Mathematical and Quantitative Methods
D |  Microeconomics
E |  Macroeconomics and Monetary Economics
F |  International Economics
G |  Financial Economics
H |  Public Economics
I |  Health, Education, and Welfare
J |  Labor and Demographic Economics
K |  Law and Economics
L |  Industrial Organization
M |  Business Administration and Business Economics / Marketing / Accounting / Personnel Economics
N |  Economic History
O |  Economic Development, Innovation, Technological Change, and Growth
P |  Economic Systems
Q |  Agricultural and Natural Resource Economics / Environmental and Ecological Economics
R |  Urban, Rural, Regional, Real Estate, and Transportation Economics
Y |  Miscellaneous Categories
Z |  Other Special Topics

One more factor influencing the variability of the number of DPs across
research fields is the area of expertise of PIs and research staff. Figure 6
shows the cumulative number of PIs within their areas of expertise and Figure
7 depicts the cumulative number of CRC research staff (in FTE) working within
same research areas for twelve years. Since the attrition of some sub-projects
and establishment of new ones influences the availability of PIs and research
staff and accordingly their expertise within the CRC life cycle, we use
cumulative numbers. We also use weights for the number of the sub-projects and
expertise areas for each PI to equalize the total time available for research.
For example, the PI who is an expert in four research areas receives 0.25 for
each JEL code and the PI who leads two sub-projects has 0.5 for the
distribution within JEL areas of each project.

Figure 6: Cumulative number of PIs (in PI years; full professors - blue,
junior professors - red, postdoctoral researchers - orange) from 2005 to 2016
(weighted by the number of research fields and sub-projects) with expertise in
corresponding JEL research fields.  Figure 7: Cumulative number of research
staff in FTE (in staff years; weighted by the number of research fields) from
2005 to 2016 working within corresponding JEL research areas.

Figures 6 and 7 show, for instance, that the area D reveals 24 years of PIs
expertise and 15 years of research staff (in FTE) work. Both figures provide
evidence that the most expertise is concentrated within the area C, followed
by E, D, G and Q. This also explains the concentration of research output
within corresponding JEL areas in Figure 5. The correlation between the number
of DPs and number of PIs specializing in the same JEL areas is 93.8% (95% for
full professors only), whereas the correlation between the number of DPs and
the amount of research staff (in FTE) working within same fields is 95.1%.

##  4 Analysis of Research Productivity

The observed time series across the same sub-projects indicate the
longitudinal or panel structure of the data. To investigate the relationship
between the input and the output variables, we use the methods designed for
panels.

###  4.1 Methodology

The basic framework for the panel data analysis shows the model (Wooldridge
2002):

$y_{i}=\beta X_{i}+u_{i},\hskip 14.22636pti=1,\ldots,K,$ (1)

where $y_{i}=(y_{i1},\ldots,y_{iT})^{\top}$ is a ($1\times T$) vector of
observations for $t=1,2,\ldots,T$,
$X_{i}=(x_{i1}^{\top},\ldots,x_{iT}^{\top})^{\top}$ is a ($K\times T$) matrix
of observations, $\beta$ is a ($K\times 1$) vector of coefficients and $u_{i}$
is a ($1\times T$) vector of unobservables.

The unobserved sub-project’s effect may contain such factors as publishing
behavior in a research field, average researchers’ abilities or skills of
principal investigators of sub-projects that should be roughly constant over
time.

We allow for arbitrary correlation between the unobserved sub-project’s
heterogeneity or fixed effects $c_{i}$ and the observed explanatory variables
$x_{it}$ and, therefore, use the fixed effects model for each $i$ (Wooldridge
2016):

$y_{it}=\beta_{1}x_{it1}+\ldots+\beta_{k}x_{itk}+c_{i}+u_{it},\hskip
14.22636ptt=1,2,\ldots,T,\hskip 14.22636pti=1,2,\ldots,K,$ (2)

where $y_{it}$ includes dependent variables and $x_{it}$ independent variables
for individual $i$ at time $t$, $\beta_{1},\ldots,\beta_{k}$ are the unknown
coefficients, $c_{i}$ is individual effect or individual heterogeneity and
$u_{it}$ are idiosyncratic errors that change across individuals $i$ and time
$t$.

The fixed effects estimator (or the within estimator) is obtained as the
pooled OLS estimator on the time-demeaned variables. The strict exogeneity
assumption on explanatory variables,
$\operatorname{E}(u_{it}|\operatorname{\textbf{X}}_{i},c_{i})=0$, provides
that the fixed effects estimator is unbiased (Wooldridge 2016). As the number
of sub-projects (clusters) is large, statistical inference after OLS should be
based on cluster-robust standard errors to account for heteroscedasticity and
within-panel serial correlation (Cameron and Miller 2015).

Next, we are interested in the pattern of sub-projects’ productivity, i.e.
number of produced discussion papers, in different time periods. For this
purpose we use time fixed effects that change over time but are constant
across sub-projects. We include the dummy variables for $T-1$ years to avoid
the multicollinearity. Usually the first year is selected as a base year. The
time fixed effects model (FE) is (Stock and Watson 2003):

$y_{it}=\beta_{1}x_{it1}+\ldots+\beta_{k}x_{itk}+\delta_{1}+\delta_{2}D_{2}+\ldots+\delta_{T}D_{T}+c_{i}+u_{it},$
(3)

where $D_{2},\ldots,D_{T}$ are time effects and $\delta_{1},\ldots,\delta_{T}$
are the parameters to estimate.

When the dependent variable involves count data, it has a Poisson distribution
instead of a normal distribution. Hausman et al. (1984) introduce a fixed
effects Poisson model (FEP) as:

$\operatorname{E}(y_{it}|x_{i},a_{i})=a_{i}\mu(x_{it},\beta_{0}),\hskip
14.22636ptt=1,2,\ldots,T,$ (4)

where $\beta_{0}$ is a ($1\times K$) vector of unknown parameters to be
estimated and $\mu$ is the conditional mean. Wooldridge (1999) further derives
a consistent estimator for FEP using a quasi-conditional maximum likelihood
estimator (QCMLE).

###  4.2 Empirical Results

Before presenting the estimates, we explain some specifications of the model.
Since the yearly staff and travel costs are in nominal Euros, a slight
increase may happen due to inflation. One possibility to deal with this is an
adjustment using a Consumer Price Index (CPI). Another way to track the effect
of real spendings is the use of a logarithmic form. The interpretation of the
estimation results is then done using the level-log model. Here we use the
second approach.

Table 7 presents the results of FE (1) and (2), and FEP (3) and (4) models for
the number of DP as a dependent variable. The parameters of interest are staff
costs $\beta_{logStaffCosts}$, travel costs $\beta_{logTravelCosts}$ and year-
specific influence $\delta_{year}$. We also include lagged variables into the
models (2) and (4), since the current number of research outputs may be
affected by the previous number of publication and invested funds in economic
sciences and mathematics (Zharova et al. 2017). The models (2) and (4)
encompass the number of DPs $\beta_{nDP_{t-1}}$, staff costs
$\beta_{logStaffCosts}$ and travel costs $\beta_{logTravelCosts}$ in the time
$t-1$. The intercept $const$ is the average of individual effects $c_{i}$
across all sub-projects that is reported by Stata. We use cluster-robust
standard errors to account for heteroscedasticity. The significance level of
all estimates decreases as a result of standard error adjustment (Wooldridge
2016).

Dependent variable: $nDP$ | FE model | FEP model
---|---|---
| (1) | (2) | (3) | (4)
$\beta_{logStaffCosts}$ | $1.38^{**}$ | $1.62^{*}$ | $0.47^{***}$ | $0.43^{**}$
| $(0.61)$ | $(0.88)$ | $(0.12)$ | $(0.19)$
$\beta_{logTravelCosts}$ | $-0.94^{*}$ | $-0.34$ | $-0.22^{**}$ | $-0.04$
| $(0.55)$ | $(0.47)$ | $(0.10)$ | $(0.09)$
$\delta_{2006}$ | $1.61$ | $1.92$ | $0.25$ | $0$
| $(1.36)$ | $(1.61)$ | $(0.26)$ | (omit.)
$\delta_{2007}$ | $-1.20$ | $-2.55$ | $-0.30$ | $-0.98^{***}$
| $(1.38)$ | $(2.46)$ | $(0.31)$ | $(0.25)$
$\delta_{2008}$ | $-0.95$ | $-2.03$ | $-0.23$ | $-0.97^{***}$
| $(1.30)$ | $(2.10)$ | $(0.32)$ | $(0.36)$
$\delta_{2009}$ | $-2.05^{*}$ | $-3.16$ | $-0.54^{*}$ | $-1.20^{***}$
| $(1.13)$ | $(1.98)$ | $(0.33)$ | $(0.23)$
$\delta_{2010}$ | $-1.93^{*}$ | $-2.13$ | $-0.51^{*}$ | $-1.03^{***}$
| $(1.14)$ | $(2.68)$ | $(0.30)$ | $(0.31)$
$\delta_{2011}$ | $1.10$ | $0$ | $0.33^{*}$ | $0$
| $(0.70)$ | (omit.) | $(0.20)$ | (omit.)
$\delta_{2012}$ | $-2.79^{*}$ | $-3.60^{*}$ | $-0.71^{**}$ | $-1.90^{***}$
| $(1.46)$ | $(1.78)$ | $(0.34)$ | $(0.20)$
$\delta_{2013}$ | $-2.98^{**}$ | $-3.18$ | $-0.80^{**}$ | $-1.32^{***}$
| $(1.30)$ | $(2.52)$ | $(0.32)$ | $(0.41)$
$\delta_{2014}$ | $-1.36$ | $-1.73$ | $-0.44$ | $-0.99^{***}$
| $(0.95)$ | $(1.61)$ | $(0.27)$ | $(0.37)$
$\delta_{2015}$ | $-2.55^{**}$ | $-1.90$ | $-0.74^{**}$ | $-1.02^{***}$
| $(1.17)$ | $(1.77)$ | $(0.33)$ | $(0.31)$
$\delta_{2016}$ | $-0.30$ | $0$ | $-0.31$ | $-0.69^{*}$
| $(1.79)$ | (omit.) | $(0.36)$ | $(0.41)$
$const$ | $-2.37$ | $0.05$ |  |
| $(5.29)$ | $(10.09)$ |  |
$\beta_{nDP_{t-1}}$ |  | $0.02$ |  | $-0.01^{*}$
|  | $(0.16)$ |  | $(0.03)$
$\beta_{logStaffCosts_{t-1}}$ |  | $-0.66$ |  | $-0.25$
|  | $(0.59)$ |  | $(0.23)$
$\beta_{logTravelCosts_{t-1}}$ |  | $-0.21$ |  | $-0.02$
|  | $(0.58)$ |  | $(0.13)$
$R^{2}$ | $0.20$ | $0.21$ |  |
AIC | $706$ | $437$ | $463$ | $253$
BIC | $742$ | $469$ | $501$ | $258$
Table 7: Estimation results for time fixed effects (within) regression (models
(1) and (2)) and fixed effects Poisson regression (models (3) and (4)) with
number of DP ($nDP$) as the dependent variable and with robust standard errors
adjusted for clusters in sub-projects. ***, ** and * indicate a statistical
significance at 1%, 5% and 10% level, respectively. Standard deviation is
provided in brackets.

In (2) and (4) two years were omitted because of collinearity. In (3) five
observations were dropped out of the analysis because there was only one
observation per group. Performing analysis on unbalanced data slightly
increases the estimated effects of considered variables, but the general idea
remains unchanged (Wooldridge 2016).

In the model (1) we see the positive, significant effect of staff costs on the
number of DPs. 1.38/100 is the unit change in $n{DP}$ when staff expenses
increase by 1%. In other words, a 100% increase in staff costs leads to an
increase in the number of DPs by 1.38. Similarly, the model (2) shows that a
100% increase in staff costs increases the number of DPs by 1.62, holding
other variables constant. The fit of the FE models in (1) and (2) in Table 7
with $nDP$ as the dependent variable is almost the same, indicating that
including lagged variables does not significantly improve the model.

The FEP estimates have a different interpretation. For instance, the
coefficient on $\beta_{logStaffCosts}$ shows that a rise of staff costs by
100% leads to an increase of the number of DPs by 47% and 43% for models (3)
and (4) correspondingly. The coefficients on staff costs estimates for four
models in Table 7 are significant at 1% to 10% level. The influence of
previous values of staff costs on the number of DPs is negative and
insignificant.

Travel costs have a diminishing effect on the number of DPs according to
estimation results of considered models. The coefficient on
$\beta_{logTravelCosts}$ implies that, if we increase the travel costs by
100%, we expect the number of DP to decrease by 0.94 DP due to FE model (1).
The Poisson coefficient in (3) means that an increase in $logTravelCosts$ by
10% decreases $nDP$ by 2% (0.22$\times$0.10).

The coefficients on the year dummy variables reveal how the average
productivity of sub-projects changes over time. As 2005 is selected as the
base year, it is not reported with a coefficient. The coefficient on
$\delta_{2006}$ in model (1) shows that, on average, 1.6 DPs are attributed to
the year effect of 2006 holding other factors fixed. In Poisson case (3) one
suggests that the expected number of DPs in 2006 is 25% higher than on
average. The coefficients on $\delta_{2006}$ and $\delta_{2011}$ indicate a
positive increase in the number of DPs even without changing expenses. The
omission of year dummies would lead to the attribution of this positive
effects to the effects of costs change.

One can see that the year effects have a negative impact on the number of DPs
in the majority of years for all models. The project’s life cycle could
explain this. Research projects generally have five main stages: proposal
development, funding review, project start-up, performing research and
finalization of the project. We map the estimates of coefficients of the
models and fit the stages of life cycles in Figures 8 and 9. Proposal
development and funding review take place before 2005 and are not depicted in
these Figures.

Figure 8: Estimates of coefficients on the year dummy variables for the time
fixed effects (within) regression (models (1) and (2)). The lower part of the
figure shows the corresponding stage of the research project life cycle.
Figure 9: Estimates of coefficients on the year dummy variables for the fixed
effects Poisson regression (models (3) and (4)). The lower part of the figure
shows the corresponding stage of the research project life cycle.

A highly demanding application for a CRC requires extensive preliminary
research. The results of this preliminary research are published as DPs in the
first year 2005, thus, creating a specific bias towards later research outputs
produced during the CRC’s life time. The three following increases in the
number of DPs take place mainly in the finalization stage caused by the
publishing of research results at the final stage of projects. The research
outputs of the last phase in 2016 show part of the positive trend. In fact, 28
DPs were published in 2017, after the CRC was officially finished and
financing ended. Three major declines could be explained by a theoretical and
empirical stage of the research in the middle of each project life cycle. In
summary, the joint depiction of the time fixed effects and the research
project’s life cycle allows a better understanding of the development of the
number of DPs over time.

##  5 Conclusions

Our findings show that the performance indicators suitable for the
intermediate or final evaluation of a CRC facilitate a better understanding of
the dependence structure between research productivity and financial inputs,
and provide relevant information for successful decision and policy making.

As a result of semantic analysis of the text from proposals for the CRC
submitted to the DFG and the abstracts from published DPs, we find out that
two word clouds standing for goals and results use 50% of the same words.
Aiming to visualize a further career path of young researchers that received
their Ph.D. within the CRC, we use mosaic plot with dimensions gender,
location and area of work. We show that almost 37% are females and 70% of
young researchers found a job in academia.

We describe the interdisciplinary structure of research results with the help
of the network analysis. We show that such fields as mathematical and
quantitative methods, financial economics, macroeconomics and monetary
economics and microeconomics are the most reflected in the published DPs.
These fields correspond to the primary research areas of the CRC. Moreover,
the most of research output takes place in the areas that have more PIs with
corresponding expertise. Additionally, the sub-projects with more research
staff are expected to produce more DPs. The network visualization provides
also the evidence that one of the main goals of the interdisciplinary research
center – interdisciplinarity – is achieved.

Using time fixed effects panel data model and fixed effects Poisson model, we
show that increasing staff costs by 100% raises the number of DPs of a sub-
project by 1.62 or 43% according to the estimates of FE and FEP models
correspondingly. Travel costs have diminishing effect on the number of DPs
according to our estimation results. We analyse the change in productivity of
the CRC over time for reasons not captured by the other independent variables
using the dummy variables for years. We depict the estimates of coefficients
for years and show the possible association between the trend and the stages
of a project’s life cycle. For instance, the major declines in the number of
DPs take place during the stage of theoretical and empirical research, whereas
the finalization stage may correspond to the growth in the number of published
DPs.

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11institutetext: Facultad de Ciencias FÃsico-MatemÃ¡ticas, Universidad
Michoacana de San NicolÃ¡s de Hidalgo, Avenida Francisco J. MÃjica, 58060
Morelia, MichoacÃ¡n, Mexico

# Pair creation in electric fields

Christian Schubert 11<EMAIL_ADDRESS>
###### Abstract

Sauter-Schwinger pair creation in electromagnetic fields is a fundamental
prediction of QED and one of the motivations for the present efforts in
constructing super-strong lasers. I will give a historical review of the
subject, and then focus on two recent developments. The first one is the
worldline instanton formalism, a sophisticated version of the WKB
approximation that makes it possible to calculate the pair creation rate for
complicated field configurations. The second one is an adaptation of the
Dirac-Heisenberg-Wigner formalism suitable for a detailed study of the
formation of real particles in time and space.

##  1 Historical review

In his famous 1931 paper “Über das Verhalten eines Elektrons im homogenen
elektrischen Feld nach der relativistischen Theorie Diracs 222On the behavior
of an electron in a homogeneous electric field according to Dirac’s theory."
sauter  Sauter solved the Dirac equation in a constant electric field and
found a certain probability for “a transition from positive to negative
impulses”. This was later to be interpreted as electron-positron pair creation
by the field, or, from a modern field-theory point of view, as “vacuum
tunneling” of a virtual to a real pair: due to a statistical fluctuation
governed by the time-energy uncertainty relation, a virtual pair separates out
far enough to draw its rest mass energy from the field and become real (Fig.
1).

Figure 1: Vacuum tunneling of an electron-positron pair in an electric field.

Twenty years later, Schwinger schwinger51  computed the pair-production rate
$P$ for the constant-field case using his novel effective action techniques.
Assuming that the rate is small, it can be approximately computed from the
imaginary part of the action,

$\displaystyle P\approx 2{\rm Im}\Gamma(E)\,.$ (1)

In the constant-field case, the effective action is just the space-time
integral of the Euler-Heisenberg Lagrangian ${\cal L}_{EH}(E)$, obtained by
Heisenberg and Euler in 1936 eulhei , for whose imaginary part Schwinger found
the well-known expansion

$\displaystyle{\rm Im}{{\cal L}_{EH}}(E)$ $\displaystyle=$
$\displaystyle\frac{m^{4}}{8\pi^{3}}\beta^{2}\,\sum_{k=1}^{\infty}\frac{1}{k^{2}}\,\exp\left[-\frac{\pi
k}{\beta}\right]$ (2)

($\beta=eE/m^{2}$). We note that

  * •

${\rm Im}{\cal L}(E)$ depends on $E$ non-perturbatively, which lends support
to the tunneling picture.

  * •

The total pair creation rate is given by the leading $k=1$ term (seeing this
requires an analysis of the corrections to (1)).

  * •

The $k\geq 2$ terms carry information on the coherent creation of $k$ pairs in
one Compton volume.

  * •

The corresponding formula for scalar QED differs from (2) only by the
normalization and sign changes from the difference between Bose-Einstein and
Fermi-Dirac statistics,

$\displaystyle{\rm Im}{{\cal L}_{\rm scal}}(E)$ $\displaystyle=$
$\displaystyle\frac{m^{4}}{16\pi^{3}}\beta^{2}\,\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{2}}\,\exp\left[-\frac{\pi
k}{\beta}\right]\,.$ (3)

For a constant field the pair creation rate is exponentially small for field
strengths below the critical field strength $E_{\rm crit}$,

$\displaystyle E_{\rm crit}=\frac{m^{2}}{e}\approx 10^{18}{\rm V}/{\rm m}\,.$
(4)

This critical field strength is such that an electron will collect its rest
energy from the field on a distance of one Compton wave length. Present-day
lasers are still two to three orders of magnitude away from this (for reviews
of the experimental situation, see LUXE ; FACET ; fikkstt ). To have any
chance at seeing pair creation soon, complicated laser configurations will
have to be used to lower the pair creation threshold. To mention here just two
of the many proposals that have been put forward over the last fifteen years,
counterpropagating linearly polarized lasers were proposed by Ruf et al. rmmhc
and superimposing a plane-wave X-ray beam with a strongly focused optical
laser pulse by Dunne et al. dugisc  (for an overview see fikkstt ).

##  2 Approximation methods for Schwinger pair creation

For the field configurations corresponding to most of these proposals an exact
calculation of the pair-creation rate is out of the question; reliable
approximation methods are called for. Until the eighties, virtually the only
method available in this context was WKB keldysh65 ; breitz70 ; narnik70 ;
popov72 . A more sophisticated version of WKB, better adapted to the
relativistic nature of the pair-creation process, is the worldline instanton
formalism. It was introduced by Affleck et al. afalma  in 1982 for scalar
pair-production in a constant field, but gained popularity only following its
generalization to fermions and non-constant fields by G.V. Dunne and the
author 63 .

For purely time-dependent fields, the quantum kinetic approach was developed
in the early nineties, based on a Vlasov-type equation kescm1 ; kescm2 ;
sbrspt ; healgi .

Even more recently, it has been found that the Dirac-Heisenberg-Wigner
formalism, invented by Wigner in 1932 wigner32  and further developed in
vagyel ; bigora ; ochhei  can provide a more detailed picture of pair creation
process healgi2 ; hebenstreit ; ilmaza ; kohlfuerst ; dialko .

In the following, we will discuss each of these three approaches in turn.

##  3 Worldline instantons

The worldline instanton formalism is based on Feynman’s “worldline
representation” of the QED effective action. For scalar QED it reads
feynman1950

$\displaystyle\Gamma_{\rm scal}[A]$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}{dT\over T}\,{\rm e}^{-{m^{2}}T}\int{\cal
D}x(\tau)\,{\rm e}^{-S[x(\tau)]}\,,$ $\displaystyle S[x(\tau)]$
$\displaystyle=$ $\displaystyle\int_{0}^{T}d\tau\Bigl{(}{\dot{x}^{2}\over
4}+ieA\cdot\dot{x}\Bigr{)}\,.$

Here $m$ and $T$ are the mass and proper time of the loop scalar, and the path
integral $\int{\cal D}x(\tau)$ is over closed trajectories in Euclidean
spacetime.

In 1982, Affleck, Alvarez and Manton afalma  used this representation for an
elegant rederivation of Schwinger’s formula (3) by simply replacing the path
integral by a single stationary trajectory, the worldline instanton. For a
constant electric field pointing into the $z$ direction,
${\vec{E}}=(0,0,E)=const.$, this trajectory is given simply by a circle in the
$z-t$ place,

$\displaystyle x^{\rm cl}_{k}(u)$ $\displaystyle=$ $\displaystyle{m\over
eE}\,\Bigl{(}x_{1},x_{2},{\rm cos}(2k\pi u),{\rm sin}(2k\pi u)\Bigr{)}$ (6)

and it carries an index $k$ for the number of times that the circle is
traversed. The worldline action evaluated on the instanton reproduces the
exponent of the $k$th term in (3),

$\displaystyle S[x^{\rm cl}]$ $\displaystyle=$ $\displaystyle
k\pi\,{m^{2}\over eE}$ (7)

and the prefactor determinant can (with a bit more of work) shown to provide
the correct normalization.

While in the constant-field case this method provides the exact answer
(because here the path integral is gaussian), for arbitrary electric fields it
has the character of a semi-classical approximation, closely related to the
WKB approximation kimpage1 ; kimpage2 ; kimpage3 . It can be summarized in the
remarkably simple formula

$\displaystyle{\displaystyle\int_{x(T)=x(0)=x^{(0)}}}{\cal D}x(\tau)\,{\rm
e}^{-S[x(\tau)]}\approx{\rm e}^{i\theta}{\rm e}^{-S[x^{\rm
cl}](T)}\frac{1}{(4\pi T)^{2}}\sqrt{\biggl{|}{\rm
det}\Bigl{[}\eta^{(\lambda)}_{\mu,{\rm
free}}(T)\Bigr{]}\biggr{|}\over\biggl{|}{\rm
det}\Bigl{[}\eta^{(\lambda)}_{\mu}(T)\Bigr{]}\biggr{|}}$ (8)

where the extremal action trajectory $x^{\rm cl}(u)$ is a periodic solution of
the (euclidean) Lorentz force equation, the $\eta_{\mu}^{(\lambda)}$ are zero
modes of the Hessian fluctuation operator around it, and the phase factor
${\rm e}^{i\theta}$ is related to the Morse index of this operator.

For the well-studied cases of a time-like or space-like Sauter field the
worldline instantons can still be given in closed form. The time-dependent
Sauter case, defined by the two-parameter single-bump field $E(t)=E\,{\rm
sech}^{2}(\omega\,t)$, has the simple trigonometric instanton solution

$\displaystyle x_{k3}^{\rm cl}(u)$ $\displaystyle=$
$\displaystyle-\frac{1}{\omega}\,\frac{1}{\sqrt{1+\gamma^{2}}}\,{\rm
arcsinh}\left[\gamma\,\cos\left(2k\pi u\right)\right]$ $\displaystyle
x_{k4}^{\rm cl}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{\omega}\,\arcsin\left[\frac{\gamma}{\sqrt{1+\gamma^{2}}}\,\sin\left(2k\pi\,u\right)\right]$

conveniently written in terms of the “adiabaticity parameter”
$\gamma\equiv\frac{m\omega}{eE}$. In Fig. 2 we plot them for various values of
this parameter.

Figure 2: Plot of the worldline instanton paths in the $(x_{3},x_{4})$ plane
for $E(t)=E{\rm sech}^{2}(\omega t)$. The paths are shown for various values
of the adiabaticity parameter $\gamma$. $x_{3,4}$ have been expressed in units
of $\frac{m}{eE}$.

We note that the instanton exists for any positive $\gamma$. In the limit
$\gamma\to 0$ it turns into the circular one of the constant-field case above,
while with increasing $\gamma$ it shrinks and becomes elongated in the $x_{4}$
direction. Calculating the stationary worldline action one finds

$\displaystyle S[x_{k}^{\rm cl}]$ $\displaystyle=$ $\displaystyle
k\,\frac{m^{2}\pi}{eE}\left(\frac{2}{1+\sqrt{1+\gamma^{2}}}\right)\,.$ (10)

The action decreases with increasing $\gamma$, therefore the pair creation
rate increases.

The space-like Sauter case, which we define by $E(x_{3})=E{\rm
sech}^{2}(x_{3}/d)$, is mathematically similar, but physically totally
different. The instanton solutions now involve hyperbolic functions:

$\displaystyle x_{k3}^{\rm cl}(u)$ $\displaystyle=$ $\displaystyle{m\over
eE}{1\over\tilde{\gamma}}\,{\rm
arcsinh}\biggl{(}{\tilde{\gamma}\over\sqrt{1-\tilde{\gamma}^{2}}}\,\sin(2k\pi
u)\biggr{)}$ $\displaystyle x_{k4}^{\rm cl}(u)$ $\displaystyle=$
$\displaystyle{m\over
eE}{1\over\tilde{\gamma}\sqrt{1-\tilde{\gamma}^{2}}}\,{\rm
arcsin}\bigl{(}\tilde{\gamma}\cos(2k\pi u)\bigr{)}$ (11)

with an “inhomogeneity parameter” $\tilde{\gamma}\equiv\frac{m}{eEd}$. The
trajectories are shown in Fig. 3.

Figure 3: Plot of the worldline instanton paths in the $(x_{3},x_{4})$ plane
for $E(x_{3})=E{\rm sech}^{2}(x_{3}/d)$. The paths are shown for various
values of the parameter $\tilde{\gamma}$. $x_{3,4}$ have been expressed in
units of ${m\over eE}$.

Again they approach the constant-field instantons for small $\tilde{\gamma}$,
but with increasing $\tilde{\gamma}$ they grow instead of shrinking. Most
importantly, they cease to exist for $\tilde{\gamma}\geq 1$! A simple
calculation shows that the limiting case $\tilde{\gamma}=1$ corresponds to the
virtual particles having to run from $x=0$ all the way to $x=\pm\infty$ to
extract their rest-mass energies from the field. For larger $\tilde{\gamma}$
this becomes impossible, so that there is no pair production, no matter how
much energy the field may contain; it cannot be dispersed (note the analogy
with the photoelectric effect). Thus this example provides a non-trivial
prediction as well as confirmation of the vacuum-tunneling picture. The
instanton action is similar to (10), but with a crucial sign change:

$\displaystyle S[x_{k}^{\rm cl}]$ $\displaystyle=$ $\displaystyle
k\,\frac{m^{2}\pi}{eE}\left(\frac{2}{1+\sqrt{1-\tilde{\gamma}^{2}}}\right)\,.$
(12)

Thus it increases with increasing $\tilde{\gamma}$, implying a decrease of the
pair creation rate.

The Sauter field is considered a benchmark case for pair-production since
there are other representations for the exact pair-production rate suitable
for numerical evaluation nikishov ; giekli . Fig. 4 (taken from 64 ) shows a
comparison of results for the imaginary part of the effective action,
normalized by the weak-field limit of the “locally constant field
approximation”, obtained by the worldline instanton approximation, a numerical
evaluation of Nikishov’s representation, and a direct numerical evaluation of
the worldline path integral. Although the instanton approach has the character
of a large-mass approximation, it turns out to work well in the full range of
$\tilde{\gamma}$, which also leaves little doubt that the vanishing of the
pair-production rate for $\tilde{\gamma}\geq 1$ is an exact result and not an
artefact of our semi-classical approximation.

Figure 4: Comparison of the imaginary part of the effective action for the
space-like Sauter field obtained by the worldline instanton method (dotted
line), a numerical evaluation of Nikishov’s integral representation (dashed
line), and the “Worldline Monte Carlo" method (circles).

The worldline instanton approach has also been applied to the following
combination of a strong space-like with a weak time-like Sauter field schsch-
schwinger :

$\displaystyle{\bf
E}(t,x)=\biggl{(}\frac{E}{\cosh^{2}(kx)}+\frac{E^{\prime}}{\cosh^{2}(\omega
t)}\biggr{)}\,{\bf e}_{x}$ (13)

where $E^{\prime}\ll E\ll E_{\rm crit}$. As seen in Fig. 5, for sufficiently
large $\omega$ the weak temporal pulse squeezes the worldline instanton in the
$x_{0}$ direction, leading to a significant enhancement of the pair creation
rate.

Figure 5: Worldline instanton for the combination of a space-like and a time-
like Sauter field. (a) shows the instanton for $\omega<\omega_{\rm crit}$ and
(b) for $\omega>\omega_{\rm crit}$ (for the definition of $\omega_{\rm crit}$
see schsch-schwinger ).

This example also demonstrates a feature of the worldline instanton approach
that is very useful for the treatment of complicated fields involving a
superposition of several components: to see whether adding one more component
will lead to an enhancement of the pair-creation probability, in a first
approximation one might just check whether it tends to diminish or enlarge the
instanton trajectory.

Moreover, this formalism has led to the following rules 63  that previously
apparently were not understood in full generality: (i) Inhomogeneity in space
tends to reduce the pair creation rate. An insufficiently extended field, no
matter how intense, will not pair-produce. (ii) Inhomogeneity in time tends to
enhance the pair creation rate. A purely time-dependent field will always give
a non-zero pair creation rate.

One of the advantages of the worldline instanton formalism over the closely
related WKB approximation is that it makes it straightforward to incorporate
loop corrections involving the internal exchange of photons. In afalma
Affleck et al. already took advantage of this to arrive, with very little
effort, at the following generalization of the scalar QED Schwinger formula
(3), which now is restricted to the weak-field limit, but takes an arbitrary
number of photon exchanges between the two nascent particles into account:

$\displaystyle{\rm Im}{\cal L}^{({\rm all-
loop})}(E)\,\,\,\,{\stackrel{{\scriptstyle\beta\to 0}}{{\sim}}}\,\,\,\,{\rm
Im}{\cal L}^{(1)}(E)\,{\rm e}^{\alpha\pi}\,.$ (14)

This result may seem counter-intuitive since it indicates an enhancement
while, naively, one might think that the attractive force between the two
oppositely charged particles ought to make it more difficult for them to
separate out and thereby reduce the pair-creation rate. It suggests that the
interaction between the particles should not been taken into account before
they have turned from virtual to real. And indeed, in 1984 Lebedev and Ritus
lebrit  showed that, in the tunnelling picture, exactly the same enhancement
is obtained by evaluating the Coulomb interaction between the particles at the
critical separation, and interpreting it as an effective lowering of the
energy that needs to be drawn from the field in the pair-creation process.
With other words, the separating out of the pair along the field lines would
have to be seen as a pure statistical fluctuation unrelated to the equations
of motion.

##  4 The Vlasov equation

For purely time-dependent fields, there is the following Vlasov-type equation
describing the time-evolution of the density of created pairs ${\cal N}_{\bf
k}$ with fixed momentum $\bf k$ kescm1 ; kescm2 ; sbrspt

$\displaystyle\frac{d}{dt}(1+2{\cal N}_{\bf k}(t))$ $\displaystyle=$
$\displaystyle\Omega^{(-)}(t)\int_{t_{0}}^{t}dt^{\prime}\Bigl{[}\Omega^{(-)}(t^{\prime})(1+2{\cal
N}_{\bf
k}(t^{\prime}))\cos(\int_{t^{\prime}}^{t}dt^{\prime\prime}\Omega^{(+)}(t^{\prime\prime}))\Bigr{]}$
(15)

where

$\displaystyle\Omega^{(\pm)}_{k}(t):=\frac{\omega_{k}^{2}(t)\pm\omega_{k}^{2}(t_{0})}{\omega_{k}(t_{0})}\,,$
(16)

and

$\displaystyle\omega_{k}^{2}(t)$ $\displaystyle=$
$\displaystyle(k_{\parallel}-qA_{\parallel}(t))^{2}+{\bf
k}_{\perp}^{2}+m^{2}.$ (17)

Here $t_{0}$ is the initial time (usually $-\infty$) and the projections
$\parallel,\perp$ refer to the field direction.

The integro-differential equation (15) has shown itself to be very amenable to
numerical evaluation, but is usually hopeless for attempts at an exact
calculation. An exception is 84 ; 101 ; 147  where an infinite family of
analytic solutions was found related to the well-known solitonic solutions of
the Korteweg-de-Vries equation. The simplest one has the gauge potential

$\displaystyle
qA_{\parallel}(t)=\tilde{k}_{\parallel}-\sqrt{\tilde{k}_{\parallel}^{2}+\frac{2\tilde{\omega}_{0}^{2}}{\cosh^{2}(\tilde{\omega}_{0}t)}}$
(18)

where $\tilde{k}$ is a fixed reference momentum and
$\tilde{\omega}_{0}=\sqrt{\tilde{\bf k}^{2}+m^{2}}$. Like any purely time-
dependent field these “solitonic” fields have non-vanishing total pair
creation rates, but there is no pair creation at that particular momentum
$\tilde{\bf k}$. At intermediate times ${\cal N}_{\tilde{\bf k}}(t)$ is non-
zero, but it returns to zero for $t\to\infty$. The external field seems to
excite the vacuum, but no particles materialize.

##  5 How particles are born

The Dirac-Heisenberg-Wigner formalism, a previously little-used phase-space
approach to QED based on a gauge-invariant density operator, has in the last
few years turned out to be capable of providing new insights into the details
of the pair-creation process healgi2 ; hebenstreit ; ilmaza ; kohlfuerst ;
dialko . Here we can only show the following example of a time-like Sauter
field in 1 + 1 dimensions, localized in the space direction:

$\displaystyle|e|{\bf E}(t,x)=E_{0}\,{\rm sech}^{2}(t/\tau){\rm
exp}\Bigl{(}-\frac{x^{2}}{2\lambda^{2}}\Bigr{)}{\bf e}_{x}$ (19)

where $\tau$, $E_{0}$ and $\lambda$ are constants. A detailed study in
momentum space dialko  shows that there are three time scales
$T_{1},T_{2},T_{3}$ involved in the pair-creation process. The process starts
with an early build-up of a narrow peak at $p=0$. Around time $T_{1}$, a side
peak appears. Around time $T_{2}$, the side peak wave packet (now called “pre-
particle”) starts to follow the classical trajectory. Around time $T_{3}$ the
central peak has faded away, which concludes the pair-creation process. Fig. 6
333I thank R. Alkofer and C. Kohlfürst for providing this figure. shows
snapshots of the particle momentum density $f_{p}$, particle space density
$f_{x}$, charge momentum density $q_{p}$ and charge space density $q_{x}$ for
two different times. The separating out into two oppositely charged particles
is clearly visible.

Figure 6: Particle and charge momentum and space densities at two different
times for the field (19).

##  6 Summary and outlook

I have discussed the Sauter-Schwinger pair-creation process with an emphasis
on the question of the interaction of the particles in formation with each
other and the external field, an issue that is not easily accessible in the
standard formulation of QFT. The recent work of kohlfuerst ; dialko  on scalar
QED demonstrates that the Dirac-Heisenberg-Wigner formalism is capable of
providing a detailed space-time picture of this process, involving three
different time scales and a complex gradual transition from virtual to real
particles. It would be of great interest to extend these studies to more
general theories, in particular to gravity. Since conservation laws require
the interaction between the nascent particles and the field to be reciprocal,
this might also throw new light on the question why straightforward attempts
at calculating the contribution of the QFT vacuum to the cosmological constant
so far generally have failed by many orders of magnitude.

Acknowledgements: I thank Naser Ahmadiniaz and Christian Kohlfürst for
discussions and correspondence.

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