Official Recommendations
The American Academy of Pediatrics publishes policy statements on this topic. The most recent recommendations for the age range in question (school-aged children: 5-18yo) are from Oct 2016.
Their recommendations for school-aged children don't included a max recommended number of hours of screen time per day. Instead, they emphasize that use of screens has different risks and benefits depending on what children are doing with screens and in what context. They emphasize the importance of balancing screen activities with more physical activities, and teaching children about the (in)appropriate use of screens (dangers of cyberbullying, sexting, etc.). They also discuss some specific risks to be aware of, such as addictive video game playing.
The whole statement is worth reading, in my opinion, (and it's pretty short), but here are their specific recommendations for families of school aged children:
Develop, consistently follow, and routinely revisit a Family Media Use
plan (see the plan from the American Academy of Pediatrics at
www.HealthyChildren.org/MediaUsePlan).
Address what type of and how much media are used and what media
behaviors are appropriate for each child or teenager, and for parents.
Place consistent limits on hours per day of media use as well as types
of media used.
Promote that children and adolescents get the recommended amount of
daily physical activity (1 hour) and adequate sleep (8–12 hours,
depending on age).
Recommend that children not sleep with devices in their bedrooms,
including TVs, computers, and smartphones. Avoid exposure to devices
or screens for 1 hour before bedtime.
Discourage entertainment media while doing homework.
Designate media-free times together (eg, family dinner) and media-free
locations (eg, bedrooms) in homes. Promote activities that are likely
to facilitate development and health, including positive parenting
activities, such as reading, teaching, talking, and playing together.
Communicate guidelines to other caregivers, such as babysitters or
grandparents, so that media rules are followed consistently.
Engage in selecting and co-viewing media with your child, through
which your child can use media to learn and be creative, and share
these experiences with your family and your community.
Have ongoing communication with children about online citizenship and
safety, including treating others with respect online and offline,
avoiding cyberbullying and sexting, being wary of online solicitation,
and avoiding communications that can compromise personal privacy and
safety.
Actively develop a network of trusted adults (eg, aunts, uncles,
coaches, etc) who can engage with children through social media and to
whom children can turn when they encounter challenges.
Relevant Research
Regarding the other part of your question, asking for research on the topic: You might be surprised about how little is really known for sure. To start, I recommend you check out this short statement summarizing the (lack of) evidence for the AAP guidelines for screen exposure during childhood. The references listed at the end will give you more details about the state of knowledge on this topic.
In particular, there has been some recent attention on the recommendations for children under 2 (which AAP changed last year, citing insufficient evidence for their previous 0-hours recommendation). Children's media --- and everyone else's media, TBH ;) --- has changed rapidly over the last several years, and the scientific knowledge hasn't caught up. Part of this has to do with the difficulty of gathering the necessary evidence (you can't randomly assign children to high and low media exposure homes for obvious ethical reasons; longitudinal research takes years to collect let alone analyze, interpret, and iterate).
That said, there definitely is some research relevant to this question. There are a few posts on this site summarizing some key studies: are there any peer reviewed studies showing the benefits of video games for kids (especially girls)?; What exactly is the problem with the Television?; Any Research on the Effects of Mobile/Cellphone Usage on Childhood Development?
