What is the correct way to add a has_and_belongs_to_many relationship to a sphinx index?

Question

Ruby on Rails

Hello everyone)) In the project, sphinx is used as a search. There are entities

class Catalog::Product < ActiveRecord::Base
  has_and_belongs_to_many :writers, class_name: 'Catalog::Author', join_table: "catalog_authors_products"
end

class Catalog::Author < ActiveRecord::Base
  has_and_belongs_to_many :products, class_name: 'Catalog::Product', join_table: 'catalog_authors_products'
end

Index file

ThinkingSphinx::Index.define 'catalog/product', with: :active_record do
  indexes :title, facet: true
  indexes :id
  indexes :sku, as: :sku_field
  indexes writers(:first_name), as: :writer_first_name

  has created_at
  has available
  has price
  has sku
end

when adding

indexes writers(:first_name), as: :writer_first_name

and when trying to do a rake ts:rebuild an error pops up ERROR: index 'catalog_product_core': sql_range_query: Unknown column 'catalog_authors.product_id' in 'on clause' (DSN=mysql://root:***@localhost:3306/byblos_dev) . Googled, everyone writes the habtm connection in this format

indexes writers(:first_name), as: :writer_first_name

or so , but the error is the same. In the development.sphinx.conf fileindexes writers.first_name, as: :writer_first_name
sql_query = SELECT SQL_NO_CACHE catalog_products.id * 2 + 0 AS id, catalog_products.title AS title, catalog_products.authors AS authors, catalog_products.id AS id, catalog_products.sku AS sku_field, GROUP_CONCAT(DISTINCT catalog_authors.first_name SEPARATOR ' ') AS writer_first_name , catalog_products.id AS sphinx_internal_id, 'Catalog::Product' AS sphinx_internal_class, 0 AS sphinx_deleted, UNIX_TIMESTAMP(catalog_products.created_at) AS created_at, catalog_products.available AS available, catalog_products.price AS price, catalog_products.sku AS sku FROM catalog_products LEFT OUTER JOIN catalog_authors ON catalog_authors.product_id = catalog_products.id WHERE (catalog_products.id BETWEEN $start AND $end) GROUP BY catalog_products.id, catalog_products.title, catalog_products.authors, catalog_products.id, catalog_products.sku,catalog_products.id, catalog_products.created_at, catalog_products.available, catalog_products.price, catalog_products.sku ORDER BY NULL
there is a string

LEFT OUTER JOIN catalog_authors ON catalog_authors.product_id = catalog_products.id

apparently the sphinx does not correctly determine the connection!?
How to correctly add the has_and_belongs_to_many relation to the index?
Thanks in advance))